120
8 Curved Space and Gravity
But we know from Chap. 7 that the metric perturbation must be related to the classical
potential by (7.23)
φ =
1
2
c
2 h 00 .
(8.46)
Thus we obtain
∇
2
φ = −
c
2
2
Cρ.
(8.47)
We do indeed get the Poisson equation in the classical limit, and by comparison with
Poisson’s equation for classical gravity (8.33) we find the value of the constant C to
be
C = −8π G/c
2
, energy-momentum tensor using mass density.
(8.48)
This completes our task: we have shown that the field equations of relativity reduce
to the classical Poisson equation for the Newtonian gravitational potential, and the
constant in the field equations is determined in (8.48). Note that the constant C is
negative; this is the price we pay for our choice of the overall metric sign as we
discussed in Part I. See also Exercise 8.10.
It is often more convenient to use an energy-momentum tensor with units of energy
density, in which case the constant contains an additional factor of 1/c
2 ,
C = −8π G/c
4
, energy-momentum tensor using energy density.
(8.49)
An important comment is in order concerning the energy-momentum tensor,
which is the source of gravity. We noted that the Einstein field equations force it
to be symmetric and have a zero divergence. The zero divergence condition means
generally that the source is conserved. This is most easily shown for the simplest case
of the dust tensor (8.39). For slowly moving material the 4-velocity is u
α
= (1,
v/c)
and the zero-divergence condition for μ = 0 is
T
0ν ,ν = T
00 ,0 + T
0k ,k =
1
c
∂ρ
∂t
+ ∇ · ρ
v
= 0.
(8.50)
The last expression implies that mass is conserved: the time change of mass density
is balanced by the mass flowing out of a small volume. It is an elegant facet of general
relativity that the Einstein equations imply conservation of the source, whatever it
might be. We will return to this in later chapters on cosmology when we discuss the
energy-momentum tensor of a perfect fluid.
8 Curved Space and Gravity
But we know from Chap. 7 that the metric perturbation must be related to the classical
potential by (7.23)
φ =
1
2
c
2 h 00 .
(8.46)
Thus we obtain
∇
2
φ = −
c
2
2
Cρ.
(8.47)
We do indeed get the Poisson equation in the classical limit, and by comparison with
Poisson’s equation for classical gravity (8.33) we find the value of the constant C to
be
C = −8π G/c
2
, energy-momentum tensor using mass density.
(8.48)
This completes our task: we have shown that the field equations of relativity reduce
to the classical Poisson equation for the Newtonian gravitational potential, and the
constant in the field equations is determined in (8.48). Note that the constant C is
negative; this is the price we pay for our choice of the overall metric sign as we
discussed in Part I. See also Exercise 8.10.
It is often more convenient to use an energy-momentum tensor with units of energy
density, in which case the constant contains an additional factor of 1/c
2 ,
C = −8π G/c
4
, energy-momentum tensor using energy density.
(8.49)
An important comment is in order concerning the energy-momentum tensor,
which is the source of gravity. We noted that the Einstein field equations force it
to be symmetric and have a zero divergence. The zero divergence condition means
generally that the source is conserved. This is most easily shown for the simplest case
of the dust tensor (8.39). For slowly moving material the 4-velocity is u
α
= (1,
v/c)
and the zero-divergence condition for μ = 0 is
T
0ν ,ν = T
00 ,0 + T
0k ,k =
1
c
∂ρ
∂t
+ ∇ · ρ
v
= 0.
(8.50)
The last expression implies that mass is conserved: the time change of mass density
is balanced by the mass flowing out of a small volume. It is an elegant facet of general
relativity that the Einstein equations imply conservation of the source, whatever it
might be. We will return to this in later chapters on cosmology when we discuss the
energy-momentum tensor of a perfect fluid.