Spherical EM Wave: Origin at t=0, S & S' Agree?

[Pezz]

Consider two frames: S and S', with S' moving to the right along the positive x-axis or S at a relative velocity v. The origins of S and S' coincide at t = 0.

A spherical electromagnetic wave leaves the origin of S the moment S and S' coincide, or at t = 0.
If we consider the transformation equations for how far from the origin of each frame the wave has travelled, would S and S' agree that x' = x? In other words I mean to ask, that if the wave has traveled 2m in the x direction in S in some time, and S detects S' 1m away from the wavefront in the S frame, regardless of relative velocity S' in his own frame would detect the wavefront in his own frame also 2m away from his own origin in that same time?

 

[pervect]

Consider two frames: S and S', with S' moving to the right along the positive x-axis or S at a relative velocity v. The origins of S and S' coincide at t = 0.

A spherical electromagnetic wave leaves the origin of S the moment S and S' coincide, or at t = 0.
If we consider the transformation equations for how far from the origin of each frame the wave has travelled, would S and S' agree that x' = x? In other words I mean to ask, that if the wave has traveled 2m in the x direction in S in some time, and S detects S' 1m away from the wavefront in the S frame, regardless of relative velocity S' in his own frame would detect the wavefront in his own frame also 2m away from his own origin in that same time?

If I'm understanding the question right, the answer is no.

The transformation equations in question are the Lorentz transforms. See http://en.wikipedia.org/wiki/Lorentz_transformation#Boost_in_the_x-direction for the transform equations for the situation you describe, which is called a "boost in the x direction".

We will denote the coordinates in frame S by the variables (t,x), and in the frame S' the coordinates are (t', x').

Using the transform equations from the Wikki article, note that x' does not equal x, nor does t' equal t. However $(x^2 - c t^2) = (x'^2 - ct'^2) = 0$. For right moving waves x = ct, x = -ct, or (x+ct) = 0. If $(x^2 - c t^2) = (x'^2 - ct'^2) = 0$. then either x=ct or x= -ct, so that equation describes a spherical wavefront which can be either left-moving or right-moving.