ABSTRACT
Proof. Necessity. Let the scaling coefficients α(..)(.) and λ (..) (.) be determined for
the case when the arbitrary point P moves with the velocity of light:
v (.) P = c
(.) (.) = c
(.) (.)u, i.e. v
(.) P = c
(.) (.). (22.1)
Let the scaling coefficient μi satisfy (11.1). Let the point P start moving from O = Oi0 at the initial instant t(.)0 = 0. In view of (22.1),
rP t(.) ≡ rL(t(.)) = c(.)(.)t(.)u. (22.2)
The position vectors can be expressed also in terms of their (algebraic, i.e. scalar) values,
r(..) t(.) = r(..)
t(.) u, (..) ∈ {G,L, P, PR, PSU}. (22.3)
Let the scaling coefficients αji , α i j , α
i j = α
j i , λ
j i and λ
i j , λ
i j = λ
j i , obey (11.2)
through (11.6) so that they together with (11.1) imply (11.7). The equations (11.1), (22.2) and (22.3) give the next scalar forms to the equations (11.4) and (11.5):
rL(ti) = λ i j [rL(tj) + v
j jitj ] = c
i iti = c
i iμit, (22.4)
rL(tj) = λ j i [rL(ti)− vijiti] = c
j jtj = c
j jμjt. (22.5)
After replacing rL(tj) by c j jtj , (22.2), and by using both ti = μit and tj = μjt,
rL(ti) ≡ λij [c j jtj + v
j jitj ] ≡ λ
1 +
ciitj ≡
1 +
1 +
μj μi rL(ti). (22.6)
The solution of (22.6) for λij is:
. (22.7)
By applying the same procedure to the equation (22.5) we get:
. (22.8)
By combining (11.1), (11.2) and (22.2) we find:
ti ≡ αij
1 +
qjwj
μj μi ti.
