Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Pmb on 01/08/2009 02:05:00

In order to keep myself from getting rusty right in relativity I'm working through the book Introduction to Tensor Calculus, Relativity and Cosmology, by D. F Lawson. One of the problems has me a bit stumped. It reads
Two light pulses are moving in the positive direction along the xaxis of the frame S, the distance between them being d. Show that, as measured in S', the distance between the pulses is
d*sqrt[(c+v)/(cv)]
The only part covered up to this point is the Lorentz transformation (LT) I assume that the author wants the reader to use the LT to arrive at this expression. Can someone you give me a hint how to start?
Here is what I'm thinking; define the distance betwen the two pulses by how long to takes both pulses to pass by the origin. Call that time T. Then d = cT. dt = T = d/c. Do the samething in frame S'. This gives
dt' = gamma(dt  vdx/c^2)
dx = 0
dt' = gamma(dt) = gamma (d/c)
d' = cdt' = d*gamma
This gives the wrong result. Hence my question.

The nice thing about forums like this is that sometimes merely asking the question to someone else stimulates the mind and the solution comes to you. After I asked the question last night I started thinking about the way I asked it. That led me to better understand how to solve the problem, although the solution still evades me.
Let two light pulses, separated by a distance d, be moving parallel to the xaxis. At t = 0 let the leading pulse be at x = 0 and the trailing pulse be at x = d. After a time delta t = T the trailing light pulse reaches S’s origin clock, which has moved a distance vT in the mean time.
Define the distance between the pulses as measured in S’ to be d’ = cT’ where T’ is the time it takes for the second pulse to reach the clock from its initial position. Determine T’ by the Lorentz transformation. Define the following two events
Event 1: First light pulse arrives at clock. Coordinates of event (x,t) = (0,0)
Event 2: Second light pulse arrives at clock. Coordinates of event (x,t) = (vT, T)
The temporal displacement between these two events is delta t = T. The spatial displacement between these two events is delta x = vT. We wish to find d’ = cT’. The Lorentz transformation for time is
t’ = gamma(t – vx/c^2)
The temporal difference between the two events is then
delta t’ = gamma(delta t – v delta x/c^2)
Substitute delta t’ = T’, delta t = T and delta x = vT
T’ = gamma(T – v(vT)/c^2) = gamma(T – v^2T/c^2) = gamma*T*(1v^2/c^2)
d’ = cT’ = gamma*cT*(1v^2/c^2)
cT = vT + d > T = d/(c – v)
d’ = cT’ = gamma*c[d/(c – v)](1 – v^2/c^2) = d*gamma*(1 – v^2/c^2)
d’ = d*sqrt[(1v/c)/(1+v/c)]
which is the wrong answer. [:'(]

I now see my mistake. The solution is as follows. From
d’ = cT’ = gamma*c[d/(c  v)](1  v^2/c^2)
Factor out (1  v^2/c^2) to (1+v/c)(1v/c)
d’ = gamma*[d/(1 – v/c)] (1+v/c)(1v/c) = d*gamma(1+v/c)
This reduces to
d’ = d*sqrt[(c+v)/(cv)]
Which is the right answer.

That must have been very satisfying  you didn't have to deal with any loony answers. Every post made sense. (Except this, last one)