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I've tried proving the invariance of the spacetime interval from Lorentz transformations 3 times now, but every time I end up with two extra terms that don't cancel! Could I have some help?
I'm trying to visualize your calculations via my paranormal abilities, but somehow I fail.
Ok, we try to show thatMy LaTeX isn't so good, but substituting:
[tex]x'=\gamma(x+vt)[/tex]
[tex]t'=\gamma(t+\frac{vx}{c^2})[/tex]
into
[tex](x')^2+(y')^2+(z')^2-c^2(t')^2[/tex]
trying to get
[tex]x^2+y^2+z^2-c^2t^2[/tex]
i.e. invariant interval
Yes - thanks, it was the [tex](1-\beta^2)[/tex] factor I was missing when re-arrangingDid you manage to do the calculation?
True, but this requires the acquisition of a working knowledge of matrix algebra. I did learn a lot of that years ago but have lost my working knowledge of this long ago and have no current knowledge of matrix operations. Thus, using the basic algebra is a little simpler for me. To wit, I don't even know what [itex](\Lambda(x-y))^T\eta\Lambda(x-y)[/itex] means yet I can still derive the basic hyperbolic relationship between t and x without it.All of these things get easier when you're used to working with matrices. We want to prove that [itex](x-y)^T\eta(x-y)[/itex] is invariant, i.e. that it's equal to [itex](\Lambda(x-y))^T\eta\Lambda(x-y)[/itex]. So we stare at it for two seconds and realize that the equality follows immediately from the definition of a Lorentz transformation and a trivial fact about the transpose of a product.
I have said before that I think you can learn SR and matrices in less time than you can learn just SR, and I still think that's right.