12.1 The Field Equations and Energy-Momentum Conservation
195
(Terms of third order and higher in v/c are neglected.) Use of the product rule and
rearrangement gives
v
j ∂ρ
∂t
+ ρ
∂v
j
∂t
+ v
j ∂
∂ x k
ρv
k
+
ρv
k
∂v
j
∂ x k
= v
j
∂ρ
∂t
+
∂
∂ x k
ρv
k
+ ρ
∂v
j
∂t
+ v
k ∂v
j
∂ x k
= 0.
(12.6)
But the first bracket of this is zero by conservation of mass in (12.4), so the second
bracket is also zero and we find
∂v
j
∂t
+ v
k ∂v
j
∂ x k =
∂v
j
∂t
+
∂ x
k
∂t
∂v
j
∂ x k ≡
dv
j
dt
= 0, cons. of momentum.
(12.7)
This last equation states that if the velocity field is viewed as a function of time
v
j
t, x
k
(t)
then the total time derivative, or Euler derivative, of the flow velocity
as defined in (12.7) is zero. That is, an element of fluid is not accelerated and its
momentum is conserved.
In summary we see that the zero-divergence condition on the dust energymomentum tensor may be interpreted in the classical limit as expressing conservation of energy and momentum. We naturally generalize this to the Reimann space
of general relativity, by saying that the zero divergence of the energy-momentum
tensor (12.2b) implies conservation of energy-momentum. Any source of gravity in
the Einstein equations has a zero divergence because the Einstein tensor does, and
its energy and momentum are thus conserved.
This property of energy-momentum conservation must be considered an extraordinarily elegant feature of general relativity. It is consistent with a fundamental
assumption of the theory that gravity couples to everything that has energy and
momentum.
12.2 Field Equations and the Cosmic Fluid Source
So far we have used only the energy-momentum tensor of dust, that is a fluid characterized by only a mass density and a flow velocity, and in particular with no internal
pressure. A perfect fluid is more general and more realistic, in that it also has an
internal pressure, but no viscosity or other fluid properties. Many real systems are
rather well described as perfect fluids, for example the diffuse “gas” of galaxies that
makes up the present universe on a cosmological scale, the electromagnetic radiation that dominated the universe in its early years, and the quark-gluon plasma that
dominated it in its early seconds. We will discuss the energy-momentum tensor for
a perfect fluid in this section. As in the previous section we will make use of the
classical limit for clarity and simplicity.
195
(Terms of third order and higher in v/c are neglected.) Use of the product rule and
rearrangement gives
v
j ∂ρ
∂t
+ ρ
∂v
j
∂t
+ v
j ∂
∂ x k
ρv
k
+
ρv
k
∂v
j
∂ x k
= v
j
∂ρ
∂t
+
∂
∂ x k
ρv
k
+ ρ
∂v
j
∂t
+ v
k ∂v
j
∂ x k
= 0.
(12.6)
But the first bracket of this is zero by conservation of mass in (12.4), so the second
bracket is also zero and we find
∂v
j
∂t
+ v
k ∂v
j
∂ x k =
∂v
j
∂t
+
∂ x
k
∂t
∂v
j
∂ x k ≡
dv
j
dt
= 0, cons. of momentum.
(12.7)
This last equation states that if the velocity field is viewed as a function of time
v
j
t, x
k
(t)
then the total time derivative, or Euler derivative, of the flow velocity
as defined in (12.7) is zero. That is, an element of fluid is not accelerated and its
momentum is conserved.
In summary we see that the zero-divergence condition on the dust energymomentum tensor may be interpreted in the classical limit as expressing conservation of energy and momentum. We naturally generalize this to the Reimann space
of general relativity, by saying that the zero divergence of the energy-momentum
tensor (12.2b) implies conservation of energy-momentum. Any source of gravity in
the Einstein equations has a zero divergence because the Einstein tensor does, and
its energy and momentum are thus conserved.
This property of energy-momentum conservation must be considered an extraordinarily elegant feature of general relativity. It is consistent with a fundamental
assumption of the theory that gravity couples to everything that has energy and
momentum.
12.2 Field Equations and the Cosmic Fluid Source
So far we have used only the energy-momentum tensor of dust, that is a fluid characterized by only a mass density and a flow velocity, and in particular with no internal
pressure. A perfect fluid is more general and more realistic, in that it also has an
internal pressure, but no viscosity or other fluid properties. Many real systems are
rather well described as perfect fluids, for example the diffuse “gas” of galaxies that
makes up the present universe on a cosmological scale, the electromagnetic radiation that dominated the universe in its early years, and the quark-gluon plasma that
dominated it in its early seconds. We will discuss the energy-momentum tensor for
a perfect fluid in this section. As in the previous section we will make use of the
classical limit for clarity and simplicity.