The Twin not-really Paradox

Everyone says you can't go faster than light, but can you do the math?

I can't. But I can sure draw some pictures! Today we're going to talk about Special Relativity and how things behave near the speed of light. I'm going to use a lot of diagrams. Diagrams can give you a good sense of what's going on in a subject. They're not as good as working through a semester's worth of physics homework, but they're better than saying "Well, um, you can't go faster than light, and I think time slows down, right?" Which is, let's be honest, where most of us get stuck.

We'll start and end with the famous "Twin Paradox". That's the one where somebody takes a long space journey and comes back younger than his stay-at-home twin. But I've got way more diagrams lined up than just that one.

(I will keep calling it the Twin Paradox even though it's not a paradox at all. That's the point. It's a totally normal thing that looks weird to us because we grew up throwing rocks and running footraces at very low speeds. Low compared to the speed of light, that is. If you could run at 90% of lightspeed, this would all be obvious to you.)

(In case you're wondering, this is Special Relativity because we're ignoring mass and gravity. We're only going to consider speed and distance. To be really rigorous, you want to run the Twin Paradox experiment out in deep space, so that the mass of the Earth and Sun don't complicate the results. But we'll just pretend that planets and stars are unusually fluffy.)

Enough footnotes. Let's start in.

The fridge

Here we have our twins, Alia and Bart, sitting on the couch. Bart stands up and walks across to the fridge, six meters away. This takes him ten seconds. He extracts a refreshing beverage, walks back, and sits down. (We assume that sitting, standing, and beverage-snarfing are brief actions; they don't take enough time to show on the graph.)

I hope this diagram is obvious, but I'll go over it anyway. We're going to see a lot of these and I want the principles to be rock-solid.

Green lines are seconds. Blue lines are meters horizontally. (We're only graphing one dimension in space -- call it "east-west" if that's easier.) Black spots are events, not places. A place -- that is, an object at rest in a particular place -- is a vertical line. The purple line shows Alia and the couch; the grey line shows the fridge. They're vertical because they sit still as time passes. Bart's path has two legs (in brown); they're diagonals because he's moving through space as time passes. Bart bounces from the couch to the fridge and back.

(No object can be drawn as a horizontal line, because that would be many different places at the same time. We leave that sort of thing to the gods, who get cranky about being pinned down to space-time diagrams.)

The first rule of graphing is "count the grid lines." Both Alia and Bart experience twenty seconds between Bart standing up and sitting down. Both their paths cross twenty green lines. Bart travels six meters (blue lines) one way and then six meters back; Alia sits still.

Okay, this is how we expect life to work. It's when we head out to interstellar space that life gets weird.

Barnard's Star

Same diagram, different labels. Bart travels six light-years, picks up a refreshing Barnard's Cola, and returns. Alia sees 20 years pass. But Bart, upon touchdown, insists that only 16 years have elapsed. His watch is running four years slow.

What's changed? The two diagrams look the same. Why are we getting different results? It's not just because space travel is longer or boring-er.

No, it's because space travel is faster. Hint: the first diagram is not to scale.

I drew the second diagram in "natural units" -- natural to a physicist, that is: years versus light-years. In such a diagram, the speed of light is a 45-degree slope. Bart's speed is 0.6 of that (six light-years in ten years), which is a slope of about 31 degrees from vertical.

The light cone

In this diagram, the diagonal grey lines are rays of light emitted from Earth at the same time as Bart. The flash of his engines firing up, perhaps. One ray reaches Barnard's Star four years before Bart does, which makes sense.

If I'd drawn the first diagram in natural units, it would have been seconds versus light-seconds (not meters). A light-second is 300 km, give or take, which is much farther than the run to the fridge. It would have looked like this:

The fridge (natural units)

...only even skinnier, really.

Special Relativity does have an effect in this situation. But the angle is very slight so the effect is very small. When Bart returns from the fridge, his watch will be running slow by something like a trillionth of a trillionth of a second.

Yeah, this happens to you every time you walk across the room. You never notice. So let's go back to the space trip; the clocks are easier to read.

I'm going to simplify the problem and only look at the first leg of the trip.

Outbound trip

Why? Because that gives us a nice straight path for Bart -- no acceleration, no change in direction. (We might have to assume that Bart was born on the spaceship travelling at this speed, whereas Alia was born at rest on the couch. This would have been biologically challenging for their mother, but we'll call her Kara Zor-El and assume she was up to it.)

Now a basic principle of relativity is that motion is relative. They named it that, right? Bart can consider that he is standing still and everything else -- Barnard's Star, the couch, Alia, the lot -- is drifting back in the opposite direction.

Let's show that by tilting the diagram. Rotate the whole thing 31 degrees counterclockwise.

Outbound trip (Bart's frame?)

Bart sees the universe in his own frame of reference. What's a frame of reference? A redefinition of space and time along different axes. I've drawn them in orange and red. (The old blue-green grid, Alia's frame of reference, is still visible in the background.)

Bart measures time with the orange lines and distance with the red lines. Note that Bart sees his journey as zero (red) light-years -- that's because he sees himself standing still.

The graph tells us that Bart reaches Barnard's Star after about 11.5 years, his time. Hang on. The distance he's travelled (red lines between Earth and Barnard's Star) is 7 light-years, which is... greater than... length anti-dilation? Wait. Stop.

We're getting all the wrong answers here! This diagram is wrong. (So don't email me about it, please.)

We wanted to make Bart's path vertical, and rotating the whole diagram seemed like a good way to do that. But that was a mistake. We need a different transformation. That is, we need to redefine space and time in a different way which fits the facts better.

What other transformations make sense? Let's go back to our ordinary Newtonian world of couches and refrigerators for a minute. Bart can see himself as sitting still on this trip too. Perhaps he's sitting on a skateboard. Let's draw the first leg of the trip in this (Newtonian) frame of reference.

The count (Bart's frame)

This looks a lot like the previous diagram, but it's not a rotation. We've skewed the vertical (blue) lines to the left, but the horizontal (green) lines haven't changed at all. The old blue-green squares have become parallelograms.

This is how everybody thought physics worked before Einstein.

Alia and Bart still agree that you count time by the green lines. They disagree about the other axis -- Alia is measuring distance by the blue lines, Bart by the red lines -- but they get the same answers. (Go ahead, count.) They both say that this trip takes ten seconds and covers six meters.

Rotation vs Galilean skew

This skew can be called the "Galilean transform", because Galileo wrapped his head around it four hundred years ago. (Although he didn't have these nice diagrams.) It describes our ordinary understanding that uniform motion feels just like sitting still, as long as you remember to oil the skateboard wheels.

Back to space. (Space!) Special Relativity doesn't use the Galileian transform. It uses a different transformation, called the Lorentz transform.

Outbound trip (Bart's frame)

What we're doing here is taking figure 5 and stretching it diagonally until the brown line is just vertical.

The math of the Lorentz transform isn't nice (read the Wikipedia page or, better, don't) but it's pretty easy to visualize. You just take the space-time grid and do this scissor maneuver, like an elevator gate. (Not quite like an elevator gate. The squares aren't just folding flat, they're also stretching out.)

This animation at Wikipedia shows the scissoring-stretching effect nice and clear.

Rotation vs Lorentz transform

Now that we're using the right transform (figure 9), we get the right answers. Bart measures eight years to reach Barnard's Star (counting orange lines). Or rather, it takes eight years for the star, drifting in at 0.6 of lightspeed, to reach Bart. Bart sees Earth and Barnard's Star as being 4.8 light-years apart (red lines).

Alia, as we recall, measures ten years and six light-years because she's still using the green and blue lines. So time is passing slower for Bart, and Bart sees distances as being contracted. This is (finally!) what we expect from our sci-fi reading.

Now, since this is relativity, you'd think this should be a symmetrical problem. From Bart's point of view, it's Alia who is on a journey. Alia is flying away (along with the Earth) at 0.6 of lightspeed. So, by symmetry, Bart should see Alia's trip as lasting ten years, while Alia sees herself as taking eight years.

But that doesn't make sense. Is time passing faster for Bart or for Alia? How can each see the other as slowed down?

(This is the puzzle that is really called "the Twin Paradox". Time dilation is weird, but one twin running both faster and slower than the other -- because of symmetry -- seems like a contradiction. We will solve the puzzle! But it'll take a few steps.)

Here's where simultaneity rears its ugly heads. Two heads, because it's about the difference between two observers.

Let's redraw figure 5 and add Bart's frame of reference, the red-and-orange grid.

Outbound trip

Alia looks around at year 5 and says, "Ooh, Bart must be halfway to Barnard's Star by now. I wonder how he's doing." So she peers across space with the mind-expanding powers of the spice...

...Well, no. Magical psi powers aren't going to tell us how the real world works. You can't perceive what's happening three light-years away (or is it 2.4?) right now.

Alia could use a telescope, but the light from Bart's jubilatory mid-trip fireworks display won't reach Earth for another three years. Telescopes don't tell you where a distant object is right now, only where it was a while ago. Alia will just have to calculate where Bart is right now (Alia's year 5). She draws this diagram and she's all set.

A year later she find the diagram in a drawer. Where is Bart now? He's travelled another 0.6 light-years, that's easy. But how much time has passed for Bart? His watch is ticking by the orange lines. Alia can project her year over to Bart's world-line, and she sees that it spans a bit less than one (orange) year. In fact, it's exactly 0.8 of Bart's years.

Now we return to Bart's frame.

Outbound trip (Bart's frame)

Bart draws his own diagram, marks off a year of his life (the orange segment), and projects it over to Alia's world-line. (Orange dotted lines.) Alia's watch ticks by the green lines, and Bart can see that his year covers 0.8 of Alia's (green) years.

So why the contradiction? Hopefully it's now obvious: they're comparing different things! Alia's purple year-segment is farther up; it doesn't even overlap the year that Bart projected.

The two twins have different views of the universe. In particular, they have different notions of what "at the same time" means. Each of them is calculating what the other one is doing "right now" (gotta calculate, telescopes don't help!) -- but Alia thinks "now" means projecting along the green lines, and Bart thinks it's along the orange lines.

This is the problem with mind-expanding psionic drugs that let space-wizards reach across the universe "instantly". Nobody can agree on what "instantly" means!

(What does everybody agree on? I'll give you a hint: both diagrams show the grey line, the light rays, running at a 45-degree angle. The Lorentz transform changed every angle but that one.)

So now, finally, we can draw Bart's full journey and make sense of it.

The whole trip

Bart's outward trip takes eight years (in his first orange frame). When he reaches Barnard's Star, Bart computes that 6.4 years have elapsed for Alia back on Earth. Then he (briskly) turns around. New frame of reference! New definition of "right now"! Bart's new diagram shows that 13.6 years have elapsed on Earth. And then Bart spends another eight years coming come.

Here's the trip from Bart's two reference frames. I'm not putting them on the same grid. As you see above, the two orange-red grids are stretched at opposite angles.

Whole trip (Bart's frames)

Alia's diagram shows that Bart's watch is ticking slow for his entire journey: 20 years pass on Earth, only 16 for Bart. And Bart's diagrams (two of them!) show that Alia's watch is also ticking slow. 6.4 years pass for Alia during Bart's first eight-year leg, and 6.4 more Earth years pass during Bart's second eight-year leg. Each twin computes that the other is living at 8/10ths normal speed.

But what happened to the missing 7.2 years on Earth? Nothing, really. Bart's definition of "right now on Earth" jumped forward as he was changing course. This isn't weird because it's just a mathematical projection! Bart swung his dotted line around and it hit a new spot on the purple line. Big whoop.

This is the real lesson: trying to figure out what someone is doing far away is a mug's game. Simultaneity at a distance is something we have to make up. It doesn't correspond to anything solid, so you shouldn't be surprised when it bounces around or behaves "paradoxically".

And, of course, Bart's journey experience is really not symmetrical with Alia's at all. Bart changes velocity (and reference frame) at Barnard's Star. (Probably he changed velocity at launch and landing too.) Both twins agree that he's the one who accelerated; he's got the bruises to prove it.

That's the story of the Twin Paradox.

I've got one more trick to show you. You've probably heard that "faster-than-light travel is the same as time travel." This sounds mysterious, but it's easy to illustrate once you've got these frame-of-reference diagrams. It would be a shame to skip it.

So imagine that Alia, tired of waiting on her couch on Earth through this whole essay, decides to really show Bart up. She fires up the Sheewash Drive and zips off to Barnard's Star at twice the speed of light.

FTL trip

Or she dials a stargate, or folds space with the power of her mind... it actually doesn't matter. The point is that Alia leaves Earth at the same time as Bart, and arrives at Barnard's Star three years later. (Seven years before Bart; three years before the flash of light of her departure.) I've drawn her path as a straight line, but only the endpoints are important.

We know this violates the laws o' physics, but it's not obvious from this diagram how it violates them. We've drawn a purple line. The purple pencil didn't catch fire.

Let's keep going. We apply our favorite trick -- we try to transform this diagram to Alia's frame of reference.


That... didn't work. Oops! No matter how much we push the transform, Alia's path won't go vertical. Remember the scissor-gate motion. If you really stretch the heck out of the grid, everything winds up on that 45-degree slope -- or very near to it. But that's the limit.

No Lorentz transform can turn a faster-than-light path into a slower-than-light path, or vice versa. This is the math underlying a rule you've always known. Cool, right?

Well, we can draw Bart's frame of reference, no problem.

FTL trip (Bart's frame)

In Bart's frame, Alia reaches Barnard's Star 8.75 (orange) years before Bart. That's nine months before they both left Earth! The purple line slants down, meaning that Alia has travelled backwards in time.

I've already said that simultaneity at a distance is sort of a made-up concept -- it's purely an artifact of our frame of reference. Well, this is the flip side. For faster-than-light travellers, the difference between "before" and "after" is a made-up concept! By choosing the right combinations of FTL jumps and speed changes, Alia can exploit her hyperdrive to go anywhere in time and space.

Have a fantastic journey, Alia.


End-notes: (or, here's the math, in case you want it)

The speed of light is 299,792,458 meters per second. We haven't really used that value in this essay, but I'd feel stupid if I left it out.

When you read about Special Relativity, you'll quickly run into the symbol γ (gamma). This is basically the time-dilation factor for a given speed. The formula for γ is:

γ = 1 / √(1 − v²)

...where v is the speed in natural units -- a fraction of lightspeed. If v is 0, γ is 1. As v increases towards 1 (the speed of light), γ increases towards infinity.

In this essay, we've always set v to 0.6, so γ has always been 1.25. That's why Alia's watch ticks 1.25 times as fast as Bart's... or vice versa, depending on who you ask!

(Older texts sometimes talked about τ (tau), the reciprocal of γ. Thus Poul Anderson's SF classic Tau Zero. If τ is zero, then v is 1. The title is just a fancy way of saying you're travelling at the speed of light! But modern texts always use γ.)

To apply a Lorentz transform to a graph, you pick your velocity (v) and then apply this formula to each point (x,t):

x' = γx − vγt
t' = γt − vγx

Updated September 19, 2015.

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