General Relativity and Cosmology
397
related to the metric g mv . The discussion of these relations is beyond the scope
of this book (interested reader may refer to Ref. 11). We only note that these
field equation determines the metric for a given energy-momentum distribution.
In the limiting case of weak gravitational field φ, one has T 00 = ρc
2
, g 00 =
1 + 2
2 ,
c
φ R 00 =
1
2
R
−
=
2
2
1
c
− ∇ φ with which Eq. (11.14) to first order in φ,
reduces to the Poisson equation for Newton’s gravitational potential, ∇
2
f = 4πGρ.
Geodesics
The path followed by a particle in the presence of gravitational forces is
determined by the geometry of the equivalent metric space. To obtain the relation
between the path and the metric, it is noted that in 3-dimensional Euclidean
space, a particle moves in a straight line, i.e. it chooses a path which corresponds
to the shortest distance between any two points. However, the ordinary length
is not invariant even in special relativity, and is not a suitable quantity for
determining the trajectory in the general case.
The proper time τ, defined in Eqs. (11.6) and (11.12), is invariant under
general transformations, and the allowed paths may be regarded as corresponding
to extrema of τ. It turns out that τ is actually a maximum for the allowed trajectory
in the case of special relativity [because of the negative sign of spatial terms in
Eq. (11.10)]. For example, the proper time corresponding to points (0, 0) and
(t, 0), with metric v
g
α
µ (Eq. 11.10), is
τ 1 = t
(11.15)
whereas that corresponding to two segments (0, 0) → (t 1 , x 1 ) and (t 1 , x 1 ) → (t, 0)
is
τ 2 =
1/2
1/2
2
2
2
2
1
1
1
1
2
2
1
1
(
)
t
x
t t
x
c
c
−
+
−
−
(11.16)
which for x 1 ≠ 0 is less than t (note that the intermediate point has to be taken so
that each proper-time interval is real). Thus, the straight line between (0, 0) and
(t, 0) corresponds to the maximum proper time. The general situation may be
covered by the requirement that the total proper time
τ AB =
B
A
d τ
∫
(11.17)
=
1/2
B
v
v
A
dx dx
g
d s
ds ds
µ
µ
∫
397
related to the metric g mv . The discussion of these relations is beyond the scope
of this book (interested reader may refer to Ref. 11). We only note that these
field equation determines the metric for a given energy-momentum distribution.
In the limiting case of weak gravitational field φ, one has T 00 = ρc
2
, g 00 =
1 + 2
2 ,
c
φ R 00 =
1
2
R
−
=
2
2
1
c
− ∇ φ with which Eq. (11.14) to first order in φ,
reduces to the Poisson equation for Newton’s gravitational potential, ∇
2
f = 4πGρ.
Geodesics
The path followed by a particle in the presence of gravitational forces is
determined by the geometry of the equivalent metric space. To obtain the relation
between the path and the metric, it is noted that in 3-dimensional Euclidean
space, a particle moves in a straight line, i.e. it chooses a path which corresponds
to the shortest distance between any two points. However, the ordinary length
is not invariant even in special relativity, and is not a suitable quantity for
determining the trajectory in the general case.
The proper time τ, defined in Eqs. (11.6) and (11.12), is invariant under
general transformations, and the allowed paths may be regarded as corresponding
to extrema of τ. It turns out that τ is actually a maximum for the allowed trajectory
in the case of special relativity [because of the negative sign of spatial terms in
Eq. (11.10)]. For example, the proper time corresponding to points (0, 0) and
(t, 0), with metric v
g
α
µ (Eq. 11.10), is
τ 1 = t
(11.15)
whereas that corresponding to two segments (0, 0) → (t 1 , x 1 ) and (t 1 , x 1 ) → (t, 0)
is
τ 2 =
1/2
1/2
2
2
2
2
1
1
1
1
2
2
1
1
(
)
t
x
t t
x
c
c
−
+
−
−
(11.16)
which for x 1 ≠ 0 is less than t (note that the intermediate point has to be taken so
that each proper-time interval is real). Thus, the straight line between (0, 0) and
(t, 0) corresponds to the maximum proper time. The general situation may be
covered by the requirement that the total proper time
τ AB =
B
A
d τ
∫
(11.17)
=
1/2
B
v
v
A
dx dx
g
d s
ds ds
µ
µ
∫