General Relativity Lecture 4: Spacetimes, SO(1,3), Spacetime Diagrams and Causality

Lecture from 2021 senior/graduate level course in general relativity in physics at Colorado School of Mines. You can follow along at:

[muttleycrew]

49:30 there is some question about an equation for lines parallel to x'. After some confusion the result is given as cτ' = k = cτ cosh(ϕ) + constant.
*This result cannot possibly be right unless x is constant, which it isn't* since more generally we had cτ' = cτ cosh(ϕ) - x sinh(ϕ).

instead, for lines of constant time cτ' = k, we must have,

cτ' = k = cτ cosh(ϕ) - x sinh(ϕ)

which we would surely want to solve as a linear equation in terms of the unprimed, rectangular coordinates as in the previous examples, instead of trying to solve in the skewed, primed coordinates. We have, then,

cτ cosh(ϕ) = x sinh(ϕ) + k

so,

cτ = x tanh(ϕ) + k/cosh(ϕ) = x tanh(ϕ) + k sech(ϕ)

And since for a given value of ϕ the term k sech(ϕ) is a constant we have located the cτ intercept, it is of course a constant, call it K,

k sech(ϕ) = K

and so,

cτ = x tanh(ϕ) + K

Should be what we're looking for. This has to be the case from grinding through the algebra as above but it's easier to see as a geometric argument - these lines of constant time, cτ' = k, must have the same slope as the x'-axis, which was given in the old, unprimed coordinates as cτ = tanh(ϕ) x. But tanh(ϕ) is definitely the slope of the x'-axis. The only thing remaining would be to evaluate the cτ intercept when, which was K = cτ' sech(ϕ).

Nota Bene: this comment was first posted containing some brand new errors, which have now also been corrected.

[warriorofdreams]

why does changing the -1 at the top left of the minkowski metric to +1 so we get the regular 4x4 identity matrix make is such that we no longer have 3 boosts, but 3 rotations involving time? I'm not quite sure I understand why this change happens...

[tawe]

12:40 please, whoever is behind the camera stop tapping