8.3 The Einstein Equations for the Gravitational Field in Vacuum
117
The zero divergence follows trivially from (8.30). Note also that the Einstein tensor
is clearly symmetric.
A simple theorem is the key to the new form of the field equations.
Theorem 5 The Einstein tensor is zero if and only if the Ricci tensor is zero. The
proof is the simple Exercise 8.6.
Thus the field equation (8.28) may also be written as
G
μν
= 0 zero divergence form of field equations.
(8.32)
The fact that the Einstein tensor has zero divergence will prove very useful when we
add matter and energy to the picture, especially in the study of cosmology.
In this section we have tried to motivate the field equations (8.27) or (8.32) heuristically as the natural covariant generalization of classical gravity. Of course the test
of their correctness is to solve them for physically interesting cases and compare the
result to experiment, as we will do in the next chapters.
8.4 The Non-vacuum Field Equations
We have so far considered only gravity in free space, that is in vacuum. Now we
want to obtain the field equations in the presence of matter or energy, such as in the
interior of a star or in the large-scale universe. In classical theory this involves going
from the Laplace equation to the Poisson equation (7.6). That is
∇
2
φ = 0
vacuum
→ ∇
2
φ = 4π Gρ
matter
, ρ = mass density.
(8.33)
That is, in classical theory we simply place a quantity representing matter on the
right side of the equation. We will do precisely the same thing for general relativity.
We will take the field equations for vacuum (8.32) and replace the zero on the right
side with an object that represents the mass and energy density in space.
G
μv
= 0
vacuum
→ G
μv
= CT
μν
mass energy
, T
μν
= energy-momentum.
(8.34)
The tensor on the right side is the source of the gravitational field. It is called the
energy-momentum tensor for reasons that will become apparent when we consider
some special cases; C is a constant to be determined, but we expect it to be
proportional to Newton’s constant G.
The field equations (8.34) are so general as to not mean much yet, since we have
not discussed the nature of the energy-momentum tensor. There are two properties
that the energy-momentum tensor must have, however. First it must be symmetric
since the Einstein tensor is symmetric. Second it must have zero divergence, since
the Einstein tensor has zero divergence, as we discussed in Sect. 8.3. That is
117
The zero divergence follows trivially from (8.30). Note also that the Einstein tensor
is clearly symmetric.
A simple theorem is the key to the new form of the field equations.
Theorem 5 The Einstein tensor is zero if and only if the Ricci tensor is zero. The
proof is the simple Exercise 8.6.
Thus the field equation (8.28) may also be written as
G
μν
= 0 zero divergence form of field equations.
(8.32)
The fact that the Einstein tensor has zero divergence will prove very useful when we
add matter and energy to the picture, especially in the study of cosmology.
In this section we have tried to motivate the field equations (8.27) or (8.32) heuristically as the natural covariant generalization of classical gravity. Of course the test
of their correctness is to solve them for physically interesting cases and compare the
result to experiment, as we will do in the next chapters.
8.4 The Non-vacuum Field Equations
We have so far considered only gravity in free space, that is in vacuum. Now we
want to obtain the field equations in the presence of matter or energy, such as in the
interior of a star or in the large-scale universe. In classical theory this involves going
from the Laplace equation to the Poisson equation (7.6). That is
∇
2
φ = 0
vacuum
→ ∇
2
φ = 4π Gρ
matter
, ρ = mass density.
(8.33)
That is, in classical theory we simply place a quantity representing matter on the
right side of the equation. We will do precisely the same thing for general relativity.
We will take the field equations for vacuum (8.32) and replace the zero on the right
side with an object that represents the mass and energy density in space.
G
μv
= 0
vacuum
→ G
μv
= CT
μν
mass energy
, T
μν
= energy-momentum.
(8.34)
The tensor on the right side is the source of the gravitational field. It is called the
energy-momentum tensor for reasons that will become apparent when we consider
some special cases; C is a constant to be determined, but we expect it to be
proportional to Newton’s constant G.
The field equations (8.34) are so general as to not mean much yet, since we have
not discussed the nature of the energy-momentum tensor. There are two properties
that the energy-momentum tensor must have, however. First it must be symmetric
since the Einstein tensor is symmetric. Second it must have zero divergence, since
the Einstein tensor has zero divergence, as we discussed in Sect. 8.3. That is