From: Xela@yabbs
To: JasonLee@yabbs
Subject: IV
Date: Mon Mar  7 22:54:21 1994

Now the theory gets into inner product spaces...

1 second after the origins of S and S' coincide, the space-time 
coordinates of S' (moving along the x-axis at velocity v), relative to S 
and C are [v, 0, 0, 1].  The space-time coord. for the origin of S' 
relative to S' and C' is [0, 0, 0, t'] for t' > 0.

Therefore T(v)[v, 0, 0, 1] = [0, 0, 0, t'] for some t' > 0.

Let A = { 1 0 0  0 }
        { 0 1 0  0 }
        { 0 0 1  0 }
        { 0 0 0 -1 }, this matrix acts as a basis for the sphere equation 
earlier.  On theory, T*(v) L(A) T(v) = L(A)). 
, so...

<T*(v) L(A) T(v) [v, 0, 0, 1], [v, 0, 0, 1]> =
                            <L(A) [v, 0, 0, 1], [v, 0, 0, 1]> = v^2 - 1

also...

 <T*(v) L(A) T(v) [v, 0, 0, 1], [v, 0, 0, 1]> = 
                            <L(A) T(v) [v, 0, 0, 1], T(v) [v, 0, 0, 1]>

by definition of adjoint transformations in inner product spaces, which 
equals...
                = <L(A)[0, 0, 0, t'], [0, 0, 0, t']> = -(t')^2

So v^2 - 1 = -(t')^2, or t' = (1- v^2) ^(1/2)

Conclussion next...