Special Theory of Relativity
7
Taking into account the possibility that time may not be a universal variable,
t′ = γ (t–βx)
(1.14)
is for the transformation of the time coordinate.
Let an electromagnetic signal be emitted at t = 0, from the origin of F, which
also coincides with the origin of F′ at that time. Since the speed of light is the
same in all inertial frames, the wavefront is described in the two frames by the
equations
x
2
+ y
2
+ z
2
= c
2
t
2
(1.15)
and
x′
2
+ y′
2
+ z′
2
= c
2
t′
2
(1.16)
respectively. Substituting Eqs. (1.12) - (1.14) in Eq. (1.16)
(α
2
–c
2
γ
2
β
2
) x
2
+ y
2
+ z
2
= (c
2
γ
2
– v
2
α
2
) t
2
+ 2 (vα
2
+ c
2
β
2
) xt
(1.17)
This relation is consistent with Eq. (1.15) provided
α 2 – c 2 γ 2 β 2 = 1
2
2
2
2
γ −
α
v
c
= 1
(1.18)
vα
2
= c
2
βγ
2
= 0
These equations are solved by first eliminating α
2
by using the third
equation, and then eliminating γ. The final solutions are:
2
2
1/ 2
2
2
1
1
v
c
v
c
β =
α = γ =


−




(1.19)
which lead to the transformations
1/ 2
2
2
1
x vt
x
v
c
−
′ =


−




y′ = y
z′ = z
(1.20)
2
1/ 2
2
2
1
v
t
x
c
t
v
c
−
′ =


−



