|
cfrogue, atyy already presented some pretty convincing math, but let me try two other ways:
1) Lorentz transform approach:
We have the standard form of the Lorentz transform
1a) t' = ( t - vx/c^2 )γ
1b) x' = ( x - vt )γ
And we have any arbitrary equation in the primed frame
1c) x' = ct'
To obtain the corresponding equation in the unprimed frame we simply substitute 1a) and 1b) into 1c)
1d) ( x - vt )γ = c(( t - vx/c^2 )γ)
Which simplifies to
1e) x = ct
2) First principles approach:
We know that the second postulate is that c is the same in all reference frames, so we can immediately write that the speed of the light pulse is c. This implies
2a) x = ct + B
Since we know that the origins coincided with the flash we know that x=0 and t=0 is a point on the light pulse, so we can use that to solve for B
2b) 0 = c0 + B
2c) B = 0
Substituting 2c into 2a gives
2d) x = ct
Note that approach 1 is a general approach that will work for any equation that you care to write. Approach 2 is specific to this problem since we are dealing with light pulses and will not work in general. I would typically recommend approach 1.
|