# I get mail: Twin Paradox Edition

Greetings, true believers! Apologies for the relative infrequency in posting, but as most of you know, I’ve been working hard on my upcoming book. I’m over the hump, and sent the first half to my editor a couple of weeks ago. I’m now working on Chapter 6, which will tentatively be entitled something like, “Why can’t you build an ansible?” While my old io9 article on the subject dealt with issues related to quantum entanglement, this chapter will talk about the relativity side of things, why simultaneity doesn’t exist in a relativistic universe, where $E=mc^2$ comes from, and much more!

In other news, you may have noticed that I got quoted in Natalie Angier’s column on Emmy Noether in the New York Times a couple weeks ago. That was very exciting. Natalie contacted me after seeing my own column/rough draft for the introductory material for Chapter 5.

But enough of that. Today I’d like to tell you about an email I got from a very earnest reader named Eduardo from Brazil. Eduardo wrote to ask me about the “Twin Paradox.” As this entry is labeled “technical” (meaning that there will be a few equations and an assumption that you remember some of your freshman level physics), you probably know the Twin Paradox already, but if not, here’s a short excerpt from the User’s Guide:

There are two twin sisters, Emily and Bonnie, who are both 30 years old. Emily decides to set out for a distant star system, so she gets in her spaceship, and flies out at 99% the speed of light. After a year, she gets a bit bored and lonely, and returns to Earth, again at 99% the speed of light.

But from Bonnie’s perspective, Emily’s clock (and watch, and heartbeat, and everything else) has been running slow. Emily hasn’t been gone for 2 years; she’s been gone for 14! This is true however you look at it. Bonnie will be 44; Emily will be 32. You can even think of traveling close to the speed of light as a sort of time machine – except it only works going forward and not backwards.

Here’s where the paradox comes in. When Emily stepped off her ship back on Earth after traveling to Wolf 359 and back, everyone agrees that she’s only aged 2 years in the same time that Bonnie has aged 14. That is totally inconsistent with pretty much everything we just told you, because we immediately know that Emily was the one who “moved” and not Bonnie, and the first rule was that you could never tell who was moving and who was sitting still. So how do we resolve it?

I’ve written a lot about the time dilation of moving ships: here (in which I figure out how long it will seem to take to get to Gliese 581g), here (in which I compute the amount of energy required using a matter-antimatter engine), here (in which I derive the general relations for the flow of time under constant acceleration) and here (where I do a simpler write-up for io9). The image at top is also a movie illustrating the effect for a trip under constant acceleration and deceleration.

The “paradox” part of this will be clear when we read Eduardo’s question. After some very kind words about how awesome I am, he gets right to it:

I refer to the paradox of *why* is the Earth twin that gets older, and not the spaceship twin, if, during the non accelerated stages of the trip, both are in inertial frameworks, by definition equivalent according to SR.

Yes, I know that the spaceship twin accelerates and decelerates, and feels the acceleration and deceleration, four times during the trip, and the Earth twin remains inertial all the time (disregarding Earth movements).

But acceleration is not part of SR.

And in a ‘thought experiment’ we can have arbitrarily high accelerations and decelerations, in order to have the non inertial stages of the trip arbitrarily short, with the trip essentially being performed in constant speed for practically all its duration.

In this case, why only the Earth twin is older? All slower aging happens to the spaceship twin in the extremely short accelerated/decelerated stages? Or the differential aging continues to happen in the inertial stages by some kind of ‘memory’ of the non-inertial stages?

I know the slower aging actually happens to an object that is moving with relation to an observer that is at rest. I know that experiments with decaying mesons, GPS satellites and atomic clocks in airplanes proved that.

But what the explanation for that *using exclusivelly the SR*, *without using* GR and acceleration arguments?

Why the inertial frameworks for the twins are different? Or, if they are equivalent, why only the Earth twin is older, if the non inertial stages of the trip can be of arbitrarily negligible duration? Why is not the spaceship twin older in this case? Or why both are not with the same age when they met each other after the trip?

Because I’m a member of an internet science discussion list in Brazil, with about a thousand members, and we are having now a ‘flamewar’ on the Twin Paradox.

I didn’t ask Eduardo for the link to the board — who wants to get into the middle of that? — but it’s a very good question. Here’s my answer.

In many ways, however, you have hit at the heart of the problem. The Twin Paradox can’t be resolved by understood by Special Relativity because it doesn’t satisfy the postulates of Special Relativity. Einstein laid out his assumptions in his original 1905 paper. As he put it:

1. [T]he same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.
2. [L]ight is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.

The first postulate is generally understood to mean that all laws, not just electromagnetism, will remain invariant in all inertial reference frames. And what happens if the reference frame is NOT inertial? The theory can’t say a thing. The short answer is that there is no answer “*using exclusivelly the SR*, *without using* GR” as you put it. It simply isn’t a special relativistic question.

You raise a couple of points. You ask, for instance, what happens if the accelerations are nearly instantaneous. To put that in mathematical terms, if the cruising momentum of the spaceship is p:

$p=FDelta t$

What happens if the force is extremely high and the time is extremely short?

Instead of thrusters, we could consider what would happen if we did the same sort of acceleration using a potential well. Suppose (for ease of calculation and for general familiarity with the final result) that the final speed of the ship was enough less than the speed of light that we can use non-relativistic approximations. You are free to generalize later, and I’ll assure you it won’t make a difference. What happens then?

Well, if you fall into a potential well $Phi$ and speed up to a speed $v$:

$Delta Phi=frac{Delta U}{m}gamma=-frac{1}{2}v^2$

This is nothing more than the statement that if you decrease your potential energy, you increase your kinetic energy. As I mentioned above, this is in the non-relativistic limit. For a deeper potential well (nearing the event horizon of a black hole, for instance), it’s more complicated, but the overall result is the same.

In a practical sense, the $v$ in the equation is the escape speed from a planet, star, or whatever body you’d like to imagine. Notice that we don’t make any mention of how long it takes you to fall into the potential well. It can be as steep as you like. However, once you’re in a deep well, General Relativity says that time runs slower near massive bodies (at negative potentials) than they do in deep space. The ratio — in the small potential limit– is:

$frac{t_{gravwell}}{t_{deep space}}simeq 1-frac{Phi}{c^2}$

Since $Phi$ is negative, more time passes near a planet than in deep space. Plugging in the potential, we get:

$frac{t_{gravwell}}{t_{deep space}}simeq 1+frac{v^2}{2c^2}$

For v/c << 1[/latex], this is simply a Taylor expansion of:

[latex size=2] frac{t_{gravwell}}{t_{deep space}}simeq gammaequivfrac{1}{sqrt{1-v^2/c^2}}

Check by plugging in $v=0.2c$ or similar numbers into both expressions if you don’t believe me.

In other words, you can imagine the trip as though the traveling twin falls into a very deep potential well, hangs out for a while, climbs out when she reaches her destination, falls in again to return, and climbs out again on reaching earth.

As I said, I did this under the assumption that the relativistic effects were relatively mild. You are welcome to try using the full:
$p=mvgamma$ relation and the Schwarzschild metric if you like. You’ll get the same result, but it will be mathematically hairier.

Hope that helps and doesn’t just add fire to your flame war.

-Dave

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