relativistic speed time travel

Is Time Travel Possible?

We all travel in time! We travel one year in time between birthdays, for example. And we are all traveling in time at approximately the same speed: 1 second per second.

We typically experience time at one second per second. Credit: NASA/JPL-Caltech

NASA's space telescopes also give us a way to look back in time. Telescopes help us see stars and galaxies that are very far away . It takes a long time for the light from faraway galaxies to reach us. So, when we look into the sky with a telescope, we are seeing what those stars and galaxies looked like a very long time ago.

However, when we think of the phrase "time travel," we are usually thinking of traveling faster than 1 second per second. That kind of time travel sounds like something you'd only see in movies or science fiction books. Could it be real? Science says yes!

This image from the Hubble Space Telescope shows galaxies that are very far away as they existed a very long time ago. Credit: NASA, ESA and R. Thompson (Univ. Arizona)

How do we know that time travel is possible?

More than 100 years ago, a famous scientist named Albert Einstein came up with an idea about how time works. He called it relativity. This theory says that time and space are linked together. Einstein also said our universe has a speed limit: nothing can travel faster than the speed of light (186,000 miles per second).

Einstein's theory of relativity says that space and time are linked together. Credit: NASA/JPL-Caltech

What does this mean for time travel? Well, according to this theory, the faster you travel, the slower you experience time. Scientists have done some experiments to show that this is true.

For example, there was an experiment that used two clocks set to the exact same time. One clock stayed on Earth, while the other flew in an airplane (going in the same direction Earth rotates).

After the airplane flew around the world, scientists compared the two clocks. The clock on the fast-moving airplane was slightly behind the clock on the ground. So, the clock on the airplane was traveling slightly slower in time than 1 second per second.

Credit: NASA/JPL-Caltech

Can we use time travel in everyday life?

We can't use a time machine to travel hundreds of years into the past or future. That kind of time travel only happens in books and movies. But the math of time travel does affect the things we use every day.

For example, we use GPS satellites to help us figure out how to get to new places. (Check out our video about how GPS satellites work .) NASA scientists also use a high-accuracy version of GPS to keep track of where satellites are in space. But did you know that GPS relies on time-travel calculations to help you get around town?

GPS satellites orbit around Earth very quickly at about 8,700 miles (14,000 kilometers) per hour. This slows down GPS satellite clocks by a small fraction of a second (similar to the airplane example above).

GPS satellites orbit around Earth at about 8,700 miles (14,000 kilometers) per hour. Credit: GPS.gov

However, the satellites are also orbiting Earth about 12,550 miles (20,200 km) above the surface. This actually speeds up GPS satellite clocks by a slighter larger fraction of a second.

Here's how: Einstein's theory also says that gravity curves space and time, causing the passage of time to slow down. High up where the satellites orbit, Earth's gravity is much weaker. This causes the clocks on GPS satellites to run faster than clocks on the ground.

The combined result is that the clocks on GPS satellites experience time at a rate slightly faster than 1 second per second. Luckily, scientists can use math to correct these differences in time.

If scientists didn't correct the GPS clocks, there would be big problems. GPS satellites wouldn't be able to correctly calculate their position or yours. The errors would add up to a few miles each day, which is a big deal. GPS maps might think your home is nowhere near where it actually is!

In Summary:

Yes, time travel is indeed a real thing. But it's not quite what you've probably seen in the movies. Under certain conditions, it is possible to experience time passing at a different rate than 1 second per second. And there are important reasons why we need to understand this real-world form of time travel.

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5.4: Time Dilation

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Learning Objectives

By the end of this section, you will be able to:

• Explain how time intervals can be measured differently in different reference frames.
• Describe how to distinguish a proper time interval from a dilated time interval.
• Describe the significance of the muon experiment.
• Explain why the twin paradox is not a contradiction.
• Calculate time dilation given the speed of an object in a given frame.

The analysis of simultaneity shows that Einstein’s postulates imply an important effect: Time intervals have different values when measured in different inertial frames. Suppose, for example, an astronaut measures the time it takes for a pulse of light to travel a distance perpendicular to the direction of his ship’s motion (relative to an earthbound observer), bounce off a mirror, and return (Figure \(\PageIndex{1a}\)). How does the elapsed time that the astronaut measures in the spacecraft compare with the elapsed time that an earthbound observer measures by observing what is happening in the spacecraft?

Examining this question leads to a profound result. The elapsed time for a process depends on which observer is measuring it. In this case, the time measured by the astronaut (within the spaceship where the astronaut is at rest) is smaller than the time measured by the earthbound observer (to whom the astronaut is moving). The time elapsed for the same process is different for the observers, because the distance the light pulse travels in the astronaut’s frame is smaller than in the earthbound frame, as seen in Figure \(\PageIndex{1b}\). Light travels at the same speed in each frame, so it takes more time to travel the greater distance in the earthbound frame.

Definition: Time Dilation

Time dilation is the lengthening of the time interval between two events for an observer in an inertial frame that is moving with respect to the rest frame of the events (in which the events occur at the same location).

To quantitatively compare the time measurements in the two inertial frames, we can relate the distances in Figure \(\PageIndex{1b}\) to each other, then express each distance in terms of the time of travel (respectively either \(\Delta t\) or \(\Delta \tau\)) of the pulse in the corresponding reference frame. The resulting equation can then be solved for \(\Delta t\) in terms of \(\Delta \tau\).

The lengths \(D\) and \(L\) in Figure \(\PageIndex{1c}\) are the sides of a right triangle with hypotenuse \(s\). From the Pythagorean theorem ,

\[s^2 = D^2 + L^2. \nonumber \]

The lengths \(2s\) and \(2L\) are, respectively, the distances that the pulse of light and the spacecraft travel in time \(\Delta t\) in the earthbound observer’s frame. The length \(D\) is the distance that the light pulse travels in time \(\Delta \tau\) in the astronaut’s frame. This gives us three equations:

\[\begin{align*} 2s &= c\Delta t \\[4pt] 2L &= v\Delta t; \\[4pt] 2D &= c\Delta \tau. \end{align*} \nonumber \]

Note that we used Einstein’s second postulate by taking the speed of light to be c in both inertial frames. We substitute these results into the previous expression from the Pythagorean theorem:

\[ \begin{align*} s^2 &= D^2 + L^2 \\[4pt] \left(c\dfrac{\Delta t}{2}\right)^2 &= \left(c\dfrac{\Delta \tau}{2}\right)^2 + \left(v\dfrac{\Delta t}{2}\right)^2 \end{align*} \nonumber \]

Then we rearrange to obtain

\[(c\Delta t)^2 - (v\Delta t)^2 = (c\Delta \tau)^2. \nonumber \]

Finally, solving for \(\Delta t\) in terms of \(\Delta \tau\) gives us

\[\Delta t = \dfrac{\Delta \tau}{\sqrt{1 - (v/c)^2}}. \nonumber \]

This is equivalent to

\[\Delta t = \gamma \Delta \tau, \label{timedilation} \]

where \(\gamma\) is the relativistic factor (often called the Lorentz factor ) given by

\[\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}} \nonumber \]

and \(v\) and \(c\) are the speeds of the moving observer and light, respectively.

Note the asymmetry between the two measurements. Only one of them is a measurement of the time interval between two events—the emission and arrival of the light pulse—at the same position. It is a measurement of the time interval in the rest frame of a single clock. The measurement in the earthbound frame involves comparing the time interval between two events that occur at different locations. The time interval between events that occur at a single location has a separate name to distinguish it from the time measured by the earthbound observer, and we use the separate symbol \(\Delta \tau\) to refer to it throughout this chapter.

Definition: Proper Time

The proper time interval \(\Delta \tau\) between two events is the time interval measured by an observer for whom both events occur at the same location.

The equation relating \(\delta t\) and \(\Delta \tau\) is truly remarkable. First, as stated earlier, elapsed time is not the same for different observers moving relative to one another, even though both are in inertial frames. A proper time interval \(\Delta \tau\) for an observer who, like the astronaut, is moving with the apparatus, is smaller than the time interval for other observers. It is the smallest possible measured time between two events. The earthbound observer sees time intervals within the moving system as dilated (i.e., lengthened) relative to how the observer moving relative to Earth sees them within the moving system. Alternatively, according to the earthbound observer, less time passes between events within the moving frame. Note that the shortest elapsed time between events is in the inertial frame in which the observer sees the events (e.g., the emission and arrival of the light signal) occur at the same point.

This time effect is real and is not caused by inaccurate clocks or improper measurements. Time-interval measurements of the same event differ for observers in relative motion. The dilation of time is an intrinsic property of time itself. All clocks moving relative to an observer, including biological clocks, such as a person’s heartbeat, or aging, are observed to run more slowly compared with a clock that is stationary relative to the observer.

Note that if the relative velocity is much less than the speed of light (v << c), then \(v^2/c^2\) is extremely small, and the elapsed times \(\Delta t\) and \(\Delta \tau\) are nearly equal. At low velocities, physics based on modern relativity approaches classical physics—everyday experiences involve very small relativistic effects. However, for speeds near the speed of light, \(v^2/c^2\) is close to one, so \(\sqrt{1 - v^2/c^2}\) is very small and \(\Delta t\) becomes significantly larger than \(\Delta \tau\).

Half-Life of a Muon

There is considerable experimental evidence that the equation \(\Delta t = \gamma \Delta \tau\) is correct. One example is found in cosmic ray particles that continuously rain down on Earth from deep space. Some collisions of these particles with nuclei in the upper atmosphere result in short-lived particles called muons . The half-life (amount of time for half of a material to decay) of a muon is 1.52 μs when it is at rest relative to the observer who measures the half-life. This is the proper time interval \(\Delta \tau\). This short time allows very few muons to reach Earth’s surface and be detected if Newtonian assumptions about time and space were correct. However, muons produced by cosmic ray particles have a range of velocities, with some moving near the speed of light. It has been found that the muon’s half-life as measured by an earthbound observer (\(\Delta t\)) varies with velocity exactly as predicted by the equation \(\Delta t = \gamma \Delta \tau\). The faster the muon moves, the longer it lives. We on Earth see the muon last much longer than its half-life predicts within its own rest frame. As viewed from our frame, the muon decays more slowly than it does when at rest relative to us. A far larger fraction of muons reach the ground as a result.

Before we present the first example of solving a problem in relativity, we state a strategy you can use as a guideline for these calculations.

PROBLEM-SOLVING STRATEGY: RELATIVITY

• Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Look in particular for information on relative velocity v .
• Identify exactly what needs to be determined in the problem (identify the unknowns).
• Make certain you understand the conceptual aspects of the problem before making any calculations (express the answer as an equation). Decide, for example, which observer sees time dilated or length contracted before working with the equations or using them to carry out the calculation. If you have thought about who sees what, who is moving with the event being observed, who sees proper time, and so on, you will find it much easier to determine if your calculation is reasonable.
• Determine the primary type of calculation to be done to find the unknowns identified above (do the calculation). You will find the section summary helpful in determining whether a length contraction, relativistic kinetic energy, or some other concept is involved.

Note that you should not round off during the calculation . As noted in the text, you must often perform your calculations to many digits to see the desired effect. You may round off at the very end of the problem solution, but do not use a rounded number in a subsequent calculation. Also, check the answer to see if it is reasonable: Does it make sense? This may be more difficult for relativity, which has few everyday examples to provide experience with what is reasonable. But you can look for velocities greater than c or relativistic effects that are in the wrong direction (such as a time contraction where a dilation was expected).

Example \(\PageIndex{1A}\): Time Dilation in a High-Speed Vehicle

The Hypersonic Technology Vehicle 2 (HTV-2) is an experimental rocket vehicle capable of traveling at 21,000 km/h (5830 m/s). If an electronic clock in the HTV-2 measures a time interval of exactly 1-s duration, what would observers on Earth measure the time interval to be?

Apply the time dilation formula to relate the proper time interval of the signal in HTV-2 to the time interval measured on the ground.

• Identify the knowns: \(\Delta \tau = 1 \, s\); \(v = 5830m/s.\)
• Identify the unknown: \(\Delta t\).

\[\Delta t = \gamma \Delta \tau = \dfrac{\Delta \tau}{\sqrt{1 - \dfrac{v^2}{c^2}}}. \nonumber \]

\[\begin{align*} \Delta t &= \dfrac{1 \, s}{\sqrt{1 - \left(\dfrac{5830 \, m/s}{3.00 \times 10^8 m/s}\right)^2}} \\[4pt] &= 1.000000000189 \, s \\[4pt] &= 1 \, s + 1.89 \times 10^{-10}s. \end{align*} \nonumber \]

Significance

The very high speed of the HTV-2 is still only 10 -5 times the speed of light. Relativistic effects for the HTV-2 are negligible for almost all purposes, but are not zero.

What Speeds are Relativistic?

How fast must a vehicle travel for 1 second of time measured on a passenger’s watch in the vehicle to differ by 1% for an observer measuring it from the ground outside?

Use the time dilation formula to find v/c for the given ratio of times.

\[\dfrac{\Delta \tau}{\Delta t} = \dfrac{1}{1.01}. \nonumber \]

• Identify the unknown: v/c .

\[ \begin{align*} \Delta t &= \gamma \Delta \tau \\[4pt] &= \dfrac{1}{\sqrt{1 - v^2/c^2}}\Delta \tau \\[4pt] \dfrac{\Delta \tau}{\Delta t} &= \sqrt{1 - v^2/c^2} \\[4pt] \left(\dfrac{\Delta \tau}{\Delta t}\right)^2 &= 1 - \dfrac{v^2}{c^2} \\[4pt] \dfrac{v}{c} &= \sqrt{1 - (\Delta \tau/\Delta t)^2}. \end{align*} \nonumber \]

\[\dfrac{v}{c} = \sqrt{1 - (1/1.01)^2} = 0.14. \nonumber \]

The result shows that an object must travel at very roughly 10% of the speed of light for its motion to produce significant relativistic time dilation effects.

Calculating \(\Delta t\) for a Relativistic Event

Suppose a cosmic ray colliding with a nucleus in Earth’s upper atmosphere produces a muon that has a velocity \(v = 0.950c\). The muon then travels at constant velocity and lives 2.20 μs as measured in the muon’s frame of reference. (You can imagine this as the muon’s internal clock.) How long does the muon live as measured by an earthbound observer (Figure \(\PageIndex{2}\))?

As we will discuss later, in the muon’s reference frame, it travels a shorter distance than measured in Earth’s reference frame.

A clock moving with the muon measures the proper time of its decay process, so the time we are given is \(\Delta \tau = 2.20 \mu s\). The earthbound observer measures \(\Delta t\) as given by the equation \(\Delta t = \gamma \Delta \tau\). Because the velocity is given, we can calculate the time in Earth’s frame of reference.

• Identify the knowns: \(v = 0.950c\); \(\delta \tau = 2.20 \mu s\).

\[\Delta t = \gamma \Delta \tau. \nonumber \]

\[\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}}. \nonumber \]

\[\begin{align*} \Delta t &= \gamma \Delta \tau. \\[4pt] &=\dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}}\delta \tau \\[4pt] &=\dfrac{2.20 \mu s}{\sqrt{1 - (0.950)^2}} \\[4pt] &= 7.05 \, \mu s.\end{align*} \nonumber \]

One implication of this example is that because \(\gamma = 3.20\) at 95.0% of the speed of light (\(v = 0.950c\)), the relativistic effects are significant. The two time intervals differ by a factor of 3.20, when classically they would be the same. Something moving at 0.950 c is said to be highly relativistic.

Example \(\PageIndex{1B}\): Relativistic Television

A non-flat screen, older-style television display (Figure \(\PageIndex{3}\)) works by accelerating electrons over a short distance to relativistic speed, and then using electromagnetic fields to control where the electron beam strikes a fluorescent layer at the front of the tube. Suppose the electrons travel at \(6.00 \times 10^7 m/s\) through a distance of 0.200m0.200m from the start of the beam to the screen.

• What is the time of travel of an electron in the rest frame of the television set?
• What is the electron’s time of travel in its own rest frame?

Strategy for (a)

(a) Calculate the time from \(vt = d\). Even though the speed is relativistic, the calculation is entirely in one frame of reference, and relativity is therefore not involved.

\[v = 6.00 \times 10^7 m/s \, d = 0.200 \, m. \nonumber \]

• Identify the unknown: the time of travel \(\Delta t\).

\[\Delta t = \dfrac{d}{v}. \nonumber \]

\[ \begin{align*} t &= \dfrac{0.200 \, m}{6.00 \times 10^7 \, m/s} \\[4pt] &= 3.33 \times 10^{-9} \, s. \end{align*} \nonumber \]

The time of travel is extremely short, as expected. Because the calculation is entirely within a single frame of reference, relativity is not involved, even though the electron speed is close to c .

Strategy for (b)

(b) In the frame of reference of the electron, the vacuum tube is moving and the electron is stationary. The electron-emitting cathode leaves the electron and the front of the vacuum tube strikes the electron with the electron at the same location. Therefore we use the time dilation formula to relate the proper time in the electron rest frame to the time in the television frame.

\[\Delta t = 3.33 \times 10^{-9} \, s; \, v = 6.00 \times 10^7 \, m/s; \, d = 0.200 \, m. \nonumber \]

• Identify the unknown: \(\tau\).

\[\Delta t = \gamma \Delta \tau = \dfrac{\Delta \tau}{\sqrt{1 - v^2/c^2}}. \nonumber \]

\[\begin{align*} \Delta \tau &= (3.33 \times 10^{-9}s)\sqrt{1 - \left(\dfrac{6.00 \times 10^7 m/s}{3.00 \times 10^8 m/s}\right)^2} \\[4pt] &= 3.26 \times 10^{-9}s. \end{align*} \nonumber \]

The time of travel is shorter in the electron frame of reference. Because the problem requires finding the time interval measured in different reference frames for the same process, relativity is involved. If we had tried to calculate the time in the electron rest frame by simply dividing the 0.200 m by the speed, the result would be slightly incorrect because of the relativistic speed of the electron.

Exercise \(\PageIndex{1}\)

What is \(\gamma\) if \(v = 0.650c\)?

\[\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}} = \dfrac{1}{\sqrt{1 - \dfrac{(0.650c)}{c^2}}} = 1.32 \nonumber \]

The Twin Paradox

An intriguing consequence of time dilation is that a space traveler moving at a high velocity relative to Earth would age less than the astronaut’s earthbound twin. This is often known as the twin paradox . Imagine the astronaut moving at such a velocity that \(\gamma = 30.0\), as in Figure \(\PageIndex{4}\). A trip that takes 2.00 years in her frame would take 60.0 years in the earthbound twin’s frame. Suppose the astronaut travels 1.00 year to another star system, briefly explores the area, and then travels 1.00 year back. An astronaut who was 40 years old at the start of the trip would be would be 42 when the spaceship returns. Everything on Earth, however, would have aged 60.0 years. The earthbound twin, if still alive, would be 100 years old.

The situation would seem different to the astronaut in Figure \(\PageIndex{4}\). Because motion is relative, the spaceship would seem to be stationary and Earth would appear to move. (This is the sensation you have when flying in a jet.) Looking out the window of the spaceship, the astronaut would see time slow down on Earth by a factor of \(\gamma = 30.0\). Seen from the spaceship, the earthbound sibling will have aged only 2/30, or 0.07, of a year, whereas the astronaut would have aged 2.00 years.

The paradox here is that the two twins cannot both be correct. As with all paradoxes, conflicting conclusions come from a false premise. In fact, the astronaut’s motion is significantly different from that of the earthbound twin. The astronaut accelerates to a high velocity and then decelerates to view the star system. To return to Earth, she again accelerates and decelerates. The spacecraft is not in a single inertial frame to which the time dilation formula can be directly applied. That is, the astronaut twin changes inertial references. The earthbound twin does not experience these accelerations and remains in the same inertial frame. Thus, the situation is not symmetric, and it is incorrect to claim that the astronaut observes the same effects as her twin. The lack of symmetry between the twins will be still more evident when we analyze the journey later in this chapter in terms of the path the astronaut follows through four-dimensional space-time.

In 1971, American physicists Joseph Hafele and Richard Keating verified time dilation at low relative velocities by flying extremely accurate atomic clocks around the world on commercial aircraft. They measured elapsed time to an accuracy of a few nanoseconds and compared it with the time measured by clocks left behind. Hafele and Keating’s results were within experimental uncertainties of the predictions of relativity. Both special and general relativity had to be taken into account, because gravity and accelerations were involved as well as relative motion.

Exercise \(\PageIndex{2A}\)

a. A particle travels at \(1.90 \times 10^8 \, m/s\) and lives \(2.1 \times 10^8 \, s\) when at rest relative to an observer. How long does the particle live as viewed in the laboratory?

\[\Delta t = \dfrac{\Delta \tau}{\sqrt{1 - \dfrac{v^2}{c^2}}} = \dfrac{2.10 \times 10^{-8}s}{\sqrt{1 - \dfrac{(1.90 \times 10^8 \, m/s)^2}{(3.00 \times 10^8 \, m/s)^2}}} = 2.71 \times 10^{-8} \, s. \nonumber \]

Exercise \(\PageIndex{2B}\)

Spacecraft A and B pass in opposite directions at a relative speed of \(4.00 \times 10^7 \, m/s\). An internal clock in spacecraft A causes it to emit a radio signal for 1.00 s. The computer in spacecraft B corrects for the beginning and end of the signal having traveled different distances, to calculate the time interval during which ship A was emitting the signal. What is the time interval that the computer in spacecraft B calculates?

Only the relative speed of the two spacecraft matters because there is no absolute motion through space. The signal is emitted from a fixed location in the frame of reference of A , so the proper time interval of its emission is \(\tau = 1.00 \, s\). The duration of the signal measured from frame of reference B is then

\[\Delta t = \dfrac{\Delta \tau}{\sqrt{1 - \dfrac{v^2}{c^2}}} = \dfrac{1.00 \, s}{\sqrt{1 - \dfrac{(4.00 \times 10^7 \, m/s)^2}{(3.00 \times 10^8 \, m/s)^2}}} = 1.01 \, s. \nonumber \]

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Would you really age more slowly on a spaceship at close to light speed?

• Neel V. Patel archive page

Every week, the readers of our space newsletter, The Airlock , send in their questions for space reporter Neel V. Patel to answer. This week: time dilation during space travel. 

I heard that time dilation affects high-speed space travel and I am wondering the magnitude of that affect. If we were to launch a round-trip flight to a nearby exoplanet—let's say 10 or 50 light-years away––how would that affect time for humans on the spaceship versus humans on Earth? When the space travelers came back, will they be much younger or older relative to people who stayed on Earth? —Serge

Time dilation is a concept that pops up in lots of sci-fi, including Orson Scott Card’s Ender’s Game , where one character ages only eight years in space while 50 years pass on Earth. This is precisely the scenario outlined in the famous thought experiment the Twin Paradox : an astronaut with an identical twin at mission control makes a journey into space on a high-speed rocket and returns home to find that the twin has aged faster.

Time dilation goes back to Einstein’s theory of special relativity, which teaches us that motion through space actually creates alterations in the flow of time. The faster you move through the three dimensions that define physical space, the more slowly you’re moving through the fourth dimension, time––at least relative to another object. Time is measured differently for the twin who moved through space and the twin who stayed on Earth. The clock in motion will tick more slowly than the clocks we’re watching on Earth. If you’re able to travel near the speed of light, the effects are much more pronounced. 

Unlike the Twin Paradox, time dilation isn’t a thought experiment or a hypothetical concept––it’s real. The 1971 Hafele-Keating experiments proved as much, when two atomic clocks were flown on planes traveling in opposite directions. The relative motion actually had a measurable impact and created a time difference between the two clocks. This has also been confirmed in other physics experiments (e.g., fast-moving muon particles take longer to decay ). 

So in your question, an astronaut returning from a space journey at “relativistic speeds” (where the effects of relativity start to manifest—generally at least one-tenth the speed of light ) would, upon return, be younger than same-age friends and family who stayed on Earth. Exactly how much younger depends on exactly how fast the spacecraft had been moving and accelerating, so it’s not something we can readily answer. But if you’re trying to reach an exoplanet 10 to 50 light-years away and still make it home before you yourself die of old age, you’d have to be moving at close to light speed. 

There’s another wrinkle here worth mentioning: time dilation as a result of gravitational effects. You might have seen Christopher Nolan’s movie Interstellar , where the close proximity of a black hole causes time on another planet to slow down tremendously (one hour on that planet is seven Earth years).

This form of time dilation is also real, and it’s because in Einstein’s theory of general relativity, gravity can bend spacetime, and therefore time itself. The closer the clock is to the source of gravitation, the slower time passes; the farther away the clock is from gravity, the faster time will pass. (We can save the details of that explanation for a future Airlock.)

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While traveling deep into space is still something out of science fiction movies like Star Trek and Star Wars, the tremendous progress made by private space companies so far seems very promising. Someday, space travel (or even interstellar travel) might be accessible to everyone!

It's never too early to start planning for a trip of a lifetime (or several lifetimes). You can also plan your own space trip and celebrate World Space Week in your own special way!

This space travel calculator is a comprehensive tool that allows you to estimate many essential parameters in theoretical interstellar space travel . Have you ever wondered how fast we can travel in space, how much time it will take to get to the nearest star or galaxy, or how much fuel it requires? In the following article, using a relativistic rocket equation, we'll try to answer questions like "Is interstellar travel possible?" , and "Can humans travel at the speed of light?"

Explore the world of light-speed travel of (hopefully) future spaceships with our relativistic space travel calculator!

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One small step for man, one giant leap for humanity

Although human beings have been dreaming about space travel forever, the first landmark in the history of space travel is Russia's launch of Sputnik 2 into space in November 1957. The spacecraft carried the first earthling, the Russian dog Laika , into space.

Four years later, on 12 April 1961, Soviet cosmonaut Yuri A. Gagarin became the first human in space when his spacecraft, the Vostok 1, completed one orbit of Earth.

The first American astronaut to enter space was Alan Shepard (May 1961). During the Apollo 11 mission in July 1969, Neil Armstrong and Buzz Aldrin became the first men to land on the moon. Between 1969 and 1972, a total of 12 astronauts walked the moon, marking one of the most outstanding achievements for NASA.

In recent decades, space travel technology has seen some incredible advancements. Especially with the advent of private space companies like SpaceX, Virgin Galactic, and Blue Origin, the dream of space tourism is looking more and more realistic for everyone!

However, when it comes to including women, we are yet to make great strides. So far, 566 people have traveled to space. Only 65 of them were women .

Although the first woman in space, a Soviet astronaut Valentina Tereshkova , who orbited Earth 48 times, went into orbit in June 1963. It was only in October 2019 that the first all-female spacewalk was completed by NASA astronauts Jessica Meir and Christina Koch.

Women's access to space is still far from equal, but there are signs of progress, like NASA planning to land the first woman and first person of color on the moon by 2024 with its Artemis missions. World Space Week is also celebrating the achievements and contributions of women in space this year!

In the following sections, we will explore the feasibility of space travel and its associated challenges.

How fast can we travel in space? Is interstellar travel possible?

Interstellar space is a rather empty place. Its temperature is not much more than the coldest possible temperature, i.e., an absolute zero. It equals about 3 kelvins – minus 270 °C or minus 455 °F. You can't find air there, and therefore there is no drag or friction. On the one hand, humans can't survive in such a hostile place without expensive equipment like a spacesuit or a spaceship, but on the other hand, we can make use of space conditions and its emptiness.

The main advantage of future spaceships is that, since they are moving through a vacuum, they can theoretically accelerate to infinite speeds! However, this is only possible in the classical world of relatively low speeds, where Newtonian physics can be applied. Even if it's true, let's imagine, just for a moment, that we live in a world where any speed is allowed. How long will it take to visit the Andromeda Galaxy, the nearest galaxy to the Milky Way?

We will begin our intergalactic travel with a constant acceleration of 1 g (9.81 m/s² or 32.17 ft/s²) because it ensures that the crew experiences the same comfortable gravitational field as the one on Earth. By using this space travel calculator in Newton's universe mode, you can find out that you need about 2200 years to arrive at the nearest galaxy! And, if you want to stop there, you need an additional 1000 years . Nobody lives for 3000 years! Is intergalactic travel impossible for us, then? Luckily, we have good news. We live in a world of relativistic effects, where unusual phenomena readily occur.

Can humans travel at the speed of light? – relativistic space travel

In the previous example, where we traveled to Andromeda Galaxy, the maximum velocity was almost 3000 times greater than the speed of light c = 299,792,458 m/s , or about c = 3 × 10 8 m/s using scientific notation.

However, as velocity increases, relativistic effects start to play an essential role. According to special relativity proposed by Albert Einstein, nothing can exceed the speed of light. How can it help us with interstellar space travel? Doesn't it mean we will travel at a much lower speed? Yes, it does, but there are also a few new relativistic phenomena, including time dilation and length contraction, to name a few. The former is crucial in relativistic space travel.

Time dilation is a difference of time measured by two observers, one being in motion and the second at rest (relative to each other). It is something we are not used to on Earth. Clocks in a moving spaceship tick slower than the same clocks on Earth ! Time passing in a moving spaceship T T T and equivalent time observed on Earth t t t are related by the following formula:

where γ \gamma γ is the Lorentz factor that comprises the speed of the spaceship v v v and the speed of light c c c :

where β = v / c \beta = v/c β = v / c .

For example, if γ = 10 \gamma = 10 γ = 10 ( v = 0.995 c v = 0.995c v = 0.995 c ), then every second passing on Earth corresponds to ten seconds passing in the spaceship. Inside the spacecraft, events take place 90 percent slower; the difference can be even greater for higher velocities. Note that both observers can be in motion, too. In that case, to calculate the relative relativistic velocity, you can use our velocity addition calculator .

Let's go back to our example again, but this time we're in Einstein's universe of relativistic effects trying to reach Andromeda. The time needed to get there, measured by the crew of the spaceship, equals only 15 years ! Well, this is still a long time, but it is more achievable in a practical sense. If you would like to stop at the destination, you should start decelerating halfway through. In this situation, the time passed in the spaceship will be extended by about 13 additional years .

Unfortunately, this is only a one-way journey. You can, of course, go back to Earth, but nothing will be the same. During your interstellar space travel to the Andromeda Galaxy, about 2,500,000 years have passed on Earth. It would be a completely different planet, and nobody could foresee the fate of our civilization.

A similar problem was considered in the first Planet of the Apes movie, where astronauts crash-landed back on Earth. While these astronauts had only aged by 18 months, 2000 years had passed on Earth (sorry for the spoilers, but the film is over 50 years old at this point, you should have seen it by now). How about you? Would you be able to leave everything you know and love about our galaxy forever and begin a life of space exploration?

Space travel calculator – relativistic rocket equation

Now that you know whether interstellar travel is possible and how fast we can travel in space, it's time for some formulas. In this section, you can find the "classical" and relativistic rocket equations that are included in the relativistic space travel calculator.

There could be four combinations since we want to estimate how long it takes to arrive at the destination point at full speed as well as arrive at the destination point and stop. Every set contains distance, time passing on Earth and in the spaceship (only relativity approach), expected maximum velocity and corresponding kinetic energy (on the additional parameters section), and the required fuel mass (see Intergalactic travel — fuel problem section for more information). The notation is:

• a a a — Spaceship acceleration (by default 1   g 1\rm\, g 1 g ). We assume it is positive a > 0 a > 0 a > 0 (at least until halfway) and constant.
• m m m — Spaceship mass. It is required to calculate kinetic energy (and fuel).
• d d d — Distance to the destination. Note that you can select it from the list or type in any other distance to the desired object.
• T T T — Time that passed in a spaceship, or, in other words, how much the crew has aged.
• t t t — Time that passed in a resting frame of reference, e.g., on Earth.
• v v v — Maximum velocity reached by the spaceship.
• K E \rm KE KE — Maximum kinetic energy reached by the spaceship.

The relativistic space travel calculator is dedicated to very long journeys, interstellar or even intergalactic, in which we can neglect the influence of the gravitational field, e.g., from Earth. We didn't include our closest celestial bodies, like the Moon or Mars, in the destination list because it would be pointless. For them, we need different equations that also take into consideration gravitational force.

Newton's universe — arrive at the destination at full speed

It's the simplest case because here, T T T equals t t t for any speed. To calculate the distance covered at constant acceleration during a certain time, you can use the following classical formula:

Since acceleration is constant, and we assume that the initial velocity equals zero, you can estimate the maximum velocity using this equation:

and the corresponding kinetic energy:

Newton's universe — arrive at the destination and stop

In this situation, we accelerate to the halfway point, reach maximum velocity, and then decelerate to stop at the destination point. Distance covered during the same time is, as you may expect, smaller than before:

Acceleration remains positive until we're halfway there (then it is negative – deceleration), so the maximum velocity is:

and the kinetic energy equation is the same as the previous one.

Einstein's universe — arrive at the destination at full speed

The relativistic rocket equation has to consider the effects of light-speed travel. These are not only speed limitations and time dilation but also how every length becomes shorter for a moving observer, which is a phenomenon of special relativity called length contraction. If l l l is the proper length observed in the rest frame and L L L is the length observed by a crew in a spaceship, then:

What does it mean? If a spaceship moves with the velocity of v = 0.995 c v = 0.995c v = 0.995 c , then γ = 10 \gamma = 10 γ = 10 , and the length observed by a moving object is ten times smaller than the real length. For example, the distance to the Andromeda Galaxy equals about 2,520,000 light years with Earth as the frame of reference. For a spaceship moving with v = 0.995 c v = 0.995c v = 0.995 c , it will be "only" 252,200 light years away. That's a 90 percent decrease or a 164 percent difference!

Now you probably understand why special relativity allows us to intergalactic travel. Below you can find the relativistic rocket equation for the case in which you want to arrive at the destination point at full speed (without stopping). You can find its derivation in the book by Messrs Misner, Thorne ( Co-Winner of the 2017 Nobel Prize in Physics ) and Wheller titled Gravitation , section §6.2. Hyperbolic motion. More accessible formulas are in the mathematical physicist John Baez's article The Relativistic Rocket :

• Time passed on Earth:
• Time passed in the spaceship:
• Maximum velocity:
• Relativistic kinetic energy remains the same:

The symbols sh ⁡ \sh sh , ch ⁡ \ch ch , and th ⁡ \th th are, respectively, sine, cosine, and tangent hyperbolic functions, which are analogs of the ordinary trigonometric functions. In turn, sh ⁡ − 1 \sh^{-1} sh − 1 and ch ⁡ − 1 \ch^{-1} ch − 1 are the inverse hyperbolic functions that can be expressed with natural logarithms and square roots, according to the article Inverse hyperbolic functions on Wikipedia.

Einstein's universe – arrive at destination point and stop

Most websites with relativistic rocket equations consider only arriving at the desired place at full speed. If you want to stop there, you should start decelerating at the halfway point. Below, you can find a set of equations estimating interstellar space travel parameters in the situation when you want to stop at the destination point :

Intergalactic travel – fuel problem

So, after all of these considerations, can humans travel at the speed of light, or at least at a speed close to it? Jet-rocket engines need a lot of fuel per unit of weight of the rocket. You can use our rocket equation calculator to see how much fuel you need to obtain a certain velocity (e.g., with an effective exhaust velocity of 4500 m/s).

Hopefully, future spaceships will be able to produce energy from matter-antimatter annihilation. This process releases energy from two particles that have mass (e.g., electron and positron) into photons. These photons may then be shot out at the back of the spaceship and accelerate the spaceship due to the conservation of momentum. If you want to know how much energy is contained in matter, check out our E = mc² calculator , which is about the famous Albert Einstein equation.

Now that you know the maximum amount of energy you can acquire from matter, it's time to estimate how much of it you need for intergalactic travel. Appropriate formulas are derived from the conservation of momentum and energy principles. For the relativistic case:

where e x e^x e x is an exponential function, and for classical case:

Remember that it assumes 100% efficiency! One of the promising future spaceships' power sources is the fusion of hydrogen into helium, which provides energy of 0.008 mc² . As you can see, in this reaction, efficiency equals only 0.8%.

Let's check whether the fuel mass amount is reasonable for sending a mass of 1 kg to the nearest galaxy. With a space travel calculator, you can find out that, even with 100% efficiency, you would need 5,200 tons of fuel to send only 1 kilogram of your spaceship . That's a lot!

So can humans travel at the speed of light? Right now, it seems impossible, but technology is still developing. For example, a photonic laser thruster is a good candidate since it doesn't require any matter to work, only photons. Infinity and beyond is actually within our reach!

How do I calculate the travel time to other planets?

To calculate the time it takes to travel to a specific star or galaxy using the space travel calculator, follow these steps:

• Choose the acceleration : the default mode is 1 g (gravitational field similar to Earth's).
• Enter the spaceship mass , excluding fuel.
• Select the destination : pick the star, planet, or galaxy you want to travel to from the dropdown menu.
• The distance between the Earth and your chosen stars will automatically appear. You can also input the distance in light-years directly if you select the Custom distance option in the previous dropdown.
• Define the aim : select whether you aim to " Arrive at destination and stop " or “ Arrive at destination at full speed ”.
• Pick the calculation mode : opt for either " Einstein's universe " mode for relativistic effects or " Newton's universe " for simpler calculations.
• Time passed in spaceship : estimated time experienced by the crew during the journey. (" Einstein's universe " mode)
• Time passed on Earth : estimated time elapsed on Earth during the trip. (" Einstein's universe " mode)
• Time passed : depends on the frame of reference, e.g., on Earth. (" Newton's universe " mode)
• Required fuel mass : estimated fuel quantity needed for the journey.
• Maximum velocity : maximum speed achieved by the spaceship.

How long does it take to get to space?

It takes about 8.5 minutes for a space shuttle or spacecraft to reach Earth's orbit, i.e., the limit of space where the Earth's atmosphere ends. This dividing line between the Earth's atmosphere and space is called the Kármán line . It happens so quickly because the shuttle goes from zero to around 17,500 miles per hour in those 8.5 minutes .

How fast does the space station travel?

The International Space Station travels at an average speed of 28,000 km/h or 17,500 mph . In a single day, the ISS can make several complete revolutions as it circumnavigates the globe in just 90 minutes . Placed in orbit at an altitude of 350 km , the station is visible to the naked eye, looking like a dot crossing the sky due to its very bright solar panels.

How do I reach the speed of light?

To reach the speed of light, you would have to overcome several obstacles, including:

Mass limit : traveling at the speed of light would mean traveling at 299,792,458 meters per second. But, thanks to Einstein's theory of relativity, we know that an object with non-zero mass cannot reach this speed.

Energy : accelerating to the speed of light would require infinite energy.

Effects of relativity : from the outside, time would slow down, and you would shrink.

Why can't sound travel in space?

Sound can’t travel in space because it is a mechanical wave that requires a medium to propagate — this medium can be solid, liquid, or gas. In space, there is no matter, or at least not enough for sound to propagate. The density of matter in space is of the order 1 particle per cubic centimeter . While on Earth , it's much denser at around 10 20 particles per cubic centimeter .

Dreaming of traveling into space? 🌌 Plan your interstellar travel (even to a Star Trek destination) using this calculator 👨‍🚀! Estimate how fast you can reach your destination and how much fuel you would need 🚀

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ol{padding-top:0px;}.css-4okk7a ul:not(:first-child),.css-4okk7a ol:not(:first-child){padding-top:4px;} Spaceship and destination 👩‍🚀👨‍🚀

Spaceship acceleration

Spaceship mass

Mass of spaceship excluding fuel.

Destination

Select a destination from the list or type in distance by hand.

Which star/galaxy?

If you want to input your own distance, select the 'Custom destination' option in the 'Which star/galaxy?' field.

Calculation options

Do you want to stop at destination point? If yes, the spaceship will start decelerating once it reaches the halfway point.

Calculations mode

You can compare Einstein's special relativity with non-relativistic Newton's physics. Remember that at near-light speeds only the former is correct!

Travel details 🚀

Time passed in spaceship

Time passed on Earth

Time passed in the resting frame of reference. It could be an observer on Earth.

Required fuel mass

Assuming 100% efficiency.

Maximum velocity

Note that our calculator may round velocity to the speed of light if it is really close to it.

Additional parameters

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Time Travel and Modern Physics

Time travel has been a staple of science fiction. With the advent of general relativity it has been entertained by serious physicists. But, especially in the philosophy literature, there have been arguments that time travel is inherently paradoxical. The most famous paradox is the grandfather paradox: you travel back in time and kill your grandfather, thereby preventing your own existence. To avoid inconsistency some circumstance will have to occur which makes you fail in this attempt to kill your grandfather. Doesn’t this require some implausible constraint on otherwise unrelated circumstances? We examine such worries in the context of modern physics.

1. Paradoxes Lost?

2. topology and constraints, 3. the general possibility of time travel in general relativity, 4. two toy models, 5. slightly more realistic models of time travel, 6. the possibility of time travel redux, 7. even if there are constraints, so what, 8. computational models, 9. quantum mechanics to the rescue, 10. conclusions, other internet resources, related entries.

• Supplement: Remarks and Limitations on the Toy Models

Modern physics strips away many aspects of the manifest image of time. Time as it appears in the equations of classical mechanics has no need for a distinguished present moment, for example. Relativity theory leads to even sharper contrasts. It replaces absolute simultaneity, according to which it is possible to unambiguously determine the time order of distant events, with relative simultaneity: extending an “instant of time” throughout space is not unique, but depends on the state of motion of an observer. More dramatically, in general relativity the mathematical properties of time (or better, of spacetime)—its topology and geometry—depend upon how matter is arranged rather than being fixed once and for all. So physics can be, and indeed has to be, formulated without treating time as a universal, fixed background structure. Since general relativity represents gravity through spacetime geometry, the allowed geometries must be as varied as the ways in which matter can be arranged. Alongside geometrical models used to describe the solar system, black holes, and much else, the scope of variation extends to include some exotic structures unlike anything astrophysicists have observed. In particular, there are spacetime geometries with curves that loop back on themselves: closed timelike curves (CTCs), which describe the possible trajectory of an observer who returns exactly back to their earlier state—without any funny business, such as going faster than the speed of light. These geometries satisfy the relevant physical laws, the equations of general relativity, and in that sense time travel is physically possible.

Yet circular time generates paradoxes, familiar from science fiction stories featuring time travel: [ 1 ]

• Consistency: Kurt plans to murder his own grandfather Adolph, by traveling along a CTC to an appropriate moment in the past. He is an able marksman, and waits until he has a clear shot at grandpa. Normally he would not miss. Yet if he succeeds, there is no way that he will then exist to plan and carry out the mission. Kurt pulls the trigger: what can happen?
• Underdetermination: Suppose that Kurt first travels back in order to give his earlier self a copy of How to Build a Time Machine. This is the same book that allows him to build a time machine, which he then carries with him on his journey to the past. Who wrote the book?
• Easy Knowledge: A fan of classical music enhances their computer with a circuit that exploits a CTC. This machine efficiently solves problems at a higher level of computational complexity than conventional computers, leading (among other things) to finding the smallest circuits that can generate Bach’s oeuvre—and to compose new pieces in the same style. Such easy knowledge is at odds with our understanding of our epistemic predicament. (This third paradox has not drawn as much attention.)

The first two paradoxes were once routinely taken to show that solutions with CTCs should be rejected—with charges varying from violating logic, to being “physically unreasonable”, to undermining the notion of free will. Closer analysis of the paradoxes has largely reversed this consensus. Physicists have discovered many solutions with CTCs and have explored their properties in pursuing foundational questions, such as whether physics is compatible with the idea of objective temporal passage (starting with Gödel 1949). Philosophers have also used time travel scenarios to probe questions about, among other things, causation, modality, free will, and identity (see, e.g., Earman 1972 and Lewis’s seminal 1976 paper).

We begin below with Consistency , turning to the other paradoxes in later sections. A standard, stone-walling response is to insist that the past cannot be changed, as a matter of logic, even by a time traveler (e.g., Gödel 1949, Clarke 1977, Horwich 1987). Adolph cannot both die and survive, as a matter of logic, so any scheme to alter the past must fail. In many of the best time travel fictions, the actions of a time traveler are constrained in novel and unexpected ways. Attempts to change the past fail, and they fail, often tragically, in just such a way that they set the stage for the time traveler’s self-defeating journey. The first question is whether there is an analog of the consistent story when it comes to physics in the presence of CTCs. As we will see, there is a remarkable general argument establishing the existence of consistent solutions. Yet a second question persists: why can’t time-traveling Kurt kill his own grandfather? Doesn’t the necessity of failures to change the past put unusual and unexpected constraints on time travelers, or objects that move along CTCs? The same argument shows that there are in fact no constraints imposed by the existence of CTCs, in some cases. After discussing this line of argument, we will turn to the palatability and further implications of such constraints if they are required, and then turn to the implications of quantum mechanics.

Wheeler and Feynman (1949) were the first to claim that the fact that nature is continuous could be used to argue that causal influences from later events to earlier events, as are made possible by time travel, will not lead to paradox without the need for any constraints. Maudlin (1990) showed how to make their argument precise and more general, and argued that nonetheless it was not completely general.

Imagine the following set-up. We start off having a camera with a black and white film ready to take a picture of whatever comes out of the time machine. An object, in fact a developed film, comes out of the time machine. We photograph it, and develop the film. The developed film is subsequently put in the time machine, and set to come out of the time machine at the time the picture is taken. This surely will create a paradox: the developed film will have the opposite distribution of black, white, and shades of gray, from the object that comes out of the time machine. For developed black and white films (i.e., negatives) have the opposite shades of gray from the objects they are pictures of. But since the object that comes out of the time machine is the developed film itself it we surely have a paradox.

However, it does not take much thought to realize that there is no paradox here. What will happen is that a uniformly gray picture will emerge, which produces a developed film that has exactly the same uniform shade of gray. No matter what the sensitivity of the film is, as long as the dependence of the brightness of the developed film depends in a continuous manner on the brightness of the object being photographed, there will be a shade of gray that, when photographed, will produce exactly the same shade of gray on the developed film. This is the essence of Wheeler and Feynman’s idea. Let us first be a bit more precise and then a bit more general.

For simplicity let us suppose that the film is always a uniform shade of gray (i.e., at any time the shade of gray does not vary by location on the film). The possible shades of gray of the film can then be represented by the (real) numbers from 0, representing pure black, to 1, representing pure white.

Let us now distinguish various stages in the chronological order of the life of the film. In stage \(S_1\) the film is young; it has just been placed in the camera and is ready to be exposed. It is then exposed to the object that comes out of the time machine. (That object in fact is a later stage of the film itself). By the time we come to stage \(S_2\) of the life of the film, it has been developed and is about to enter the time machine. Stage \(S_3\) occurs just after it exits the time machine and just before it is photographed. Stage \(S_4\) occurs after it has been photographed and before it starts fading away. Let us assume that the film starts out in stage \(S_1\) in some uniform shade of gray, and that the only significant change in the shade of gray of the film occurs between stages \(S_1\) and \(S_2\). During that period it acquires a shade of gray that depends on the shade of gray of the object that was photographed. In other words, the shade of gray that the film acquires at stage \(S_2\) depends on the shade of gray it has at stage \(S_3\). The influence of the shade of gray of the film at stage \(S_3\), on the shade of gray of the film at stage \(S_2\), can be represented as a mapping, or function, from the real numbers between 0 and 1 (inclusive), to the real numbers between 0 and 1 (inclusive). Let us suppose that the process of photography is such that if one imagines varying the shade of gray of an object in a smooth, continuous manner then the shade of gray of the developed picture of that object will also vary in a smooth, continuous manner. This implies that the function in question will be a continuous function. Now any continuous function from the real numbers between 0 and 1 (inclusive) to the real numbers between 0 and 1 (inclusive) must map at least one number to itself. One can quickly convince oneself of this by graphing such functions. For one will quickly see that any continuous function \(f\) from \([0,1]\) to \([0,1]\) must intersect the line \(x=y\) somewhere, and thus there must be at least one point \(x\) such that \(f(x)=x\). Such points are called fixed points of the function. Now let us think about what such a fixed point represents. It represents a shade of gray such that, when photographed, it will produce a developed film with exactly that same shade of gray. The existence of such a fixed point implies a solution to the apparent paradox.

Let us now be more general and allow color photography. One can represent each possible color of an object (of uniform color) by the proportions of blue, green and red that make up that color. (This is why television screens can produce all possible colors.) Thus one can represent all possible colors of an object by three points on three orthogonal lines \(x, y\) and \(z\), that is to say, by a point in a three-dimensional cube. This cube is also known as the “Cartesian product” of the three line segments. Now, one can also show that any continuous map from such a cube to itself must have at least one fixed point. So color photography can not be used to create time travel paradoxes either!

Even more generally, consider some system \(P\) which, as in the above example, has the following life. It starts in some state \(S_1\), it interacts with an object that comes out of a time machine (which happens to be its older self), it travels back in time, it interacts with some object (which happens to be its younger self), and finally it grows old and dies. Let us assume that the set of possible states of \(P\) can be represented by a Cartesian product of \(n\) closed intervals of the reals, i.e., let us assume that the topology of the state-space of \(P\) is isomorphic to a finite Cartesian product of closed intervals of the reals. Let us further assume that the development of \(P\) in time, and the dependence of that development on the state of objects that it interacts with, is continuous. Then, by a well-known fixed point theorem in topology (see, e.g., Hocking & Young 1961: 273), no matter what the nature of the interaction is, and no matter what the initial state of the object is, there will be at least one state \(S_3\) of the older system (as it emerges from the time travel machine) that will influence the initial state \(S_1\) of the younger system (when it encounters the older system) so that, as the younger system becomes older, it develops exactly into state \(S_3\). Thus without imposing any constraints on the initial state \(S_1\) of the system \(P\), we have shown that there will always be perfectly ordinary, non-paradoxical, solutions, in which everything that happens, happens according to the usual laws of development. Of course, there is looped causation, hence presumably also looped explanation, but what do you expect if there is looped time?

Unfortunately, for the fan of time travel, a little reflection suggests that there are systems for which the needed fixed point theorem does not hold. Imagine, for instance, that we have a dial that can only rotate in a plane. We are going to put the dial in the time machine. Indeed we have decided that if we see the later stage of the dial come out of the time machine set at angle \(x\), then we will set the dial to \(x+90\), and throw it into the time machine. Now it seems we have a paradox, since the mapping that consists of a rotation of all points in a circular state-space by 90 degrees does not have a fixed point. And why wouldn’t some state-spaces have the topology of a circle?

However, we have so far not used another continuity assumption which is also a reasonable assumption. So far we have only made the following demand: the state the dial is in at stage \(S_2\) must be a continuous function of the state of the dial at stage \(S_3\). But, the state of the dial at stage \(S_2\) is arrived at by taking the state of the dial at stage \(S_1\), and rotating it over some angle. It is not merely the case that the effect of the interaction, namely the state of the dial at stage \(S_2\), should be a continuous function of the cause, namely the state of the dial at stage \(S_3\). It is additionally the case that path taken to get there, the way the dial is rotated between stages \(S_1\) and \(S_2\) must be a continuous function of the state at stage \(S_3\). And, rather surprisingly, it turns out that this can not be done. Let us illustrate what the problem is before going to a more general demonstration that there must be a fixed point solution in the dial case.

Forget time travel for the moment. Suppose that you and I each have a watch with a single dial neither of which is running. My watch is set at 12. You are going to announce what your watch is set at. My task is going to be to adjust my watch to yours no matter what announcement you make. And my actions should have a continuous (single valued) dependence on the time that you announce. Surprisingly, this is not possible! For instance, suppose that if you announce “12”, then I achieve that setting on my watch by doing nothing. Now imagine slowly and continuously increasing the announced times, starting at 12. By continuity, I must achieve each of those settings by rotating my dial to the right. If at some point I switch and achieve the announced goal by a rotation of my dial to the left, I will have introduced a discontinuity in my actions, a discontinuity in the actions that I take as a function of the announced angle. So I will be forced, by continuity, to achieve every announcement by rotating the dial to the right. But, this rotation to the right will have to be abruptly discontinued as the announcements grow larger and I eventually approach 12 again, since I achieved 12 by not rotating the dial at all. So, there will be a discontinuity at 12 at the latest. In general, continuity of my actions as a function of announced times can not be maintained throughout if I am to be able to replicate all possible settings. Another way to see the problem is that one can similarly reason that, as one starts with 12, and imagines continuously making the announced times earlier, one will be forced, by continuity, to achieve the announced times by rotating the dial to the left. But the conclusions drawn from the assumption of continuous increases and the assumption of continuous decreases are inconsistent. So we have an inconsistency following from the assumption of continuity and the assumption that I always manage to set my watch to your watch. So, a dial developing according to a continuous dynamics from a given initial state, can not be set up so as to react to a second dial, with which it interacts, in such a way that it is guaranteed to always end up set at the same angle as the second dial. Similarly, it can not be set up so that it is guaranteed to always end up set at 90 degrees to the setting of the second dial. All of this has nothing to do with time travel. However, the impossibility of such set ups is what prevents us from enacting the rotation by 90 degrees that would create paradox in the time travel setting.

Let us now give the positive result that with such dials there will always be fixed point solutions, as long as the dynamics is continuous. Let us call the state of the dial before it interacts with its older self the initial state of the dial. And let us call the state of the dial after it emerges from the time machine the final state of the dial. There is also an intermediate state of the dial, after it interacts with its older self and before it is put into the time machine. We can represent the initial or intermediate states of the dial, before it goes into the time machine, as an angle \(x\) in the horizontal plane and the final state of the dial, after it comes out of the time machine, as an angle \(y\) in the vertical plane. All possible \(\langle x,y\rangle\) pairs can thus be visualized as a torus with each \(x\) value picking out a vertical circular cross-section and each \(y\) picking out a point on that cross-section. See figure 1 .

Figure 1 [An extended description of figure 1 is in the supplement.]

Suppose that the dial starts at angle \(i\) which picks out vertical circle \(I\) on the torus. The initial angle \(i\) that the dial is at before it encounters its older self, and the set of all possible final angles that the dial can have when it emerges from the time machine is represented by the circle \(I\) on the torus (see figure 1 ). Given any possible angle of the emerging dial, the dial initially at angle \(i\) will develop to some other angle. One can picture this development by rotating each point on \(I\) in the horizontal direction by the relevant amount. Since the rotation has to depend continuously on the angle of the emerging dial, circle \(I\) during this development will deform into some loop \(L\) on the torus. Loop \(L\) thus represents all possible intermediate angles \(x\) that the dial is at when it is thrown into the time machine, given that it started at angle \(i\) and then encountered a dial (its older self) which was at angle \(y\) when it emerged from the time machine. We therefore have consistency if \(x=y\) for some \(x\) and \(y\) on loop \(L\). Now, let loop \(C\) be the loop which consists of all the points on the torus for which \(x=y\). Ring \(I\) intersects \(C\) at point \(\langle i,i\rangle\). Obviously any continuous deformation of \(I\) must still intersect \(C\) somewhere. So \(L\) must intersect \(C\) somewhere, say at \(\langle j,j\rangle\). But that means that no matter how the development of the dial starting at \(I\) depends on the angle of the emerging dial, there will be some angle for the emerging dial such that the dial will develop exactly into that angle (by the time it enters the time machine) under the influence of that emerging dial. This is so no matter what angle one starts with, and no matter how the development depends on the angle of the emerging dial. Thus even for a circular state-space there are no constraints needed other than continuity.

Unfortunately there are state-spaces that escape even this argument. Consider for instance a pointer that can be set to all values between 0 and 1, where 0 and 1 are not possible values. That is, suppose that we have a state-space that is isomorphic to an open set of real numbers. Now suppose that we have a machine that sets the pointer to half the value that the pointer is set at when it emerges from the time machine.

Figure 2 [An extended description of figure 2 is in the supplement.]

Suppose the pointer starts at value \(I\). As before we can represent the combination of this initial position and all possible final positions by the line \(I\). Under the influence of the pointer coming out of the time machine the pointer value will develop to a value that equals half the value of the final value that it encountered. We can represent this development as the continuous deformation of line \(I\) into line \(L\), which is indicated by the arrows in figure 2 . This development is fully continuous. Points \(\langle x,y\rangle\) on line \(I\) represent the initial position \(x=I\) of the (young) pointer, and the position \(y\) of the older pointer as it emerges from the time machine. Points \(\langle x,y\rangle\) on line \(L\) represent the position \(x\) that the younger pointer should develop into, given that it encountered the older pointer emerging from the time machine set at position \(y\). Since the pointer is designed to develop to half the value of the pointer that it encounters, the line \(L\) corresponds to \(x=1/2 y\). We have consistency if there is some point such that it develops into that point, if it encounters that point. Thus, we have consistency if there is some point \(\langle x,y\rangle\) on line \(L\) such that \(x=y\). However, there is no such point: lines \(L\) and \(C\) do not intersect. Thus there is no consistent solution, despite the fact that the dynamics is fully continuous.

Of course if 0 were a possible value, \(L\) and \(C\) would intersect at 0. This is surprising and strange: adding one point to the set of possible values of a quantity here makes the difference between paradox and peace. One might be tempted to just add the extra point to the state-space in order to avoid problems. After all, one might say, surely no measurements could ever tell us whether the set of possible values includes that exact point or not. Unfortunately there can be good theoretical reasons for supposing that some quantity has a state-space that is open: the set of all possible speeds of massive objects in special relativity surely is an open set, since it includes all speeds up to, but not including, the speed of light. Quantities that have possible values that are not bounded also lead to counter examples to the presented fixed point argument. And it is not obvious to us why one should exclude such possibilities. So the argument that no constraints are needed is not fully general.

An interesting question of course is: exactly for which state-spaces must there be such fixed points? The arguments above depend on a well-known fixed point theorem (due to Schauder) that guarantees the existence of a fixed point for compact, convex state spaces. We do not know what subsequent extensions of this result imply regarding fixed points for a wider variety of systems, or whether there are other general results along these lines. (See Kutach 2003 for more on this issue.)

A further interesting question is whether this line of argument is sufficient to resolve Consistency (see also Dowe 2007). When they apply, these results establish the existence of a solution, such as the shade of uniform gray in the first example. But physicists routinely demand more than merely the existence of a solution, namely that solutions to the equations are stable—such that “small” changes of the initial state lead to “small” changes of the resulting trajectory. (Clarifying the two senses of “small” in this statement requires further work, specifying the relevant topology.) Stability in this sense underwrites the possibility of applying equations to real systems given our inability to fix initial states with indefinite precision. (See Fletcher 2020 for further discussion.) The fixed point theorems guarantee that for an initial state \(S_1\) there is a solution, but this solution may not be “close” to the solution for a nearby initial state, \(S'\). We are not aware of any proofs that the solutions guaranteed to exist by the fixed point theorems are also stable in this sense.

Time travel has recently been discussed quite extensively in the context of general relativity. General relativity places few constraints on the global structure of space and time. This flexibility leads to a possibility first described in print by Hermann Weyl:

Every world-point is the origin of the double-cone of the active future and the passive past [i.e., the two lobes of the light cone]. Whereas in the special theory of relativity these two portions are separated by an intervening region, it is certainly possible in the present case [i.e., general relativity] for the cone of the active future to overlap with that of the passive past; so that, in principle, it is possible to experience events now that will in part be an effect of my future resolves and actions. Moreover, it is not impossible for a world-line (in particular, that of my body), although it has a timelike direction at every point, to return to the neighborhood of a point which it has already once passed through. (Weyl 1918/1920 [1952: 274])

A time-like curve is simply a space-time trajectory such that the speed of light is never equaled or exceeded along this trajectory. Time-like curves represent possible trajectories of ordinary objects. In general relativity a curve that is everywhere timelike locally can nonetheless loop back on itself, forming a CTC. Weyl makes the point vividly in terms of the light cones: along such a curve, the future lobe of the light cone (the “active future”) intersects the past lobe of the light cone (the “passive past”). Traveling along such a curve one would never exceed the speed of light, and yet after a certain amount of (proper) time one would return to a point in space-time that one previously visited. Or, by staying close to such a CTC, one could come arbitrarily close to a point in space-time that one previously visited. General relativity, in a straightforward sense, allows time travel: there appear to be many space-times compatible with the fundamental equations of general relativity in which there are CTC’s. Space-time, for instance, could have a Minkowski metric everywhere, and yet have CTC’s everywhere by having the temporal dimension (topologically) rolled up as a circle. Or, one can have wormhole connections between different parts of space-time which allow one to enter “mouth \(A\)” of such a wormhole connection, travel through the wormhole, exit the wormhole at “mouth \(B\)” and re-enter “mouth \(A\)” again. CTCs can even arise when the spacetime is topologically \(\mathbb{R}^4\), due to the “tilting” of light cones produced by rotating matter (as in Gödel 1949’s spacetime).

General relativity thus appears to provide ample opportunity for time travel. Note that just because there are CTC’s in a space-time, this does not mean that one can get from any point in the space-time to any other point by following some future directed timelike curve—there may be insurmountable practical obstacles. In Gödel’s spacetime, it is the case that there are CTCs passing through every point in the spacetime. Yet these CTCs are not geodesics, so traversing them requires acceleration. Calculations of the minimal fuel required to travel along the appropriate curve should discourage any would-be time travelers (Malament 1984, 1985; Manchak 2011). But more generally CTCs may be confined to smaller regions; some parts of space-time can have CTC’s while other parts do not. Let us call the part of a space-time that has CTC’s the “time travel region” of that space-time, while calling the rest of that space-time the “normal region”. More precisely, the “time travel region” consists of all the space-time points \(p\) such that there exists a (non-zero length) timelike curve that starts at \(p\) and returns to \(p\). Now let us turn to examining space-times with CTC’s a bit more closely for potential problems.

In order to get a feeling for the sorts of implications that closed timelike curves can have, it may be useful to consider two simple models. In space-times with closed timelike curves the traditional initial value problem cannot be framed in the usual way. For it presupposes the existence of Cauchy surfaces, and if there are CTCs then no Cauchy surface exists. (A Cauchy surface is a spacelike surface such that every inextendable timelike curve crosses it exactly once. One normally specifies initial conditions by giving the conditions on such a surface.) Nonetheless, if the topological complexities of the manifold are appropriately localized, we can come quite close. Let us call an edgeless spacelike surface \(S\) a quasi-Cauchy surface if it divides the rest of the manifold into two parts such that

• every point in the manifold can be connected by a timelike curve to \(S\), and
• any timelike curve which connects a point in one region to a point in the other region intersects \(S\) exactly once.

It is obvious that a quasi-Cauchy surface must entirely inhabit the normal region of the space-time; if any point \(p\) of \(S\) is in the time travel region, then any timelike curve which intersects \(p\) can be extended to a timelike curve which intersects \(S\) near \(p\) again. In extreme cases of time travel, a model may have no normal region at all (e.g., Minkowski space-time rolled up like a cylinder in a time-like direction), in which case our usual notions of temporal precedence will not apply. But temporal anomalies like wormholes (and time machines) can be sufficiently localized to permit the existence of quasi-Cauchy surfaces.

Given a timelike orientation, a quasi-Cauchy surface unproblematically divides the manifold into its past (i.e., all points that can be reached by past-directed timelike curves from \(S)\) and its future (ditto mutatis mutandis ). If the whole past of \(S\) is in the normal region of the manifold, then \(S\) is a partial Cauchy surface : every inextendable timelike curve which exists to the past of \(S\) intersects \(S\) exactly once, but (if there is time travel in the future) not every inextendable timelike curve which exists to the future of \(S\) intersects \(S\). Now we can ask a particularly clear question: consider a manifold which contains a time travel region, but also has a partial Cauchy surface \(S\), such that all of the temporal funny business is to the future of \(S\). If all you could see were \(S\) and its past, you would not know that the space-time had any time travel at all. The question is: are there any constraints on the sort of data which can be put on \(S\) and continued to a global solution of the dynamics which are different from the constraints (if any) on the data which can be put on a Cauchy surface in a simply connected manifold and continued to a global solution? If there is time travel to our future, might we we able to tell this now, because of some implied oddity in the arrangement of present things?

It is not at all surprising that there might be constraints on the data which can be put on a locally space-like surface which passes through the time travel region: after all, we never think we can freely specify what happens on a space-like surface and on another such surface to its future, but in this case the surface at issue lies to its own future. But if there were particular constraints for data on a partial Cauchy surface then we would apparently need to have to rule out some sorts of otherwise acceptable states on \(S\) if there is to be time travel to the future of \(S\). We then might be able to establish that there will be no time travel in the future by simple inspection of the present state of the universe. As we will see, there is reason to suspect that such constraints on the partial Cauchy surface are non-generic. But we are getting ahead of ourselves: first let’s consider the effect of time travel on a very simple dynamics.

The simplest possible example is the Newtonian theory of perfectly elastic collisions among equally massive particles in one spatial dimension. The space-time is two-dimensional, so we can represent it initially as the Euclidean plane, and the dynamics is completely specified by two conditions. When particles are traveling freely, their world lines are straight lines in the space-time, and when two particles collide, they exchange momenta, so the collision looks like an “\(X\)” in space-time, with each particle changing its momentum at the impact. [ 2 ] The dynamics is purely local, in that one can check that a set of world-lines constitutes a model of the dynamics by checking that the dynamics is obeyed in every arbitrarily small region. It is also trivial to generate solutions from arbitrary initial data if there are no CTCs: given the initial positions and momenta of a set of particles, one simply draws a straight line from each particle in the appropriate direction and continues it indefinitely. Once all the lines are drawn, the worldline of each particle can be traced from collision to collision. The boundary value problem for this dynamics is obviously well-posed: any set of data at an instant yields a unique global solution, constructed by the method sketched above.

What happens if we change the topology of the space-time by hand to produce CTCs? The simplest way to do this is depicted in figure 3 : we cut and paste the space-time so it is no longer simply connected by identifying the line \(L-\) with the line \(L+\). Particles “going in” to \(L+\) from below “emerge” from \(L-\) , and particles “going in” to \(L-\) from below “emerge” from \(L+\).

Figure 3: Inserting CTCs by Cut and Paste. [An extended description of figure 3 is in the supplement.]

How is the boundary-value problem changed by this alteration in the space-time? Before the cut and paste, we can put arbitrary data on the simultaneity slice \(S\) and continue it to a unique solution. After the change in topology, \(S\) is no longer a Cauchy surface, since a CTC will never intersect it, but it is a partial Cauchy surface. So we can ask two questions. First, can arbitrary data on \(S\) always be continued to a global solution? Second, is that solution unique? If the answer to the first question is \(no\), then we have a backward-temporal constraint: the existence of the region with CTCs places constraints on what can happen on \(S\) even though that region lies completely to the future of \(S\). If the answer to the second question is \(no\), then we have an odd sort of indeterminism, analogous to the unwritten book: the complete physical state on \(S\) does not determine the physical state in the future, even though the local dynamics is perfectly deterministic and even though there is no other past edge to the space-time region in \(S\)’s future (i.e., there is nowhere else for boundary values to come from which could influence the state of the region).

In this case the answer to the first question is yes and to the second is no : there are no constraints on the data which can be put on \(S\), but those data are always consistent with an infinitude of different global solutions. The easy way to see that there always is a solution is to construct the minimal solution in the following way. Start drawing straight lines from \(S\) as required by the initial data. If a line hits \(L-\) from the bottom, just continue it coming out of the top of \(L+\) in the appropriate place, and if a line hits \(L+\) from the bottom, continue it emerging from \(L-\) at the appropriate place. Figure 4 represents the minimal solution for a single particle which enters the time-travel region from the left:

Figure 4: The Minimal Solution. [An extended description of figure 4 is in the supplement.]

The particle “travels back in time” three times. It is obvious that this minimal solution is a global solution, since the particle always travels inertially.

But the same initial state on \(S\) is also consistent with other global solutions. The new requirement imposed by the topology is just that the data going into \(L+\) from the bottom match the data coming out of \(L-\) from the top, and the data going into \(L-\) from the bottom match the data coming out of \(L+\) from the top. So we can add any number of vertical lines connecting \(L-\) and \(L+\) to a solution and still have a solution. For example, adding a few such lines to the minimal solution yields:

Figure 5: A Non-Minimal Solution. [An extended description of figure 5 is in the supplement.]

The particle now collides with itself twice: first before it reaches \(L+\) for the first time, and again shortly before it exits the CTC region. From the particle’s point of view, it is traveling to the right at a constant speed until it hits an older version of itself and comes to rest. It remains at rest until it is hit from the right by a younger version of itself, and then continues moving off, and the same process repeats later. It is clear that this is a global model of the dynamics, and that any number of distinct models could be generating by varying the number and placement of vertical lines.

Knowing the data on \(S\), then, gives us only incomplete information about how things will go for the particle. We know that the particle will enter the CTC region, and will reach \(L+\), we know that it will be the only particle in the universe, we know exactly where and with what speed it will exit the CTC region. But we cannot determine how many collisions the particle will undergo (if any), nor how long (in proper time) it will stay in the CTC region. If the particle were a clock, we could not predict what time it would indicate when exiting the region. Furthermore, the dynamics gives us no handle on what to think of the various possibilities: there are no probabilities assigned to the various distinct possible outcomes.

Changing the topology has changed the mathematics of the situation in two ways, which tend to pull in opposite directions. On the one hand, \(S\) is no longer a Cauchy surface, so it is perhaps not surprising that data on \(S\) do not suffice to fix a unique global solution. But on the other hand, there is an added constraint: data “coming out” of \(L-\) must exactly match data “going in” to \(L+\), even though what comes out of \(L-\) helps to determine what goes into \(L+\). This added consistency constraint tends to cut down on solutions, although in this case the additional constraint is more than outweighed by the freedom to consider various sorts of data on \({L+}/{L-}\).

The fact that the extra freedom outweighs the extra constraint also points up one unexpected way that the supposed paradoxes of time travel may be overcome. Let’s try to set up a paradoxical situation using the little closed time loop above. If we send a single particle into the loop from the left and do nothing else, we know exactly where it will exit the right side of the time travel region. Now suppose we station someone at the other side of the region with the following charge: if the particle should come out on the right side, the person is to do something to prevent the particle from going in on the left in the first place. In fact, this is quite easy to do: if we send a particle in from the right, it seems that it can exit on the left and deflect the incoming left-hand particle.

Carrying on our reflection in this way, we further realize that if the particle comes out on the right, we might as well send it back in order to deflect itself from entering in the first place. So all we really need to do is the following: set up a perfectly reflecting particle mirror on the right-hand side of the time travel region, and launch the particle from the left so that— if nothing interferes with it —it will just barely hit \(L+\). Our paradox is now apparently complete. If, on the one hand, nothing interferes with the particle it will enter the time-travel region on the left, exit on the right, be reflected from the mirror, re-enter from the right, and come out on the left to prevent itself from ever entering. So if it enters, it gets deflected and never enters. On the other hand, if it never enters then nothing goes in on the left, so nothing comes out on the right, so nothing is reflected back, and there is nothing to deflect it from entering. So if it doesn’t enter, then there is nothing to deflect it and it enters. If it enters, then it is deflected and doesn’t enter; if it doesn’t enter then there is nothing to deflect it and it enters: paradox complete.

But at least one solution to the supposed paradox is easy to construct: just follow the recipe for constructing the minimal solution, continuing the initial trajectory of the particle (reflecting it the mirror in the obvious way) and then read of the number and trajectories of the particles from the resulting diagram. We get the result of figure 6 :

Figure 6: Resolving the “Paradox”. [An extended description of figure 6 is in the supplement.]

As we can see, the particle approaching from the left never reaches \(L+\): it is deflected first by a particle which emerges from \(L-\). But it is not deflected by itself , as the paradox suggests, it is deflected by another particle. Indeed, there are now four particles in the diagram: the original particle and three particles which are confined to closed time-like curves. It is not the leftmost particle which is reflected by the mirror, nor even the particle which deflects the leftmost particle; it is another particle altogether.

The paradox gets it traction from an incorrect presupposition. If there is only one particle in the world at \(S\) then there is only one particle which could participate in an interaction in the time travel region: the single particle would have to interact with its earlier (or later) self. But there is no telling what might come out of \(L-\): the only requirement is that whatever comes out must match what goes in at \(L+\). So if you go to the trouble of constructing a working time machine, you should be prepared for a different kind of disappointment when you attempt to go back and kill yourself: you may be prevented from entering the machine in the first place by some completely unpredictable entity which emerges from it. And once again a peculiar sort of indeterminism appears: if there are many self-consistent things which could prevent you from entering, there is no telling which is even likely to materialize. This is just like the case of the unwritten book: the book is never written, so nothing determines what fills its pages.

So when the freedom to put data on \(L-\) outweighs the constraint that the same data go into \(L+\), instead of paradox we get an embarrassment of riches: many solution consistent with the data on \(S\), or many possible books. To see a case where the constraint “outweighs” the freedom, we need to construct a very particular, and frankly artificial, dynamics and topology. Consider the space of all linear dynamics for a scalar field on a lattice. (The lattice can be though of as a simple discrete space-time.) We will depict the space-time lattice as a directed graph. There is to be a scalar field defined at every node of the graph, whose value at a given node depends linearly on the values of the field at nodes which have arrows which lead to it. Each edge of the graph can be assigned a weighting factor which determines how much the field at the input node contributes to the field at the output node. If we name the nodes by the letters a , b , c , etc., and the edges by their endpoints in the obvious way, then we can label the weighting factors by the edges they are associated with in an equally obvious way.

Suppose that the graph of the space-time lattice is acyclic , as in figure 7 . (A graph is Acyclic if one can not travel in the direction of the arrows and go in a loop.)

Figure 7: An Acyclic Lattice. [An extended description of figure 7 is in the supplement.]

It is easy to regard a set of nodes as the analog of a Cauchy surface, e.g., the set \(\{a, b, c\}\), and it is obvious if arbitrary data are put on those nodes the data will generate a unique solution in the future. [ 3 ] If the value of the field at node \(a\) is 3 and at node \(b\) is 7, then its value at node \(d\) will be \(3W_{ad}\) and its value at node \(e\) will be \(3W_{ae} + 7W_{be}\). By varying the weighting factors we can adjust the dynamics, but in an acyclic graph the future evolution of the field will always be unique.

Let us now again artificially alter the topology of the lattice to admit CTCs, so that the graph now is cyclic. One of the simplest such graphs is depicted in figure 8 : there are now paths which lead from \(z\) back to itself, e.g., \(z\) to \(y\) to \(z\).

Figure 8: Time Travel on a Lattice. [An extended description of figure 8 is in the supplement.]

Can we now put arbitrary data on \(v\) and \(w\), and continue that data to a global solution? Will the solution be unique?

In the generic case, there will be a solution and the solution will be unique. The equations for the value of the field at \(x, y\), and \(z\) are:

Solving these equations for \(z\) yields

which gives a unique value for \(z\) in the generic case. But looking at the space of all possible dynamics for this lattice (i.e., the space of all possible weighting factors), we find a singularity in the case where \(1-W_{zx}W_{xz} - W_{zy}W_{yz} = 0\). If we choose weighting factors in just this way, then arbitrary data at \(v\) and \(w\) cannot be continued to a global solution. Indeed, if the scalar field is everywhere non-negative, then this particular choice of dynamics puts ironclad constraints on the value of the field at \(v\) and \(w\): the field there must be zero (assuming \(W_{vx}\) and \(W_{wy}\) to be non-zero), and similarly all nodes in their past must have field value zero. If the field can take negative values, then the values at \(v\) and \(w\) must be so chosen that \(vW_{vx}W_{xz} = -wW_{wy}W_{yz}\). In either case, the field values at \(v\) and \(w\) are severely constrained by the existence of the CTC region even though these nodes lie completely to the past of that region. It is this sort of constraint which we find to be unlike anything which appears in standard physics.

Our toy models suggest three things. The first is that it may be impossible to prove in complete generality that arbitrary data on a partial Cauchy surface can always be continued to a global solution: our artificial case provides an example where it cannot. The second is that such odd constraints are not likely to be generic: we had to delicately fine-tune the dynamics to get a problem. The third is that the opposite problem, namely data on a partial Cauchy surface being consistent with many different global solutions, is likely to be generic: we did not have to do any fine-tuning to get this result.

This third point leads to a peculiar sort of indeterminism, illustrated by the case of the unwritten book: the entire state on \(S\) does not determine what will happen in the future even though the local dynamics is deterministic and there are no other “edges” to space-time from which data could influence the result. What happens in the time travel region is constrained but not determined by what happens on \(S\), and the dynamics does not even supply any probabilities for the various possibilities. The example of the photographic negative discussed in section 2, then, seems likely to be unusual, for in that case there is a unique fixed point for the dynamics, and the set-up plus the dynamical laws determine the outcome. In the generic case one would rather expect multiple fixed points, with no room for anything to influence, even probabilistically, which would be realized. (See the supplement on

Remarks and Limitations on the Toy Models .

It is ironic that time travel should lead generically not to contradictions or to constraints (in the normal region) but to underdetermination of what happens in the time travel region by what happens everywhere else (an underdetermination tied neither to a probabilistic dynamics nor to a free edge to space-time). The traditional objection to time travel is that it leads to contradictions: there is no consistent way to complete an arbitrarily constructed story about how the time traveler intends to act. Instead, though, it appears that the more significant problem is underdetermination: the story can be consistently completed in many different ways.

Echeverria, Klinkhammer, and Thorne (1991) considered the case of 3-dimensional single hard spherical ball that can go through a single time travel wormhole so as to collide with its younger self.

Figure 9 [An extended description of figure 9 is in the supplement.]

The threat of paradox in this case arises in the following form. Consider the initial trajectory of a ball as it approaches the time travel region. For some initial trajectories, the ball does not undergo a collision before reaching mouth 1, but upon exiting mouth 2 it will collide with its earlier self. This leads to a contradiction if the collision is strong enough to knock the ball off its trajectory and deflect it from entering mouth 1. Of course, the Wheeler-Feynman strategy is to look for a “glancing blow” solution: a collision which will produce exactly the (small) deviation in trajectory of the earlier ball that produces exactly that collision. Are there always such solutions? [ 4 ]

Echeverria, Klinkhammer & Thorne found a large class of initial trajectories that have consistent “glancing blow” continuations, and found none that do not (but their search was not completely general). They did not produce a rigorous proof that every initial trajectory has a consistent continuation, but suggested that it is very plausible that every initial trajectory has a consistent continuation. That is to say, they have made it very plausible that, in the billiard ball wormhole case, the time travel structure of such a wormhole space-time does not result in constraints on states on spacelike surfaces in the non-time travel region.

In fact, as one might expect from our discussion in the previous section, they found the opposite problem from that of inconsistency: they found underdetermination. For a large class of initial trajectories there are multiple different consistent “glancing blow” continuations of that trajectory (many of which involve multiple wormhole traversals). For example, if one initially has a ball that is traveling on a trajectory aimed straight between the two mouths, then one obvious solution is that the ball passes between the two mouths and never time travels. But another solution is that the younger ball gets knocked into mouth 1 exactly so as to come out of mouth 2 and produce that collision. Echeverria et al. do not note the possibility (which we pointed out in the previous section) of the existence of additional balls in the time travel region. We conjecture (but have no proof) that for every initial trajectory of \(A\) there are some, and generically many, multiple-ball continuations.

Friedman, Morris, et al. (1990) examined the case of source-free non-self-interacting scalar fields traveling through such a time travel wormhole and found that no constraints on initial conditions in the non-time travel region are imposed by the existence of such time travel wormholes. In general there appear to be no known counter examples to the claim that in “somewhat realistic” time-travel space-times with a partial Cauchy surface there are no constraints imposed on the state on such a partial Cauchy surface by the existence of CTC’s. (See, e.g., Friedman & Morris 1991; Thorne 1994; Earman 1995; Earman, Smeenk, & Wüthrich 2009; and Dowe 2007.)

How about the issue of constraints in the time travel region \(T\)? Prima facie , constraints in such a region would not appear to be surprising. But one might still expect that there should be no constraints on states on a spacelike surface, provided one keeps the surface “small enough”. In the physics literature the following question has been asked: for any point \(p\) in \(T\), and any space-like surface \(S\) that includes \(p\) is there a neighborhood \(E\) of \(p\) in \(S\) such that any solution on \(E\) can be extended to a solution on the whole space-time? With respect to this question, there are some simple models in which one has this kind of extendability of local solutions to global ones, and some simple models in which one does not have such extendability, with no clear general pattern. The technical mathematical problems are amplified by the more conceptual problem of what it might mean to say that one could create a situation which forces the creation of closed timelike curves. (See, e.g., Yurtsever 1990; Friedman, Morris, et al. 1990; Novikov 1992; Earman 1995; and Earman, Smeenk, & Wüthrich 2009). What are we to think of all of this?

The toy models above all treat billiard balls, fields, and other objects propagating through a background spacetime with CTCs. Even if we can show that a consistent solution exists, there is a further question: what kind of matter and dynamics could generate CTCs to begin with? There are various solutions of Einstein’s equations with CTCs, but how do these exotic spacetimes relate to the models actually used in describing the world? In other words, what positive reasons might we have to take CTCs seriously as a feature of the actual universe, rather than an exotic possibility of primarily mathematical interest?

We should distinguish two different kinds of “possibility” that we might have in mind in posing such questions (following Stein 1970). First, we can consider a solution as a candidate cosmological model, describing the (large-scale gravitational degrees of freedom of the) entire universe. The case for ruling out spacetimes with CTCs as potential cosmological models strikes us as, surprisingly, fairly weak. Physicists used to simply rule out solutions with CTCs as unreasonable by fiat, due to the threat of paradoxes, which we have dismantled above. But it is also challenging to make an observational case. Observations tell us very little about global features, such as the existence of CTCs, because signals can only reach an observer from a limited region of spacetime, called the past light cone. Our past light cone—and indeed the collection of all the past light cones for possible observers in a given spacetime—can be embedded in spacetimes with quite different global features (Malament 1977, Manchak 2009). This undercuts the possibility of using observations to constrain global topology, including (among other things) ruling out the existence of CTCs.

Yet the case in favor of taking cosmological models with CTCs seriously is also not particularly strong. Some solutions used to describe black holes, which are clearly relevant in a variety of astrophysical contexts, include CTCs. But the question of whether the CTCs themselves play an essential representational role is subtle: the CTCs arise in the maximal extensions of these solutions, and can plausibly be regarded as extraneous to successful applications. Furthermore, many of the known solutions with CTCs have symmetries, raising the possibility that CTCs are not a stable or robust feature. Slight departures from symmetry may lead to a solution without CTCs, suggesting that the CTCs may be an artifact of an idealized model.

The second sense of possibility regards whether “reasonable” initial conditions can be shown to lead to, or not to lead to, the formation of CTCs. As with the toy models above, suppose that we have a partial Cauchy surface \(S\), such that all the temporal funny business lies to the future. Rather than simply assuming that there is a region with CTCs to the future, we can ask instead whether it is possible to create CTCs by manipulating matter in the initial, well-behaved region—that is, whether it is possible to build a time machine. Several physicists have pursued “chronology protection theorems” aiming to show that the dynamics of general relativity (or some other aspects of physics) rules this out, and to clarify why this is the case. The proof of such a theorem would justify neglecting solutions with CTCs as a source of insight into the nature of time in the actual world. But as of yet there are several partial results that do not fully settle the question. One further intriguing possibility is that even if general relativity by itself does protect chronology, it may not be possible to formulate a sensible theory describing matter and fields in solutions with CTCs. (See SEP entry on Time Machines; Smeenk and Wüthrich 2011 for more.)

There is a different question regarding the limitations of these toy models. The toy models and related examples show that there are consistent solutions for simple systems in the presence of CTCs. As usual we have made the analysis tractable by building toy models, selecting only a few dynamical degrees of freedom and tracking their evolution. But there is a large gap between the systems we have described and the time travel stories they evoke, with Kurt traveling along a CTC with murderous intentions. In particular, many features of the manifest image of time are tied to the thermodynamical properties of macroscopic systems. Rovelli (unpublished) considers a extremely simple system to illustrate the problem: can a clock move along a CTC? A clock consists of something in periodic motion, such as a pendulum bob, and something that counts the oscillations, such as an escapement mechanism. The escapement mechanism cannot work without friction; this requires dissipation and increasing entropy. For a clock that counts oscillations as it moves along a time-like trajectory, the entropy must be a monotonically increasing function. But that is obviously incompatible with the clock returning to precisely the same state at some future time as it completes a loop. The point generalizes, obviously, to imply that anything like a human, with memory and agency, cannot move along a CTC.

Since it is not obvious that one can rid oneself of all constraints in realistic models, let us examine the argument that time travel is implausible, and we should think it unlikely to exist in our world, in so far as it implies such constraints. The argument goes something like the following. In order to satisfy such constraints one needs some pre-established divine harmony between the global (time travel) structure of space-time and the distribution of particles and fields on space-like surfaces in it. But it is not plausible that the actual world, or any world even remotely like ours, is constructed with divine harmony as part of the plan. In fact, one might argue, we have empirical evidence that conditions in any spatial region can vary quite arbitrarily. So we have evidence that such constraints, whatever they are, do not in fact exist in our world. So we have evidence that there are no closed time-like lines in our world or one remotely like it. We will now examine this argument in more detail by presenting four possible responses, with counterresponses, to this argument.

Response 1. There is nothing implausible or new about such constraints. For instance, if the universe is spatially closed, there has to be enough matter to produce the needed curvature, and this puts constraints on the matter distribution on a space-like hypersurface. Thus global space-time structure can quite unproblematically constrain matter distributions on space-like hypersurfaces in it. Moreover we have no realistic idea what these constraints look like, so we hardly can be said to have evidence that they do not obtain.

Counterresponse 1. Of course there are constraining relations between the global structure of space-time and the matter in it. The Einstein equations relate curvature of the manifold to the matter distribution in it. But what is so strange and implausible about the constraints imposed by the existence of closed time-like curves is that these constraints in essence have nothing to do with the Einstein equations. When investigating such constraints one typically treats the particles and/or field in question as test particles and/or fields in a given space-time, i.e., they are assumed not to affect the metric of space-time in any way. In typical space-times without closed time-like curves this means that one has, in essence, complete freedom of matter distribution on a space-like hypersurface. (See response 2 for some more discussion of this issue). The constraints imposed by the possibility of time travel have a quite different origin and are implausible. In the ordinary case there is a causal interaction between matter and space-time that results in relations between global structure of space-time and the matter distribution in it. In the time travel case there is no such causal story to be told: there simply has to be some pre-established harmony between the global space-time structure and the matter distribution on some space-like surfaces. This is implausible.

Response 2. Constraints upon matter distributions are nothing new. For instance, Maxwell’s equations constrain electric fields \(\boldsymbol{E}\) on an initial surface to be related to the (simultaneous) charge density distribution \(\varrho\) by the equation \(\varrho = \text{div}(\boldsymbol{E})\). (If we assume that the \(E\) field is generated solely by the charge distribution, this conditions amounts to requiring that the \(E\) field at any point in space simply be the one generated by the charge distribution according to Coulomb’s inverse square law of electrostatics.) This is not implausible divine harmony. Such constraints can hold as a matter of physical law. Moreover, if we had inferred from the apparent free variation of conditions on spatial regions that there could be no such constraints we would have mistakenly inferred that \(\varrho = \text{div}(\boldsymbol{E})\) could not be a law of nature.

Counterresponse 2. The constraints imposed by the existence of closed time-like lines are of quite a different character from the constraint imposed by \(\varrho = \text{div}(\boldsymbol{E})\). The constraints imposed by \(\varrho = \text{div}(\boldsymbol{E})\) on the state on a space-like hypersurface are:

• local constraints (i.e., to check whether the constraint holds in a region you just need to see whether it holds at each point in the region),
• quite independent of the global space-time structure,
• quite independent of how the space-like surface in question is embedded in a given space-time, and
• very simply and generally stateable.

On the other hand, the consistency constraints imposed by the existence of closed time-like curves (i) are not local, (ii) are dependent on the global structure of space-time, (iii) depend on the location of the space-like surface in question in a given space-time, and (iv) appear not to be simply stateable other than as the demand that the state on that space-like surface embedded in such and such a way in a given space-time, do not lead to inconsistency. On some views of laws (e.g., David Lewis’ view) this plausibly implies that such constraints, even if they hold, could not possibly be laws. But even if one does not accept such a view of laws, one could claim that the bizarre features of such constraints imply that it is implausible that such constraints hold in our world or in any world remotely like ours.

Response 3. It would be strange if there are constraints in the non-time travel region. It is not strange if there are constraints in the time travel region. They should be explained in terms of the strange, self-interactive, character of time travel regions. In this region there are time-like trajectories from points to themselves. Thus the state at such a point, in such a region, will, in a sense, interact with itself. It is a well-known fact that systems that interact with themselves will develop into an equilibrium state, if there is such an equilibrium state, or else will develop towards some singularity. Normally, of course, self-interaction isn’t true instantaneous self-interaction, but consists of a feed-back mechanism that takes time. But in time travel regions something like true instantaneous self-interaction occurs. This explains why constraints on states occur in such time travel regions: the states “ ab initio ” have to be “equilibrium states”. Indeed in a way this also provides some picture of why indeterminism occurs in time travel regions: at the onset of self-interaction states can fork into different equi-possible equilibrium states.

Counterresponse 3. This is explanation by woolly analogy. It all goes to show that time travel leads to such bizarre consequences that it is unlikely that it occurs in a world remotely like ours.

Response 4. All of the previous discussion completely misses the point. So far we have been taking the space-time structure as given, and asked the question whether a given time travel space-time structure imposes constraints on states on (parts of) space-like surfaces. However, space-time and matter interact. Suppose that one is in a space-time with closed time-like lines, such that certain counterfactual distributions of matter on some neighborhood of a point \(p\) are ruled out if one holds that space-time structure fixed. One might then ask

Why does the actual state near \(p\) in fact satisfy these constraints? By what divine luck or plan is this local state compatible with the global space-time structure? What if conditions near \(p\) had been slightly different?

And one might take it that the lack of normal answers to these questions indicates that it is very implausible that our world, or any remotely like it, is such a time travel universe. However the proper response to these question is the following. There are no constraints in any significant sense. If they hold they hold as a matter of accidental fact, not of law. There is no more explanation of them possible than there is of any contingent fact. Had conditions in a neighborhood of \(p\) been otherwise, the global structure of space-time would have been different. So what? The only question relevant to the issue of constraints is whether an arbitrary state on an arbitrary spatial surface \(S\) can always be embedded into a space-time such that that state on \(S\) consistently extends to a solution on the entire space-time.

But we know the answer to that question. A well-known theorem in general relativity says the following: any initial data set on a three dimensional manifold \(S\) with positive definite metric has a unique embedding into a maximal space-time in which \(S\) is a Cauchy surface (see, e.g., Geroch & Horowitz 1979: 284 for more detail), i.e., there is a unique largest space-time which has \(S\) as a Cauchy surface and contains a consistent evolution of the initial value data on \(S\). Now since \(S\) is a Cauchy surface this space-time does not have closed time like curves. But it may have extensions (in which \(S\) is not a Cauchy surface) which include closed timelike curves, indeed it may be that any maximal extension of it would include closed timelike curves. (This appears to be the case for extensions of states on certain surfaces of Taub-NUT space-times. See Earman, Smeenk, & Wüthrich 2009). But these extensions, of course, will be consistent. So properly speaking, there are no constraints on states on space-like surfaces. Nonetheless the space-time in which these are embedded may or may not include closed time-like curves.

Counterresponse 4. This, in essence, is the stonewalling answer which we indicated in section 1. However, whether or not you call the constraints imposed by a given space-time on distributions of matter on certain space-like surfaces “genuine constraints”, whether or not they can be considered lawlike, and whether or not they need to be explained, the existence of such constraints can still be used to argue that time travel worlds are so bizarre that it is implausible that our world or any world remotely like ours is a time travel world.

Suppose that one is in a time travel world. Suppose that given the global space-time structure of this world, there are constraints imposed upon, say, the state of motion of a ball on some space-like surface when it is treated as a test particle, i.e., when it is assumed that the ball does not affect the metric properties of the space-time it is in. (There is lots of other matter that, via the Einstein equation, corresponds exactly to the curvature that there is everywhere in this time travel worlds.) Now a real ball of course does have some effect on the metric of the space-time it is in. But let us consider a ball that is so small that its effect on the metric is negligible. Presumably it will still be the case that certain states of this ball on that space-like surface are not compatible with the global time travel structure of this universe.

This means that the actual distribution of matter on such a space-like surface can be extended into a space-time with closed time-like lines, but that certain counterfactual distributions of matter on this space-like surface can not be extended into the same space-time. But note that the changes made in the matter distribution (when going from the actual to the counterfactual distribution) do not in any non-negligible way affect the metric properties of the space-time. (Recall that the changes only effect test particles.) Thus the reason why the global time travel properties of the counterfactual space-time have to be significantly different from the actual space-time is not that there are problems with metric singularities or alterations in the metric that force significant global changes when we go to the counterfactual matter distribution. The reason that the counterfactual space-time has to be different is that in the counterfactual world the ball’s initial state of motion starting on the space-like surface, could not “meet up” in a consistent way with its earlier self (could not be consistently extended) if we were to let the global structure of the counterfactual space-time be the same as that of the actual space-time. Now, it is not bizarre or implausible that there is a counterfactual dependence of manifold structure, even of its topology, on matter distributions on spacelike surfaces. For instance, certain matter distributions may lead to singularities, others may not. We may indeed in some sense have causal power over the topology of the space-time we live in. But this power normally comes via the Einstein equations. But it is bizarre to think that there could be a counterfactual dependence of global space-time structure on the arrangement of certain tiny bits of matter on some space-like surface, where changes in that arrangement by assumption do not affect the metric anywhere in space-time in any significant way . It is implausible that we live in such a world, or that a world even remotely like ours is like that.

Let us illustrate this argument in a different way by assuming that wormhole time travel imposes constraints upon the states of people prior to such time travel, where the people have so little mass/energy that they have negligible effect, via the Einstein equation, on the local metric properties of space-time. Do you think it more plausible that we live in a world where wormhole time travel occurs but it only occurs when people’s states are such that these local states happen to combine with time travel in such a way that nobody ever succeeds in killing their younger self, or do you think it more plausible that we are not in a wormhole time travel world? [ 5 ]

An alternative approach to time travel (initiated by Deutsch 1991) abstracts away from the idealized toy models described above. [ 6 ] This computational approach considers instead the evolution of bits (simple physical systems with two discrete states) through a network of interactions, which can be represented by a circuit diagram with gates corresponding to the interactions. Motivated by the possibility of CTCs, Deutsch proposed adding a new kind of channel that connects the output of a given gate back to its input —in essence, a backwards-time step. More concretely, given a gate that takes \(n\) bits as input, we can imagine taking some number \(i \lt n\) of these bits through a channel that loops back and then do double-duty as inputs. Consistency requires that the state of these \(i\) bits is the same for output and input. (We will consider an illustration of this kind of system in the next section.) Working through examples of circuit diagrams with a CTC channel leads to similar treatments of Consistency and Underdetermination as the discussion above (see, e.g., Wallace 2012: § 10.6). But the approach offers two new insights (both originally due to Deutsch): the Easy Knowledge paradox, and a particularly clear extension to time travel in quantum mechanics.

A computer equipped with a CTC channel can exploit the need to find consistent evolution to solve remarkably hard problems. (This is quite different than the first idea that comes to mind to enhance computational power: namely to just devote more time to a computation, and then send the result back on the CTC to an earlier state.) The gate in a circuit incorporating a CTC implements a function from the input bits to the output bits, under the constraint that the output and input match the i bits going through the CTC channel. This requires, in effect, finding the fixed point of the relevant function. Given the generality of the model, there are few limits on the functions that could be implemented on the CTC circuit. Nature has to solve a hard computational problem just to ensure consistent evolution. This can then be extended to other complex computational problems—leading, more precisely, to solutions of NP -complete problems in polynomial time (see Aaronson 2013: Chapter 20 for an overview and further references). The limits imposed by computational complexity are an essential part of our epistemic situation, and computers with CTCs would radically change this.

We now turn to the application of the computational approach to the quantum physics of time travel (see Deutsch 1991; Deutsch & Lockwood 1994). By contrast with the earlier discussions of constraints in classical systems, they claim to show that time travel never imposes any constraints on the pre-time travel state of quantum systems. The essence of this account is as follows. [ 7 ]

A quantum system starts in state \(S_1\), interacts with its older self, after the interaction is in state \(S_2\), time travels while developing into state \(S_3\), then interacts with its younger self, and ends in state \(S_4\) (see figure 10 ).

Figure 10 [An extended description of figure 10 is in the supplement.]

Deutsch assumes that the set of possible states of this system are the mixed states, i.e., are represented by the density matrices over the Hilbert space of that system. Deutsch then shows that for any initial state \(S_1\), any unitary interaction between the older and younger self, and any unitary development during time travel, there is a consistent solution, i.e., there is at least one pair of states \(S_2\) and \(S_3\) such that when \(S_1\) interacts with \(S_3\) it will change to state \(S_2\) and \(S_2\) will then develop into \(S_3\). The states \(S_2, S_3\) and \(S_4\) will typically be not be pure states, i.e., will be non-trivial mixed states, even if \(S_1\) is pure. In order to understand how this leads to interpretational problems let us give an example. Consider a system that has a two dimensional Hilbert space with as a basis the states \(\vc{+}\) and \(\vc{-}\). Let us suppose that when state \(\vc{+}\) of the young system encounters state \(\vc{+}\) of the older system, they interact and the young system develops into state \(\vc{-}\) and the old system remains in state \(\vc{+}\). In obvious notation:

Similarly, suppose that:

Let us furthermore assume that there is no development of the state of the system during time travel, i.e., that \(\vc{+}_2\) develops into \(\vc{+}_3\), and that \(\vc{-}_2\) develops into \(\vc{-}_3\).

Now, if the only possible states of the system were \(\vc{+}\) and \(\vc{-}\) (i.e., if there were no superpositions or mixtures of these states), then there is a constraint on initial states: initial state \(\vc{+}_1\) is impossible. For if \(\vc{+}_1\) interacts with \(\vc{+}_3\) then it will develop into \(\vc{-}_2\), which, during time travel, will develop into \(\vc{-}_3\), which inconsistent with the assumed state \(\vc{+}_3\). Similarly if \(\vc{+}_1\) interacts with \(\vc{-}_3\) it will develop into \(\vc{+}_2\), which will then develop into \(\vc{+}_3\) which is also inconsistent. Thus the system can not start in state \(\vc{+}_1\).

But, says Deutsch, in quantum mechanics such a system can also be in any mixture of the states \(\vc{+}\) and \(\vc{-}\). Suppose that the older system, prior to the interaction, is in a state \(S_3\) which is an equal mixture of 50% \(\vc{+}_3\) and 50% \(\vc{-}_3\). Then the younger system during the interaction will develop into a mixture of 50% \(\vc{+}_2\) and 50% \(\vc{-}_2\), which will then develop into a mixture of 50% \(\vc{+}_3\) and 50% \(\vc{-}_3\), which is consistent! More generally Deutsch uses a fixed point theorem to show that no matter what the unitary development during interaction is, and no matter what the unitary development during time travel is, for any state \(S_1\) there is always a state \(S_3\) (which typically is not a pure state) which causes \(S_1\) to develop into a state \(S_2\) which develops into that state \(S_3\). Thus quantum mechanics comes to the rescue: it shows in all generality that no constraints on initial states are needed!

One might wonder why Deutsch appeals to mixed states: will superpositions of states \(\vc{+}\) and \(\vc{-}\) not suffice? Unfortunately such an idea does not work. Suppose again that the initial state is \(\vc{+}_1\). One might suggest that that if state \(S_3\) is

one will obtain a consistent development. For one might think that when initial state \(\vc{+}_1\) encounters the superposition

it will develop into superposition

and that this in turn will develop into

as desired. However this is not correct. For initial state \(\vc{+}_1\) when it encounters

will develop into the entangled state

In so far as one can speak of the state of the young system after this interaction, it is in the mixture of 50% \(\vc{+}_2\) and 50% \(\vc{-}_2\), not in the superposition

So Deutsch does need his recourse to mixed states.

This clarification of why Deutsch needs his mixtures does however indicate a serious worry about the simplifications that are part of Deutsch’s account. After the interaction the old and young system will (typically) be in an entangled state. Although for purposes of a measurement on one of the two systems one can say that this system is in a mixed state, one can not represent the full state of the two systems by specifying the mixed state of each separate part, as there are correlations between observables of the two systems that are not represented by these two mixed states, but are represented in the joint entangled state. But if there really is an entangled state of the old and young systems directly after the interaction, how is one to represent the subsequent development of this entangled state? Will the state of the younger system remain entangled with the state of the older system as the younger system time travels and the older system moves on into the future? On what space-like surfaces are we to imagine this total entangled state to be? At this point it becomes clear that there is no obvious and simple way to extend elementary non-relativistic quantum mechanics to space-times with closed time-like curves: we apparently need to characterize not just the entanglement between two systems, but entanglement relative to specific spacetime descriptions.

How does Deutsch avoid these complications? Deutsch assumes a mixed state \(S_3\) of the older system prior to the interaction with the younger system. He lets it interact with an arbitrary pure state \(S_1\) younger system. After this interaction there is an entangled state \(S'\) of the two systems. Deutsch computes the mixed state \(S_2\) of the younger system which is implied by this entangled state \(S'\). His demand for consistency then is just that this mixed state \(S_2\) develops into the mixed state \(S_3\). Now it is not at all clear that this is a legitimate way to simplify the problem of time travel in quantum mechanics. But even if we grant him this simplification there is a problem: how are we to understand these mixtures?

If we take an ignorance interpretation of mixtures we run into trouble. For suppose that we assume that in each individual case each older system is either in state \(\vc{+}_3\) or in state \(\vc{-}_3\) prior to the interaction. Then we regain our paradox. Deutsch instead recommends the following, many worlds, picture of mixtures. Suppose we start with state \(\vc{+}_1\) in all worlds. In some of the many worlds the older system will be in the \(\vc{+}_3\) state, let us call them A -worlds, and in some worlds, B -worlds, it will be in the \(\vc{-}_3\) state. Thus in A -worlds after interaction we will have state \(\vc{-}_2\) , and in B -worlds we will have state \(\vc{+}_2\). During time travel the \(\vc{-}_2\) state will remain the same, i.e., turn into state \(\vc{-}_3\), but the systems in question will travel from A -worlds to B -worlds. Similarly the \(\vc{+}\) \(_2\) states will travel from the B -worlds to the A -worlds, thus preserving consistency.

Now whatever one thinks of the merits of many worlds interpretations, and of this understanding of it applied to mixtures, in the end one does not obtain genuine time travel in Deutsch’s account. The systems in question travel from one time in one world to another time in another world, but no system travels to an earlier time in the same world. (This is so at least in the normal sense of the word “world”, the sense that one means when, for instance, one says “there was, and will be, only one Elvis Presley in this world.”) Thus, even if it were a reasonable view, it is not quite as interesting as it may have initially seemed. (See Wallace 2012 for a more sympathetic treatment, that explores several further implications of accepting time travel in conjunction with the many worlds interpretation.)

We close by acknowledging that Deutsch’s starting point—the claim that this computational model captures the essential features of quantum systems in a spacetime with CTCs—has been the subject of some debate. Several physicists have pursued a quite different treatment of evolution of quantum systems through CTC’s, based on considering the “post-selected” state (see Lloyd et al. 2011). Their motivations for implementing the consistency condition in terms of the post-selected state reflects a different stance towards quantum foundations. A different line of argument aims to determine whether Deutsch’s treatment holds as an appropriate limiting case of a more rigorous treatment, such as quantum field theory in curved spacetimes. For example, Verch (2020) establishes several results challenging the assumption that Deutsch’s treatment is tied to the presence of CTC’s, or that it is compatible with the entanglement structure of quantum fields.

What remains of the grandfather paradox in general relativistic time travel worlds is the fact that in some cases the states on edgeless spacelike surfaces are “overconstrained”, so that one has less than the usual freedom in specifying conditions on such a surface, given the time-travel structure, and in some cases such states are “underconstrained”, so that states on edgeless space-like surfaces do not determine what happens elsewhere in the way that they usually do, given the time travel structure. There can also be mixtures of those two types of cases. The extent to which states are overconstrained and/or underconstrained in realistic models is as yet unclear, though it would be very surprising if neither obtained. The extant literature has primarily focused on the problem of overconstraint, since that, often, either is regarded as a metaphysical obstacle to the possibility time travel, or as an epistemological obstacle to the plausibility of time travel in our world. While it is true that our world would be quite different from the way we normally think it is if states were overconstrained, underconstraint seems at least as bizarre as overconstraint. Nonetheless, neither directly rules out the possibility of time travel.

If time travel entailed contradictions then the issue would be settled. And indeed, most of the stories employing time travel in popular culture are logically incoherent: one cannot “change” the past to be different from what it was, since the past (like the present and the future) only occurs once. But if the only requirement demanded is logical coherence, then it seems all too easy. A clever author can devise a coherent time-travel scenario in which everything happens just once and in a consistent way. This is just too cheap: logical coherence is a very weak condition, and many things we take to be metaphysically impossible are logically coherent. For example, it involves no logical contradiction to suppose that water is not molecular, but if both chemistry and Kripke are right it is a metaphysical impossibility. We have been interested not in logical possibility but in physical possibility. But even so, our conditions have been relatively weak: we have asked only whether time-travel is consistent with the universal validity of certain fundamental physical laws and with the notion that the physical state on a surface prior to the time travel region be unconstrained. It is perfectly possible that the physical laws obey this condition, but still that time travel is not metaphysically possible because of the nature of time itself. Consider an analogy. Aristotle believed that water is homoiomerous and infinitely divisible: any bit of water could be subdivided, in principle, into smaller bits of water. Aristotle’s view contains no logical contradiction. It was certainly consistent with Aristotle’s conception of water that it be homoiomerous, so this was, for him, a conceptual possibility. But if chemistry is right, Aristotle was wrong both about what water is like and what is possible for it. It can’t be infinitely divided, even though no logical or conceptual analysis would reveal that.

Similarly, even if all of our consistency conditions can be met, it does not follow that time travel is physically possible, only that some specific physical considerations cannot rule it out. The only serious proof of the possibility of time travel would be a demonstration of its actuality. For if we agree that there is no actual time travel in our universe, the supposition that there might have been involves postulating a substantial difference from actuality, a difference unlike in kind from anything we could know if firsthand. It is unclear to us exactly what the content of possible would be if one were to either maintain or deny the possibility of time travel in these circumstances, unless one merely meant that the possibility is not ruled out by some delineated set of constraints. As the example of Aristotle’s theory of water shows, conceptual and logical “possibility” do not entail possibility in a full-blooded sense. What exactly such a full-blooded sense would be in case of time travel, and whether one could have reason to believe it to obtain, remain to us obscure.

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How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• Adlam, Emily, unpublished, “ Is There Causation in Fundamental Physics? New Insights from Process Matrices and Quantum Causal Modelling ”, 2022, arXiv: 2208.02721. doi:10.48550/ARXIV.2208.02721
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Time travel: Is it possible?

Science says time travel is possible, but probably not in the way you're thinking.

Albert Einstein's theory

• General relativity and GPS
• Wormhole travel
• Alternate theories

Science fiction

Is time travel possible? Short answer: Yes, and you're doing it right now — hurtling into the future at the impressive rate of one second per second. 

You're pretty much always moving through time at the same speed, whether you're watching paint dry or wishing you had more hours to visit with a friend from out of town. 

But this isn't the kind of time travel that's captivated countless science fiction writers, or spurred a genre so extensive that Wikipedia lists over 400 titles in the category "Movies about Time Travel." In franchises like " Doctor Who ," " Star Trek ," and "Back to the Future" characters climb into some wild vehicle to blast into the past or spin into the future. Once the characters have traveled through time, they grapple with what happens if you change the past or present based on information from the future (which is where time travel stories intersect with the idea of parallel universes or alternate timelines). 

Related: The best sci-fi time machines ever

Although many people are fascinated by the idea of changing the past or seeing the future before it's due, no person has ever demonstrated the kind of back-and-forth time travel seen in science fiction or proposed a method of sending a person through significant periods of time that wouldn't destroy them on the way. And, as physicist Stephen Hawking pointed out in his book " Black Holes and Baby Universes" (Bantam, 1994), "The best evidence we have that time travel is not possible, and never will be, is that we have not been invaded by hordes of tourists from the future."

Science does support some amount of time-bending, though. For example, physicist Albert Einstein 's theory of special relativity proposes that time is an illusion that moves relative to an observer. An observer traveling near the speed of light will experience time, with all its aftereffects (boredom, aging, etc.) much more slowly than an observer at rest. That's why astronaut Scott Kelly aged ever so slightly less over the course of a year in orbit than his twin brother who stayed here on Earth. 

Related: Controversially, physicist argues that time is real

There are other scientific theories about time travel, including some weird physics that arise around wormholes , black holes and string theory . For the most part, though, time travel remains the domain of an ever-growing array of science fiction books, movies, television shows, comics, video games and more. 

Einstein developed his theory of special relativity in 1905. Along with his later expansion, the theory of general relativity , it has become one of the foundational tenets of modern physics. Special relativity describes the relationship between space and time for objects moving at constant speeds in a straight line. 

The short version of the theory is deceptively simple. First, all things are measured in relation to something else — that is to say, there is no "absolute" frame of reference. Second, the speed of light is constant. It stays the same no matter what, and no matter where it's measured from. And third, nothing can go faster than the speed of light.

From those simple tenets unfolds actual, real-life time travel. An observer traveling at high velocity will experience time at a slower rate than an observer who isn't speeding through space. 

While we don't accelerate humans to near-light-speed, we do send them swinging around the planet at 17,500 mph (28,160 km/h) aboard the International Space Station . Astronaut Scott Kelly was born after his twin brother, and fellow astronaut, Mark Kelly . Scott Kelly spent 520 days in orbit, while Mark logged 54 days in space. The difference in the speed at which they experienced time over the course of their lifetimes has actually widened the age gap between the two men.

"So, where[as] I used to be just 6 minutes older, now I am 6 minutes and 5 milliseconds older," Mark Kelly said in a panel discussion on July 12, 2020, Space.com previously reported . "Now I've got that over his head."

General relativity and GPS time travel

The difference that low earth orbit makes in an astronaut's life span may be negligible — better suited for jokes among siblings than actual life extension or visiting the distant future — but the dilation in time between people on Earth and GPS satellites flying through space does make a difference. 

Read more: Can we stop time?

The Global Positioning System , or GPS, helps us know exactly where we are by communicating with a network of a few dozen satellites positioned in a high Earth orbit. The satellites circle the planet from 12,500 miles (20,100 kilometers) away, moving at 8,700 mph (14,000 km/h). 

According to special relativity, the faster an object moves relative to another object, the slower that first object experiences time. For GPS satellites with atomic clocks, this effect cuts 7 microseconds, or 7 millionths of a second, off each day, according to the American Physical Society publication Physics Central .  

Read more: Could Star Trek's faster-than-light warp drive actually work?

Then, according to general relativity, clocks closer to the center of a large gravitational mass like Earth tick more slowly than those farther away. So, because the GPS satellites are much farther from the center of Earth compared to clocks on the surface, Physics Central added, that adds another 45 microseconds onto the GPS satellite clocks each day. Combined with the negative 7 microseconds from the special relativity calculation, the net result is an added 38 microseconds. 

This means that in order to maintain the accuracy needed to pinpoint your car or phone — or, since the system is run by the U.S. Department of Defense, a military drone — engineers must account for an extra 38 microseconds in each satellite's day. The atomic clocks onboard don’t tick over to the next day until they have run 38 microseconds longer than comparable clocks on Earth.

Given those numbers, it would take more than seven years for the atomic clock in a GPS satellite to un-sync itself from an Earth clock by more than a blink of an eye. (We did the math: If you estimate a blink to last at least 100,000 microseconds, as the Harvard Database of Useful Biological Numbers does, it would take thousands of days for those 38 microsecond shifts to add up.) 

This kind of time travel may seem as negligible as the Kelly brothers' age gap, but given the hyper-accuracy of modern GPS technology, it actually does matter. If it can communicate with the satellites whizzing overhead, your phone can nail down your location in space and time with incredible accuracy. 

Can wormholes take us back in time?

General relativity might also provide scenarios that could allow travelers to go back in time, according to NASA . But the physical reality of those time-travel methods is no piece of cake. 

Wormholes are theoretical "tunnels" through the fabric of space-time that could connect different moments or locations in reality to others. Also known as Einstein-Rosen bridges or white holes, as opposed to black holes, speculation about wormholes abounds. But despite taking up a lot of space (or space-time) in science fiction, no wormholes of any kind have been identified in real life. 

Related: Best time travel movies

"The whole thing is very hypothetical at this point," Stephen Hsu, a professor of theoretical physics at the University of Oregon, told Space.com sister site Live Science . "No one thinks we're going to find a wormhole anytime soon."

Primordial wormholes are predicted to be just 10^-34 inches (10^-33 centimeters) at the tunnel's "mouth". Previously, they were expected to be too unstable for anything to be able to travel through them. However, a study claims that this is not the case, Live Science reported . 

The theory, which suggests that wormholes could work as viable space-time shortcuts, was described by physicist Pascal Koiran. As part of the study, Koiran used the Eddington-Finkelstein metric, as opposed to the Schwarzschild metric which has been used in the majority of previous analyses.

In the past, the path of a particle could not be traced through a hypothetical wormhole. However, using the Eddington-Finkelstein metric, the physicist was able to achieve just that.

Koiran's paper was described in October 2021, in the preprint database arXiv , before being published in the Journal of Modern Physics D.

Alternate time travel theories

While Einstein's theories appear to make time travel difficult, some researchers have proposed other solutions that could allow jumps back and forth in time. These alternate theories share one major flaw: As far as scientists can tell, there's no way a person could survive the kind of gravitational pulling and pushing that each solution requires.

Infinite cylinder theory

Astronomer Frank Tipler proposed a mechanism (sometimes known as a Tipler Cylinder ) where one could take matter that is 10 times the sun's mass, then roll it into a very long, but very dense cylinder. The Anderson Institute , a time travel research organization, described the cylinder as "a black hole that has passed through a spaghetti factory."

After spinning this black hole spaghetti a few billion revolutions per minute, a spaceship nearby — following a very precise spiral around the cylinder — could travel backward in time on a "closed, time-like curve," according to the Anderson Institute. 

The major problem is that in order for the Tipler Cylinder to become reality, the cylinder would need to be infinitely long or be made of some unknown kind of matter. At least for the foreseeable future, endless interstellar pasta is beyond our reach.

Time donuts

Theoretical physicist Amos Ori at the Technion-Israel Institute of Technology in Haifa, Israel, proposed a model for a time machine made out of curved space-time — a donut-shaped vacuum surrounded by a sphere of normal matter.

"The machine is space-time itself," Ori told Live Science . "If we were to create an area with a warp like this in space that would enable time lines to close on themselves, it might enable future generations to return to visit our time."

Amos Ori is a theoretical physicist at the Technion-Israel Institute of Technology in Haifa, Israel. His research interests and publications span the fields of general relativity, black holes, gravitational waves and closed time lines.

There are a few caveats to Ori's time machine. First, visitors to the past wouldn't be able to travel to times earlier than the invention and construction of the time donut. Second, and more importantly, the invention and construction of this machine would depend on our ability to manipulate gravitational fields at will — a feat that may be theoretically possible but is certainly beyond our immediate reach.

Time travel has long occupied a significant place in fiction. Since as early as the "Mahabharata," an ancient Sanskrit epic poem compiled around 400 B.C., humans have dreamed of warping time, Lisa Yaszek, a professor of science fiction studies at the Georgia Institute of Technology in Atlanta, told Live Science .  

Every work of time-travel fiction creates its own version of space-time, glossing over one or more scientific hurdles and paradoxes to achieve its plot requirements. 

Some make a nod to research and physics, like " Interstellar ," a 2014 film directed by Christopher Nolan. In the movie, a character played by Matthew McConaughey spends a few hours on a planet orbiting a supermassive black hole, but because of time dilation, observers on Earth experience those hours as a matter of decades. 

Others take a more whimsical approach, like the "Doctor Who" television series. The series features the Doctor, an extraterrestrial "Time Lord" who travels in a spaceship resembling a blue British police box. "People assume," the Doctor explained in the show, "that time is a strict progression from cause to effect, but actually from a non-linear, non-subjective viewpoint, it's more like a big ball of wibbly-wobbly, timey-wimey stuff." 

Long-standing franchises like the "Star Trek" movies and television series, as well as comic universes like DC and Marvel Comics, revisit the idea of time travel over and over. 

Related: Marvel movies in order: chronological & release order

Here is an incomplete (and deeply subjective) list of some influential or notable works of time travel fiction:

Books about time travel:

• Rip Van Winkle (Cornelius S. Van Winkle, 1819) by Washington Irving
• A Christmas Carol (Chapman & Hall, 1843) by Charles Dickens
• The Time Machine (William Heinemann, 1895) by H. G. Wells
• A Connecticut Yankee in King Arthur's Court (Charles L. Webster and Co., 1889) by Mark Twain
• The Restaurant at the End of the Universe (Pan Books, 1980) by Douglas Adams
• A Tale of Time City (Methuen, 1987) by Diana Wynn Jones
• The Outlander series (Delacorte Press, 1991-present) by Diana Gabaldon
• Harry Potter and the Prisoner of Azkaban (Bloomsbury/Scholastic, 1999) by J. K. Rowling
• Thief of Time (Doubleday, 2001) by Terry Pratchett
• The Time Traveler's Wife (MacAdam/Cage, 2003) by Audrey Niffenegger
• All You Need is Kill (Shueisha, 2004) by Hiroshi Sakurazaka

Movies about time travel:

• Planet of the Apes (1968)
• Superman (1978)
• Time Bandits (1981)
• The Terminator (1984)
• Back to the Future series (1985, 1989, 1990)
• Star Trek IV: The Voyage Home (1986)
• Bill & Ted's Excellent Adventure (1989)
• Groundhog Day (1993)
• Galaxy Quest (1999)
• The Butterfly Effect (2004)
• 13 Going on 30 (2004)
• The Lake House (2006)
• Meet the Robinsons (2007)
• Hot Tub Time Machine (2010)
• Midnight in Paris (2011)
• Looper (2012)
• X-Men: Days of Future Past (2014)
• Edge of Tomorrow (2014)
• Interstellar (2014)
• Doctor Strange (2016)
• A Wrinkle in Time (2018)
• The Last Sharknado: It's About Time (2018)
• Avengers: Endgame (2019)
• Tenet (2020)
• Palm Springs (2020)
• Zach Snyder's Justice League (2021)
• The Tomorrow War (2021)

Television about time travel:

• Doctor Who (1963-present)
• The Twilight Zone (1959-1964) (multiple episodes)
• Star Trek (multiple series, multiple episodes)
• Samurai Jack (2001-2004)
• Lost (2004-2010)
• Phil of the Future (2004-2006)
• Steins;Gate (2011)
• Outlander (2014-2023)
• Loki (2021-present)

Games about time travel:

• Chrono Trigger (1995)
• TimeSplitters (2000-2005)
• Kingdom Hearts (2002-2019)
• Prince of Persia: Sands of Time (2003)
• God of War II (2007)
• Ratchet and Clank Future: A Crack In Time (2009)
• Sly Cooper: Thieves in Time (2013)
• Dishonored 2 (2016)
• Titanfall 2 (2016)
• Outer Wilds (2019)

Additional resources

Explore physicist Peter Millington's thoughts about Stephen Hawking's time travel theories at The Conversation . Check out a kid-friendly explanation of real-world time travel from NASA's Space Place . For an overview of time travel in fiction and the collective consciousness, read " Time Travel: A History " (Pantheon, 2016) by James Gleik. 

Join our Space Forums to keep talking space on the latest missions, night sky and more! And if you have a news tip, correction or comment, let us know at: [email protected].

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Ailsa is a staff writer for How It Works magazine, where she writes science, technology, space, history and environment features. Based in the U.K., she graduated from the University of Stirling with a BA (Hons) journalism degree. Previously, Ailsa has written for Cardiff Times magazine, Psychology Now and numerous science bookazines. 

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5.3 Time Dilation

Learning objectives.

By the end of this section, you will be able to:

• Explain how time intervals can be measured differently in different reference frames.
• Describe how to distinguish a proper time interval from a dilated time interval.
• Describe the significance of the muon experiment.
• Explain why the twin paradox is not a contradiction.
• Calculate time dilation given the speed of an object in a given frame.

The analysis of simultaneity shows that Einstein’s postulates imply an important effect: Time intervals have different values when measured in different inertial frames. Suppose, for example, an astronaut measures the time it takes for a pulse of light to travel a distance perpendicular to the direction of his ship’s motion (relative to an earthbound observer), bounce off a mirror, and return ( Figure 5.4 ). How does the elapsed time that the astronaut measures in the spacecraft compare with the elapsed time that an earthbound observer measures by observing what is happening in the spacecraft?

Examining this question leads to a profound result. The elapsed time for a process depends on which observer is measuring it. In this case, the time measured by the astronaut (within the spaceship where the astronaut is at rest) is smaller than the time measured by the earthbound observer (to whom the astronaut is moving). The time elapsed for the same process is different for the observers, because the distance the light pulse travels in the astronaut’s frame is smaller than in the earthbound frame, as seen in Figure 5.4 . Light travels at the same speed in each frame, so it takes more time to travel the greater distance in the earthbound frame.

Time Dilation

Time dilation is the lengthening of the time interval between two events for an observer in an inertial frame that is moving with respect to the rest frame of the events (in which the events occur at the same location).

To quantitatively compare the time measurements in the two inertial frames, we can relate the distances in Figure 5.4 to each other, then express each distance in terms of the time of travel (respectively either Δ t Δ t or Δ τ Δ τ ) of the pulse in the corresponding reference frame. The resulting equation can then be solved for Δ t Δ t in terms of Δ τ . Δ τ .

The lengths D and L in Figure 5.4 are the sides of a right triangle with hypotenuse s . From the Pythagorean theorem,

The lengths 2 s and 2 L are, respectively, the distances that the pulse of light and the spacecraft travel in time Δ t Δ t in the earthbound observer’s frame. The length D is the distance that the light pulse travels in time Δ τ Δ τ in the astronaut’s frame. This gives us three equations:

Note that we used Einstein’s second postulate by taking the speed of light to be c in both inertial frames. We substitute these results into the previous expression from the Pythagorean theorem:

Then we rearrange to obtain

Finally, solving for Δ t Δ t in terms of Δ τ Δ τ gives us

This is equivalent to

where γ γ is the relativistic factor (often called the Lorentz factor ) given by

and v and c are the speeds of the moving observer and light, respectively.

Note the asymmetry between the two measurements. Only one of them is a measurement of the time interval between two events—the emission and arrival of the light pulse—at the same position. It is a measurement of the time interval in the rest frame of a single clock. The measurement in the earthbound frame involves comparing the time interval between two events that occur at different locations. The time interval between events that occur at a single location has a separate name to distinguish it from the time measured by the earthbound observer, and we use the separate symbol Δ τ Δ τ to refer to it throughout this chapter.

Proper Time

The proper time interval Δ τ Δ τ between two events is the time interval measured by an observer for whom both events occur at the same location.

The equation relating Δ t Δ t and Δ τ Δ τ is truly remarkable. First, as stated earlier, elapsed time is not the same for different observers moving relative to one another, even though both are in inertial frames. A proper time interval Δ τ Δ τ for an observer who, like the astronaut, is moving with the apparatus, is smaller than the time interval for other observers. It is the smallest possible measured time between two events. The earthbound observer sees time intervals within the moving system as dilated (i.e., lengthened) relative to how the observer moving relative to Earth sees them within the moving system. Alternatively, according to the earthbound observer, less time passes between events within the moving frame. Note that the shortest elapsed time between events is in the inertial frame in which the observer sees the events (e.g., the emission and arrival of the light signal) occur at the same point.

This time effect is real and is not caused by inaccurate clocks or improper measurements. Time-interval measurements of the same event differ for observers in relative motion. The dilation of time is an intrinsic property of time itself. All clocks moving relative to an observer, including biological clocks, such as a person’s heartbeat, or aging, are observed to run more slowly compared with a clock that is stationary relative to the observer.

Note that if the relative velocity is much less than the speed of light ( v < < c ) , ( v < < c ) , then v 2 / c 2 v 2 / c 2 is extremely small, and the elapsed times Δ t Δ t and Δ τ Δ τ are nearly equal. At low velocities, physics based on modern relativity approaches classical physics—everyday experiences involve very small relativistic effects. However, for speeds near the speed of light, v 2 / c 2 v 2 / c 2 is close to one, so 1 − v 2 / c 2 1 − v 2 / c 2 is very small and Δ t Δ t becomes significantly larger than Δ τ . Δ τ .

Half-Life of a Muon

There is considerable experimental evidence that the equation Δ t = γ Δ τ Δ t = γ Δ τ is correct. One example is found in cosmic ray particles that continuously rain down on Earth from deep space. Some collisions of these particles with nuclei in the upper atmosphere result in short-lived particles called muons . The half-life (amount of time for half of a material to decay) of a muon is 1.52 μs when it is at rest relative to the observer who measures the half-life. This is the proper time interval Δ τ . Δ τ . This short time allows very few muons to reach Earth’s surface and be detected if Newtonian assumptions about time and space were correct. However, muons produced by cosmic ray particles have a range of velocities, with some moving near the speed of light. It has been found that the muon’s half-life as measured by an earthbound observer ( Δ t Δ t ) varies with velocity exactly as predicted by the equation Δ t = γ Δ τ . Δ t = γ Δ τ . The faster the muon moves, the longer it lives. We on Earth see the muon last much longer than its half-life predicts within its own rest frame. As viewed from our frame, the muon decays more slowly than it does when at rest relative to us. A far larger fraction of muons reach the ground as a result.

Before we present the first example of solving a problem in relativity, we state a strategy you can use as a guideline for these calculations.

Problem-Solving Strategy

• Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Look in particular for information on relative velocity v .
• Identify exactly what needs to be determined in the problem (identify the unknowns).
• Make certain you understand the conceptual aspects of the problem before making any calculations (express the answer as an equation). Decide, for example, which observer sees time dilated or length contracted before working with the equations or using them to carry out the calculation. If you have thought about who sees what, who is moving with the event being observed, who sees proper time, and so on, you will find it much easier to determine if your calculation is reasonable.
• Determine the primary type of calculation to be done to find the unknowns identified above (do the calculation). You will find the section summary helpful in determining whether a length contraction, relativistic kinetic energy, or some other concept is involved.

Note that you should not round off during the calculation . As noted in the text, you must often perform your calculations to many digits to see the desired effect. You may round off at the very end of the problem solution, but do not use a rounded number in a subsequent calculation. Also, check the answer to see if it is reasonable: Does it make sense? This may be more difficult for relativity, which has few everyday examples to provide experience with what is reasonable. But you can look for velocities greater than c or relativistic effects that are in the wrong direction (such as a time contraction where a dilation was expected).

Example 5.1

Time dilation in a high-speed vehicle.

• Identify the knowns: Δ τ = 1 s; v = 5830 m/s. Δ τ = 1 s; v = 5830 m/s.
• Identify the unknown: Δ t . Δ t .
• Express the answer as an equation: Δ t = γ Δ τ = Δ τ 1 − v 2 c 2 . Δ t = γ Δ τ = Δ τ 1 − v 2 c 2 .
• Do the calculation. Use the expression for γ γ to determine Δ t Δ t from Δ τ Δ τ : Δ t = 1 s 1 − ( 5830 m/s 3.00 × 10 8 m/s ) 2 = 1.000000000189 s = 1 s + 1.89 × 10 −10 s. Δ t = 1 s 1 − ( 5830 m/s 3.00 × 10 8 m/s ) 2 = 1.000000000189 s = 1 s + 1.89 × 10 −10 s.

Significance

Example 5.2, what speeds are relativistic.

• Identify the known: Δ τ Δ t = 1 1.01 . Δ τ Δ t = 1 1.01 .
• Identify the unknown: v/c .
• Express the answer as an equation: Δ t = γ Δ τ = 1 1 − v 2 / c 2 Δ τ Δ τ Δ t = 1 − v 2 / c 2 ( Δ τ Δ t ) 2 = 1 − v 2 c 2 v c = 1 − ( Δ τ / Δ t ) 2 . Δ t = γ Δ τ = 1 1 − v 2 / c 2 Δ τ Δ τ Δ t = 1 − v 2 / c 2 ( Δ τ Δ t ) 2 = 1 − v 2 c 2 v c = 1 − ( Δ τ / Δ t ) 2 .
• Do the calculation: v c = 1 − ( 1 / 1.01 ) 2 = 0.14 . v c = 1 − ( 1 / 1.01 ) 2 = 0.14 .

Example 5.3

Calculating δ t δ t for a relativistic event.

As we will discuss later, in the muon’s reference frame, it travels a shorter distance than measured in Earth’s reference frame.

• Identify the knowns: v = 0.950 c , Δ τ = 2.20 μ s. v = 0.950 c , Δ τ = 2.20 μ s.
• Express the answer as an equation. Use: Δ t = γ Δ τ Δ t = γ Δ τ with γ = 1 1 − v 2 c 2 . γ = 1 1 − v 2 c 2 .
• Do the calculation. Use the expression for γ γ to determine Δ t Δ t from Δ τ Δ τ Δ t = γ Δ τ = 1 1 − v 2 c 2 Δ τ = 2.20 μ s 1 − ( 0.950 ) 2 = 7.05 μ s. Δ t = γ Δ τ = 1 1 − v 2 c 2 Δ τ = 2.20 μ s 1 − ( 0.950 ) 2 = 7.05 μ s. Remember to keep extra significant figures until the final answer.

Example 5.4

Relativistic television, strategy for (a).

• Identify the knowns: v = 6.00 × 10 7 m/s; d = 0.200 m . v = 6.00 × 10 7 m/s; d = 0.200 m .
• Identify the unknown: the time of travel Δ t . Δ t .
• Express the answer as an equation: Δ t = d v . Δ t = d v .
• Do the calculation: t = 0.200 m 6.00 × 10 7 m/s = 3.33 × 10 −9 s. t = 0.200 m 6.00 × 10 7 m/s = 3.33 × 10 −9 s.

Strategy for (b)

• Identify the knowns (from part a): Δ t = 3.33 × 10 −9 s; v = 6.00 × 10 7 m/s; d = 0.200 m. Δ t = 3.33 × 10 −9 s; v = 6.00 × 10 7 m/s; d = 0.200 m.
• Identify the unknown: τ . τ .
• Express the answer as an equation: Δ t = γ Δ τ = Δ τ 1 − v 2 / c 2 Δ τ = Δ t 1 − v 2 / c 2 . Δ t = γ Δ τ = Δ τ 1 − v 2 / c 2 Δ τ = Δ t 1 − v 2 / c 2 .
• Do the calculation: Δ τ = ( 3.33 × 10 −9 s ) 1 − ( 6.00 × 10 7 m/s 3.00 × 10 8 m/s ) 2 = 3.26 × 10 −9 s. Δ τ = ( 3.33 × 10 −9 s ) 1 − ( 6.00 × 10 7 m/s 3.00 × 10 8 m/s ) 2 = 3.26 × 10 −9 s.

Check Your Understanding 5.2

What is γ γ if v = 0.650 c ? v = 0.650 c ?

The Twin Paradox

An intriguing consequence of time dilation is that a space traveler moving at a high velocity relative to Earth would age less than the astronaut’s earthbound twin. This is often known as the twin paradox . Imagine the astronaut moving at such a velocity that γ = 30.0 , γ = 30.0 , as in Figure 5.7 . A trip that takes 2.00 years in her frame would take 60.0 years in the earthbound twin’s frame. Suppose the astronaut travels 1.00 year to another star system, briefly explores the area, and then travels 1.00 year back. An astronaut who was 40 years old at the start of the trip would be would be 42 when the spaceship returns. Everything on Earth, however, would have aged 60.0 years. The earthbound twin, if still alive, would be 100 years old.

The situation would seem different to the astronaut in Figure 5.7 . Because motion is relative, the spaceship would seem to be stationary and Earth would appear to move. (This is the sensation you have when flying in a jet.) Looking out the window of the spaceship, the astronaut would see time slow down on Earth by a factor of γ = 30.0 . γ = 30.0 . Seen from the spaceship, the earthbound sibling will have aged only 2/30, or 0.07, of a year, whereas the astronaut would have aged 2.00 years.

The paradox here is that the two twins cannot both be correct. As with all paradoxes, conflicting conclusions come from a false premise. In fact, the astronaut’s motion is significantly different from that of the earthbound twin. The astronaut accelerates to a high velocity and then accelerates opposite to the motion to view the star system. To return to Earth, she again accelerates and decelerates. The spacecraft is not in a single inertial frame to which the time dilation formula can be directly applied. That is, the astronaut twin changes inertial references. The earthbound twin does not experience these accelerations and remains in the same inertial frame. Thus, the situation is not symmetric, and it is incorrect to claim that the astronaut observes the same effects as her twin. The lack of symmetry between the twins will be still more evident when we analyze the journey later in this chapter in terms of the path the astronaut follows through four-dimensional space-time.

In 1971, American physicists Joseph Hafele and Richard Keating verified time dilation at low relative velocities by flying extremely accurate atomic clocks around the world on commercial aircraft. They measured elapsed time to an accuracy of a few nanoseconds and compared it with the time measured by clocks left behind. Hafele and Keating’s results were within experimental uncertainties of the predictions of relativity. Both special and general relativity had to be taken into account, because gravity and accelerations were involved as well as relative motion.

Check Your Understanding 5.3

a. A particle travels at 1.90 × 10 8 m/s 1.90 × 10 8 m/s and lives 2.10 × 10 −8 s 2.10 × 10 −8 s when at rest relative to an observer. How long does the particle live as viewed in the laboratory?

b. Spacecraft A and B pass in opposite directions at a relative speed of 4.00 × 10 7 m/s . 4.00 × 10 7 m/s . An internal clock in spacecraft A causes it to emit a radio signal for 1.00 s. The computer in spacecraft B corrects for the beginning and end of the signal having traveled different distances, to calculate the time interval during which ship A was emitting the signal. What is the time interval that the computer in spacecraft B calculates?

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How Special Relativity Works

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Time Travel

Now that you have been introduced to the concepts of the theory, let's take a quick look at the relation between time travel and Special Relativity. If you remember the result from the twin paradox, you should agree that traveling into the future is possible, even at the speeds that our astronauts travel. Granted they would probably only be gaining a few nanoseconds, but when they return, the time on earth is ahead of their system time. Thus, they have returned to the future. As far as travelling back in time, Special Relativity is not as gracious as it is with moving forward. Let's take a look at this approach…

Many creative minds have wondered that since time slows down as you approach the speed of light, if you could find a way to travel faster than the speed of light, could you travel back in time? If I am to believe that special relativity is correct, then I am also to believe that the following events would occur. In order to travel faster than the speed of light, I assume that you would at some point have to travel at exactly the speed of light. For example, you can not travel 51 miles/hour without having traveled 50 miles/hour at some point, of course, this is providing that you were traveling 50 miles/hour or less to begin with. Now SR tells us that at the speed of light, time stops, your length contracts to nothing, and your resistance to acceleration becomes infinite requiring infinite energy (as observed by a frame of reference that is not in motion with the system). These conditions do not sound very conducive to life. Thus, I conclude that time travel into the past, using the concepts of SR, has some severe issues to overcome.

Please copy/paste the following text to properly cite this HowStuffWorks.com article:

A beginner's guide to time travel

Learn exactly how Einstein's theory of relativity works, and discover how there's nothing in science that says time travel is impossible.

Everyone can travel in time . You do it whether you want to or not, at a steady rate of one second per second. You may think there's no similarity to traveling in one of the three spatial dimensions at, say, one foot per second. But according to Einstein 's theory of relativity , we live in a four-dimensional continuum — space-time — in which space and time are interchangeable.

Einstein found that the faster you move through space, the slower you move through time — you age more slowly, in other words. One of the key ideas in relativity is that nothing can travel faster than the speed of light — about 186,000 miles per second (300,000 kilometers per second), or one light-year per year). But you can get very close to it. If a spaceship were to fly at 99% of the speed of light, you'd see it travel a light-year of distance in just over a year of time. 

That's obvious enough, but now comes the weird part. For astronauts onboard that spaceship, the journey would take a mere seven weeks. It's a consequence of relativity called time dilation , and in effect, it means the astronauts have jumped about 10 months into the future. 

Traveling at high speed isn't the only way to produce time dilation. Einstein showed that gravitational fields produce a similar effect — even the relatively weak field here on the surface of Earth . We don't notice it, because we spend all our lives here, but more than 12,400 miles (20,000 kilometers) higher up gravity is measurably weaker— and time passes more quickly, by about 45 microseconds per day. That's more significant than you might think, because it's the altitude at which GPS satellites orbit Earth, and their clocks need to be precisely synchronized with ground-based ones for the system to work properly. 

The satellites have to compensate for time dilation effects due both to their higher altitude and their faster speed. So whenever you use the GPS feature on your smartphone or your car's satnav, there's a tiny element of time travel involved. You and the satellites are traveling into the future at very slightly different rates.

But for more dramatic effects, we need to look at much stronger gravitational fields, such as those around black holes , which can distort space-time so much that it folds back on itself. The result is a so-called wormhole, a concept that's familiar from sci-fi movies, but actually originates in Einstein's theory of relativity. In effect, a wormhole is a shortcut from one point in space-time to another. You enter one black hole, and emerge from another one somewhere else. Unfortunately, it's not as practical a means of transport as Hollywood makes it look. That's because the black hole's gravity would tear you to pieces as you approached it, but it really is possible in theory. And because we're talking about space-time, not just space, the wormhole's exit could be at an earlier time than its entrance; that means you would end up in the past rather than the future.

Trajectories in space-time that loop back into the past are given the technical name "closed timelike curves." If you search through serious academic journals, you'll find plenty of references to them — far more than you'll find to "time travel." But in effect, that's exactly what closed timelike curves are all about — time travel

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There's another way to produce a closed timelike curve that doesn't involve anything quite so exotic as a black hole or wormhole: You just need a simple rotating cylinder made of super-dense material. This so-called Tipler cylinder is the closest that real-world physics can get to an actual, genuine time machine. But it will likely never be built in the real world, so like a wormhole, it's more of an academic curiosity than a viable engineering design.

Yet as far-fetched as these things are in practical terms, there's no fundamental scientific reason — that we currently know of — that says they are impossible. That's a thought-provoking situation, because as the physicist Michio Kaku is fond of saying, "Everything not forbidden is compulsory" (borrowed from T.H. White's novel, "The Once And Future King"). He doesn't mean time travel has to happen everywhere all the time, but Kaku is suggesting that the universe is so vast it ought to happen somewhere at least occasionally. Maybe some super-advanced civilization in another galaxy knows how to build a working time machine, or perhaps closed timelike curves can even occur naturally under certain rare conditions.

This raises problems of a different kind — not in science or engineering, but in basic logic. If time travel is allowed by the laws of physics, then it's possible to envision a whole range of paradoxical scenarios . Some of these appear so illogical that it's difficult to imagine that they could ever occur. But if they can't, what's stopping them? 

Thoughts like these prompted Stephen Hawking , who was always skeptical about the idea of time travel into the past, to come up with his "chronology protection conjecture" — the notion that some as-yet-unknown law of physics prevents closed timelike curves from happening. But that conjecture is only an educated guess, and until it is supported by hard evidence, we can come to only one conclusion: Time travel is possible.

A party for time travelers 

Hawking was skeptical about the feasibility of time travel into the past, not because he had disproved it, but because he was bothered by the logical paradoxes it created. In his chronology protection conjecture, he surmised that physicists would eventually discover a flaw in the theory of closed timelike curves that made them impossible. 

In 2009, he came up with an amusing way to test this conjecture. Hawking held a champagne party (shown in his Discovery Channel program), but he only advertised it after it had happened. His reasoning was that, if time machines eventually become practical, someone in the future might read about the party and travel back to attend it. But no one did — Hawking sat through the whole evening on his own. This doesn't prove time travel is impossible, but it does suggest that it never becomes a commonplace occurrence here on Earth.

The arrow of time 

One of the distinctive things about time is that it has a direction — from past to future. A cup of hot coffee left at room temperature always cools down; it never heats up. Your cellphone loses battery charge when you use it; it never gains charge. These are examples of entropy , essentially a measure of the amount of "useless" as opposed to "useful" energy. The entropy of a closed system always increases, and it's the key factor determining the arrow of time.

It turns out that entropy is the only thing that makes a distinction between past and future. In other branches of physics, like relativity or quantum theory, time doesn't have a preferred direction. No one knows where time's arrow comes from. It may be that it only applies to large, complex systems, in which case subatomic particles may not experience the arrow of time.

Time travel paradox 

If it's possible to travel back into the past — even theoretically — it raises a number of brain-twisting paradoxes — such as the grandfather paradox — that even scientists and philosophers find extremely perplexing.

Killing Hitler

A time traveler might decide to go back and kill him in his infancy. If they succeeded, future history books wouldn't even mention Hitler — so what motivation would the time traveler have for going back in time and killing him?

Killing your grandfather

Instead of killing a young Hitler, you might, by accident, kill one of your own ancestors when they were very young. But then you would never be born, so you couldn't travel back in time to kill them, so you would be born after all, and so on … 

A closed loop

Suppose the plans for a time machine suddenly appear from thin air on your desk. You spend a few days building it, then use it to send the plans back to your earlier self. But where did those plans originate? Nowhere — they are just looping round and round in time.

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Andrew May holds a Ph.D. in astrophysics from Manchester University, U.K. For 30 years, he worked in the academic, government and private sectors, before becoming a science writer where he has written for Fortean Times, How It Works, All About Space, BBC Science Focus, among others. He has also written a selection of books including Cosmic Impact and Astrobiology: The Search for Life Elsewhere in the Universe, published by Icon Books.

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Why does time change when traveling close to the speed of light? A physicist explains

Assistant Professor of Physics and Astronomy, Rochester Institute of Technology

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Why does time change when traveling close to the speed of light? – Timothy, age 11, Shoreview, Minnesota

Imagine you’re in a car driving across the country watching the landscape. A tree in the distance gets closer to your car, passes right by you, then moves off again in the distance behind you.

Of course, you know that tree isn’t actually getting up and walking toward or away from you. It’s you in the car who’s moving toward the tree. The tree is moving only in comparison, or relative, to you – that’s what we physicists call relativity . If you had a friend standing by the tree, they would see you moving toward them at the same speed that you see them moving toward you.

In his 1632 book “ Dialogue Concerning the Two Chief World Systems ,” the astronomer Galileo Galilei first described the principle of relativity – the idea that the universe should behave the same way at all times, even if two people experience an event differently because one is moving in respect to the other.

If you are in a car and toss a ball up in the air, the physical laws acting on it, such as the force of gravity, should be the same as the ones acting on an observer watching from the side of the road. However, while you see the ball as moving up and back down, someone on the side of the road will see it moving toward or away from them as well as up and down.

Special relativity and the speed of light

Albert Einstein much later proposed the idea of what’s now known as special relativity to explain some confusing observations that didn’t have an intuitive explanation at the time. Einstein used the work of many physicists and astronomers in the late 1800s to put together his theory in 1905, starting with two key ingredients: the principle of relativity and the strange observation that the speed of light is the same for every observer and nothing can move faster. Everyone measuring the speed of light will get the same result, no matter where they are or how fast they are moving.

Let’s say you’re in the car driving at 60 miles per hour and your friend is standing by the tree. When they throw a ball toward you at a speed of what they perceive to be 60 miles per hour, you might logically think that you would observe your friend and the tree moving toward you at 60 miles per hour and the ball moving toward you at 120 miles per hour. While that’s really close to the correct value, it’s actually slightly wrong.

This discrepancy between what you might expect by adding the two numbers and the true answer grows as one or both of you move closer to the speed of light. If you were traveling in a rocket moving at 75% of the speed of light and your friend throws the ball at the same speed, you would not see the ball moving toward you at 150% of the speed of light. This is because nothing can move faster than light – the ball would still appear to be moving toward you at less than the speed of light. While this all may seem very strange, there is lots of experimental evidence to back up these observations.

Time dilation and the twin paradox

Speed is not the only factor that changes relative to who is making the observation. Another consequence of relativity is the concept of time dilation , whereby people measure different amounts of time passing depending on how fast they move relative to one another.

Each person experiences time normally relative to themselves. But the person moving faster experiences less time passing for them than the person moving slower. It’s only when they reconnect and compare their watches that they realize that one watch says less time has passed while the other says more.

This leads to one of the strangest results of relativity – the twin paradox , which says that if one of a pair of twins makes a trip into space on a high-speed rocket, they will return to Earth to find their twin has aged faster than they have. It’s important to note that time behaves “normally” as perceived by each twin (exactly as you are experiencing time now), even if their measurements disagree.

You might be wondering: If each twin sees themselves as stationary and the other as moving toward them, wouldn’t they each measure the other as aging faster? The answer is no, because they can’t both be older relative to the other twin.

The twin on the spaceship is not only moving at a particular speed where the frame of references stay the same but also accelerating compared with the twin on Earth. Unlike speeds that are relative to the observer, accelerations are absolute. If you step on a scale, the weight you are measuring is actually your acceleration due to gravity. This measurement stays the same regardless of the speed at which the Earth is moving through the solar system, or the solar system is moving through the galaxy or the galaxy through the universe.

Neither twin experiences any strangeness with their watches as one moves closer to the speed of light – they both experience time as normally as you or I do. It’s only when they meet up and compare their observations that they will see a difference – one that is perfectly defined by the mathematics of relativity.

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April 26, 2023

Is Time Travel Possible?

The laws of physics allow time travel. So why haven’t people become chronological hoppers?

By Sarah Scoles

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In the movies, time travelers typically step inside a machine and—poof—disappear. They then reappear instantaneously among cowboys, knights or dinosaurs. What these films show is basically time teleportation .

Scientists don’t think this conception is likely in the real world, but they also don’t relegate time travel to the crackpot realm. In fact, the laws of physics might allow chronological hopping, but the devil is in the details.

Time traveling to the near future is easy: you’re doing it right now at a rate of one second per second, and physicists say that rate can change. According to Einstein’s special theory of relativity, time’s flow depends on how fast you’re moving. The quicker you travel, the slower seconds pass. And according to Einstein’s general theory of relativity , gravity also affects clocks: the more forceful the gravity nearby, the slower time goes.

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“Near massive bodies—near the surface of neutron stars or even at the surface of the Earth, although it’s a tiny effect—time runs slower than it does far away,” says Dave Goldberg, a cosmologist at Drexel University.

If a person were to hang out near the edge of a black hole , where gravity is prodigious, Goldberg says, only a few hours might pass for them while 1,000 years went by for someone on Earth. If the person who was near the black hole returned to this planet, they would have effectively traveled to the future. “That is a real effect,” he says. “That is completely uncontroversial.”

Going backward in time gets thorny, though (thornier than getting ripped to shreds inside a black hole). Scientists have come up with a few ways it might be possible, and they have been aware of time travel paradoxes in general relativity for decades. Fabio Costa, a physicist at the Nordic Institute for Theoretical Physics, notes that an early solution with time travel began with a scenario written in the 1920s. That idea involved massive long cylinder that spun fast in the manner of straw rolled between your palms and that twisted spacetime along with it. The understanding that this object could act as a time machine allowing one to travel to the past only happened in the 1970s, a few decades after scientists had discovered a phenomenon called “closed timelike curves.”

“A closed timelike curve describes the trajectory of a hypothetical observer that, while always traveling forward in time from their own perspective, at some point finds themselves at the same place and time where they started, creating a loop,” Costa says. “This is possible in a region of spacetime that, warped by gravity, loops into itself.”

“Einstein read [about closed timelike curves] and was very disturbed by this idea,” he adds. The phenomenon nevertheless spurred later research.

Science began to take time travel seriously in the 1980s. In 1990, for instance, Russian physicist Igor Novikov and American physicist Kip Thorne collaborated on a research paper about closed time-like curves. “They started to study not only how one could try to build a time machine but also how it would work,” Costa says.

Just as importantly, though, they investigated the problems with time travel. What if, for instance, you tossed a billiard ball into a time machine, and it traveled to the past and then collided with its past self in a way that meant its present self could never enter the time machine? “That looks like a paradox,” Costa says.

Since the 1990s, he says, there’s been on-and-off interest in the topic yet no big breakthrough. The field isn’t very active today, in part because every proposed model of a time machine has problems. “It has some attractive features, possibly some potential, but then when one starts to sort of unravel the details, there ends up being some kind of a roadblock,” says Gaurav Khanna of the University of Rhode Island.

For instance, most time travel models require negative mass —and hence negative energy because, as Albert Einstein revealed when he discovered E = mc 2 , mass and energy are one and the same. In theory, at least, just as an electric charge can be positive or negative, so can mass—though no one’s ever found an example of negative mass. Why does time travel depend on such exotic matter? In many cases, it is needed to hold open a wormhole—a tunnel in spacetime predicted by general relativity that connects one point in the cosmos to another.

Without negative mass, gravity would cause this tunnel to collapse. “You can think of it as counteracting the positive mass or energy that wants to traverse the wormhole,” Goldberg says.

Khanna and Goldberg concur that it’s unlikely matter with negative mass even exists, although Khanna notes that some quantum phenomena show promise, for instance, for negative energy on very small scales. But that would be “nowhere close to the scale that would be needed” for a realistic time machine, he says.

These challenges explain why Khanna initially discouraged Caroline Mallary, then his graduate student at the University of Massachusetts Dartmouth, from doing a time travel project. Mallary and Khanna went forward anyway and came up with a theoretical time machine that didn’t require negative mass. In its simplistic form, Mallary’s idea involves two parallel cars, each made of regular matter. If you leave one parked and zoom the other with extreme acceleration, a closed timelike curve will form between them.

Easy, right? But while Mallary’s model gets rid of the need for negative matter, it adds another hurdle: it requires infinite density inside the cars for them to affect spacetime in a way that would be useful for time travel. Infinite density can be found inside a black hole, where gravity is so intense that it squishes matter into a mind-bogglingly small space called a singularity. In the model, each of the cars needs to contain such a singularity. “One of the reasons that there's not a lot of active research on this sort of thing is because of these constraints,” Mallary says.

Other researchers have created models of time travel that involve a wormhole, or a tunnel in spacetime from one point in the cosmos to another. “It's sort of a shortcut through the universe,” Goldberg says. Imagine accelerating one end of the wormhole to near the speed of light and then sending it back to where it came from. “Those two sides are no longer synced,” he says. “One is in the past; one is in the future.” Walk between them, and you’re time traveling.

You could accomplish something similar by moving one end of the wormhole near a big gravitational field—such as a black hole—while keeping the other end near a smaller gravitational force. In that way, time would slow down on the big gravity side, essentially allowing a particle or some other chunk of mass to reside in the past relative to the other side of the wormhole.

Making a wormhole requires pesky negative mass and energy, however. A wormhole created from normal mass would collapse because of gravity. “Most designs tend to have some similar sorts of issues,” Goldberg says. They’re theoretically possible, but there’s currently no feasible way to make them, kind of like a good-tasting pizza with no calories.

And maybe the problem is not just that we don’t know how to make time travel machines but also that it’s not possible to do so except on microscopic scales—a belief held by the late physicist Stephen Hawking. He proposed the chronology protection conjecture: The universe doesn’t allow time travel because it doesn’t allow alterations to the past. “It seems there is a chronology protection agency, which prevents the appearance of closed timelike curves and so makes the universe safe for historians,” Hawking wrote in a 1992 paper in Physical Review D .

Part of his reasoning involved the paradoxes time travel would create such as the aforementioned situation with a billiard ball and its more famous counterpart, the grandfather paradox : If you go back in time and kill your grandfather before he has children, you can’t be born, and therefore you can’t time travel, and therefore you couldn’t have killed your grandfather. And yet there you are.

Those complications are what interests Massachusetts Institute of Technology philosopher Agustin Rayo, however, because the paradoxes don’t just call causality and chronology into question. They also make free will seem suspect. If physics says you can go back in time, then why can’t you kill your grandfather? “What stops you?” he says. Are you not free?

Rayo suspects that time travel is consistent with free will, though. “What’s past is past,” he says. “So if, in fact, my grandfather survived long enough to have children, traveling back in time isn’t going to change that. Why will I fail if I try? I don’t know because I don’t have enough information about the past. What I do know is that I’ll fail somehow.”

If you went to kill your grandfather, in other words, you’d perhaps slip on a banana en route or miss the bus. “It's not like you would find some special force compelling you not to do it,” Costa says. “You would fail to do it for perfectly mundane reasons.”

In 2020 Costa worked with Germain Tobar, then his undergraduate student at the University of Queensland in Australia, on the math that would underlie a similar idea: that time travel is possible without paradoxes and with freedom of choice.

Goldberg agrees with them in a way. “I definitely fall into the category of [thinking that] if there is time travel, it will be constructed in such a way that it produces one self-consistent view of history,” he says. “Because that seems to be the way that all the rest of our physical laws are constructed.”

No one knows what the future of time travel to the past will hold. And so far, no time travelers have come to tell us about it.

Space Travel Calculator

Traveling in space: an introduction, before einstein: non-relativistic space travel, how to calculate the travel time: speed of light as ultimate speed limit, travel in a relativistic spaceship: calculations for time and speed, fuel calculator for space travel: astronomical pit-stop.

Humans are barely a spacefaring civilization, as we only entered our spatial neighborhood: our space travel calculator will answer the question "what if..."

• What if I board a ship that travels in space at constant acceleration?
• What if I can ignore the speed of light in calculating the travel time in space?
• What if Einstein was right (he is) and space travel is relativistic?

And much more.

Traveling in space is a whole different kettle of fish. No air means no friction, the ideal rocket equation rules undisputed, and usually, your destination is not exactly behind the corner.

Spaceflight is hard: humanity ventured as far as the Moon (slightly beyond if you consider the orbits around it) and did so only six times between 1969 and 1972. Since then, we have only ventured into Earth's orbit. However, the push for exploration didn't make vane; we are limited by technology and physics!

In this tool, we will consider what would happen to a spaceship that travels in space at constant acceleration . The good news is that since there is no friction up there, we don't have to burn fuel to maintain a constant speed. If our engine is on, we are accelerating (in fact, most of the time spent in space by a craft consists of coasting , engines off, and patiently waiting to reach the time for a correction in the trajectory).

Input the spacecraft mass, your destination (trust us on the directions), and what you want to do precisely: a fly-by or a full stop (in this case, we will calculate your space travel in two parts, the latter at a constant deceleration that would bring you at destination with zero speed, à la Expanse ).

🙋 Feel free to input a destination of your choice by inserting any distance in the proper variable's field.

The last choice before the departure: is your universe following the rules of Newton or Einstein? We'll see the differences in a second. Board the spaceship Calculator , buckle up and wait for the countdown.

🔎 To explain our space travel calculator, we will assume a constant 1 g 1g 1 g acceleration (the most comfortable for a human) and an empty spacecraft mass of 1.000  t 1.000\ \text{t} 1.000   t . The destination we chose for our spaceship calculator is the center of our galaxy , a supermassive black hole 27 , 900 27,900 27 , 900 light years away.

Gravity rules Newton's universe alone. There is no speed limit and no one of the weird relativistic effects we will meet shortly. We calculate your space travel using the equation for motion in a purely classic framework.

If you choose to arrive at your destination at the maximum speed possible, then we input your acceleration in space in the formula:

• a a a — The acceleration ;
• t t t — The time of flight ; and
• v f v_{\text{f}} v f ​ — The final speed .

To calculate the time, we use the distance d d d :

If you plan on visiting Sagittarius A, then you need to decelerate. In this case, the final speed is $$v_{\text{ f}} = 0$$, obviously, and the time of flight changes accordingly:

The time required to travel such a distance is... astronomical . As you can see in our constant acceleration space travel calculator:

• For a maximum speed flyby, the time is 232.5  y 232.5\ \text{y} 232.5   y ; and
• To stop at destination, 328.8  y 328.8\ \text{y} 328.8   y .

The maximum velocity in the first case is 240 240 240 times the speed of light. If Einstein could hear this, he would be utterly disappointed. To right this wrong, we will calculate the travel time if the speed of light genuinely represent an impenetrable barrier.

We enter the territory of relativistic effects . Relativistic space travel calculations are a bit more complicated. In layman's terms, the faster you go, the slower time passes for you, and the perceived length for you, the traveler, also reduces. These two effects, described by the theory of special relativity, are coded in two equations:

γ \gamma γ is the Lorentz factor :

Where β \beta β is the ratio, always smaller than 1 1 1 , between the spacecraft's speed and the light's speed.

To find the time required to reach a given destination in a universe ruled by Einstein's relativity theory, with constant acceleration in space, the formula we've seen before must be changed and split: time is relative, and because of this, the trip will have two durations.

For a maximum speed fly-by from the perspective of a stationary observer:

The duration of the journey as experienced by our astronauts is:

In these equations, d d d is the distance. In this relativistic framework, we calculate it with the formula:

Lastly, we can calculate the maximum velocity in relativistic space travel without deceleration:

In these formulas, we used the hyperbolic functions : visit our hyperbolic functions calculator to learn more about them.

For a visit to Sagittarius A*, the times required for relativistic travel at constant 1 g 1g 1 g acceleration would be:

The difference is noticeable , to say the least. The maximum speed would be 0.4 0.4 0.4 parts per billion smaller than the speed of light: the dilation effects would be extreme.

The formulas would change slightly if we wanted to stop at our destination. From the observer's point of view, the time passed is:

In our example, t = 27 , 902  y t=27,902\ \text{y} t = 27 , 902   y . From the perspective of the travelers, the time is:

Corresponding to 20  y 20\ \text{y} 20   y . The perceived time is much longer than before: almost two times. This is because the astronauts would not "enjoy" a noticeable time dilation during the initial and final parts of the journey.

For distance and maximum velocity, we apply the following formulas:

You can use our space travel calculator also to find the kinetic energy of an object moving at such speeds. You won't be surprised to learn that the kinetic energy of an object moving almost at the speed of light is astronomical .

Rocketry is another word for mastery in fuel economy : you can learn everything about it with our rocket thrust calculator . Imagining an interstellar journey using chemical, ionic, or nuclear rockets is wishful thinking. To even have a shot to the stars, we need to learn how to control the mass to energy conversion . The annihilation reaction between matter and antimatter would have a perfect yield, converting all the mass involved into energy .

Assuming this 100 % 100\% 100% efficiency, we can compute the required mass for our journey both in the classic and relativistic case:

The results of these equations are disheartening: to send our ship to the center of our galaxy and stop there, the required fuel in the relativistic case is almost 830 830 830 billion tons.

Will humans ever reach the star? Will Enterprises and Millenium Falcons cross the darkness between other Suns? With the technology of today, it's unlikely. But things change quickly, and what looks impossible today may be tomorrow's science. Be hopeful and keep dreaming about touching the sky.

Gravitational time dilation

Schwarzschild radius, velocity addition.

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Most amateurs only think of ball position relative to the feet. Here’s why that’s wrong

Too many players only think of ball position relative to their feet.

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Welcome to Shaving Strokes , a GOLF.com series in which we’re sharing improvements, learnings and takeaways from amateur golfers just like you — including some of the speed bumps and challenges they faced along the way.

Ball position in golf is one of the most important factors in hitting a good shot — yet so many amateurs still overlook it during their pre-shot process.

For instance, most mid-to-high handicappers know it’s recommended to position the golf ball on the inside of their lead foot when using a driver. This allows the player to swing up on the ball, giving them a better opportunity to launch it.

Rethink ball position to pure crucial approach shots from 75-100 yards

But next time you go to the driving range, look around at how many golfers still don’t do this, and just set up to the ball with it (typically) in the center or even the back of their stance.

Thing is, using your feet isn’t the only way to determine what your ball position should actually be, as the body plays a big role in it, too.

To understand this, take a look below at what GOLF Teacher to Watch Kelvin Kelley has to say, as he explains why your golf ball position should also be relative to your body.

Why using your body is important to determine golf ball position

The concept of golf ball position being only relative to the feet — with the idea that, after taking your stance, the ball should be in the middle of the feet with the irons, and the more towards the front foot with the driver — is partially true, but there’s more to it.

Instead of only thinking about ball position using your feet, take into account where your mass (spine and upper body) is positioned relative to the ball at address. This is most critical when looking for solid contact. It’s why the angles of your body are so important.

The photo below shows what the spine and mass look like at the top of the swing. Notice how they’re behind the golf ball.

By starting this way, your swing is much more efficient, as the body can move forward and around toward the target in the downswing. This also allows you to hit with the entirety of your mass and body going in the optimal direction.

To picture this, think of a boxer ready to punch a boxing bag. The boxer would have their body positioned slightly behind the bag to punch forward into the bag. A boxer that sets up in front of the bag would have no body mass to use into the punch.

This concept in golf is nothing new, as the picture below shows Jack Nicklaus using it with a driver, mid-iron, and short wedge.

Another example comes from Ben Hogan’s book, “ Five Lessons “, where he described his ball position method.

Hogan’s left foot would stay the same distance from the ball regardless of club, while his trail foot would progressively widen as the club got longer. The distance that the trail foot moved back would also move his upper body.

Another note to pay attention to is how open or closed the feet should be for each club in your bag. The image below is a good guide to follow — but remember, this is just a baseline. The ball should be moved relative to the feet when hitting a certain flighted shot.

To play with the proper angle of attack, you should slightly drop the trail foot back when using longer clubs. This promotes a more shallow attack angle, as the body can slightly close. For shorter irons, open the feet, which influences a more downward and across strike since the body will be more open.

By rethinking the ball position relative to your body at impact, you’ll allow for a better attack angle and more opportunities of hitting it flush.

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Latest in instruction, the simple way to read greens, according to a top instructor, this simple divot drill will help you compress the golf ball. here's how, scottie scheffler uses this training aid. here's why you should too, want to know your fastest potential swing speed this app will tell you.

Kelvin is a Class A PGA golf professional in San Francisco, California. He has taught at some of the top golf clubs in the Bay Area, which include The Olympic Club and Sonoma Golf Club. He is TPI certified and certified Callaway and Titleist club fitter. Kelvin has sought advice and learned under several of the top instructors in the game, including Alex Murray and Scott Hamilton.

Kelvin works with his students to develop an efficient golf swing. A swing that is repeatable, powerful and will strive under pressure. He welcomes golfers of all skill levels however is geared towards the more committed golfer. Besides swing and short game instruction, Kelvin is an expert in on and off the course coaching. This includes mental preparation, developing practice routines and course strategizing.

Nick Dimengo

Related articles, don't let a disastrous hole wreck you. do this to bounce back, facing a long par-3 why it's time to rethink your club choice, stuck with a loose lie top teacher shares tips for a quick recovery, 10 ways to simplify greenside bunker shots, practicing on range mats here are the benefits and drawbacks, upwind vs. downwind: how to master these tricky shots.

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Midwest tornadoes cause severe damage in Omaha suburbs

The Associated Press

Gopala Penmetsa walks past his house after it was leveled by a tornado near Omaha, Neb., on Friday. Chris Machian/Omaha World-Herald via AP hide caption

Gopala Penmetsa walks past his house after it was leveled by a tornado near Omaha, Neb., on Friday.

OMAHA, Neb. — A tornado plowed through suburban Omaha, Nebraska, on Friday afternoon, damaging hundreds of homes and other structures as the twister tore for miles along farmland and into subdivisions. Injuries were reported but it wasn't yet clear if anyone was killed in the storm.

Multiple tornadoes were reported in Nebraska but the most destructive storm moved from a largely rural area into suburbs northwest of Omaha, a city of 485,000 people.

Photos on social media showed heavily damaged homes and shredded trees. Video showed homes with roofs stripped of shingles, in a rural area near Omaha. Law enforcement were blocking off roads in the area.

Hundreds of houses sustained damage in Omaha, mostly in the Elkhorn area in the western part of the city, police Lt. Neal Bonacci said.

Police and firefighters are now going door-to-door helping people who are trapped.

Omaha Fire Chief Kathy Bossman said crews had gone to the "hardest hit area" and had a plan to search anywhere someone could be trapped.

"They're going to be putting together a strategic plan for a detailed search of the area, starting with the properties with most damage," Bossman said. "We'll be looking throughout properties in debris piles, we'll be looking in basements, trying to find any victims and make sure everybody is rescued who needs assistance."

Damaged houses are seen after a tornado passed through the area near Omaha, Neb., on Friday. Chris Machian/Omaha World-Herald via AP hide caption

Damaged houses are seen after a tornado passed through the area near Omaha, Neb., on Friday.

Omaha police Lt. Neal Bonacci said many homes were destroyed or severely damaged.

"You definitely see the path of the tornado," Bonacci said.

In one area of Elkhorn, dozens of newly built, large homes were damaged. At least six were destroyed, including one that was leveled, while others had the top half ripped off.

There were dozens of emergency vehicles in the area.

"We watched it touch down like 200 yards over there and then we took shelter," said Pat Woods, who lives in Elkhorn. "We could hear it coming through. When we came up our fence was gone and we looked to the northwest and the whole neighborhood's gone."

His wife, Kim Woods added, "The whole neighborhood just to the north of us is pretty flattened."

Dhaval Naik, who said he works with the man whose house was demolished, said three people, including a child, were in the basement when the tornado hit. They got out safely.

KETV-TV video showed one woman being removed from a demolished home on a stretcher in Blair, a city just north of Omaha.

Omaha Police Chief Todd Schmaderer said there appeared to be few serious injuries, in part because people had plenty of warning that storms were likely.

The exact link between tornadoes and climate change is hard to draw. Here's why

"We not upon by a sudden storm," Schmaderer said. "People had warnings of this and that saved lives."

The tornado warning was issued in the Omaha area on Friday afternoon just as children were due to be released from school. Many schools had students shelter in place until the storm passed. Hours later, buses were still transporting students home.

Another tornado hit an area on the eastern edge of Omaha, passing directly through parts of Eppley Airfield, the city's airport. Officials closed the airport to aircraft operations to access damage but then reopened the facility, Omaha Airport Authority Chief Strategy Officer Steve McCoy said.

Severe weather damage to Eppley Airfield in Omaha, Neb., can be seen from the Lewis and Clark Monument in Council Bluffs, Iowa on Friday Anna Reed/Omaha World-Herald via AP hide caption

Severe weather damage to Eppley Airfield in Omaha, Neb., can be seen from the Lewis and Clark Monument in Council Bluffs, Iowa on Friday

The passenger terminal wasn't hit by the tornado but people rushed to storm shelters until the twister passed, McCoy said.

Flight delays are expected Friday evening.

After passing through the airport, the tornado crossed the Missouri River and into Iowa, north of Council Bluffs.

Nebraska Emergency Management Agency spokesperson Katrina Sperl said damage is just now being reported. Taylor Wilson, a spokesperson for the University of Nebraska Medical Center, said they hadn't seen any injuries yet.

Before the tornado hit the Omaha area, three workers in an industrial plant were injured Friday afternoon when a tornado struck an industrial plant in Lancaster County, sheriff's officials said in an update on the damage.

The building just northeast of the state capital of Lincoln had collapsed with about 70 employees inside and several people trapped, sheriff's officials said. Everyone was evacuated, and three people had injuries that were considered not life-threatening, authorities said.

Sheriff's officials say they also had reports of a tipped-over train near Waverly, also in Lancaster County.

Two people who were injured when the tornado passed through Lancaster County were being treated at the trauma center at Bryan Medical Center West Campus in Lincoln, the facility said in a news release. It said the patients were in triage and no details were released on their condition.

The Omaha Public Power District reported that nearly 10,000 customers were without power in the Omaha area.

Daniel Fienhold, manager of the Pink Poodle Steakhouse in Crescent, Iowa, said he was outside watching the weather with his daughter and restaurant employees. He said "it looked like a pretty big tornado was forming" northeast of town.

"It started raining, and then it started hailing, and then all the clouds started to kind of swirl and come together, and as soon as the wind started to pick up, that's when I headed for the basement, but we never saw it," Fienhold said.

The Weather Service also issued tornado watches across parts of Iowa, Kansas, Missouri, Oklahoma and Texas. And forecasters warned that large hail and damaging wind gusts were possible.