# Relativity Question with Lorentz tranformations

1. Sep 24, 2007

### sr6622

1. The problem statement, all variables and given/known data

Observer O sees a red flash of light at the origin at t=0 and a blue flash of light at x = 3.26km at a time t = 7.63 $$\mu$$ seconds. What are the distance and the time interval between the flasheds according to ovserver O', who moes relative to O in the direction of increasing x with a speed of 0.625c? Assume that the origins of the two coordinate systems line up at t = t' = 0.

2. Relevant equations

x' = $$\gamma$$(x - ut)

t' = $$\gamma$$t - (u/c^2)x

3. The attempt at a solution

I solved for t' and got the answer 1.73 $$\mu$$seconds. From the back of my textbook, this seems to be the correct answer.

When I put x = 3.26km into the first equation above, I always get the answer to be 2.34 km. The answer fromt the back of my textbook says $$\Delta$$ x' = -1.90 km.

I'm wondering what i'm doing wrong...

2. Sep 24, 2007

### dynamicsolo

Just to make it a little clearer, this equation should be written

t' = $$\gamma$$[t - (u/c^2)x]

Are you sure you have the answer for the right problem? (I'm hoping they didn't change the numbers in the problem between editions and forget to change the answers -- I've seen that a little too often...)

Something about their answer doesn't seem right because the spacetime interval should be

(delta-s)^2 = (c delta-t)^2 - (delta-x)^2 = (c delta-t')^2 - (delta-x')^2 ,

which doesn't happen with their values.

I have $$\gamma$$ = 1.2810 . I guess we're agreed that x' = 0 , t' = 0 for the red flash. I also agree with you that, for the blue flash,

x' = (1.281) [ (3.26 km) - (0.625)(300,000 km/sec)(7.63e^-6 sec) ] = 2.343 km .

However, I get

t' = (1.281) [ (7.63e^-6) - (0.625)(3.26 km/300,000 km/sec) ]

= 1.074e^-6 sec or

ct' = 0.322 km.

In O's frame,

(delta-s)^ 2 = [ (300,000 km/sec)(7.63e^-6 - 0 sec) ]^2 - ( 3.26 - 0 km )^2
= -5.388 km^2

A negative space-time separation makes sense because the light-travel time in O's frame from the location of the blue flash is 3.26 km/300,000 km/sec = 10.9 microseconds, which is longer than the time separation between the flashes.

In O' 's frame,

(delta-s)^ 2 = [ (300,000 km/sec)(1.074e^-6 - 0 sec) ]^2 - ( 2.343 - 0 km )^2
= -5.386 km^2 (the difference is likely "round-off" error) .

You do not get this value with the book's answers. Unless we're missing some point about the problem description, I find their answer suspect.

Last edited: Sep 24, 2007