Show that c^2(t^2) - x^2 - y^2 - z^2 is invariant under a change of frame

[phosgene]

[itex][/itex]

Homework Statement

Show that the quantity

[itex]T = c^2(Δt)^2 - (Δx)^2 - (Δy)^2 - (Δz)^2[/itex]

is invariant under a change of frame

Homework Equations

Lorentz transformations

[itex]Δx' = \gamma(Δx - vΔt)[/itex]

[itex]Δt' = \gamma(Δt - vΔx/c^2)[/itex]

[itex]Δy' = Δy[/itex]

[itex]Δz' = Δz[/itex]

The Attempt at a Solution

I know that the way to do this is to substitute the Lorentz transformations into the original invariant and then do some algebra, but I end up with the following:

[itex]T' = c^2(Δt')^2 - (Δx')^2 - (Δy')^2 - (Δz')^2[/itex]

[itex]=c^2(\gamma(Δt - vΔx/c^2))^2 - (\gamma(Δx - vΔt))^2 - (Δy)^2 - (Δz)^2[/itex]

After doing some algebra I get

[itex]T' = \gamma^2c^2Δt^2 + \gamma^2v^2Δx^2/c^2 - \gamma^2Δx^2 - \gamma^2v^2Δt^2 - (Δy)^2 - (Δz)^2[/itex]

I also try to express gamma in terms of

[itex]\gamma=\frac{1}{\sqrt{1-v^2/c^2}}[/itex]

But the equation gets much worse without any apparent progress. I only see it stated in textbooks and websites that plugging in the formula leads to the answer, but no actual steps in between. I'm really confused here and would greatly appreciate help.

 

[frogjg2003]

Try grouping all the Δx² and Δy² terms, see what you get then.

 

[phosgene]

Ahh, so the gamma terms cancel out the v and c squared terms. Thanks for that, I've been trying to get it for ages!