How Do Lorentz Transformations Prove Light Pulse Symmetry?

[physicyst]

Homework Statement

Frame S' has an x component of velocity u relative to the frame S and at t=t'=0 the two frames coincide. A light pulse with a spherical wave front at the origin of S' at t'=0. Its distance x' from the origin after a time t' is given by x'^2=(c^2)(t'^2). Transform this to an equation in x and t, showing that the result is x^2=(c^2)(t^2).

Homework Equations

I think the relevant equations are:

x'=(gamma)(x-ut) and
t' = (gamma)(t-[ux/c^2])

The Attempt at a Solution

x'^2=(gamma)([x^2]-(2xut)+([ut]^2))
t'^2=(gamma)([t^2]-(2tux/c^2)+([ux/c^2]^2))

plugged into the equation x'^2=(c^2)(t'^2) and reduced somewhat...

(x^2)-(2xut)+((ut)^2) = ((c^2)(t^2))-2tux+((c^2)((ux/c^2)^2)

I think my algebra is crud somewhere, but the farthest I can seem to get this down to is...

(x^2)(1-(u^2)) = (t^2)((c^2)-(u^2))

Thank you for looking at my problem! I hope I put enough clear information down, this is my first time here, please let me know if I need to clear anything up, and thanks again!

 

[nrqed]

Homework Statement

Frame S' has an x component of velocity u relative to the frame S and at t=t'=0 the two frames coincide. A light pulse with a spherical wave front at the origin of S' at t'=0. Its distance x' from the origin after a time t' is given by x'^2=(c^2)(t'^2). Transform this to an equation in x and t, showing that the result is x^2=(c^2)(t^2).

Homework Equations

I think the relevant equations are:

x'=(gamma)(x-ut) and
t' = (gamma)(t-[ux/c^2])

The Attempt at a Solution

x'^2=(gamma)([x^2]-(2xut)+([ut]^2))
t'^2=(gamma)([t^2]-(2tux/c^2)+([ux/c^2]^2))

You forgot to square the gamma factor but it does not affect the rest of your calculation because this factor cancels out anyway.

plugged into the equation x'^2=(c^2)(t'^2) and reduced somewhat...

(x^2)-(2xut)+((ut)^2) = ((c^2)(t^2))-2tux+((c^2)((ux/c^2)^2)

I think my algebra is crud somewhere, but the farthest I can seem to get this down to is...

(x^2)(1-(u^2)) = (t^2)((c^2)-(u^2))

Thank you for looking at my problem! I hope I put enough clear information down, this is my first time here, please let me know if I need to clear anything up, and thanks again!

You forgot a factor of 1/c^2 on the left side (it should read $$x^2 (1 - \frac{u^2}{c^2}$$). This is because you had $$c^2 \times (\frac{ux}{c^2})^2 = \frac{u^2 x^2}{c^2}$$. With this correction, everything works out.