Prove s^2 = s'^2 using the Lorentz Transformation

[beecher]

Homework Statement

Using the Lorentz transformation, prove that s² = s'²

Homework Equations

s² = x² - (c²t²)
x' = gamma * (x-Beta*ct)
t' = gamma * (t - Beta*x/c)

The Attempt at a Solution

s² = x² - (c²t²)
Therefore
s'²= x'² - (c²t'²)

and x' = gamma * (x-Beta*ct)
So x'² = [gamma * (x-Beta*ct)]²
Using FOIL x'² = gamma² (x² - 2xBeta*ct + Beta²*c²t²)

and t' = gamma * (t - Beta*x/c)
So t'² = [gamma * (t - Beta*x/c)]²
Using FOIL t'² = gamma² (t² - 2Beta*tx/c + Beta²*x²/c²)

combining

s'² = gamma² [x² - 2Beta*cxt + Beta²*c²t² - c²t² + 2Beta*cxt - Beta²*x²
Cancel like terms
s'² = gamma² [x² + Beta²*c²t² - c²t² - Beta²*x²

This is where I get stuck and can't tell how to progress and make it equal to the original s² = x² - (c²t²).

Any insight would be greatly appreciated.

Thanks!

 

[gabbagabbahey]

$x^2-\beta^2x^2-c^2t^2+\beta^2c^2t^2=x^2(1-\beta^2)-c^2t^2(1-\beta^2)=$______?

 

[beecher]

Thanks, factoring it out like that made things more clear.
That gives 1-beta²(x² - c²t²)
This is what we want except for the factor of 1 - beta² in the front which I'm still not sure how to get rid of, or account for.

 

[gabbagabbahey]

Hint: $\gamma=$___?

 

[beecher]

gamma = (1-Beta²)^-1/2.
But I'm still confused. Possibly because I've been busting my brains on physics for many hours now. Where will the gamma get me? i need to completely remove the 1-beta² factor don't I?
I can change it into terms of gamma but then its still there in front.
gamma = (1-Beta²)^-1/2
thus 1-Beta²) can be called gamma^-2 since it is the inverse of gamma².
Im still stuck though

 

[gabbagabbahey]

Are you forgetting about the gamma^2 you had here:

Cancel like terms
s'² = gamma² [x² + Beta²*c²t² - c²t² - Beta²*x²

?

 

[beecher]

s'² = gamma²[1-Beta²(x²-c²t²)
s'² = gamma²[Gamma^-2(x²-c²t²)
s'² = x²-c²t²

Finally, finished! I feel exhilaration and also relief, I can go to bed now lol. Thank you so much for your help tonight. I can't really pay you back other then by trying my best to help others in the way you have helped me. Thanks a million

 

[gabbagabbahey]

You're welcome!...And get some sleep:zzz: