# Naked Science Forum

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

Title: How is the distance between two light pulses calculated?
Post 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
Quote
Two light pulses are moving in the positive direction along the x-axis 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)/(c-v)]
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.
Title: Re: How is the distance between two light pulses calculated?
Post by: Pmb on 01/08/2009 12:56:39
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 x-axis. 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*(1-v^2/c^2)

d = cT = gamma*cT*(1-v^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[(1-v/c)/(1+v/c)]

which is the wrong answer. [:-'(]
Title: How is the distance between two light pulses calculated?
Post by: Pmb on 02/08/2009 16:25:21
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)(1-v/c)

d = gamma*[d/(1  v/c)] (1+v/c)(1-v/c) = d*gamma(1+v/c)

This reduces to

d = d*sqrt[(c+v)/(c-v)]

Which is the right answer.
Title: How is the distance between two light pulses calculated?
Post by: lyner on 02/08/2009 16:58:10
That must have been very satisfying - you didn't have to deal with any loony answers. Every post made sense. (Except this, last one)