Special Relativity: Spacetime Diagrams
I'm having some confusion understanding the basic idea of a spacetime diagram and what it really shows. Here is (hopefully) a diagram representing two frames, S and S', where S' moves at relative velocity u to S.
Now, I'm fine with tan(theta) = u/c, since S' moves at velocity u, x = ut gives the blue ct' line.
P happens when x'=0, t'=T'
Inverse Lorentz transform says t = T = gamma[u](t' + vx'/(c^2)) = gamma[u]*T'
I'm taking this to mean that the t-coord in S, of event P, is given by this relation.
tan(theta) = u/c => gamma[u] = [1 - tan^2(theta)]^-1/2
Hence T = T'*[1 - tan^2(theta)]^-1/2
However, using basic trig on the right-angled triangle:
T = T'*cos(theta)
cos(theta) does not equal [1 - tan^2(theta)]^-1/2 !