Proving x^2-c^2*t^2 invariance

[durand]

How do you prove x²-c²t² is invariant under the lorentz transformations given that;

 

[durand]

Ive tried the obvious replacing x and t with x' and t' but i still can't get it to drop out :(

 

[George Jones]

Ive tried the obvious replacing x and t with x' and t' but i still can't get it to drop out :(

In x'^2 - ct'^2, replace x' and t' be the expressions that you gave in the original post, and factor x^2 and t^2 out of terms in which they occur.

 

[Mentz114]

Try again. I just worked it out and found that c²t²-x²=c²t'²-x'².

 

[jtbell]

You need to prove that

$$(x^{\prime})^2 - c^2 (t^{\prime})^2 = x^2 - c^2 t^2$$

After substituting the Lorentz transformation on the left side, everything should eventually cancel out.

 

[durand]

Thanks everyone! I didn't see that gamma could be taken out of the equation and cancelled. It works now :D

 

[vin300]

What you've done is a special case, in the general case, x^2-(ct)^2 is a constant and gamma does not cancel out, the calc is a bit rigorous then, you need to differentiate and substitute v

 

[durand]

Ok, I'l definitely keep that in mind! Thanks.

 

[vin300]

Oops, mistake.No differentiation.The invariance yields directly by substituting LT.
I differentiated x^2-(ct)^2=k and substituted v , but that's again a special case when v is the velocity of the object in the unprimed frame

 

[durand]

Mmm, ok. I think I might be doing that next semester at uni.