# Prove that s^2=(s')^2 using the Lorentz Transformation

1. Aug 28, 2014

### castrodisastro

1. The problem statement, all variables and given/known data I am learning special relativity and we came across the invariant quantity s = x2 - (ct)2. Our professor wants us to prove it. I admit that this is a proof and belongs in the mathematics section but I didn't see an Algebra section and this is most easily identified by those learning special relativity.

The assignment simply states

"Prove s2 = s'2"

2. Relevant equations
s2= x2-(ct)2

$\gamma$=[1-($\frac{v}{c}$)2]-1/2

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

t' = $\gamma$(t-(vx/c2)

3. The attempt at a solution

My textbook is telling me in one sentence that if we apply the lorentz transformation to x and t then s2 = s'2.....so I did that...

Applying the lorentz transformation to x' and t' our equation becomes...

s'2 = ($\gamma$(x-vt))2-(c($\gamma$(t-(vx/c2))2

Expanding what we have takes us to...

s'2 = ($\gamma$2(x2-2vt+(vt)2)-(c2$\gamma$2(t2)-2(v/c2)x+(v2/c4)x2))

If I combine some terms...

s'2 = $\gamma$2[x2(1-(v2/c4)+t2(v-1)+2v((x/c2)-t)]

From here I tried a couple of different things on scratch paper but I couldn't see particular direction that would simplify it all down. Am I just not being patient enough and not seeing that it gets worse before it gets better?

2. Aug 28, 2014

### Staff: Mentor

You made a bunch of algebra errors. The last equation you had that was correct was: s'2 = x'2-(ct')2

Chet