Elements of Modern Physics
418
In the nonrelativistic limit, this relations tends to the usual relation
ω
2
= GM/r
3
. (It is simpler to start with the geodesic equation for r.)
7. A photon may be bound in a closed orbit by the potential of a black hole.
Show that for a circular orbit
r =
2
1 / 2
3
/ and
/(3 )
d
GM c
c
r
dt
φ =
.
8. Consider the Robertson-Walker metric with k = 0. If a signal is emitted
at t and received at t 0 ,
(a) show that
( )
( )
D t
R t
= σ
=
1/3
1/3
1/3
0
12
(
)
m
c
t
t
R


−




(b) show that
v(t) =
2 ( )
3
D t
t
(c) show that
D(t) = 0
3
3
.
2 (1 /2 )
v
t
v c
+
9. In the steady state theory of the universe, the decrease in the density of
matter due to expansion of the universe is compensated by continuous
creation of matter. Using continuity equation show that the rate of creation
is given by
c
d
dt
ρ = 3ρ 0 H
Given that ρ 0 ≈ 3 × 10
–28
kg/m
3
, estimate the rate of creation in terms of
protons/m
3
/s. Argue that the steady state theory does not imply a sky with
a uniform brightness equal to that of the sun.