Intergalactic rocket race at half the speed of light

Factual Questions
1

Apologies if I’ve missed this on previous relativity discussions, but I can’t find anything about this particular point…

OK, about this whole relativity thing - imagine the following scenario (it’s probably the result of a drunken bet or a bored afternoon):

Let’s say that two long rockets start side-by-side, facing opposite directions. Stretched away in both directions are countless numbers of observers.

At the word go, both rockets streak away from each other, aiming to race at half the speed of light towards opposite points far, far away.

Question 1. When both rockets are up to speed, rocket one (getting a bit cocky) decides to broadcast a “so long, suckers!” message to rocket two.The broadcast equipment is in the centre of the rocket, and radios are stationed at either end of the rocket. Now, based on measurements by these radio operators (who will pick up the broadcast shortly after it is sent), the signal is moving at the speed of light relative to them in both directions - i.e. at one-and-a-half times the speed of light fowards (relative to a stationary point), and half the speed of light backwards.

However, the stationary observers will detect the broadcast as moving at exactly the speed of light relative to them in both directions. So, in one direction, the observers measure the broadcast moving quicker than the rocket crew measure it, and in the other direction, they see it as moving slower. Put another way, in one direction, relative to the crew, time is moving slower, whereas in the other, it is moving faster. So, if the rocket came to an abrupt stop, the people in front would have aged more than the crew, whilst the people behind will have aged less.

How can this be? All the stationary observers that the crew can see could get together and say “Hi, cool race huh?”, but cleary some can’t be grandpas while others are still kids.

Question 2. Back to the “so long, suckers!” message… The message sender sees the message leaving his ship at the speed of light relative to him - i.e. it can never reach rocket two. However, the observers lined along the route would record the message passing them at exactly the speed of light - i.e. it will reach the second ship at some determined time. In fact, the radio operator on rocket two would measure even less elapsed time between sending and receiving than the obervers do, as according to this operator’s measurements, the broadcast is moving at the speed of light relative to him - i.e. one-and-a-half times the speed of light relative to the stationary obervers, or twice the speed light relative to rocket one.

So, as far as the people on rocket one are concerned, the message never reaches rocket two, but as far as rocket two is concerned, it gets there fairly promptly.

Now, just so that the members of rocket two don’t escape these unfortunate problems with time, let’s say they reply to the broadcast as soon as they receive it. So, they beam out their “yeah, yeah, you turkeys”. Now the roles are reversed - from their point of view, the message never reaches rocket one. Yet the obervers see that it does, and in fact rocket one does receive it even quicker than the obervers think, and sends similar abuse back.

This makes my brain hurt - we have time shrinking and stretching willy-nilly, and it seems to depend on who is sending a message and who is receiving. How is it possible for rocket one to send a message that will never be received from it’s point of view, and yet still get a reply?

2

I have asked similar questions, and the answer is “time dilation”. Time slows down the faster you go, and that is the compensating factor which cleans up the conundrums you’ve described.

Note that you used the phrase “one-and-a-half times the speed of light” — and in fact, you used it twice. The cute trick which makes relativity so cool is that NOTHING can go faster than light NO MATTER WHICH frame of reference you are using.

If you are moving at half the speed of light, and broadcast light or radio, those waves are moving at exactly the speed of light, both to you in the ship, and to me on the outside. How can that be?

Let’s talk in round terms, where the speed of light is 300,000 km/sec. You are 150,000 km from Earth, and start moving at 11:59:59, going towards Earth at 150,000 km/sec. (We’re going to keep it simple, and presume that you can accelerate instantaneously.) One second later, at 12:00:00, you pass by Earth at half the speed of light, and start broadcasting your signal.

From my perspective, a second later at 12:00:01, you are 150,000 km from earth, and your signal is 300,000 km from earth. At 12:00:02, you’ve gotten 300,000 km from Earth, but the signal is going twice as fast as you are, and is 600,000 km from Earth, or 300,000 km ahead of you.

From your perspective, things look VERY different. To you, time has slowed down, and you pass the sign that says “This is the 300,000 km mark” long before your on-board clock gets to 12:00:02. At first you think that’s odd, but then you remember about the guy who traveled to a star 10 light-years away. He traveled at about 99% of the speed of light, so he arrived in only slightly over ten years, but because time had slowed down so much, it only felt like two years. (Forgive me, I don’t know the exact formulas, but you get the idea.)

The weird thing is (and perhaps the formulas are derived from this idea) that your signal does appear to be going at the speed of light relative to you. Let’s say that there is a radio on that “300,000 km” sign, and when it hears your signal, it will turn on a certain light. That is going to happen at exactly 12:00:01, according to MY clock, but time has slowed down for you, and so the light turns on some time before your clock gets to 12:00:01. And you say to yourself, “Well, sure, of course the light turned on less than a second after I passed Earth. After all, that signal is going at one-and-a-half times the speed of light!”

No, that last crack was not meant to be sarcastic. Since the signal APPEARS to be going at your own speed plus the speed of light, the signal is able to traverse 300,000 km in WHAT SEEMS TO BE less than one second.

If you understand all that on the first reading, you’re fooling yourself. Try it a few more times, and THEN apply the same logic to your questions, and you’ll see what I mean.

Good luck!

3

You should definitely get a book on relativity for the layman. I very much doubt anyone can explain it adequately in this forum for your needs.

One thing I noticed: Speeds don’t add up. 1/2 SOL (speed of light) + 1/2 SOL does not equal SOL. At sub-relativistic speeds we are used to there is practially no difference between adding and the truly correct mathematics. As soon as speeds get up to a significant fraction of SOL (also called “c”), there are significiant differences.

4

First of all, the ships are not moving at the speed of light relative to each other. Relativistic velocities do not simply add like that. They add according to the formula Vrel = (V1 + V2)/(1 + V1V2) and are actually going at V = 0.8 relative to each other. Secondly, as Keeve points out, the speed of light is constant, no matter what frame you are in. That is a basic tenant of relativity. The all observers, no matter their frame of reference, see the signal leave rocket one at the speed of light in all directions.

Now things really get weird. Not only do you have time dilation, but also Lorentz contraction. The length of along the axis of motion contracts, and observers in different frames would disagree about measurements of length. In this case, the rocket passengers would measure the distance to the other rocket shorter than the stationery observers.

There is a lot more to this. Specifically, you may want to look into simultaneity and the Twin Paradox.

5

Yeah, I meant to mention the contraction, thanks.

The Twin Paradox is very cool too. Suppose you have two twins, and one of them goes zipping around at close-to-light speed for a few years or decades while the other one stays at home. Then the traveler returns. You’d think that because he was going so fast, he would age more slowly, and end up appearing much younger than his twin. But no, they’d end up the exact same age. Same would apply to clocks, where one went traveling and the other didn’t; when they got back together they’d say the same thing.

There are two explanations:

(1) Everything is relative. Who are we to say which one was moving faster. It is entirely legitimate to say that the stay-at-home was moving fast relative to the astronaut, who stayed motionless while the universe moved around him.

(2) It’s actually a trick question, because they’re not the same age until they both show up in the same location at the same time. It turns out that in the process of slowing down and going come, he loses whatever he gained in the first part.

Yes, get a relativity-for-the-layman book. This stuff is way cool.

6

First, I just want to point out that speed in my first post is a fraction of the speed of light (their relative speed is 80% the speed of light). On second reading, I realized that might not be clear to some people.

The Twin Paradox is cool, but that’s not quite right. In the Twin Paradox, one twin is an astronaut that takes a flight at near the speed of light to a distant star and returns, while the other twin stays home. Because of time dilation, the twin on the rocket ages less than the twin on earth. However, relativity says it is equally reasonable from the rocket twin’s frame of reference to say he stays at home, while the earth-bound twin speeds away in the opposite direction and returns. Hence, the earthbound twin should be younger, which is a contradiction.

The paradox is resolved by the fact that the rocket twin turns around, and that is what makes it clear he is the one that is traveling. His reference frame changes when he reverses direction while the earthbound twin’s frame has remained the same the entire time. The rocket twin will have aged less than the earthbound twin.

Now for a real stumper: What happens if the rocket twin’s spacecraft breaks down at his destination and he is unable to return? If he only travels one-way, can we no longer say he has aged less? I understand the solution to this one, but just barely.

7

This is exactly wrong. One of the first and best pieces of evidence supporting relativity is the opposite. Two professors calibrated their clocks. One got on a round-the-world flight. So he circled Earth, returned. When they compared clocks, the traveler’s clock was slow, to the exact amount predicted by Einstein.

8

Relativity by Albert Einstein is really quite good in explaining this to the layman.

9

Right - read the book it is then.

Cheers all.

10

Let’s take a rocked speeding by a stationary observer designated S. On the rocket we will place two observers, F and R. F and R syncronize their watches and F goes to the front of the ship and R to the rear. If we emit a light pulse from the midpoint of the rocket, then F and R will say the light reached them at the same time. S however will say the light reached R before reaching F, because the light has to travel further to catch up with F. S will not be surprised that F and R report the signal reached them at them same time, because S will say that the R’s clock is set ahead of F’s clock. Nether clock is ticking faster or slower than the other. S would say they both tick at the same rate and that rate is slower that his clock. From F and R’s viewpoint, their clocks are syncornized and it is S’s clock that ticks more slowly. This brings us to this question:

Simultanety is relative. The best you can say about two clocks at different locations is whether there is an inertial reference frame where they are syncronized or not.