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

1. Jan 20, 2010

### durand

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

2. Jan 20, 2010

### durand

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

3. Jan 20, 2010

### George Jones

Staff Emeritus
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.

4. Jan 20, 2010

### Mentz114

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

5. Jan 20, 2010

### Staff: Mentor

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.

6. Jan 20, 2010

### durand

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

7. Jan 20, 2010

### 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

8. Jan 20, 2010

### durand

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

9. Jan 20, 2010

### 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

10. Jan 20, 2010

### durand

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

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