# The Twin Paradox in Special and General Relativity.

Relativity is profoundly unintuitive to humans. Our brain seem hardwired to visualize geometry in at most 3 dimensions, and 3 Euclidean dimensions at that. This is probably because we evolved in an environment where objects move at non-relativistic speeds. Similarly since we evolved in an environment where actions were much larger than the Planck constant, our brains just do not think naturally in terms of quantum mechanics. We are, at our core, creatures who think in classical physics. And that is good enough if you're a naked ape looking to hit a gnu with a rock, or even an engineer building the Hoover Dam, but that physical intuition falls apart when you get to the physics of the very fast, the very big, or the very small. And since the Universe is really quantum and relativistic, those limits are where things get fun.

But this disconnect between the physics of how things really are and the way that we can mentally model the Universe can cause apparent paradoxes. Perhaps the most famous paradox of this sort is the Twin Paradox of Special Relativity. If you're interested in physics, you've probably heard of this paradox, and probably know the solution. But recently I was thinking about this paradox in General Relativity, which turns out to have some fun consequences. So I thought I'd explain first the Special Relativistic paradox (and its solution), and then how it works in General Relativity. Also, it allows me to draw spacetime diagrams, and those are always good times.

So what is the Twin Paradox? It is an apparent logical paradox that arises from applying some of the consequences of Special Relativity, but not fully working through the problem. Once you work the problem out completely, you see that there is - of course - no logical paradox. Which is good, because the Universe and the laws of physics should not break logic, as that's equivalent to breaking math, and we don't put up with that sort of nonsense in this establishment.

The Twin Paradox comes from taking the following true statements that arise from Special Relativity.

1. All observers moving at constant velocities are justified in considering themselves stationary and the rest of the Universe moving. That is, any local experiment they perform cannot depend on their constant motion relative to some external privileged reference frame. These constant velocity observers live in an inertial reference frame. This is actually a very old principle, called Galilean Relativity. We should be very familar with it: it is the statement that if you're on a plane without turbulence, you can't tell you're moving unless you look out the window (looking out the window is a non-local experiment).
2. In order for all observers to measure the same speed of light (an experimental fact, and all of Special Relativity really just comes from requiring Galilean Relativity adapt to deal with this experimental fact), an inertial observer will measure clocks moving relative to them as ticking slower than their own clock. This isn't some trick, or optical illusion, and "clocks" include the rate at which the fundamental physics processes which drive human cognition and metabolism occur, so if someone is moving relative to you, they really are appearing to move and age slower than you are. This is called time dilation. The effect is small (but measurable) for relative speeds much less than light, but can grow to significant relative differences (approaching infinite time dilation, i.e. someone appearing not to progress forward in time) if something is moving past you close to light speed.
3. As a further consequence, inertial observers would see objects moving past them at relativistic speeds as contracted (along the direction of motion) relative to their length if measured by someone at rest with respect to the object. This is called length contraction. Again, it is a small, but measurable effect, unless the relative speeds are near the speed of light. You can pretty easily show that the factor by which time is dilated is the same as the factor by which lengths are contracted.

So consider the following thought experiment, and then we'll get the Paradox. We have a pair of twins, exactly the same age. One, Twin A, stays on Earth, never moving at relativistic speeds relative to the frame of reference that the rest of us call home. The other twin, Twin B, leaves on a relativistic rocket, on a long round trip that takes them away from Earth for many years, then they stop, turn around, and come home, again at relativistic speeds. To put some numbers on it, let's say that the rocket travels at a speed close enough to the speed of light that the time dilation factor is 10: so to Twin A, Twin B's clocks tick at 1/10 the rate as the ones on Earth. We can imagine that, according to Twin A, they see Twin B travel away from Earth for 10 years, stop, turn around, and travel back to Earth for another 10 years. Twin A then would have aged 20 years, and if they have kept an eye on their sibling using an impressively powerful telescope, they would have seen Twin B age only 2 years.

But what does Twin B see? Remember that relativity works regardless of which frame of reference you live in: everyone is justified in thinking they are in a stationary frame of reference (as long as you don't change velocity). So Twin B sees the distance over which they have to travel before stopping and turning around as length contracted: they see the distance they need to cover as 1/10th the distance as an observer on Earth would measure. So rather than taking 10 years for them to cover the distance, they think the Universe moved past them fast enough for the finish line to come to them in only 1 year. They then stop, turn around, and return to Earth, again taking 1 year. So they have aged 2 years when they get home.

But, here's the paradox: Twin B sees Twin A (and Earth in general) as moving relative to them. So they see Twin A as time dilated, and aging slower than Twin B. So Twin B thinks that only 2/10th of a year has elapsed for Twin A back home as they return (using the same relative time dilation factor of 10). So, the paradox is

Twin A thinks they are older than Twin B, but Twin B thinks they are older than Twin A. Thus a paradox: when Twin B gets off their rocket, logically only one of them can be older. Who is it?

The solution is not too hard to find, and resolves the paradox nicely. The symmetry that causes the paradox is that both observers appear to be inertial: both are justified (it seems) to say they are stationary and the other is moving. But, that symmetry is not really there: we skipped an important step. In order for Twin B to return home, they must accelerate: they must stop their rocket and turn around and reaccelerate to get back to Earth. During this, they will feel a force, they will know that they are moving relative to the frame of reference they started with. Twin A feels no such acceleration (ignore the motion of the Earth around the Sun for this). So Twin A is justified in saying they are the "stationary" observer and Twin B is the moving one. In the end, Twin A, the one who didn't travel, is the older twin, Twin B returns 18 years younger than their identical sibling.

Ok... so that's the solution, and the math works out. But where did the missing years go? It's fine to say that the deceleration and acceleration break the symmetry, but nothing I said about Twin A seeing Twin B aging 2 years and Twin B seeing Twin A aging 2/10th of a year during the trip was wrong. If each twin had powerful enough telescopes, they could watch their distant sibling, moving at 1/10th speed, aging slower than them. At some point, Twin A needs to age 19.8 years according to Twin B for the paradox to resolve itself. How does that happen?

You can see the resolution in a clever way using spacetime diagrams. What a spacetime diagram does is show how particular observers divide up space and time. The consequences of special relativity mean that other observers will divvy up space and time in a different way, but that's just how the Universe works. A spacetime diagram looks like this:

Since I can't draw all of three dimensional space, I've made the simplification of just drawing one coordinate (call it x). I scale my coordinates so that light travels at 45 degree angles on this plane (Pretend my drawing abilities are up to the task). One important consequence of the fact that everyone measures light to move at the speed of light is that everyone measures light to move at 45 degrees on their spacetime diagram. As you'll see, time will change, space will change, but light always moves on these 45 degree lines, which we call (for various reasons) "null lines."

Some areas of the spacetime diagram are far enough away from a particular moment in space and time that light from that could not have reached those other regions at a given time. Events in these regions are spacelike separated from the reference event. Events that you can reach by traveling slower than light are called timelike separated. Spacetime diagrams make it easy to tell which is which.

Now, let's specialize my spacetime diagram to the Twin Paradox. I have three events I care about: the moment Twin B leaves Earth, the moment Twin B stops and turns around, and the moment that Twin B disembarks and greets their wizened sibling back on Earth. In between these events, Twin B must be seen by Twin A as traveling first away from, and then back towards Earth. Since Twin B cannot travel faster than light, their path on the spacetime diagram is "above" the null lines: they are crossing space "slower" than they are crossing time (as we all are). Twin A doesn't leave Earth, so they stay traveling in time at the spatial origin.

But relativity is relativity, and any inertial frame can be justified in drawing a spacetime diagram in which that person remains at their spatial origin. So Twin B can draw their own spacetime diagram, in which they stay stationary and Twin A moves away from them (to the left, if Twin A sees Twin B moving to the right), and then towards them. However Twin B must draw two such frames, and imagine "jumping" from one frame to the other as they decelerate and then accelerate homewards at Event 2.

Since both twins see the other moving, if you ignore the fact that Twin A stays in one frame all the time while Twin B switches, you have the Twin Paradox.

And now we know the solution to that paradox.

But let's find the missing years. To do this, I need to draw what Twin B counts as "lines of constant time." Now, for Twin A, lines of constant time are easy, they are lines parallel to the space axes. So we need the space axes for the two frames that Twin B will inhabit. The time axes are easy, they are just the place where Twin B is at any given time (since Twin B is always under the impression they are at the origin of their own spatial coordinates). The space axis is a bit trickier to find: it turns out to be the mirror image of the time axes flipped across the line that light travels on.

Why this is the right answer may not be clear, but it should be at least clear that just rotating Twin A's coordinate system cannot be the right answer. If you did that (keeping time and space at 90 degrees, but aligning the time axis with the trajectory of Twin B), then light would not be traveling at 45 degrees according to Twin B. That is, Twin B would measure light to not be traveling at the speed of light. So something more complicated must happen. At the very least, the rule I just described (which you can prove without too much trouble) means that null lines are equally separated from the time and space axis, as they must be.

So now having done the hard work of finding the space axis, I can easily just draw lines of constant time, according to Twin B. Again, pretend my drawing skills are good enough that all these lines are parallel to each other. But now we see the times Twin B will assign to events in Twin A's life.

And now you can visually see the "missing years" that make up the resolution to the paradox. And according to this diagram, Twin B never sees Twin A age over that time. No wonder then Twin B gets off the rocket thinking Twin A has aged only 0.2 years, when in reality they've aged 20.

But in drawing a simplified version of how Twin B moves, we've imagined them jumping instantaneously from "moving away from Earth" to "moving towards Earth." In reality, they must slow down gradually, then speed back up. So rather than having a sharp inflection, their spacetime path arcs gradually, and their lines of constant time sweep out through the "missing years." But notice what this means: Twin B sees all the missing years of their sibling's life occur in the short time they are accelerating. In our example, that missing time is 19.8 years. So Twin B sees Twin A age 0.1 year in the year they are moving away from Earth, and in the arbitrarily short period of time they turn around, their Twin suddenly ages in high speed, gaining 19.8 years, and then ages the final 0.1 years in Twin B's journey home.

Well, I say Twin B "sees this" as they turn around. Really, by time they turn around, they are far from Earth, and light takes time to reach them.

So they actually see the images of 19.9 years of their homebody sibling's life during their trip home. But if Twin B worked out when in their own life the light from these images was emitted, they'd see that 19.8 years worth occurred during the turn-around. That's what we mean by Twin B seeing the missing years during the moment they switch frames - not that they literally see all the light from those years hit them at that moment, but they would later identify the emission of that light with the time in which they turned around.

So, that's the Twin Paradox in Special Relativity. If you're interested enough to read this far, you've probably heard at least this much. What else is there to say?

Notice that, in Special Relativity, the entire resolution requires one twin to be the one who is clearly moving: they accelerate, switching frames, and that tells you which one was "stationary" and which is traveling. But what if you could set things up so that the person who is traveling can get home without ever having to stop and turn around?

How can you possibly travel in a straight line, never accelerating, and get back home? On a plane, it is impossible. But everyone who grew up in the 80's knows of a universe in which you can travel in a straight line and come back home without changing directions.

In the game Asteroids, you fly off the top of the screen and wrap around to the bottom. Leave from the left and return to the right. The universe is topologically flat (unlike a sphere, which is another way you can head off and come back without changing direction). We've just identified the top and the bottom edges as being "the same" and likewise with the left and right sides.

Amusingly, what this means is that the Asteroids universe is a donut, or, more mathematically a torusIn the same way, it is possible that we could live in a Universe that was toroidal: travel in one direction and you'll wrap around again and come home. I'm not saying our real Universe is toroidal, just that there is no logical reason it can't be, and thought experiments like the Twin Paradox better not break physics in such a universe. So we might learn something by considering how a toroid universe deals with the Twins.

So mathematically, how can we talk about this torus? The universe you fly through in Asteroids is a flat sheet: there is no bending or anything (if there was, you'd think the ship moved faster or slower as it approached the "edges" of the screen, as a bent space was projected onto the flat one). So what the torus is is just a sheet, with the edges identified with each other. You hit one edge, and we declare, by fiat, that this is the same as the other edge. Obviously, you can't do this with a flat 2D sheet in 3D space without bending the sheet, but mathematically (and in our computer game), you don't need to introduce any curvature. I make a big deal out of this point because it's important to realize there is no flexing of space (or spacetime) anywhere on this torus. Everything is exactly like the space of special relativity, just with these strange boundary conditions.

So I'm drawing this both as a flat sheet and a curved donut, but mathematically the torus is more like the sheet than the donut in important respects.

We know from long experience flying through Asteroids that if you head off in some random direction on the torus, you'll just wind around the space, over and over again. And now you know what that looks like.

There is a periodicity to the spacetime now: if you move a certain distance over in space, you'll return to where you came from.

The way to resolve the Twin Paradox on the torus is to realize that the fact that there is this periodicity picks out a special frame of reference. It breaks the symmetry which allows all observers in Special Relativity to say "no, I'm not moving, you're moving" with equal validity. Some special observers on the donut can see that they are not moving with respect to the coordinates which define this periodic relation.

One way to see that is imagine what happens to light rays on the donut: they wrap around, just like every other moving object. So if you send off a pulse of light, you'll see it return (which means you can see yourself in the past, by the way).

There is a special frame of reference "at rest" with respect to the torus, which sees light rays sent off simultaneously return simultaneously. People in this frame of reference are justified in saying they are stationary relative to the torus.

People moving with respect to this frame can tell they are moving. They'll see light sent in front of them return later than light sent behind them. So, because of this global nature of spacetime, the special symmetry that defines Special Relativity doesn't exist. It isn't that physics is magically different in the torus rest frame, it is just that there happens to be a frame where a very specific thing happens to be true.

So you might guess that the twin who "isn't moving with respect to this frame is the one who ages "more" than the twin who takes a quick spin around the donut and comes back. And you'd be right. But how do we prove that? What goes wrong with our picture of special relativity for the torus?

The problem is that our picture of the flat screen on which we play Asteroids has edges, and a torus doesn't. If you think about it, you could easily imagine moving the "center" of the screen during a game of Asteroids and nothing would change: the rock coming at your ship would still be coming towards it, your vector through the space would be the same. Just where you arbitrarily called "the center" changes. The sheet of paper, with edges doesn't have that.

So we need to cut the edges off the sheet. But that presents us with a problem: how do we identify "the edges" on the top and bottom and left and right if there are no edges?

What you do is use more than one sheet of paper (without edges). Instead of waiting till you run off the side of the sheet, you overlay a sheet. So for a while, you can identify the location of an object on the torus by more than one coordinate system. That's fine, either will do, and redundancy doesn't hurt. Then, you leave one of the coordinate systems behind (flying off the sheet), and use only the new coordinates. Then, you go overlay the new coordinates with the old ones again, and voila, you have the exact same topology of a torus. The two descriptions must be identical.

Here I'm showing the overlay only over one edge of the sheet that will make up a torus, since as you can probably tell, my drawing skills are a bit shaky.

Also, at this point if you turn this description into more mathematical language, you've got yourself a manifold. A manifold allows us to do all the things we want to like we live in flat space, which over small enough regions we do (this is what is allowing us to say that all experiments give the same answer if done locally), but still keep track of the global, or large scale, properties of our spacetime.

This is what that mapping would look like curved around part of the donut (again, I'm suppressing the left-right identification so you can see a bit better what's going on. The place where you overlap is arbitrary, and any version will work (and you can use more than two sheets, if you like. Two is just the minimum that works to map the top and the bottom, you'll need at least two more for the left and right).

Now you should be able to se the difference between the two observers in the twin experiment clearly: one doesn't have to ever worry about changing which coordinate system they live on. The other must, as they are wrapping around the spacetime. It is during the readjustment of the coordinates as you sew the two sheets together than the difference in time would become apparent when you compare the journey through spacetime of Twin A and Twin B on the torus.

Turns out that this adjustment is kind of a pain, so there's even an easier way to do it, mathematically. You just say that the person moving is going from some starting point to some ending point, somewhere else in space (and time).

You do this by just putting a sheet above the one you started with (or to the side, if moving in that direction). Nothing wrong with that, and no edges to worry about crossing.

The trick is though, when the traveler gets to this new point, you yell "surprise" and identify the new point with the point in the original sheet using the symmetry of the torus.

So the poor sap thinks they're getting away from their twin, only for you to change the nice new spot they picked out for themselves to the place they started from. It's a mathematical sleight-of-hand, but it is the easiest way to see exactly who is younger in the Twin Paradox on the torus.

As you can see, In General Relativity, all the rules of special relativity apply, but with these added possible quirks that you have to care about what the spacetime you're moving through is doing: that is, what we call the metric, the way we measure distances, can be non-trival and can matter (after all, all the torus is is a declaration that points at the top of a sheet of paper are "near" the bottom of the sheet. That is, the distance between those points is shorter than it would be if space -- or spacetime -- were flat).

So the Twin Paradox continues to have a satisfactory resolution in General Relativity, but only when you start thinking not just about the local properties of spacetime, but how those little patches that "look like" Special Relativistic flat spacetime get sewn together to create the whole. We probably don't live in a torus-like universe, but by thinking about how physics would have to work if we did, we can learn something about how the laws of physics work in the Universe we do live in.