Elements of Modern Physics
408
H(t) =
( )
( )
R t
R t
(11.66)
Thus, Hubble’s constant, in general, is a function of time. At present it has
a value of about 1.8 × 10
–18
s
–1
.
Red Shifts
The Robertson-Walker metric provides the proper framework for the description
of cosmological red shifts. Since the proper time for the propagation of radiation
is zero, ∆τ = 0, the time interval for the propagation is given by
∆t =
2 1/2
( )
(1
)
R t
c
k
∆σ
− σ
(11.67)
Consider now two crests emitted from a galaxy at times t e and t e + ∆t e ,
which are received by another galaxy at times t 0 and t 0 + ∆t 0 respectively,
0
( )
( )
e
t
t
d t
R t
∫
=
0
2 1/2
1
(1
)
e
d
c
k
σ
σ
σ
− σ
∫
(11.68)
0
0
( )
e
e
t
t
t
t
dt
R t
+ ∆
+ ∆
∫
=
0
2 1/2
1
(1
)
e
d
c
k
σ
σ
σ
− σ
∫
(11.69)
From these two relations, it follows that
0
0
0
( )
t
t
t
dt
R t
+ ∆
∫
=
( )
e
e
e
t
t
t
dt
R t
+ ∆
∫
(11.70)
For the short intervals under consideration,
0
e
t
t
∆
∆
=
0
( )
( )
e
R t
R t
(11.71)
and since wavelengths are proportional to the time intervals,
0
e
λ
λ
=
0
( )
( )
e
R t
R t
(11.72)
In the case of an expanding universe, R(t 0 ) > R(t e ), which would explain the
observed red shifts. Expanding R(t e ) at t 0 , the red shift z, is
z =
0
1
e
λ −
λ
=
0
2
2
0
0
0
0
0
0
( )
1
1
( ) [1
(
)
(
) ...]
2
e
e
R t
R t
H t t
q H t t
−
+
− −
−
+