116
8 Curved Space and Gravity
The path to the field equations is now clear. The condition (8.24) that the Riemann
tensor be zero corresponds to flat space and no gravity. If we weaken the condition
on the classical potential from φ ,i, j = 0 by summing over i = j we get φ ,i,i = 0,
Laplace’s equation, which is the correct equation for the classical potential in vacuum!
Thus we are led to contract the Reimann tensor in exactly the same way and postulate
for the gravitational field in vacuum,
R
α μαν = R μν = 0, vacuum field equations,
R μν ≡
β
βν,μ −
β
μν,β +
β
τ μ
τ
βν −
β
τβ
τ
μν .
(8.27)
The contracted Riemann tensor defined in (8.27) is called the Ricci tensor.
The Ricci tensor has several interesting properties. First it might seem that there
are 6 different ways to contract the Riemann tensor, but the symmetries discussed
in the last section imply that the different ways either give zero or the same result
up to a sign. That is, the Ricci tensor is really the only independent contraction of
the Riemann tensor. Secondly the Ricci tensor is symmetric, as may easily be shown
(see Exercise 8.5). Thus it has only 10 independent components in 4 dimensions, so
the field equation (8.27) are a set of 10 partial differential equations, the right number
to determine the 10 components of the symmetric metric tensor.
There is another equivalent form for the field equation (8.27) that is mathematically interesting and will prove useful when we study the gravitational field in
nonvacuum regions of space, that is where there is matter and energy present. This
involves a tensor with zero divergence known as the Einstein tensor.
To get the alternative form we first calculate the divergence of the Ricci tensor,
that is R
α
η;α . Using the Bianchi identities (8.22) we raise an index to find
R
αη βγ ;δ + R
αη γ δ;β + R
αη δβ;γ = 0.
(8.28)
Then we contract α with β and η with γ to get
R
αη αη;δ + R
αη ηδ;α + R
αη δα;η = 0, or R
η η;δ − R
α δ;α − R
η δ;η = 0.
(8.29)
Next we denote the contracted Ricci tensor, or Riemann scalar, as R = R
η η , and
relabel indices in (8.29) to obtain the divergence of the Ricci tensor
R
α δ;α =
1
2
R
η η;δ =
1
2
R ;δ , or R
μν ;ν =
1
2
g
μν R ;ν .
(8.30)
Having obtained the divergence of the Ricci tensor in (8.30) we may define a tensor
with a zero divergence, called the Einstein tensor, as
G
μν
= R
μν
−
1
2
g
μν R.
(8.31)
8 Curved Space and Gravity
The path to the field equations is now clear. The condition (8.24) that the Riemann
tensor be zero corresponds to flat space and no gravity. If we weaken the condition
on the classical potential from φ ,i, j = 0 by summing over i = j we get φ ,i,i = 0,
Laplace’s equation, which is the correct equation for the classical potential in vacuum!
Thus we are led to contract the Reimann tensor in exactly the same way and postulate
for the gravitational field in vacuum,
R
α μαν = R μν = 0, vacuum field equations,
R μν ≡
β
βν,μ −
β
μν,β +
β
τ μ
τ
βν −
β
τβ
τ
μν .
(8.27)
The contracted Riemann tensor defined in (8.27) is called the Ricci tensor.
The Ricci tensor has several interesting properties. First it might seem that there
are 6 different ways to contract the Riemann tensor, but the symmetries discussed
in the last section imply that the different ways either give zero or the same result
up to a sign. That is, the Ricci tensor is really the only independent contraction of
the Riemann tensor. Secondly the Ricci tensor is symmetric, as may easily be shown
(see Exercise 8.5). Thus it has only 10 independent components in 4 dimensions, so
the field equation (8.27) are a set of 10 partial differential equations, the right number
to determine the 10 components of the symmetric metric tensor.
There is another equivalent form for the field equation (8.27) that is mathematically interesting and will prove useful when we study the gravitational field in
nonvacuum regions of space, that is where there is matter and energy present. This
involves a tensor with zero divergence known as the Einstein tensor.
To get the alternative form we first calculate the divergence of the Ricci tensor,
that is R
α
η;α . Using the Bianchi identities (8.22) we raise an index to find
R
αη βγ ;δ + R
αη γ δ;β + R
αη δβ;γ = 0.
(8.28)
Then we contract α with β and η with γ to get
R
αη αη;δ + R
αη ηδ;α + R
αη δα;η = 0, or R
η η;δ − R
α δ;α − R
η δ;η = 0.
(8.29)
Next we denote the contracted Ricci tensor, or Riemann scalar, as R = R
η η , and
relabel indices in (8.29) to obtain the divergence of the Ricci tensor
R
α δ;α =
1
2
R
η η;δ =
1
2
R ;δ , or R
μν ;ν =
1
2
g
μν R ;ν .
(8.30)
Having obtained the divergence of the Ricci tensor in (8.30) we may define a tensor
with a zero divergence, called the Einstein tensor, as
G
μν
= R
μν
−
1
2
g
μν R.
(8.31)