This article contains proof of formulas in Riemannian geometry that involve the Christoffel symbols.
Start with the Bianchi identity
R
a
b
m
n
;
l
+
R
a
b
l
m
;
n
+
R
a
b
n
l
;
m
=
0
.
Contract both sides of the above equation with a pair of metric tensors:
g
b
n
g
a
m
(
R
a
b
m
n
;
l
+
R
a
b
l
m
;
n
+
R
a
b
n
l
;
m
)
=
0
,
g
b
n
(
R
m
b
m
n
;
l
−
R
m
b
m
l
;
n
+
R
m
b
n
l
;
m
)
=
0
,
g
b
n
(
R
b
n
;
l
−
R
b
l
;
n
−
R
b
m
n
l
;
m
)
=
0
,
R
n
n
;
l
−
R
n
l
;
n
−
R
n
m
n
l
;
m
=
0.
The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a mixed Ricci tensor,
R
;
l
−
R
n
l
;
n
−
R
m
l
;
m
=
0.
The last two terms are the same (changing dummy index n to m) and can be combined into a single term which shall be moved to the right,
R
;
l
=
2
R
m
l
;
m
,
which is the same as
∇
m
R
m
l
=
1
2
∇
l
R
.
Swapping the index labels l and m yields
∇
l
R
l
m
=
1
2
∇
m
R
,
Q.E.D. (
return to article)
The last equation in Proof 1 above can be expressed as
∇
l
R
l
m
−
1
2
δ
l
m
∇
l
R
=
0
where δ is the Kronecker delta. Since the mixed Kronecker delta is equivalent to the mixed metric tensor,
δ
l
m
=
g
l
m
,
and since the covariant derivative of the metric tensor is zero (so it can be moved in or out of the scope of any such derivative), then
∇
l
R
l
m
−
1
2
∇
l
g
l
m
R
=
0.
Factor out the covariant derivative
∇
l
(
R
l
m
−
1
2
g
l
m
R
)
=
0
,
then raise the index m throughout
∇
l
(
R
l
m
−
1
2
g
l
m
R
)
=
0.
The expression in parentheses is the Einstein tensor, so
∇
l
G
l
m
=
0
,
Q.E.D. (
return to article)
this means that the covariant divergence of the Einstein tensor vanishes.
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Danielson, Donald A. (2003). Vectors and Tensors in Engineering and Physics (2/e ed.). Westview (Perseus). ISBN 978-0-8133-4080-7.
Lovelock, David; Hanno Rund (1989) [1975]. Tensors, Differential Forms, and Variational Principles. Dover. ISBN 978-0-486-65840-7.
Synge J.L., Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. ISBN 978-0-486-63612-2.
J.R. Tyldesley (1975), An introduction to Tensor Analysis: For Engineers and Applied Scientists, Longman, ISBN 0-582-44355-5
D.C. Kay (1988), Tensor Calculus, Schaum’s Outlines, McGraw Hill (USA), ISBN 0-07-033484-6
T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, ISBN 978-1107-602601
