From: Xela@yabbs
To: JasonLee@yabbs
Subject: III
Date: Mon Mar  7 22:36:55 1994

There are some assumptions about T(v) that Einstein used, called the 
"axioms of the Special Theory of Relativity."

(R1) The speed of any light beam, when measured in either coordinate 
system using a clock stationary to its coord. system, is 1.

(R2) T(v) is an isomorphism. (meaning basically that T(v) is invertible; 
it has an inverse matrix)

(R3) For any [x, y, z, t] in R^4,
    If T(v)[x, y, z, t] = [x', y', z', t'], then y' = y, and z' = z.

(R4) For T(v)[x, y, z, t] = [x', y', z', t'],
    x' and t' are independent of y and z.  This means if:
    T(v)[x, y1, z1, t] = [x', y', z', t'] and if
    T(v)[x, y2, z2, t] = [x'', y'', z'', t''] then
    x'' = x' and t'' = t'.

(R5) The origin of S moves in the neg. direction of the x'-axis of S'
at constant velocity -v < 0 as measured from S'.

So, consider the instant the origins of S and S' coincide, and say that a 
flash of light is emitted from the twocommon origins.  The event of the 
light flash when measured with respect to either S and C or S' and C' has 
space-time coord. [0, 0, 0, 0].

Let P be the set of all events whose space-time coord. equal [x, y, z, t] 
relative to S and C, such that the flash is observable from [x, y, z] at 
the time t.  Since the speed of light was defined as 1, at any time t >= 0 
the light flash is obserrvable from any point with distance   d
(sorry lag there)...from any point with distance t times 1, which equals 
t.  These points correspond to x-y-z sphere points with radius t with a 
center at the origin.  This satisfies the equation x^2 + y^2 + z^2 = t^2.

Therefore, an event lies in P if and only if its space-time coord. satisfy 
the equation x^2 + y^2 + z^2 - t^2 = 0, relative to S and C.  The same can 
be said for event P relative to S' and C': it must satisfy the equation 
(x')^2 + (y')^2 + (z')^2 - (t')^2 = 0.