ABSTRACT
Proof. Let the same time and length units hold for all integral spaces so that the time axes are equal and the same for all the frames,
Ti = Tj = T , i.e. ti = tj = t. (21.1)
Let A = B in the block diagonal matrix D, (6.22). Let v(.)G ≡ v O(.) P and (8.8)
hold in (6.22). Let the velocity v O(.) P of the arbitrary point P be arbitrary. We
use rOiP (ti) ≡ r
Oi P (t) ≡ r
Oi P (t)u, r
Oj P (t; t0) ≡ tv
Oj P , v
O(.) P ≡ v
in (6.22). The equations (8.2) through (8.6), and (21.1) transform the left-hand side of (6.22) as follows:
T D
rOiP (t)
≡ rOiP (t)u
T A O O −A
rOiP (t)u
tvOiP u
≡
≡
⎡ ⎢⎣
8 r Oj P (t)
92 + 2r
v0jit
2−
2 + 2v
⎤ ⎥⎦uTAu ≡
≡
⎡ ⎢⎣
8 r Oj P (t)
92 + 2v
v0jit
2−
− v Oj P t
2 − 2v0jiv
2 − v0jit
⎤ ⎥⎦ uTAu ≡
T D
r Oj P (t)
.
This holds also for the light velocity c(.) if and only if P = L, for the velocity
