
#1
Feb1013, 04:12 PM

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 :) 



#2
Feb1013, 04:20 PM

P: 2,471

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/Velocityaddition_formula 



#3
Feb1013, 04:24 PM

P: 424

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. 



#4
Feb1013, 04:29 PM

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?




#5
Feb1013, 04:29 PM

P: 5,634

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. 



#6
Feb1013, 04:32 PM

P: 5,634

You might find the details here of interest:
Frames of reference for speed? http://www.physicsforums.com/showthread.php?t=670436 



#7
Feb1013, 04:35 PM

PF Gold
P: 5,674

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. 



#8
Feb1013, 04:53 PM

Physics
Sci Advisor
PF Gold
P: 5,503

Have you had any experience doing any of the above? If not, I would suggest consulting a basic reference on special relativity. 



#9
Feb1013, 09:24 PM

P: 424





#10
Feb1113, 02:11 AM

P: 11





#11
Feb1113, 04:59 AM

P: 97





#12
Feb1113, 05:16 AM

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 



#13
Feb1113, 05:51 AM

Emeritus
Sci Advisor
PF Gold
P: 8,989

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. $$v=\frac{wu}{1wu}.$$ 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 righthand 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##. $$w =\left\frac{\tanh\theta(u)+\tanh\theta(v)}{1 +\tanh\theta(u)\tanh\theta(v)}\right =\left\tanh(\theta(u)+\theta(v))\right<1.$$ 



#14
Feb1113, 06:35 AM

P: 2,471




#15
Feb1113, 01:40 PM

P: 11

New question: Will the light from p1 ever reach p2?




#16
Feb1113, 01:43 PM

P: 2,471





#17
Feb1113, 02:07 PM

P: 1,098

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. 



#18
Feb1113, 02:08 PM

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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