(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that the spacetime interval

-(ct)[itex]^{2}[/itex] + x[itex]^{2}[/itex] + y[itex]^{2}[/itex] + z[itex]^{2}[/itex]

is invariant.

[/itex][itex]

2. Relevant equations

Lorentz transformations

[itex]\Delta[/itex][itex]x' = \gamma(\Delta[/itex][itex]x-u\Delta[/itex][itex]t)[/itex]

[itex]\Delta[/itex][itex]y' = \Delta[/itex][itex]y[/itex]

[itex]\Delta[/itex][itex]z' = \Delta[/itex][itex]z[/itex]

[itex]\Delta[/itex][itex]t' = \gamma(\Delta[/itex][itex]t-u\Delta[/itex][itex]x/c^{2})[/itex]

3. The attempt at a solution

I have tried to prove that [itex]\Delta S = \Delta S'[/itex]

So first I said that [itex]\Delta S' = - \Delta (ct')^{2} + \Delta (x')^{2} + \Delta (y')^{2} + \Delta (z')^{2}[/itex]

And inserted all the Lorentz Transformations above into the above formula.

I end up simplyfying it to get

[itex]\gamma^{2} (x^{2} + u^{2}t^{2} - c^{2}t^{2} - \frac{u^{2}x^{2}}{c^{2}}) + y^{2} + z^{2}[/itex]

How does this equal [itex]S = - \Delta (ct)^{2} \Delta (x)^{2} + \Delta (y)^{2} + \Delta (z)^{2}[/itex] ? I can't see a way to get rid of the extra terms to get this simple function.

Any help would be really really great!

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# Spacetime Inverval Invariance using Lorentz Transformations

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