The Speed of Time: The Twin Paradox

Most of you are probably familiar with half-life. Many subatomic particles are not stable, and so, after a short time, some of them decay (change into something else). The half-life of a particle is how long the particle has to wait until it had a 50-50 chance of decaying.

decay

For example, muons have a half-life of about 2.2 microseconds. So, if you had 100 muons, 2.2 microseconds later you should only have 50 muons.

Right?

Well, actually… it depends!

The 2.2 microseconds is time that has passed for the muon. If you have 100 muons just sitting there, after 2.2 microseconds there will be about 50 left.1 But, in the last post, we introduced the idea of time dilation. If you have a friend shoot by you in a rocket ship going half the speed of light, while you think 10 seconds have passed, your friend will have only experienced 8.66 seconds.

The same thing works just as well for muons. So, if instead of just sitting there, your 100 muons are traveling in a particle accelerator at half the speed of light, after 2.2 microseconds have passed for you, only 2.2*0.866 \approx 1.9 microseconds have passed for the muons. A quick calculation2 shows that about 55 muons would still remain!

Accelerator

In your reference frame, 2.2/0.866 \approx 2.54 microseconds would need to pass before the muons experienced 2.2 microseconds. So, only after 2.54 microseconds would you have 50 muons!

This is why time dilation is one of the most thoroughly tested facts in science. We can measure the half-life of particles that are just sitting there. But, every day, particles are traveling around particle accelerators at near the speed of light. The half-lives of these particles can be compared to those sitting still. The relative half-lives line up precisely with the predictions of special relativity!

Einstein

Let’s derive a formula for this effect. Last time, we gave the metric for special relativity as ds^2 = dx^2 - dt^2. The speed of time, i.e., how fast time passes for something traveling along a path, was given by \sqrt{-ds^2} = \sqrt{dt^2 - dx^2}. If you are traveling at a constant speed v, we know that dx = v dt. (Recall that, in relativity, it’s convenient to measure speeds as a ratio of the speed of light. So v = 1/2 is half the speed of light, so dt =1 second should mean dx = 1/2.)

Rearranging this, we see that the speed of time is \sqrt{-ds^2} = dt \sqrt{1-v^2}.3 To interpret this, dt is how fast time is passing for you, while \sqrt{-ds^2} is how fast it is passing for, say, your friend in the rocket ship. So, the quantity \sqrt{1-v^2} gives how much faster time is passing for you than for your friend! Let’s call this quantity the “time dilation factor.”

Another common example of this time dilation is GPS satellites.

GPS satellites are essentially (three4) super accurate clocks. They orbit the Earth at about 14000 km/hr (0.000013 of the speed of light), while we mostly sit still.

Satellite

GPS satellites work by sending out a regular signal saying what time they think it is. If you receive a signal from four of them, they will be slightly differently delayed depending on how far away they are. If you know where they are in the sky, you can then (in theory) calculate where you are.

The problem is that the time dilation factor shows that the clocks on the satellites experience 7 microseconds less time per day than our clocks on Earth do.5 A naive calculation then shows that the GPS position should be off by kilometers per day.

That standard reasoning is slightly flawed, though. Yes, our thug friend who always travels with an atomic clock absolutely could compare his clock to the clocks on board three GPS satellites, and then be able to calculate his position.

Thug Time

You, presumably, aren’t like him and don’t have an atomic clock with you. This means that you have no way of measuring that a satellite’s clock is “wrong” compared to your atomic clock because… you don’t have one.

Sad

You’re far more likely to have a smartphone with GPS. This doesn’t bother measuring how time passes at your position–after all, it isn’t as accurate as an atomic clock, so that would be futile. Instead, it calculates your position by comparing the times from four different satellites instead of three.

The satellites are all at the same elevation above the earth and are all traveling at the same speed, and so they all experience the same time dilation. Since your position is calculated by comparing the times from different satellites, the time dilation factor cancels out, and so you would still calculate the correct position.

The time dilation would make some difference, though, in a different way. As part of the calculation, your phone needs to know where the satellites are supposed to be in the sky. The expected positions of the satellites would end up being wrong by a few cm/day, as compared to the time the satellites were transmitting.6 Over a year this adds up, and so the creators of the GPS system took this dilation into account, but this effect is still fairly small.

Next, let’s talk about the twin paradox.

Early in your life, you discovered that your twin was the evil one.

Baby Twins

To spare the world her monstrosities, you sent your twin off in a rocketship at 90% of the speed of light, putting her safely out of reach of the Earth.

Evil in Rocket

Unfortunately, due to time dilation, time for your twin passes at \sqrt{1-0.9^2} \approx 0.44 of the rate it passes for you. Nine years for you makes her only age four years.

If your twin manages to turn around the ship, and make it back at the same speed, you’d have aged 18 years, and she’d have aged only eight!

Often, this story stops here. Your evil twin still has plenty of time to wreak havoc on the world!

Evil Younger

But, if we look a little closer, a question arises.

Relative to your evil twin, you’re the one traveling at 90% of the speed of light, and so after nine years have passed for her, you should’ve only aged by four years. In other words, if she thinks eight years passed before her return, you should have aged only three and a half years!

Evil Older

So, when she gets back, which one of you actually is older?

This is the heart of the twin paradox.

At first glance, this paradox seems to spell doom for special relativity. In fact, some… well-meaning… but confused people make entire websites devoted to showing that the twin paradox shows that all of modern science is wrong. (Spoiler: It’s not.)

The confusion comes about because of a misunderstanding about simultaneity versus time dilation. Let’s draw a spacetime diagram for the twin paradox.

spacetime

Using the same definition of simultaneity as before, the events you think are simultaneous are horizontal lines. If we draw this line just before nine years have passed for you, we see your nine years is right near where your twin turned around.

And, using the metric from last time to calculate the time experienced by your twin, we can calculate that only four years have passed for her.

4years

But, the principle of relativity, one of the two assumptions of special relativity, says that the laws of physics should be the same for your twin as they are for you. So, when your twin has experienced those four years, you should have only experienced 4*0.44 \approx 1.76 years.

Impossible, right?

But the trick is to realize that you aren’t comparing the same things!

Remember how everyone disagrees on what events are simultaneous? Let’s draw your evil twin’s line of simultaneity for all events just before she seizes control of the rocket. (!!!)

twinline1

Next, let’s draw the line of simultaneity for all events just after she seizes control of the rocket (and is thus returning to Earth at 90% of the speed of light).

twinline2

While your twin is travelling at 90% of the speed of light, you both think the other is aging more slowly. The trick is that you are comparing different things! To illustrate this, let’s highlight the parts of your path that your twin is comparing her trip to.

firstandlast

At the sharp turn, your twin’s idea of simultaneity changes, and that explains why you can both think the other is aging more slowly.7

In particular, your twin has to accelerate, while you don’t. That acceleration is what breaks the symmetry, since the principle of relativity only says that the laws of physics work the same for people with constant velocity.

Now, this is a good argument, and true. But, still, some people will argue that this reasoning is flawed.

How can we be 100% sure we’re correct?

Using the ideas from our intro to relativity, it’s actually quite easy.

The important claim we made in the last post is that any path has the same length for everyone, when measured using the metric ds^2. This length has the universal interpretation as the length of time experienced by someone travelling on that path. The quantity ds^2 is something everyone can agree on, even if they disagree on which events are simultaneous.

We didn’t actually prove this in the last post8, but this fact means that everyone, in the end, agrees that you aged eighteen years and your evil twin aged only eight.

For better or worse.

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Special relativity is awesome, but general relativity is even more awesome. (After all, it’s what I do research in!) But to really talk about general relativity, we need to first introduce some concepts from geometry, curvature and geodesics. We’ll start with curvature next time.

 


  1. It’s a random process, so there may be slightly more or slightly fewer by random chance, but it’ll usually be 50 left. 
  2. This is essentially the calculation you might have done in a pre-calc class. The amount remaining is 100*(1/2)^{1.9/2.2} \approx 55
  3. Sorry for the technical mumbo-jumbo, but it was necessary. 
  4. Each GPS satellites has three atomic clocks on board. 
  5. Also, general relativity actually shows that the satellites gain 45 microseconds a day, thanks to the satellites experiencing less gravity way up there, for a total error of 38 microseconds gained a day. 
  6. This is because after, say, one orbit around the Earth, the satellites will be transmitting a time a fraction of a second in the future of what you would expect for its position. In other words, the time its telling you, you would expect it to say that time when it was a few meters in front of its actual current position. 
  7. Though, it is important to note that thinking the other is aging slowly is not the same as seeing them age slowly. If you sketch the light rays coming from your path going to her path (i.e., lines at 45 degrees up to the right), your twin sees you aging slowly till she turns around, then sees you age more quickly than herself on the return trip. Simultaneity is measured by a reflected beam, not just a one directional beam. 
  8. The proof takes two or three (not very dense) pages and takes some notation and such. It’s not that bad. My reference is Schutz’s A First Course in General Relativity, which is where I started when I learned relativity. The proof is essentially: 1. Use the universality of the speed of light to show that everyone’s measured interval is a multiple of everyone else’s, depending on their relative speed. 2. Use an easy example to show that this factor has to be 1. Thus everyone agrees! Huzzah! 
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2 thoughts on “The Speed of Time: The Twin Paradox”

  1. These are some of the clearest explanations of SR and GR that I’ve seen. Keep it up!

    One question which I see asked a lot is “what’s special about light?” I’ll pitch in an answer: it’s not light, it’s the speed, c, that’s special. The Lorentz transformations between reference frames, which can be derived from the postulates of SR, dictate that there must be one speed (let’s call it “c”,) which is the same in all reference frames. Light (and gravity) happen to move at that speed.

    Not only is c the one speed that’s universal, it’s also the maximum speed (hypothetical tachyons aside.) The math says that anything moving faster than c could violate causality. In other words, if I throw a ball and break a window, you, in a different reference frame moving faster than c, could see the window break before the ball is thrown. We can’t have that!

    Like

    1. I can live with that explanation. I mean, how SR originally came up was that some physicists rearranged Maxwell’s equations (maybe it was Maxwell, can’t remember), and found the equations for electricity and magnetism magically form a wave-like differential equation system, with wave propagation speed of the speed of light! That’s pretty special.

      In GR, null directions are obviously geometrically important, but it’s the connection with SR that makes us claim light goes in null directions. It’s pure mathematics, though, that gravitational waves must go along null curves as well.

      I will mention that there are some variations of the theory, where they have light go slightly slower than the speed of gravity.

      Like

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