# space contraction near lightspeed

I read a lot of the Time-Life Science books when I was a kid (well, mostly the pictures and captions). One of the things that always stuck in my mind was an explanation of the twin paradox, using travel to Andromeda as an example. The aging thing I already get, but what I still wonder about is this:

Andromeda is 2,538,000ly away. The near lightspeed travelling twin reaches Andromeda in something like 20 years. Since you can't travel 2+ million lightyears in 20 years, it must mean that as the twin approaches lightspeed, the space in front of him contracts so that Andromeda is about 20ly distant. Correct?

If so, wouldn't Andromeda be approaching at faster-than-light speed?

• Yes, time-dilation and length-contraction are sides of the same coin in that way. But note that asking "Is this correct?" is not a good fit for the SE model, because "Yes, that's right." is too short to even add as an answer. Apr 27 '15 at 20:30
• Speed of light is constant in all frames and is the universal speed limit so no matter how lengths contract and time dilates the galaxy cannot approach faster than light. Apr 27 '15 at 20:40
• Before I take off in the spaceship, I look at Andromeda and say "Oh, that's about 2 million ly away." I take off and as I accelerate really fast (squished to an atom-thin pancake by g's), Andromeda has gone from appearing 2m ly away to 20 ly away. How did it get so close so fast? Apr 27 '15 at 21:35
• math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html may or may not be helpful.
– user854
Mar 17 '16 at 4:06
• @JeffLowery: I am standing in Chicago, saying "Canada is 1000 miles straight ahead". I turn 90 degrees to the west, and suddenly Canada is 0 miles straight ahead, plus 1000 miles to the right. How did Canada move so far so fast? Jan 6 '17 at 2:46

Suppose Andromeda is at rest relative to earth. You are on earth, holding your ruler, which just touches Andromeda.

Now you instantly start traveling toward Andromeda at a high speed. Your ruler, of course, travels with you, still pointing toward Andromeda.

Your journey just began an instant ago, so neither you nor your ruler has yet moved appreciably. I, sitting on earth, say "I see that the end of your ruler just touches Andromeda". I also say that "In a little while, the far end of your ruler will extend way past Andromeda".

But because you're traveling relative to me, we have different notions of simultaneity. While I say that in a while your ruler will extend past Andromeda, you see that event as occurring right now.

So as far as you're concerned, Andromeda, instead of being at the end of the ruler (marked 25 million) is now at the point marked 20.

You haven't seen Andromeda move; you've just seen a change in what your ruler measures.

The laws of physics say that you can travel from here to "20" on your ruler in just a little over 20 years. Enjoy your journey.

• +1 Very elegant mix of the Twin Paradox and simultaneity, a wonderful link to make if you're a teacher. I like to think of the twin paradox as an example (mostly for those with a mathematics bent) of the shattering of the triangle inequality: I hadn't thought in your elegant terms before. Apr 28 '15 at 6:28
• Just to clear up another possible point of confusion: At first, the 25=million mark on your ruler is at Andromeda. You accelerate toward Andromeda. Now I say that the 20-mark on your ruler will eventually get to Andromeda, whereas you say that the 20-mark on your ruler is at Andromeda right now. But of course you don't see that right now; you say that it's happening right now 20 light years away, so you won't see it until 20 of your years from now. You don't see any instant change in Andromeda, but you can get there in 20 years and piece together what happened after the fact. Apr 28 '15 at 13:59
• I would put that comment into your answer. BTW Andromeda is 2,5million, not 25million light years hence. I only know this as I just looked this up to answer a question on the Andromeda paradox. Apr 29 '15 at 1:15
• Actually, the "far end of your ruler will extend way past Andromeda" isn't entirely true. As long as you're moving at a high speed, I'll never see your ruler more than 20 light years past Andromeda, since the 25 million on your ruler is now only 20.
– user854
Mar 17 '16 at 4:11

Yes, you're quite correct that in the example you describe Andromeda can be moving faster than the speed of light. But that's perfectly in accord with special relativity. It's just that the rule that nothing can move faster than the speed of light is true only for unaccelerated motion.

Although beginners are (usually) taught special relativity using Einstein's two postulates this is an approach that doesn't extend easily to accelerated motion, let alone the curved spacetimes we encounter in general relativity. A better way to understand relativity is as the spacetime geometry defined by the Minkowski metric:

$$c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2 \tag{1}$$

where $c$ is a constant that is equal to the local speed of light. This equation tells you everything about special relativity. For example it's straightforward to use it to derive time dilation and length contraction. The $(t, x, y, z)$ are your coordinates i.e. the time and space coordinates system you've set up using your rulers and clock. If something moves a small distance $dx, dy, dz$ (as measured with your rulers) in a time $dt$ (as measured with your clock), then the equation gives us the corresponding value $d\tau$, which is called the proper time. If you aren't moving (in your coordinates) then $dx = dy = dz = 0$ and equation (1) simplifies to:

$$c^2d\tau^2 = c^2dt^2$$

or:

$$d\tau = dt$$

So this mysterious quantity called the proper time is just the time measured by an observer who is stationary in the coordinate system being used.

We can easily show that equation (1) forbids faster than light travel. Since the proper time is just a time measured by a stationary observer it must be a real number so in equation (1) we must have $c^2d\tau^2 \ge 0$. If $c^2d\tau^2$ was negative when we took the square root we'd get an imaginary number. So let's put this condition into equation (1) to get:

$$c^2dt^2 - dx^2 - dy^2 - dz^2 \ge 0$$

For convenience we'll choose our axes so the $x$ axis is along the direction of motion, which means $dy = dz = 0$, and this simplifies our equation to:

$$c^2dt^2 - dx^2 \ge 0$$

or with a bit of rearrangement:

$$\left(\frac{dx}{dt}\right)^2 \le c^2$$

and since $dx/dt$ is just the velocity $v$ we get:

$$v^2 \le c^2$$

and the modulus of the velocity must be equal to or less than the speed of light (I say modulus because when we take the square root of $v^2$ the result can be $\pm v$).

By now you're probably wondering what the point of all this, and how it's relevant to the question, but now I shall reveal all! The Minkowski metric (1) only applies to a non-accelerating observer. Technically you can show this by using the metric to calculate the four-acceleration, and you'll always get the value zero i.e. not accelerating. Anyhow, if you're accelerating with an acceleration $a$ then the metric that applies to you is the Rindler metric not the Minkowski metric. This can be written in many ways, but for our purposes the simplest is:

$$c^2d\tau^2 = \left(1 + \frac{a}{c^2}x \right)^2 c^2 dt^2 - dx^2 \tag{2}$$

where I've assumed we're accelerating in the $x$ direction so $dy = dz = 0$. Let's play the same trick that we did before, and require that $c^2d\tau^2 \ge 0$, in which case we get:

$$\left(1 + \frac{a}{c^2}x \right)^2 c^2 dt^2 - dx^2 \ge 0$$

or rearranging as we did before:

$$\left(\frac{dx}{dt}\right)^2 \le \left(1 + \frac{a}{c^2}x \right)^2 c^2$$

This looks comparable to the result we got before except for that factor of $(1 + ax/c^2)^2$, where $x$ is the distance measured in the direction we're accelerating. This factor can be greater than one, and that means the maximum possible velocity can be greater than $c$. In fact we can make the maximum possible velocity arbitrarily large just by going to a big enough value of $x$. This is why in your example Andromeda can be moving towards the travelling twin at a velocity greater than $c$.

But there's a very important principle that applies even to general relativity as well as accelerated motion in special relativity - the local speed of light can never exceed $c$. By local I mean the speed measured at the observer's location. In this case the observer is positioned at $x = 0$, and if we put $x = 0$ into our equation above we find:

$$v^2 \le c^2$$

Just as we did before.

A few closing comments: you'll know that distant galaxies are moving away from us, and that if you go far enough away the galaxies will be moving away from us faster than the speed of light. This is generally explained away by saying that it's spacetime expanding rather than the galaxies moving, though that doesn't change the fact that the galaxies really are moving away from us faster than $c$. This is the same sort of thing that's going on with accelerated motion. I see Asher has made the point in one of his comments that what's really going on is space shrinking. Quite true, but the fact remains that in the example you give Andromeda really is moving towards the travelling faster than the speed of light.

• Thank so much, I appreciate the thorough answer, including the math. It's been a long time since I last used calculus, but I assume these equations would tell me why distant galaxies are moving away from me at faster than c, even though I feel no acceleration (other than g). Apr 29 '15 at 0:50
• @JeffLowery: the metric that describes the expanding universe is different again - it's called the FLRW metric. Apr 29 '15 at 5:03

The high speed of travel causes both time dilation and space contraction, so the numbers add up from both frames of reference.

For the astronaut traveling at near lightspeed, the space between him and Andromeda is contracted; at a very high speed, Andromeda would appear to be only 25ly from the Milky Way, but since the astronaut is traveling slower than light speed (by both his measurements and those of his twin on Earth) it would take him more than 25 years to travel the distance, measured by his own clocks and calendars on the ship.

For his twin back on Earth, Andromeda is still 2.5Mly away because there's no space contraction (since the motion between the Milky Way and Andromeda galaxies is relatively small) but there is also no time dilation, so if the presumably immortal Earthbound twin measures the trip he sees that the astronaut travels the full 2.5Mly to Andromeda, but it takes him longer than 2.5M years - again, because the astronaut is traveling slower than light.

So from the astronaut's perspective, he is taking a (relatively) short trip from one galaxy to another, while from the Earthbound observer's perspective, he is taking a very long trip but aging very slowly. In both cases, the measured speed of the astronaut's travel is slower than the measured speed of light, so there's no FTL paradox at play.

• When the astronaut started, Andromeda appeared 2m ly away. As he accelerates, is soon appears 25 ly away. Went from '.' to 'O' incredibly quickly. He knows he's not traveling faster than light, and that Andromeda is not traveling faster than light, but did space contract faster than light? The perception would be that Andromeda "approached" the astronaut faster than light. Apr 27 '15 at 21:38
• It wouldn't appear any wider, space contraction only acts along the direction of travel, so it would appear much closer (and thinner). It wouldn't be approaching "faster than light," however, since if the astronaut shines a flashlight out the front of his ship, the light still travels faster than he does, and if he approaches a lighthouse in Andromeda, the light will reach him before he passes the lighthouse. Light will still appear (and be) faster than even the apparent approach speed between the astronaut and the galaxy. Apr 27 '15 at 21:59
• Let's say Andromeda, at 2 million lightyears out, has a width of x arc-seconds. At 25 lys out it's going to be greater than x arc-seconds wide, because it's closer. The rate at which its width increased would determine how fast it "approached" the astronaut. I wonder if anything sophisticated enough to observe Andromeda approaching at faster-than-light would survive the acceleration needed to make that happen? No observer, no paradox. Apr 27 '15 at 22:36
• You seem to think that the perception of an observer is what causes a paradox, which is not the case. Yes, if you were on a ship making a high-speed trip to Andromeda, your perception would be "I have made a 2.5Mly trip in only 25 years, thus traveling faster than light." But you could at no point observe any object (yourself or Andromeda, for example) traveling faster than the observed speed of light. What you would observe is space shrinking faster than the speed of light, which is not forbidden in relativity theory and thus creates no paradox. Space does what it wants. Apr 27 '15 at 22:48
• "You seem to think that the perception of an observer is what causes a paradox" Well, that was a philosophical jump. "If a tree falls in a forest, does it make a sound?" "If Andromeda approaches faster than light, can anything observe it?" seemed similar quandaries. Anyway, what you seem to be saying is: as long as you feel acceleration, the rate of space contraction is not limited and objects in front of you can approach at a rate that appears to be faster-than-light. I haven't heard this stated before (if I got it right). Apr 27 '15 at 23:00