How Do Lorentz Transformations Affect Light Signal Velocities?

[JonathanT]

Homework Statement

Consider a light signal propagating in some arbitrary direction, with

v_x \neq 0
v_y \neq 0
v_z \neq 0 and

v_x² + v_y² + v_z² = c²


Use the Lorentz transformation equations for the components of velocity to show that


v'_x² + v'_y² + v'_z² = c²

Homework Equations

Combination of Velocities

v'_x = (v_x - V)/(1-v_xV/c²)

v'_y = (v_y√1-V²/c²))/(1-v_xV/c²)

v'_z = (v_z√1-V²/c²))/(1-v_xV/c²)

The Attempt at a Solution

I know this is just a simple algebra proof but for some reason I'm getting stuck on it. Maybe I'm using the wrong equations?

I would really appreciate being shown where to start for this proof. Thanks in advance for the help and/or time.

 

[nlsherrill]

You should get the transformation equations for vx, vy, vz to the prime variables and insert them into vx^2 +vy^2 +vz^2 = c^2 to show this reduces to vx'^2 +vy'^2 +vz'^2 = c^2. Since the speed of light is constant in all frames, the change in frames should be invariant under the lorentz transformation. Just be careful with your algebra...if it starts to get too messy then you probably did a simplification that led you in the wrong direction.

 

[JonathanT]

Thank you. I apparently was making a stupid algebra mistake and over complicating it. Just knowing I was heading in the right direction helped. Made me find my mistake. Thanks again.

 

[nlsherrill]

no problem!