8.2 Symmetries of the Riemann Tensor
115
R
α ηβγ ;μ + R
α ηγ μ;β + R
α ημβ;γ = 0,
(8.22)
which we see by writing out all the terms using the connections and their symmetry.
These are called the Bianchi identities. As with the algebraic symmetries we have
obtained the Bianchi identities in the geodesic coordinate system in which the connections vanish at the selected point, but they are tensor symmetries and thus hold in
all coordinate systems. They will prove useful in obtaining the Einstein gravitational
field equations.
8.3 The Einstein Equations for the Gravitational Field
in Vacuum
There is a convincing heuristic path that leads from classical gravity to the field
equations of general relativity (Adler 1975). Recall that classical gravity may be
viewed in geometric terms if we relate the metric to the classical potential by (7.23),
which we repeat here
g 00 = 1 +
2φ
c 2 , geometry ↔ classical gravity.
(8.23)
From the discussion of the Riemann tensor we see moreover that the absence of a
gravitational field corresponds to a zero Riemann tensor, for then there is a coordinate
system in which the metric is Lorentz and the gravitational potential in (8.23) must
vanish; that is φ = 0 everywhere, and all the second derivatives vanish, φ ,i, j = 0.
Thus
R αβγ δ = 0 ↔ φ ,i, j = 0, absence of gravity.
(8.24)
Indeed, using the correspondence in (8.23) we can make the above correspondence
more explicit. As before we take the classical potential divided by c
2 to be very small
and time independent, so the metric is nearly Lorentz. Working to lowest order we
may express components of the Riemann tensor in terms of the classical potential;
from the definition (8.8) we have
R
i
0 j0 =
i
0 j,0 −
i
00, j = −
i
00, j = −
1
2
h 00,i, j ,
(8.25)
where we have made use of the time independence of the metric and the connections,
and have used (8.23) to calculate the connection. Now using (8.23) and (8.25) we
obtain the important approximate relation
R
i
0 j0 = −
1
c 2 φ ,i, j classical limit.
(8.26)
115
R
α ηβγ ;μ + R
α ηγ μ;β + R
α ημβ;γ = 0,
(8.22)
which we see by writing out all the terms using the connections and their symmetry.
These are called the Bianchi identities. As with the algebraic symmetries we have
obtained the Bianchi identities in the geodesic coordinate system in which the connections vanish at the selected point, but they are tensor symmetries and thus hold in
all coordinate systems. They will prove useful in obtaining the Einstein gravitational
field equations.
8.3 The Einstein Equations for the Gravitational Field
in Vacuum
There is a convincing heuristic path that leads from classical gravity to the field
equations of general relativity (Adler 1975). Recall that classical gravity may be
viewed in geometric terms if we relate the metric to the classical potential by (7.23),
which we repeat here
g 00 = 1 +
2φ
c 2 , geometry ↔ classical gravity.
(8.23)
From the discussion of the Riemann tensor we see moreover that the absence of a
gravitational field corresponds to a zero Riemann tensor, for then there is a coordinate
system in which the metric is Lorentz and the gravitational potential in (8.23) must
vanish; that is φ = 0 everywhere, and all the second derivatives vanish, φ ,i, j = 0.
Thus
R αβγ δ = 0 ↔ φ ,i, j = 0, absence of gravity.
(8.24)
Indeed, using the correspondence in (8.23) we can make the above correspondence
more explicit. As before we take the classical potential divided by c
2 to be very small
and time independent, so the metric is nearly Lorentz. Working to lowest order we
may express components of the Riemann tensor in terms of the classical potential;
from the definition (8.8) we have
R
i
0 j0 =
i
0 j,0 −
i
00, j = −
i
00, j = −
1
2
h 00,i, j ,
(8.25)
where we have made use of the time independence of the metric and the connections,
and have used (8.23) to calculate the connection. Now using (8.23) and (8.25) we
obtain the important approximate relation
R
i
0 j0 = −
1
c 2 φ ,i, j classical limit.
(8.26)