# Relativity and Time Travel

Do the laws of nature allow for time travel? We won’t answer this question, because the answer depends on what the laws of nature really are, and that is a matter we leave to the scientists, and specifically the physicists. Still, what we do have to say here is certainly pertinent to the question. Take this as an elementary primer for a study of the actual physics of time travel.

Our modest goal is to sketch how both forward and backward time travel are permitted by *relativity theory.* The prospects of forward time travel are less complex to describe and will require fewer idealizations. Backwards time travel is much more a matter of purely theoretical consideration. After a brief introduction to relativity theory, we will describe a case of time travel to the future and a case of time travel to the past; each case is consistent with the laws of relativity theory.

## Relativity and Minkowski Spacetime

In 1895, H. G. Wells penned a depiction of machine-based time travel as a serial later republished as a novella: *The Time Machine*. In it, Wells’ main character proposes that the universe includes not just three dimensions of space, but also a fourth of time. With the advent of Einstein’s Special Theory of Relativity in 1905 and the General Theory of Relativity in 1916, space and time were ultimately intertwined in effect making scientific a bit of what Wells imagined.

It is a common practice to start discussions of relativity theory by introducing the idea of a *frame of reference*. We are brought up understanding space and time, in part, through our ordinary ways of measuring spatial properties like length (e.g., using meter sticks, tape measures, odometers, sonar, lasers, …) and temporal properties like duration (e.g., using sundials, calendars, stopwatches, atomic clocks, …). Employing frames of reference allows one to begin thinking about relativity theory using these familiar ways of making observations about space and time. Taking such an approach leads one quickly to conclusions that show that the resulting measurements are not absolute, but are relative to a frame of reference. Length and duration turn out to be *frame-dependent*.

Relativity theory, however, is a theory about the nature of the universe, and is not—in the first instance—about frames of reference. Indeed, when it comes to the possibility of time travel, frame-relative measurements are—conveniently for us—pretty incidental. Insofar as the laws of relativity do permit time travel, time travel turns out to be a *frame-independent* phenomenon. In order to present our examples of time travel, we will introduce a coordinate system for a particular frame of reference, but we do so only to allow a simple algebraic description of the key frame-independent phenomenon. In Minkowski Spacetime, the spacetime of the Special Theory of Relativity, the time measured or experienced by a person or object moving inertially between two events is proportional to the spacetime interval between the events. The spacetime interval between two events is frame-independent.

In Euclidean 3-Space, described using Cartesian coordinates, the interval between two points, (*x*_{2}*, **y*_{2}*, **z*_{2}) and (*x*_{1}*, **y*_{1}*, **z*_{1}) is defined as the square root of the sum of the squares of the separation between the points along the three spatial dimensions:

This separation between two points is the same no matter the choice of the origin or any rotation of the axes for the coordinate system; this is a defining feature of Cartesian coordinate systems.

To describe the separation between events in Minkowski Spacetime, we’ll use Lorentz coordinates. The location of a point is provided by an ordered four-tuple of coordinates, each along one of four dimensions. The interval between two points, (*t*_{2}, *x*_{2}*, **y*_{2}*, **z*_{2}) and (*t*_{1},*x*_{1}*, **y*_{1}*, **z*_{1}), is *not* given by the separation for Euclidean 4-Space:

It *is* given by:

This interval formula is the chosen formula for the separation between two events in Minkowski spacetime. The interval value between two points holds no matter the choice of the origin or any rotation of the axes of the coordinate system; this is a defining feature of Lorentz coordinate systems.

The spacetime interval, D_{M}, given by the equation displayed above has physical significance. The interval is proportional to the amount of time that would be measured by a clock traveling along an inertial path between events located at (*t*_{2}, *x*_{2}, *y*_{2}, *z*_{2}) and (*t*_{1}, *x*_{1}, *y*_{1}, *z*_{1}).^{1} Observers in different frames of reference, using a Lorentz coordinate system with a different origin or a different orientation of the axes, may disagree about the separation of the events along the *t*-axis, the *x*-axis, the *y*-axis, or the *z*-axis, but not about the spacetime interval between the events. The correlation between the spacetime interval between two events and the time elapsed along the inertial path between them has been experimentally confirmed in numerous ways. We will mention a couple of these ways below.

## A Trip to the Future

In order to present a case of forward time travel, we shall describe a simple example set in Minkowski spacetime. Suppose there is a civilization on a planet that includes events at (0, 0, 0, 0) and (10, 0, 0, 0), and at all the intermediate points between these two events along the *t*-axis of this Lorentz coordinate system. If Tim wants to time travel to the future to witness an event at (10, 0, 0 ,0), and if he wants to do it much sooner than the rest of his civilization, then Tim should take a different route through spacetime from (0, 0, 0, 0) to (10, 0, 0, 0) than does his civilization.

To make the case a little more concrete, we assume that the separation on the *t*-axis between the two events corresponds to 10 years. Suppose that Tim travels away from the planet at a very high speed arriving at a nearby star at (5, 4, 0, 0). Take him to be traveling 4 light years along the *x*-axis and 5 years along the *t*-axis. Leaving immediately upon arrival at the star, Tim heads back to the planet, arriving back at the planet at (10, 0, 0, 0). So, Tim’s departure is at (0, 0, 0, 0). He travels at 80% of the speed of light, taking only five years to go 4 light years in the positive *x *direction. Then Tim travels at 80% of the speed of light, taking five years to travel 4 light years in the negative *x* direction, arriving home at (10, 0, 0, 0).^{2}

As we will see once we complete some easy calculations, one shouldn’t take diagrams like the one above too seriously. It turns out that the sum of the spacetime interval from (0, 0, 0, 0) to (5, 4, 0, 0) and the interval from (5, 4, 0, 0) to (10, 0, 0, 0) *is less than* the interval from (0, 0, 0, 0) to (10, 0, 0, 0). This clearly is not in line with the spatial distances we see between the coordinates and the figures depicting these points in the diagram!

An application of the spacetime interval formula reveals that time passes differently for Tim than it does for his civilization. The spacetime interval between (10, 0, 0, 0) and (0, 0, 0, 0) along the path of his home planet is given by:

Checking the two intervals that make up Tim’s trip, we get:

So, 40% less time has elapsed for Tim than has passed for his friends who stayed put on the planet. If 10 years have elapsed on the planet, then only 6 years have passed for Tim. Had he traveled at even higher speeds to and from the star, he could have gotten back in even less time. He could have witnessed the event at (10, 0, 0, 0) at an even younger age.

This is not just theory. The consequence of taking different paths through spacetime on the amount of time that passes has been observed and documented often. In one of the more famous experiments, completed in 1971, scientists J.C. Hafele and Richard E. Keating placed cesium beam clocks aboard a commercial airliner headed eastward around the world. The plane was flown around the world and then compared with a reference atomic time scale at rest. Upon completion of the trip, it was found that the clocks from the flight were about 59 nanoseconds behind the clock at rest, which is almost the exact value that relativity theory predicts. Confirmation also comes from the behavior of basic particles called muons. They have a short half-life—less than two microseconds. When moving at high speeds, muons survive much longer than this short half-life would lead us to expect. With higher speeds, the particles survive longer, because they have experienced less time. For additional discussion of these and other results, see Gott (2001, p. 36), Greene (2004, p. 449), and Pickover (1998, pp. 119-121).

Has Tim time-traveled to the future? Frank Arntzenius reports being unsure about there being a clear notion of forward time travel. He downplays the interest of trips like Tim’s:

According to (special and general) relativity two clocks that travel along different world-lines from spacetime point A to spacetime point B will, almost always, measure different time intervals between A and B no matter what the structure the spacetime has … So, on a fairly natural characterization of what it is for there to be forwards time travel, forwards time travel would be ubiquitous, too ubiquitous to be interesting (2006, p. 605).

(Also see Smeenk and Wϋthrich 2011, p. 580.) Others might worry that, really, time merely slows for Tim—that he doesn’t really time travel. Still others might worry that this couldn’t be time travel because the method described includes no way for Tim to travel back to the past.

We are sympathetic with Phil Dowe’s assessment of trips like Tim’s: “This is time travel by any reasonable definition” (p. 443). (Also see Horwich 1975, p. 432.) We are so for the following reasons: In some ways, Tim’s trip is like the trip the astronauts take in *Planet of the Apes *(1968), who unbeknownst to them make a round trip to Earth of the distant future. In some ways, Tim is like Marygay of *The Forever War *who purchases a cruiser to use as a time machine (p. 264) to meet up in a timely fashion with William, who is not expected to return from battle for at least two centuries of Earth time. Perhaps, it would be clearer that Tim was time traveling had we constructed our example so that Tim’s trip took even less time for him. Wells’s time traveler and the time travelers in most sci-fi plots cover days, years, or even centuries in a matter of seconds. That could have been done in our example by having Tim travel along a different path.

Admittedly, it would be truer to standard time travel plots if Tim could just as easily travel back to the past via a similar method; this aspect of Tim’s situation does not match with one natural preconception prompted by science fiction. But it should hardly be a surprise that, when we turn our attention from a concept born and initially revealed in *science fiction* to looking for instances of the concept in the pages of science, that we find some divergence. We should respect science-fiction informed thoughts about what time travel is like, but we should not demand that they *all *apply in order to reasonably classify a scientific phenomenon as time travel. Not all preconceptions of a phenomenon are required to be met for there to be an actual discovery of the phenomenon. Tim’s is a case of time travel to the future.

## A Trip to the Past

Relativity theory also allows for ways to travel to the past. Because it connects nicely with our discussion above of Tim’s trip to the future, we will consider a form of backwards time travel associated with the work of physicist Kip Thorne, which also provides another manner of travel to the future. What we need is a wormhole time machine.

To understand what a wormhole is, think about a tunnel through a mountain. Without the tunnel, we would have to travel around or over the mountain. With the tunnel, we could get to the other side much more quickly and easily, traveling a much shorter distance. Similarly, wormholes connect more ordinary regions of spacetime, thus permitting a shortcut between them. Of course, a tunnel through a mountain is not a time machine into the past. Neither, necessarily, is a wormhole.

Thorne and his colleagues described a method for turning a wormhole into a time machine. There are lots of obstacles. For example, you must first find a wormhole—and so far no one has. The wormhole has also got to be large enough (or be made large enough) for your body to fit through. You also better make sure that the wormhole will not collapse before you are done with your trip. Thorne and his colleagues describe possible methods for doing all this—none easy to put in place—but we will not concern ourselves with these practical details. For us, the focus will be on what Thorne has to say about how the nature of the spacetime permits one end of the wormhole to be later in time than the other end, thus providing a gateway to and from earlier times.

To do so, we are going to simplify in one significant way. Minkowski Spacetime does not include wormholes. We would need to go beyond the Special Theory of Relativity to consider the General Theory of Relativity in order to describe a spacetime that allows for wormholes. That, in turn, would require us to bring in more difficult math than would be appropriate for this discussion. Instead, think of us as working in with a spacetime that’s just like Minkowski Spacetime except for the wormhole and its immediate vicinity. With this simplification, the characteristic of spacetime that permits one to turn a wormhole into a wormhole time machine is easy to describe. In fact, we have already covered most of what you need to know.

To build our wormhole time machine, we just need to separate the mouths of a wormhole in time. So, let’s have Tim take one mouth of a wormhole connecting two spatial regions on his home planet with him on his trip to the star. (How does he manage that? Never you mind! That’s another one of those practical details that we are skipping over.)

The time it takes for one to travel through the wormhole need not change no matter how far apart in ordinary spacetime the mouths are, but the wormhole mouth traveling with Tim ages more slowly than does Tim’s civilization and the mouth of the wormhole left behind there. In keeping with our earlier example, when Tim arrives back at (10, 0, 0, 0), only 6 years have passed for Tim though 10 years have passed for his civilization. There is the same difference in the time that has passed for the two wormhole ends that end being located at (10, 0, 0, 0).

In the diagram above, though the two mouths of the wormhole are the same age at (0, 0, 0, 0), at (10, 0, 0, 0), the tan end of the wormhole is 4 years older than the gray end.

With no change in the length of the wormhole, the two mouths of the wormhole will stay in sync: Tom, a friend of Tim’s, could be watching Tim through the tan mouth of the wormhole, and there need not be any discrepancy between Tom’s watch and Tim’s watch. At any point during the trip, Tim could have hopped in the gray mouth, immediately exited the tan mouth, and Tim and Tom would agree about how much time had passed since Tim started his trip. Similarly, Tom could have hopped in the tan mouth of the wormhole, exited the gray mouth, and the two friends would agree about how much time had passed.

So, what happens if Tom hops into the tan entrance of the wormhole at (6, 0, 0, 0)? Because the ends of the wormhole are in sync, he would end up where and when 6 years had passed for Tim. So, he would immediately exit at (10, 0, 0, 0), thus using the wormhole time machine to take a trip four years into the future of his civilization! The other direction is available too. Tim could jump into the gray mouth of the wormhole he’s brought with him on his trip, hopping in at (10, 0, 0, 0), and time travel four years into the past of his civilization. If Tim wants to make a bundle of money once he is back four years in time, he should stop at a library and make note of some winning lottery numbers from three or four years earlier before entering the wormhole!

## Conclusion

That is our introductory case for the conclusion that forward time travel and backward time travel are consistent with the theory of relativity. In fact, we have described two different methods for traveling to the future: (i) traveling along a different (shorter) path through spacetime than the rest of your civilization, or (ii) hopping in the earlier end of a wormhole time machine that connects two points in the history of your civilization. The wormhole time machine provides our only case of time travel to the past: Tim’s hopping in the later mouth of the wormhole connecting events in the history of his civilization. There are other possibilities permitted by relativity theory that provide for time travel to the future and time travel to the past. In *Time Travel in Einstein’s Universe* (Chapters 2 and 3), J. Richard Gott nicely describes some of these methods including a stay-at-home method of traveling to the future, a wormhole time machine, and Gott’s own two-cosmic-strings method for traveling to the past.

For those of you looking for a conclusive statement on whether the *actual *laws of nature permit time travel to the past, you’ll need to wait for a theory that goes beyond relativity theory, something like a finalized theory of quantum gravity. Thorne has recently provided an accessible sketch of his and other physicists’ doubts about whether the actual laws of nature do permit time travel in “Is Time Travel Allowed?” For those of you who are adventurous and feel ready to deal with some difficult physics, Smeenk and Wϋthrich’s “Time Travel and Time Machines” (2011) includes advanced discussion of whether time travel is permitted by our laws.

## References

Arntzenius, Frank. “Time Travel: Double Your Fun.” *Philosophy Compass* 1 (2006): 599–616.

Dowe, Phil. “The Case for Time Travel.” *Philosophy *75 (2005): 441-451.

Gott, J. Richard. *Time Travel in Einstein’s Universe*. Boston: Houghton-Mifflin, 2001.

Greene, Brian. *The Fabric of the Cosmos*. New York: Vintage Books, 2004.

Hafele, J.C. and Richard E. Keating. “Around-the-World Atomic Clocks: Observed Relativistic Time Gains” *Science* 177 (1972): 168-170.

Haldeman, Joe. *The Forever War*. New York: St. Martin’s Griffin, 1974.

Horwich, Paul. “On Some Alleged Paradoxes of Time Travel.” *Journal of Philosophy* 72 (1975): 432-444.

Maudlin, Tim. “Relativity Theory.” *Encyclopedia of Philosophy, Vol. 8. 2nd ed. *Ed. Donald M. Borchert. Detroit: Macmillan Reference USA, 2006: 345-357.

Morris, Michael; Kip Thorne; and Ulvi Yurtsever. “Wormholes, Time Machines, and the Weak Energy Condition.” *Physical Review Letters* 61 (1988): 1446–1449.

*Planet of the Apes*. Dir. Franklin Schaffer. Perf. Charlton Heston, Roddy McDowall, Kim Hunter. DVD. Twentieth Century Fox,1968.

Pickover, Clifford A. *Time: A Traveler’s Guide*. Oxford: Oxford University Press, 1998.

Smeenk, Chris and Christian Wϋthrich. “Time Travel and Time Machines.” *The Oxford Handbook of Philosophy of Time*. Ed. Craig Callender. Oxford: Oxford University Press, 2011: 577-630.

Thorne, Kip. *Black Holes and Time Warps*. New York: Norton, 1994.

Thorne, Kip. “Is Time Travel Allowed?” *Plus Magazine*. 2009. 28 Jan 2012. <http://plus.maths.org/content/time-travel-allowed>.

Wells, H. G. *The Time Machine*. New York: Tom Doherty Associates, 1992. (First published in 1895.)

For page credits, see the “Topics-Page Credits” page.

1. More generally, *the proper time* for something traveling between two events is just the path length of its worldline, which is indirectly calculated using the interval formula; for straight-line paths, this reduces to the interval between the departure and arrival events.

2. We are idealizing by assuming that both legs of the trip take place at one constant speed, setting aside matters about how a real spaceship would have to accelerate up to that speed, make the turn, and decelerate down from that speed. We are also setting aside what such acceleration and deceleration would do to Tim’s body. These details are not needed to accurately illustrate what it is about relativity theory that permits time travel to the future.