# Faster velocity than the speed of light?

by sydfremmer
Tags: faster than c, faster than light
 P: 11 Dear Forum Users, I am a graduate student in Mathematics and not physics, so please bare with me. Also, I know that a similar topic has been discussed before but i could not get a clear answer from reading the previous posts. And here is the question i have been wondering about: Suppose we have two particles, p1 and p2, travelling in opposite direction, each with speed 0.6c relative to point "A". Question 1: After 1 year, the distance from A to p1 will be 0.6 lightyear, and the distance from A to p2 will be 0.6 lightyear. Thus the distance between p1 and p2 will be 1.2 lightyear - right? Question 2: If so, p1 and p2 have travelled away from each other with speed greater than the speed of light. How can that be if, as i understand, no objects can have a relative speed larger than the speed of light? I am looking forward to see your replies :)
P: 2,162
 Quote by sydfremmer Dear Forum Users, I am a graduate student in Mathematics and not physics, so please bare with me. Also, I know that a similar topic has been discussed before but i could not get a clear answer from reading the previous posts. And here is the question i have been wondering about: Suppose we have two particles, p1 and p2, travelling in opposite direction, each with speed 0.6c relative to point "A". Question 1: After 1 year, the distance from A to p1 will be 0.6 lightyear, and the distance from A to p2 will be 0.6 lightyear. Thus the distance between p1 and p2 will be 1.2 lightyear - right? Question 2: If so, p1 and p2 have travelled away from each other with speed greater than the speed of light. How can that be if, as i understand, no objects can have a relative speed larger than the speed of light? I am looking forward to see your replies :)
If you're familiar with coordinate transformations you have to ask what speed does particle p1 see you traveling away and the answer is 0.6c by symmetry and then ask what speed does particle p1 see p2 traveling away and the answer becomes something like 0.9c (actually 0.88c) because you have to use the relativistic coordinate transformation to compute both cases.

In relativity no observer will ever see a particle or object of any type travel faster than light speed and that goes for observers on p1 and p2 thats a given borne out by experiments that have yet to prove relativity wrong.

http://en.wikipedia.org/wiki/Special_relativity

and specific to your question:

http://en.wikipedia.org/wiki/Velocity-addition_formula
 P: 334 From the point of view of a person at point A, they will have relative velocities greater than light the same way that two cars driving away from me at 60mph in opposite directions will have a relative velocity of 120mph according to me. The difference is that from the point of view of a person in either car (which is moving slow enough that SR need not be applied although it technically has an effect), the other one will be moving at 120mph whereas in SR, the two spaceships that you describe will NOT see the other moving at greater than the speed of light. To find the perceived velocity of the other ship, you need the Lorentz transformation of the velocity. Note, just as the two cars are not driving 120mph (with respect to the earth), there is no frame of reference where either ship will be moving at greater than the speed of light.
P: 11

## Faster velocity than the speed of light?

OK, so in effect the particles ARE moving away from each other with speed greater than light?
 PF Patron P: 5,489 The first paragraph of post #3 is incorrect. I suspect, syd, that you may not have noted your question 1 is posed with respect to the 'stationary' point A;question 2 is posed with respect to a different frame, say, p1. In relativity, different frames observe different results. In SR, different observers are separated not by fixed time and fixed distance but by the Lorentz transforms.
 PF Patron P: 5,489 You might find the details here of interest: Frames of reference for speed? http://www.physicsforums.com/showthread.php?t=670436
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 Quote by sydfremmer OK, so in effect the particles ARE moving away from each other with speed greater than light?
RELATIVE TO WHAT ???

You need to get straight on the fact that speed is relative. You can't just say "they are traveling at X speed" --- that is a meaningless statement. You have to pick a frame of reference and stick with it. There IS no frame of reference in which they are moving at > c relative to each other.
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 Quote by sydfremmer After 1 year, the distance from A to p1 will be 0.6 lightyear, and the distance from A to p2 will be 0.6 lightyear. Thus the distance between p1 and p2 will be 1.2 lightyear - right?
This is true in the frame in which A is at rest, but not in other frames. Distance is frame-dependent. Also, "after 1 year" is frame-dependent; you are implicitly picking out events on p1's worldline and p2's worldline which are simultaneous with 1 year having elapsed at A, but which events you pick will depend on which frame you use.

 Quote by sydfremmer If so, p1 and p2 have travelled away from each other with speed greater than the speed of light.
This "speed" is not the same as the "relative speed" that can't be greater than the speed of light. To determine "relative speed", you have to pick a frame in which, say, p1 is at rest, and then ask what distance p2 covers in that frame in a given time in that frame. Since distance and time are frame-dependent, as above, you can't just use distances and times in the frame in which A is at rest to do this. You have to properly transform distances and times from frame to frame, using the Lorentz transformation. You also have to pay attention to the relativity of simultaneity.

Have you had any experience doing any of the above? If not, I would suggest consulting a basic reference on special relativity.
P: 334
 Quote by Naty1 The first paragraph of post #3 is incorrect.
No, it is correct. SR holds that no object can move greater than the speed of light, but the distance between two objects that are moving with respect to a frame of reference can grow at greater than the speed of light with respect to that reference frame. Neither will appear to move faster than light in their own frames.

 OK, so in effect the particles ARE moving away from each other with speed greater than light?
No, they are not moving faster than the speed of light in ANY frame. The distance between them may be growing faster than the speed of light from a separate frame, but both will be measured to be moving less than the speed of light.
P: 11
 Quote by DrewD No, they are not moving faster than the speed of light in ANY frame. The distance between them may be growing faster than the speed of light from a separate frame, but both will be measured to be moving less than the speed of light.
Just to be clear. Is it correct to say that the distance between them is growing faster then the speed of light?
P: 97
 Quote by sydfremmer Just to be clear. Is it correct to say that the distance between them is growing faster then the speed of light?
In your frame of reference 'A', yes. The distance is increasing with a rate of 1.2 lightyears/year, that is 1.2c. I said rate and not speed since it's not the speed of anything. There is nothing going faster than light here.
 P: 11 Thank you all for the replies. I feel that i understand the basic concept now. By the way, can anyone recommend a good introductory physics book explaining time dilation and related topics
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 Quote by sydfremmer Thank you all for the replies. I feel that i understand the basic concept now. By the way, can anyone recommend a good introductory physics book explaining time dilation and related topics
It's not an "introductory physics" book, but I really like the presentation of SR in "A first course in general relativity" by Schutz. It doesn't go very deep into it, but it covers all the basics very well.

I don't like that book's presentation of GR however. It's considered the easiest intro to GR, but it's easy because it does everything it can to avoid explaining differential geometry.

Your question in the OP seems to have been answered, but I'll offer my thoughts anyway, since you got the answer in bits and pieces. Perhaps this will make it easier for the next person who asks this question.

The distance between the position coordinates of the two objects in the inertial coordinate system comoving with A is increasing at a rate faster than c. Some people even call that rate the "relative speed" of the two objects. Another term is "separation speed". The velocity of object p2 relative to p1 is however something completely different. This is the ##dx^i/dt## of the line in ##\mathbb R^4## (or the dx/dt of the line in ##\mathbb R^2##) that p2's world line is mapped to by the inertial coordinate system that's comoving with p1. Since you specified that the coordinate velocities of p1 and p2 in the inertial coordinate system comoving with A are in opposite directions, you can calculate the velocity of p2 in the inertial coordinate system comoving with p1 by using the velocity addition formula. In units such that c=1, it takes the form
$$w=\frac{u+v}{1+uv}.$$ In this case, u and w are known, and we're looking for v, so we solve for v.
$$v=\frac{w-u}{1-wu}.$$ Now you can just plug in the values w=0.6 and u=-0.6 to get the result v=1.2/1.36≈0.88.

I think the easiest way to prove that the right-hand side of the velocity addition formula is in the interval (-1,1) for all u,v<1 is to define ##\theta(r)=\tanh^{-1}(r)## for all ##r\in\mathbb R##, and use the identity
$$\frac{\tanh x+\tanh y}{1+\tanh x\tanh y}=\tanh(x+y)$$ and the fact that ##|\tanh x|<1## for all ##x\in \mathbb R##.
$$|w| =\left|\frac{\tanh\theta(u)+\tanh\theta(v)}{1 +\tanh\theta(u)\tanh\theta(v)}\right| =\left|\tanh(\theta(u)+\theta(v))\right|<1.$$
 P: 2,162 A free ebook to check out would be Ben Crowell's GR book: http://www.lightandmatter.com/genrel/
 P: 11 New question: Will the light from p1 ever reach p2?
P: 2,162
 Quote by sydfremmer New question: Will the light from p1 ever reach p2?
Yes it will.
P: 1,065
 Quote by jedishrfu Yes it will.
Earlier in the thread it said that in A's FoR the distance is increasing at a rate of 1.2 ly.

In FoR P1 & P2 the rate is 0.88c

in A's FoR does the light from P1 reach P2? if so how?

Never mind I see the 1.2 ly distance increase from FoR A is useless, 0.6 isn't.
 P: 11 A related topic that I have a difficulty understand is the expansion of the universy. I read that the universe is expanding such that some galaxies move away from each other faster than the speed of light. If this is the case, does it not mean that some galaxies are moving away from us faster than c?

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