Relativity Question

January 16, 2005 4:11 PM

physicsfilter: This is moderately involved, so bear with me. Let's say there are two spaceships which are situated on two planets, and the planets are about one light hour away from each other, just for the sake of argument. These ships depart from their planets, both heading for a common point between the planets. The ships both accelerate to an appreciable fraction of the speed of light above half, e.g. 3/4 or 4/5 or something. My question is, couldn't it seem then that one was traveling faster than the speed of light and one standing still? I suspect this can't be so, but I don't know why.
posted by clockzero to science & nature (22 comments total)

I've always had this same question. I hope it's not too chatty to survive.
posted by scarabic at 4:25 PM on January 16, 2005

Your suspicion is right, but to see why I suggest you take a look at the equations of special relativity, or the more down-to-earth explanation here; the faster the ships get, the greater their mass will be, and the harder it will become to approach the speed of light; in fact, it would require an infinite amount of energy for either vehicle to reach c, clearly ruling it out as a possibility.
posted by Goedel at 4:26 PM on January 16, 2005

Nope, they're not. In one frame of reference R, A is travelling at 3/4c in one direction, and B is travelling in 3/4c the other direction.

However in frame of reference R', in which A isn't moving at all, B is flying away from it at some speed larger than 3/4c and less than 1c.

You can't just add up speeds with relativity.
posted by fvw at 4:42 PM on January 16, 2005

A similar conundrum was brought up on AskMe or MeFi a while ago, so might be worth searching for.

The conundrum was that if you take a long flat backed cargo train travelling at 60mph, and put a car on the back which would drive at 60mph along the train.. you have a car doing 120mph, right? So, if we could build a spaceship that went the speed of light, and you drove a car at 100mph inside the spaceship.. then the car would be breaking the speed of light, right? As far as I recall there were lots of good answers for this, but sadly I can't remember how to find the post.

My question is, couldn't it seem then that one was traveling faster than the speed of light and one standing still?

However, my personal unscientific answer to your question is.. no. You can't perceive something that's moving away from you faster than the speed of light, by definition, so no.. it would seem the other craft does not exist. This is why we can't see the universe expanding, as the boundary is moving away from us too quickly.
posted by wackybrit at 4:44 PM on January 16, 2005

Because they are travelling at relativistic speeds, both ships will experience time dilation which will reduce the perceived approach speed of the other ship (to c, I believe, with a frequency shift).
posted by cardboard at 4:46 PM on January 16, 2005

It's been a while since modern physics, but I believe if both are going at 0.75c, then they would see the gap between them closing at

(2*0.75c)/(1+0.75^2)=0.96c.
posted by Wolfdog at 4:46 PM on January 16, 2005

A fundamental discovery of relativity is that velocities don't add.

It may seem like they do: that 30mph+30mph=60 mph. But in fact they dont. For low velocities the error is so small though that we can ignore it.

If you have a rocket going at near the speed of light and that rocket fires another rocket going at near the speed of light (relative to the first rocket) and that second rocket fires a third rocket and so on and so on...the speed of the Nth rocket will never, never exceed the speed of light.
posted by vacapinta at 4:48 PM on January 16, 2005

Special relativity teaches you that you can't trust your intuition when it comes to speed. Specifically, you can't just add relative speeds together and get the right answer.

Imagine you were sitting still, and a you saw a beam of light shoot past you. You could measure the speed of that beam of light, and you'd find that it was c, or 300000000 m/s.

Now imagine you're in a super-fast space-ship. What would happen if you zoomed off in the direction of the beam of light, trying to catch it up?

Suppose your space-ship can manage 90% of the speed of light. Your intuition about relative speeds would suggest that if you measured the speed of the beam of light, while sitting in your space-ship, you'd see it going at 10% of the usual speed, c.

This is wrong, according to special relativity. What you actually find is that the light beam is still going at c. How can this be?

Special relativity says that, when you're travelling very very fast, time ticks by slower for you than it would for someone who was sitting still. That's how you can measure the light beam speed, and still get c. Neither your first measurement nor your second measurement is the 'correct' measurement -- they're both just different, valid interpretations of the same event: a light beam.

Now consider your situation, with two space-ships coming at each other. They are both travelling at 4/5 the speed of light -- according to the measurements of an outside, stationary observer. For the people sitting in the space-ships, the situation looks very different.

Again, your intuition would suggest that someone in space-ship A would see space-ship B coming at them at a speed of 8/5 the speed of light. What actually happens is that in space-ship A, time is slowed down by a large factor, and when they measure the speed of space-ship B, they get a value that is still lower than the speed of light.

The key point is that, when you're travelling fast, time slows down for you, so your measurements of speeds are different to what you'd expect.
posted by chrismear at 4:50 PM on January 16, 2005

fvw is right, and to perhaps clarify, from the frame of reference of the ship "standing still", an observer will find that the time the approaching ship took to reach him will have taken longer, thus making it's speed (distance / time) less than the speed of light.

on preview, what chrismear said at the end there.
posted by keep the aspidistra flying at 4:57 PM on January 16, 2005

Thanks for all the great responses.
posted by clockzero at 5:00 PM on January 16, 2005

Oh, and from the frame of reference of space-ship B travelling toward the stationary space-ship A, an observer on space-ship B will measure a shorter time than a person on space-ship A, but the distance between them will also be shorter. so the speed will be less than the speed of light in this situation as well. Time dilation in one frame of reference is length contraction in the other. Confused yet?
posted by keep the aspidistra flying at 5:03 PM on January 16, 2005

chrismear is spot on. The answer is time dilation. This is also why if your twin leaves the planet at the speed of light and comes back twenty years later he's only aged ten years. I hate when this happens too.
posted by nixerman at 5:57 PM on January 16, 2005

Others in this thread have it: velocities do not simply add as they do under a theory called Galilean relativity, accurate at low velocities. I've recommended Spacetime Physics by Taylor and Wheeler before and I'll do so again; used copies can be had for cheap. If I can throw in $0.02, however, the following low velocity thought-experiment has always seemed worthwhile:

Imagine Scarabic standing in the bed of a pickup truck and Clockzero standing on the ground behind the truck. Suppose the truck starts moving forward, and attains 15 miles/hour (mph). Scarabic then throws a baseball at Clockzero. Scarabic, being Major League material can throw at 80 mph. Because he is on the truck, though, Clockzero would read the ball's speed at 65 mph (80 mph - 15 mph). If the truck were moving faster, say 80 mph, then the thrown ball would move at 0 mph according to Clockzero; it would just fall to the ground. This should be easy to see, and with these low velocities this becomes an approximation but is extremely accurate under Special Relativity.

Now the good stuff! With the truck still moving, Scarabic shines a laser beam at Clockzero. Photons move at a very fast speed, c = 186,000 miles/second as measured in a laboratory in vacuum (and not much slower in air). The photons Clockzero sees do not move at c - 15 mph (if the truck's speed is 15 mph) or c - 80 mph (if the truck's speed is 80 mph) or 0.5 c (if the truck's speed is half the speed of light).

No, in all cases, Clockzero would measure the speed of the photons to be c. Scarabic, if he also measured the speed of the photons on the truck, would also obtain a result of c.

That photons are fundamentally not like baseballs is at the very heart of Special Relativity. If one does not like the notion that measured velocities depend on the observers I offer that under Special Relativity at least one thing is absolute: the observed speed of light.

Nixerman, time dilation alone does not explain the twin paradox. The turn-around for one of the twins is where all the action is and the reason only one ages.
posted by fatllama at 8:03 PM on January 16, 2005

fatllama: Just to clarify, the constant speed of light in all reference frames is simply an axiom of the theory of relativity. This means that it is an unprovable assumption from which all the unsettling results of relativity are derived. (That and the idea that there is no such thing as absolute motion, only relative motion.)

Here is a good explanation of how you add velocities under special relativity. Note, this works at all speeds, even ones we sometimes call "non relativistic". The only thing is at low enough speeds the differences are negligible and Newton's laws are accurate enough.
posted by knave at 8:31 PM on January 16, 2005

... all reference frames ...

I should say all intertial reference frames. If you are accelerating or in a very strong gravitational field, special relativity does not apply. You need to consult general relativity at this point.
posted by knave at 8:34 PM on January 16, 2005

knave, good point. To a fault, I think about it as an experimentally testable observation though that was not what initially motivated Einstein.
posted by fatllama at 8:57 PM on January 16, 2005

The ships both accelerate to an appreciable fraction of the speed of light above half, e.g. 3/4 or 4/5 or something.

This is the problem. The sentence contains implied assumptions about speed (velocity, really; speed is the absolute value of a velocity vector) that are not in accord with reality.

Let us begin by saying that velocity is something that can be measured. Imagine 2 people. One is standing on the planet, one is riding on ship A. Each one measures the speed of ship A and ship B relative to themselves.

The difference between the two measurements will not be the same. This is a simple consequence of special relativity.
posted by ikkyu2 at 9:11 PM on January 16, 2005

special relativity does apply to accelerating frames; the maths is just harder. general relativity is the relationship between mass/energy and acceleration/geometry - so it necessarily refers to accelerating frames, but that's just a detail.

there's a famous quote from einstein (iirc) saying how if SR couldn't describe accelerating frames it would be like driving a car whose wheels fell off when it went over bumps. unfortunately my google fu is failing me...

also, wackybrit's final answer is wrong. otherwise: what everyone else is saying.
posted by andrew cooke at 4:00 AM on January 17, 2005

So, if I drive my car as fast as I can, whenever I drive, I'll live longer?
posted by TeamBilly at 12:42 PM on January 17, 2005

No, you'll live shorter, because you'll die in a wreck.
posted by kindall at 12:48 PM on January 17, 2005

TeamBilly, unfortunately, no. Time always affects you at the same rate in your reference frame. That is, you never preceive your time being altered. You may perceive others' time speeding up, however.

So if you attempt to drive your car very fast all the time, you may live longer by other people's watches standards, but your life will be a normal length by your watch. Anyone ever see Flight of the Navigator? :)
posted by knave at 2:57 PM on January 17, 2005

I remember an old Tag-Heuer watch ad with a Formula One tie-in: "Even at 200 mph, a second is still a second. Exactly."

Well, it depends on how you look at it.
posted by ikkyu2 at 6:03 AM on January 20, 2005