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:
trying to get
i.e. invariant interval
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.