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electromagnetic potential). We shall find that EinsteinвЂ™s equations are also a set of
second-order partial differential equations, but instead for the metric coefficients
g of spacetime.

8.1 The energyвЂ“momentum tensor
To construct the gravitational field equations, we must first find a properly rela-
tivistic (or covariant) way of expressing the source term. In other words, we must
identify a tensor that describes the matter distribution at each event in spacetime.
We will use our discussion of the 4-current density in Chapter 6 as a guide. Thus,
let us consider some general time-dependent distribution of (electrically neutral)
non-interacting particles, each of rest mass m0 . This is commonly called dust in
the literature. At each event P in spacetime we can characterise the distribution
completely by giving the matter density and 3-velocity u as measured in some
inertial frame. For simplicity, let us consider the fluid in its instantaneous rest
frame S at P, in which u = 0. In this frame, the (proper) density is given by

176
177
8.1 The energyвЂ“momentum tensor

l
l

l
l' = l/ Оі

Lorentz contracted in
direction of motion

Figure 8.1 The Lorentz contraction of a fluid element in the direction of motion.

= m0 n0 , where m0 is the rest mass of each particle and n0 is the number of
0
particles in a unit volume. In some other frame S , moving with speed v relative
to S, the volume containing a fixed number of particles is Lorentz contracted
along the direction of motion (see Figure 8.1). Hence, in S the number density
of particles is n = v n0 . We now have an additional effect, however, since the
mass of each particle in S is m = v m0 . Thus, the matter density in S is

= 2
v0

We may conclude that the matter density is not a scalar but does transform as
the 00-component of a rank-2 tensor. This suggests that the source term in the
gravitational field equations should be a rank-2 tensor. At each point in spacetime,
the obvious choice is

Tx = x u x вЉ—u x (8.1)
0

where 0 x is the proper density of the fluid, i.e. that measured by an observer
comoving with the local flow, and u x is its 4-velocity. The tensor T x is
called the energyвЂ“momentum tensor (or the stressвЂ“energy tensor) of the matter
distribution. We will see the reason for these names shortly. Note that from now
on we will denote the proper density simply by , i.e. without the zero subscript.
In some arbitrary coordinate system x , in which the 4-velocity of the fluid is
u , the contravariant components of (8.1) are given simply by

= uu (8.2)
T

To give a physical interpretation of the components of the energyвЂ“momentum
tensor, it is convenient to consider a local Cartesian inertial frame at P in which
178 The gravitational field equations

=
the set of components of the 4-velocity of the fluid is u c u . In this
u
frame, writing out the components in full we have

T 00 = u0 u0 = 2
c2
u

T 0i = T i0 = u0 ui = 2
cui
u

T ij = ui uj = 2
ui uj
u

Thus the physical meanings of these components in this frame are as follows:

T 00 is the energy density of the particles;
T 0i is the energy flux Г— cв€’1 in the i-direction;
T i0 is the momentum density Г— c in the i-direction;
T ij is the rate of flow of the i-component of momentum per unit area in the
j-direction.

It is because of these identifications that the tensor T is known as the energyвЂ“
momentum or stressвЂ“energy tensor.

8.2 The energyвЂ“momentum tensor of a perfect fluid
To generalise our discussion to real fluids, we have to take account of the facts that
(i) besides the bulk motion of the fluid, each particle has some random (thermal)
velocity and (ii) there may be various forces between the particles that contribute
potential energies to the total. The physical meanings of the components of the
energyвЂ“momentum tensor T give us an insight into how to generalise its form to
include these properties of real fluids.
Let us consider T at some event P and work in a local Cartesian inertial frame
S that is the IRF of the fluid at P. For dust, the only non-zero component is T 00 .
However, let us consider the components of T in the IRF for a real fluid.

вЂў T 00 is the total energy density, including any potential energy contributions from forces
between the particles and kinetic energy from their random thermal motions.
вЂў T 0i : although there is no bulk motion, energy might be transmitted by heat conduction,
so this is basically a heat conduction term in the IRF.
вЂў T i0 : again, although the particles have no bulk motion, if heat is being conducted then
the energy will carry momentum.
вЂў T ij : the random thermal motions of the particles will give rise to momentum flow,
so that T ii is the isotropic pressure in the i-direction and the T ij (with i = j) are the
viscous stresses in the fluid.

These identifications are valid for a general fluid. A perfect fluid is defined as
one for which there are no forces between the particles, and no heat conduction
179
8.3 Conservation of energy and momentum for a perfect fluid

or viscosity in the IRF. Thus, in the IRF the components of T for a perfect fluid
are given by
вЋ›2 вЋћ
0 0 0
c
вЋњ0 0вЋџ
вЋњ вЋџ
0
p
=вЋњ вЋџ (8.3)
T
вЋќ0 0вЋ
0 p
0 0 0 p

It is not hard to show that

= + p/c2 u u в€’ p (8.4)
T

However, because of the way in which we have written this equation, it must
be valid in any local Cartesian inertial frame at P. Moreover, we can obtain an
expression that is valid in an arbitrary coordinate system simply by replacing
with the metric functions g in the arbitrary system. Thus, we arrive at a
fully covariant expression for the components of the energyвЂ“momentum tensor of
a perfect fluid:

= + p/c2 u u в€’ pg (8.5)
T

We see that T is symmetric and is made up from the two scalar fields and p
and the vector field u that characterise the perfect fluid. We also see that in the
limit p в†’ 0 a perfect fluid becomes dust.
Finally, we note that it is possible to give more complicated expressions repre-
senting the energyвЂ“momentum tensors for imperfect fluids, for charged fluids and
even for the electromagnetic field. These tensors are all symmetric.

8.3 Conservation of energy and momentum for a perfect fluid
Let us investigate how to express energy and momentum conservation in a local
Cartesian inertial frame S at some event P that is represented by the local inertial
coordinates x . In these coordinates, the energyвЂ“momentum tensor takes the
form (8.4).
By analogy with the equation j = 0 for the conservation of charge, which
we derived in Chapter 6, the conservation of energy and momentum is represented
by the equation

=0 (8.6)
T

Rather than arriving at this result from first principles, which would take us into
a lengthy discussion of relativistic fluid mechanics, let us instead reverse the
180 The gravitational field equations

process and justify our assertion by arguing that it produces the correct equations
of motion and continuity for a fluid in the Newtonian limit.
Substituting the form (8.4) into (8.6) gives
+ p/c2 u u + + p/c2 u u +u в€’ =0 (8.7)
u p
Now, the 4-velocity satisfies the normalisation condition u u = c2 and differenta-
tion of this gives
u u +u =2 u u =0
u
Thus, contracting (8.7) with u , dividing through by c2 and collecting terms gives

+ p/c2 u =0 (8.8)
u

Equation (8.7) therefore simplifies to

+ p/c2 uu= в€’ u u /c2 (8.9)
p

Equations (8.8) and (8.9) are, in fact, respectively the relativistic equation of
continuity and the relativistic equation of motion for a perfect fluid in local inertial
coordinates at some event P.1 We will now show that for slowly moving fluids
and small pressures they reduce to the classical equations of Newtonian theory.
By a slowly moving fluid, we mean one for which we may neglect u/c and so
take u в‰€ 1 and u в‰€ c u ; note that the difference between the proper density
and the density disappears in this limit. By small pressures we mean that p/c2 is
negligible compared with . In these limits, equation (8.8) then reduces to
=0
u
or, in 3-vector notation,
+ В· u =0
t
which is the classical equation of continuity for a fluid. In the limit of small
pressures, equation (8.9) reduces to
uu= в€’ u u /c2 p
Moreover, in our slowly-moving approximation, the zeroth components of the
left- and right-hand sides are both zero. Thus the spatial components i = 1 2 3
satisfy
ui u = в€’ ji
jp

1
As usual, these equations may be generalised to a form valid in arbitrary coordinates by replacing by
and replacing by g .
181
8.4 The Einstein equations

+uВ· u=в€’ p
t
which is EulerвЂ™s classical equation of motion for a perfect fluid. Hence we have
shown that the relativistic continuity equation (8.8) and the equation of motion
(8.9) for a perfect fluid reduce to the appropriate Newtonian equations. If we were
to accept that a relativistic fluid were described by (8.8) and (8.9) then we could
reverse our overall argument and derive the result T = 0.
So far we have worked in local inertial coordinates in order to make contact
with the Newtonian theory. Nevertheless, we can trivially obtain the condition for
energy and momentum conservation in arbitrary coordinates by replacing by
in (8.6), which then gives

=0 (8.10)
T

This important equation is worthy of further comment. In our discussion so
far, we have not been explicit about whether our spacetime is Minkowskian or
curved. Although the form (8.10) is valid (in arbitrary coordinates) in both cases,
its interpretation differs in the two cases. If we neglect gravity and assume a
Minkowski spacetime, the relation (8.10) does indeed represent the conservation
of energy and momentum. In the presence of a gravitational field (and hence
a curved spacetime), however, the energy and momentum of the matter alone
is not conserved. In this case, (8.10) represents the equation of motion of the
matter under the influence of the gravitational field; this is discussed further in
Section 8.8. As we will see below, the condition (8.10) places a tight restriction
on the possible forms that the gravitational field equations may take.

8.4 The Einstein equations
We are now in a position to deduce the form of the gravitational field equations
proposed by Einstein. Let us begin by recalling some of our previous results.

вЂў The field equation of Newtonian gravity is

=4 G
2

вЂў If gravity is a manifestation of spacetime curvature, we showed in Chapter 7, equa-
tion (7.8), that for a weak gravitational field, in coordinates such that g = +h
(with h 1) and in which the metric is static, then
2
g00 = 1 + (8.11)
c2
182 The gravitational field equations

вЂў The correct relativistic description of matter is provided by the energyвЂ“momentum
tensor and, for a perfect fluid or dust, in the IRF we have

T00 = c2

Combining these observations suggests that, for a weak static gravitational field
in the low-velocity limit,
8G
g00 = 4 T00
2
c

EinsteinвЂ™s fundamental intuition was that the curvature of spacetime at any
event is related to the matter content at that event. The above considerations thus
suggest that the gravitational field equations should be of the form
=T (8.12)
K
where K is a rank-2 tensor related to the curvature of spacetime and we have
set = 8 G/c4 . Since the curvature of spacetime is expressed by the curvature
tensor R , the tensor K must be constructed from R and the metric tensor
g . Moreover, K should have the following properties: (i) the Newtonian limit
suggests that K should contain terms no higher than linear in the second-order
derivatives of the metric tensor; and (ii) since T is symmetric then K should
also be symmetric. The curvature tensor R is already linear in the second
derivatives of the metric, and so the most general form for K that satisfies (i)
and (ii) is
= aR + bRg +g (8.13)
K
where R is the Ricci tensor, R is the curvature scalar and a b are constants.
Let us now consider the constants a b . First, if we require that every term
in K is linear in the second derivatives of g then we see immediately that
= 0. We will relax this condition later, but for the moment we therefore have
= aR + bRg
K
To find the constants a and b we recall that the energyвЂ“momentum tensor satisfies
T = 0; thus, from (8.10), we also require
= + bRg =0
K aR
However, in Section 7.11 we showed that
в€’ 2g R = 0
1
R
= 0, we obtain
and so, remembering that g
= 2a+b R=0
1
K g
183
8.5 The Einstein equations in empty space

The quantity R will, in general, be non-zero throughout (a region of) spacetime
unless the latter is flat and hence there is no gravitational field. Thus we find that
b = в€’a/2, and so the gravitational field equations must take the form

в€’ 2g R = T
1
aR

To fix the constant a, we must compare the weak-field limit of these equations
with PoissonвЂ™s equation in Newtonian gravity. The comparison is presented in the
next section, where we show that, for consistency with the Newtonian theory, we
require a = в€’1 and so

в€’ 2g R = в€’ T
1
(8.14)
R

where = 8 G/c4 . Equation (8.14) constitutes EinsteinвЂ™s gravitational field
equations, which form the mathematical basis of the theory of general relativity.
We note that the left-hand side of (8.14) is simply the Einstein tensor G , defined
in Chapter 7.
We can obtain an alternative form of EinsteinвЂ™s equations by writing (8.14) in
terms of mixed components,

R в€’2 R=в€’ T
1

and contracting by setting = . We thus find that R = T , where T в‰Ў T .
Hence we can write EinsteinвЂ™s equations (8.14) as

=в€’ T в€’ 2 Tg
1
(8.15)
R

In four-dimensional spacetime g has 10 independent components and so in
general relativity we have 10 independent field equations. We may compare this
with Newtonian gravity, in which there is only one gravitational field equation.
Furthermore, the Einstein field equations are non-linear in the g whereas Newto-
nian gravity is linear in the field . EinsteinвЂ™s theory thus involves numerous
non-linear differential equations, and so it should come as no surprise that the
theory is complicated.

8.5 The Einstein equations in empty space
In general, T contains all forms of energy and momentum. Of course, this
includes any matter present but if there is also electromagnetic radiation, for
example, then it too must be included in T (the resulting expression is somewhat
complicated; see Exercise 8.3).
184 The gravitational field equations

A region of spacetime in which T = 0 is called empty, and such a region is
therefore not only devoid of matter but also of radiative energy and momentum. It
can be seen from (8.15) that the gravitational field equations for empty space are

=0 (8.16)
R

From this simple equation, we can immediately establish a profound result.
Consider the number of field equations as a function of the number of space-
time dimensions; then, for two, three and four dimensions, the numbers of field
equations and independent components of R are as shown in the table.

No. of spacetime dimensions 2 3 4
No. of field equations 3 6 10
No. of independent components of R 1 6 20

Thus we see that in two or three dimensions the field equations in empty space
guarantee that the full curvature tensor must vanish. In four dimensions, however,
we have 10 field equations but 20 independent components of the curvature
tensor. It is therefore possible to satisfy the field equations in empty space with
a non-vanishing curvature tensor. Remembering that a non-vanishing curvature
tensor represents a non-vanishing gravitational field, we conclude that it is only
in four dimensions or more that gravitational fields can exist in empty space.

8.6 The weak-field limit of the Einstein equations
To determine the вЂ˜weak-fieldвЂ™ limit of the Einstein equations our preliminary
discussion in Section 8.4 suggests that we need only consider their 00-component.
It is most convenient to use the form (8.15) of the equations, from which we have
R00 = в€’ T00 в€’ 2 Tg00
1
(8.17)
In the weak-field approximation, spacetime is only вЂ˜slightlyвЂ™ curved and so
there exist coordinates in which g = + h , with h 1, and the metric
is stationary. Hence in this case g00 в‰€ 1. Moreover, from the definition of the
curvature tensor we find that R00 is given by
R00 = в€’ 00 + 0в€’
0 0 0 00

In our coordinate system the are small, so we can neglect the last two
terms to first order in h . Also using the fact that the metric is stationary in our
coordinate system, we then have
R00 в‰€ в€’ i
00
i
185
8.7 The cosmological-constant term

In our discussion of the Newtonian limit in Chapter 7, however; we found that
i в‰€ 1 ij h to first order in h , and so
00 j 00
2

R00 в‰€ в€’ 2
1 ij
j h00

Substituting our approximate expressions for g00 and R00 into (8.17), in the
вЂ˜weak-fieldвЂ™ limit we thus have

в‰€ I00 в€’ 2 T
1 ij 1
(8.18)
j h00
2

To proceed further we must assume a form for the matter producing the weak
gravitational field and for simplicity we consider a perfect fluid. Most classical
matter distributions have p/c2 and so we may in fact take the energyвЂ“
momentum tensor to be that of dust, i.e.

= uu
T

which gives T = c2 . In addition, let us assume that the particles making up the
fluid have speeds u in our coordinate system that are small compared with c. We
thus make the approximation u в‰€ 1 and hence u0 в‰€ c. Therefore equation (8.18)
reduces to
в‰€ c2
1 ij 1
i j h00
2 2

We may, however, write ij i j = 2 ; furthermore, from (8.11) we have h00 =
is the gravitational potential. Thus, remembering that =
2 /c2 , where
4 , we finally obtain
8 G/c

в‰€4 G
2

which is PoissonвЂ™s equation in Newtonian gravity. This identification verifies our
earlier assertion that a = в€’1 in the derivation of EinsteinвЂ™s equations (8.14).

8.7 The cosmological-constant term
The standard Einstein gravitational field equations are

в€’ 2g R = в€’ T
1
(8.19)
R

However, these equations are not unique. In fact, shortly after Einstein derived
them he proposed a modification known as the cosmological term.
In deriving the field equations (8.14), we assumed that the tensor K that
makes up the left-hand side of the field equations,

=T
K
186 The gravitational field equations

should contain only terms that are linear in the second-order derivatives of g .
This led us to set = 0 in (8.13), i.e. to discard the term g in the tensor K .
Let us now relax this assumption.
Recalling that T = 0 we still require K = 0, but in Section 4.12 we
showed that
=0
g

Thus, we can add any constant multiple of g to the left-hand side of (8.19) and
still obtain a consistent set of field equations. It is usual to denote this multiple
by , so that the field equations become

в€’ 2g R+ g =в€’ T
1
(8.20)
R

where is some new universal constant of nature known as the cosmological
constant. By writing this equation in terms of the mixed components and contract-
ing, as we did with the standard field equations, we find that R = T + 4 .
Substituting this expression into (8.20), we obtain an alternative form of the field
equations,

=в€’ в€’ 2 Tg +g
1
(8.21)
R T

Following the procedure presented in Section 8.6, it is straightforward to show
that, in the weak-field limit, the field equation of вЂ˜NewtonianвЂ™ gravity becomes

= 4 G в€’ c2
2

For a spherical mass M, the gravitational field strength is easily found to be
c2 r
GM
g=в€’ =в€’ 2 r+ r
3
r
Thus, in this case, we see that the cosmological constant term corresponds to a
gravitational repulsion whose strength increases linearly with r.
The reason for calling the cosmological constant is historical. Einstein first
introduced this term because he was unable to construct static models of the
universe from his standard field equations (8.19). What he found (and we will
discuss this in detail in Chapter 15) was that the standard field equations predicted
a universe that was either expanding or contracting. Einstein did this work in about
1916, when people thought that our Milky Way Galaxy represented the whole
universe, which Einstein represented as a uniform distribution of вЂ˜fixed starsвЂ™.
By introducing , Einstein constructed static models of the universe (which as
we will see are actually unstable). It was later realised, however, that the Milky
Way is just one of a great many galaxies. Moreover, in 1929 Edwin Hubble
187
8.7 The cosmological-constant term

discovered the expansion of the universe by measuring distances and redshifts to
nearby external galaxies. The universe was proved to be expanding and the need
for a cosmological constant disappeared. Einstein is reputed to have said that the
introduction of the cosmological constant was his вЂ˜biggest blunderвЂ™.
Nowadays we have a rather different view of the cosmological constant. Recall
that the energyвЂ“momentum tensor of a perfect fluid is

= + p/c2 u u в€’ pg
T

Imagine some type of вЂ˜substanceвЂ™ with a strange equation of state p = в€’ c2 . This
is unlike any kind of substance that you have ever encountered because it has a
negative pressure! The energyвЂ“momentum tensor for this substance would be

= в€’pg = c2 g
T

tensor of this strange substance depends only on the metric tensor вЂ“ it is therefore
a property of the vacuum itself and we can call the energy density of the
vacuum. Second, the form of T is the same as the cosmological-constant term
in (8.20). We can therefore view the cosmological constant as a universal constant
that fixes the energy density of the vacuum,

c4
vac c =
2
(8.22)
8G

Denoting the energyвЂ“momentum tensor of the vacuum by T vac = 2g , we
vac c
can thus write the modified gravitational field equations (8.20) as

в€’ 2g R = в€’ + T vac
1
R T

where T is the energyвЂ“momentum tensor of any matter or radiation present.
How can we calculate the energy density of the vacuum? This is one of the
major unsolved problems in physics. The simplest calculation involves summing
the quantum mechanical zero-point energies of all the fields known in Nature.
This gives an answer about 120 orders of magnitude higher than the upper limits
on set by cosmological observations. This is probably the worst theoretical
prediction in the history of physics! Nobody knows how to make sense of this
result. Some physical mechanism must exist that makes the cosmological constant
very small.
Some physicists have thought that A mechanism must exist that makes exactly
equal to zero. But in the last few years there has been increasing evidence that
the cosmological constant is small but non-zero. The strongest evidence comes
188 The gravitational field equations

from observations of distant Type Ia supernovae that indicate that the expansion of
the universe is actually accelerating rather than decelerating. Normally, one would
have thought that the gravity of matter in the universe would cause the expansion
to slow down (perhaps even eventually halting the expansion and causing the
universe to collapse). But if the cosmological constant is non-zero, the negative
pressure of the vacuum can cause the universe to accelerate.
Whether these supernova observations are right or not is an area of active
research, and the theoretical problem of explaining the value of the cosmological
constant is one of the great challenges of theoretical physics. It is most likely
that we require a fully developed theory of quantum gravity (perhaps superstring
theory) before we can understand .

8.8 Geodesic motion from the Einstein equations
The Einstein equations give a quantitative description of how the energyвЂ“
momentum distribution of matter (or other fields) at any event determines the
spacetime curvature at that event. We also know that, under the influence of
gravity alone, matter moves along geodesics in the curved spacetime. We now
show that it is, in fact, unnecessary to make the separate postulate of geodesic
motion, since it follows directly from the Einstein equations themselves.
The field equations were derived partly from the requirement that the covariant
divergence of the energyвЂ“momentum tensor vanishes,

=0 (8.23)
T

As noted in Section 8.3, this relation represents the equation of motion for matter
in the curved spacetime, and in this section we explore this interpretation in more
detail. For later convenience, we may also write (8.23) as

= + +
T T T T
в€љ
1
=в€љ в€’gT + (8.24)
T
в€’g
where in the last line we have used the expression (3.26) for the contracted
, and we note that g = в€’g for a spacetime metric
connection coefficient
with signature в€’2.
Let us first consider directly the specific case of a single test particle of rest
mass m. By analogy with (8.2), the energyвЂ“momentum tensor of the particle as a
function of position x may be written as
m dz dz
x =в€љ xв€’z
4
(8.25)
T d
в€’g dd
189
8.8 Geodesic motion from the Einstein equations

is the worldline of the particle and is its proper time.2 Inserting
where z
(8.25) into (8.23) and using the result (8.24), we obtain

Л™Л™ xв€’z d+ Л™Л™ xв€’z d =0
4 4
(8.26)
zz zz
x
where the dots denote differentiation with respect to . Since 4 x в€’z depends
only on the difference x в€’ z , we can replace / x by в€’ / z where it acts
upon the delta function. Then, by noting that
d
Л™ xв€’z = xв€’z
4 4
z
z d
we may write (8.26) as
d
в€’ Л™ xв€’z d+ Л™Л™ xв€’z d =0
4 4
z zz
d
On performing the first integral on the left-hand side by parts and collecting
together terms, this becomes

z+
ВЁ Л™Л™ xв€’z d =0
4
zz

For this integral to vanish, we clearly require the first factor in the integrand to
equal zero, from which we recover directly the standard geodesic equation of
motion.
The derivation above offers an entirely new insight into the equation of motion.
The position of the particle is where the field equations become singular, but
the solution of the field equations in the empty space surrounding the singularity
determines how it should move, i.e. it obeys the same equation of motion as that of
a вЂ˜test particleвЂ™. The fact that the Einstein equations predict the equation of motion
is remarkable and should be contrasted with the situation in electrodynamics. In the
latter case, the Maxwell equations for the electromagnetic field do not contain the
corresponding equation of motion for a charged particle, which has to be postulated
separately. The origin of this distinction between gravity and electromagnetism
lies in the non-linear nature of the Einstein equations. The physical reason for
this non-linearity is that the gravitational field itself carries energyвЂ“momentum
and can therefore act as its own source, whereas electromagnetic field carries no
charge and so cannot act as its own source.

x в€’ y is defined by the relation
2 4
The four-dimensional delta function

x в€’ y d4 x =
4
x y

в€љ в€љ
where is any scalar field. Since в€’g d4 x is the invariant volume element, it follows that x в€’ y / в€’g
4

is the invariant scalar that must be used in (8.25).
190 The gravitational field equations

It is worthwhile generalising the above discussion from a single point particle
to a continuous matter distribution. As a simple example, we shall consider
a distribution of dust (i.e. a pressureless perfect fluid), for which the energyвЂ“
momentum tensor is given by
= uu
T
In this case, the equation of motion (8.23) thus reads
= u u+u u =0 (8.27)
uu
Contracting this expression with u , we have
+uu u =0
c2 (8.28)
u
where we have used the fact that u u = c2 . Using this result again, one finds
u = 0, and so the second term in (8.28) vanishes. Thus, we obtain
that u

=0
u

which is simply the general-relativistic conservation equation. Substituting this
expression back into (8.27) gives

u =0 (8.29)
u

which is the equation of motion for the dust distribution in a gravitational field.
Moreover, let us consider the worldline x of a dust particle. From (3.38) the
intrinsic derivative of the particleвЂ™s 4-velocity u along the worldline is given by
Du
= u u =0
D
where we have used (8.29) to obtain the last equality. Since the intrinsic derivative
of the 4-velocity (i.e. the tangent vector to the worldline) is zero, the dust particleвЂ™s
worldline x is a geodesic. We can show this explicitly using the expression
(3.37) for the intrinsic derivative, from which we immediately obtain the geodesic
equation
x+
ВЁ x x =0
Л™Л™

8.9 Concluding remarks
We have now completed the task commenced in Chapter 1 of formulating a
consistent relativistic theory of gravity. This has led us to the interpretation of
gravity as a manifestation of spacetime curvature induced by the presence of
matter (and other fields). This principle is embodied mathematically in the Einstein
191
Appendix 8A: Alternative relativistic theories of gravity

field equations (8.20). In the remainder of this book, apart from the final chapter,
we will explore the physical consequences of these equations in a wide variety of
astrophysical and cosmological applications. In the final chapter we will return to
the formulation of general relativity itself, rederiving the Einstein equations from
a variational principle.

Appendix 8A: Alternative relativistic theories of gravity
In Section 8.7, we described a relatively simple (but theoretically profound)
modification of the Einstein field equations. This shows that EinsteinвЂ™s field
equations are not unique. It is also worth noting that it is possible to create more
radically different theories of gravity, as follows.

Scalar theory of gravity
The simplest relativistic generalisation of Newtonian gravity is obtained by contin-
uing to represent the gravitational field by the scalar . Since matter is described
relativistically by the energyвЂ“momentum tensor T , the only scalar with the
dimensions of a mass density is T . Thus a consistent scalar relativistic theory of
gravity is given by the field equation

4G
=в€’
2
(8.30)
T
c2

However, this theory must be rejected since, when used with the appropriate
equation of motion, it predicts a retardation of the perihelion of Mercury, in
contradiction of observations. Moreover, it does not allow one to couple gravity
to electromagnetism, since T EM = 0; in such a theory we could have neither
gravitational redshift nor the deflection of light by matter.

BransвЂ“Dicke theory
A gravitational theory based on a vector field can be eliminated since such a theory
predicts that two massive particles would repel one another, rather than attract. It is,
of course, possible to construct relativistic theories of gravity in which combinations
of the three kinds of field (scalar, vector and tensor) are used. The most important
of these alternative theories is BransвЂ“Dicke theory, which we now discuss briefly.
In deriving the Einstein field equations, we started with the principle of equiv-
alence, which led us to consider gravitation as spacetime curvature, and we found
a rank-2 tensor theory of gravity that agreed with NewtonвЂ™s theory in the limit
of weak gravitational fields and small velocities. Brans and Dicke also took the
principle of equivalence as a starting point, and thus again described gravity
in terms of spacetime curvature. However, they set about finding a consistent
192 The gravitational field equations

scalarвЂ“tensor theory of gravity. Instead of treating the gravitational constant G as
a constant of nature, Brans and Dicke introduced a scalar field that determines
the strength of G, i.e. the scalar field determines the coupling strength of matter
to gravity. The key ideas of the theory are thus:

вЂў matter, represented by the energyвЂ“momentum tensor T M , and a coupling constant
fix the scalar field ;
вЂў the scalar field fixes the value of G;
вЂў the gravitational field equations relate the curvature to the energyвЂ“momentum tensors
of the scalar field and matter.

The coupled equations for the scalar field and the gravitational field in this
theory are therefore

= в€’4
2
TM
(8.31)
1 8
в€’ g R=в€’ 4 +T
M
R T
2 c

where T M is the energyвЂ“momentum tensor of the matter and T is the energyвЂ“
momentum tensor of the scalar field (the form of T is rather complicated). It
is usual (for historical reasons) to write the coupling constant as = 2/ 3 + 2 .
In the limit в†’ we have в†’ 0, so is not affected by the matter distribution
and can be set equal to a constant = 1/G. In this case, T vanishes, and hence
BransвЂ“Dicke theory reduces to EinsteinвЂ™s theory in the limit в†’ .
The BransвЂ“Dicke theory is interesting because it shows that it is possible to
construct alternative theories that are consistent with the principle of equivalence.
EinsteinвЂ™s theory is beautiful and simple, but it is not unique. One must therefore
look to experiment to find out which theory is correct. One of the features of
the BransвЂ“Dicke model is that the effective gravitational вЂ˜constantвЂ™ G varies with
time because it is determined by the scalar field . A variation in G would affect
the orbits of the planets, altering, for example, the dates of solar eclipses (which
can be checked against historical records). A reasonably conservative conclusion
from experiments is that в‰Ґ 500, so EinsteinвЂ™s theory does seem to be the correct
theory of gravity, at least at low energies.

Torsion theories
Throughout our discussion of curved spacetimes we have assumed that the mani-
fold is torsionless. This is not a requirement, and we can generalise our discussion
to spacetimes with a non-zero torsion tensor,

= в€’
T
193
Exercises

Typically, torsion is generated by the (quantum-mechanical) spin of particles.
Such theories are rather complicated mathematically, since we must make the
distinction between affine and metric connections and geodesics. Gravitational
theories that include spacetime torsion are often described as EinsteinвЂ“Cartan
theories and have been extensively investigated. We will not discuss these theories
any further, however.

Appendix 8B: Sign conventions
There is no accepted system of sign conventions in general relativity. Different
books use different sign conventions for the metric tensor, for the curvature tensors
and for the field equations. We can summarize these sign conventions in terms
of three sign factors S1, S2 and S3. These are defined as follows:
= S1 в€’1 +1 +1 +1
= S2 в€’ + в€’
R
8G
= S3
G T
c4
= S2 S3 R
R
In this text we have used a convention that matches that of both R. dвЂ™Inverno,
An Introduction to EinsteinвЂ™s Relativity, Oxford University Press, 1992, and
W. Rindler, Relativity: Special, General and Cosmological, Oxford University
Press, 2001, but this differs from the convention used by, for example, Misner,
Thorne and Wheeler, Gravitation, Freeman (1973) or Weinberg, Gravitation and
Cosmology, Wiley, (1972). Here is a summary of the sign conventions used in
the various books:
Present text dвЂ™Inverno, Rindler MTW Weinberg
в€’ в€’ + +
S1
+ + + в€’
S2
в€’ в€’ + в€’
S3

Exercises
8.1 Show that the components of the energyвЂ“momentum tensor of a perfect fluid in its
instantaneous rest frame can be written as in (8.3):
= + p/c2 u u в€’ p
T
Can the components be written in any other covariant form?
8.2 Show that, for any fluid,
u =0
u
194 The gravitational field equations

Hence show that a perfect fluid in a gravitational field must satisfy the equations
p
+ u =0
u
c2
p uu
+ u= g в€’
u p
c2 c2
Obtain the equation of motion for the worldline x of a particle in a perfect fluid
with pressure, and hence show that the particle is вЂ˜pushed offвЂ™ geodesics by the
8.3 The electromagnetic field in vacuo has an energyвЂ“momentum tensor Tem . By analogy
with the energyвЂ“momentum tensor for dust, we require that (i) Tem is symmetric;
Tem = 0; (iii) Tem must be quadratic in the dynamical variable F . Hence
(ii)
show that
Tem = в€’ 4g F F
1
F F
00
where is a constant. By examining the component Tem in local Cartesian inertial
coordinates, show that the constant = в€’1/ 0 .
8.4 Consider a cloud of charged dust particles. Show that the equation of motion of such
a fluid is
u=F
u u

where is the proper matter density of the fluid, is its proper charge density and
u is the fluid 4-velocity. Define an energyвЂ“momentum tensor Td = u u , where
is the proper density of the fluid. Hence show that

Td = F j
Tem = в€’F j

where j = u is the 4-current density. Thus write down the energyвЂ“momentum
tensor for charged dust, T = Td + Tem .
8.5 The energyвЂ“momentum tensor of an electromagnetic field interacting with a source
satisfies Tem = в€’F J , where J is the 4-current density of the source. Hence
show that the worldline of a particle of charge q in an electromagnetic field satisfies
q
z+
ВЁ zz =
Л™Л™ Л™
F z
m
and interpret this result physically.
8.6 The weak energy condition (WEC) states that any energyвЂ“momentum tensor must
satisfy
T t t в‰Ґ0

for all timelike vectors t . Show that for a perfect fluid the WEC implies that

в‰Ґ0 c2 + p в‰Ґ 0
and
195
Exercises

8.7 The strong energy condition (SEC) states that any energyвЂ“momentum tensor must
satisfy
T t t в‰Ґ 2T t t
1

for all timelike vectors t . Show that for a perfect fluid the SEC implies that

c2 + p в‰Ґ 0 c2 + 3p в‰Ґ 0
and

Does the SEC imply the WEC in Exercise 8.7? Show further that, from the Einstein
equations, the SEC implies that R t t в‰Ґ 0, where R is the Ricci tensor.
8.8 The equation-of-state parameter w is defined by w = p/ c2 . If one restricts
oneself to sources for which в‰Ґ 0, show that both the weak and strong energy
conditions in Exercises 8.6 and 8.7 imply that w в‰Ґ в€’1.
8.9 Write down the form of the energyвЂ“momentum tensor for a perfect fluid with
4-velocity u with respect to some Cartesian inertial frame S. Show that for the
energyвЂ“momentum tensor to be invariant under a Lorentz transformation to any
other inertial frame one requires p = в€’ c2 . Compare this result with that for the
energyвЂ“momentum tensor of the vacuum.
8.10 Find the most general tensor which can be constructed from the curvature tensor
and the metric tensor and which contains terms no higher than quadratic in the
second-order derivatives of g . Hence write down the most general form of the
gravitational field equations in such a theory.
8.11 In the Newtonian limit of weak gravitational fields, for a slowly moving perfect fluid
c2 show that the 00-component of the Einstein field equations
with pressure p
with a non-zero cosmological constant reduces to

= 4 G в€’ c2
2

where 2 = ij i j and is the proper density of the fluid. Hence show that the
corresponding Newtonian gravitational potential of a spherically symmetric mass
M centred at the origin can be written as
c2 r 2
GM
=в€’ в€’
6
r
where r 2 = ij xi xj . Give a physical interpretation of this result.
8.12 In the scalar theory of gravity (8.30) show that, in any inertial frame, the gravitational
potential produced by a perfect fluid at some event P satisfies
12 3p
в€’ = в€’4 G в€’
2
c2 t 2 c2
where and p are the density and isotropic pressure as measured in the instantaneous
rest frame of the fluid at P. Hence show that the theory reduces to Newtonian
gravity in the non-relativistic limit. How might a cosmological constant be included
in the theory?
9
The Schwarzschild geometry

We now consider how to solve the Einstein field equations and so discover the
metric functions g in any given physical situation. Clearly, the high degree of
non-linearity in the field equations means that a general solution for an arbitrary
matter distribution is analytically intractable. The problem becomes easier if we
look for special solutions, for example those representing spacetimes possessing
symmetries. The first exact solution to EinsteinвЂ™s equations was found by Karl
Schwarzschild in 1916.1 As we shall see, the Schwarzschild solution represents
the spacetime geometry outside a spherically symmetric matter distribution.

9.1 The general static isotropic metric
Schwarzschild sought the metric g representing the static spherically symmetric
gravitational field in the empty space surrounding some massive spherical object
such as a star. Thus, a good starting point for us is to construct the most general
form of the metric for a static spatially isotropic spacetime.
A static spacetime is one for which some timelike coordinate x0 (say) with the
following properties: (i) all the metric components g are independent of x0 ; and
(ii) the line element ds2 is invariant under the transformation x0 в†’ в€’x0 . Note that
(i) does not necessarily imply (ii), as is made clear by the example of a rotating
star: time reversal changes the sense of rotation, but the metric components are
constant in time. A spacetime that satisfies (i) but not (ii) is called stationary.
Thus, starting from the general expression for the line element
ds2 = g dx dx
we wish to find a set of coordinates x in which the g do not depend on the
timelike coordinate x0 and the line element ds2 is invariant under x0 в†’ в€’x0 , i.e.

1
Astonishingly, Schwarzschild derived the solution while in the trenches on the Eastern Front during the First
World War but sadly he did not survive the conflict.

196
197
9.1 The general static isotropic metric

the metric is static, and in which ds2 depends only on rotational invariants of the
spacelike coordinates xi and their differentials, i.e. the metric is isotropic.
In fact, it is only slightly more complicated to derive the general form of the
spatially isotropic metric without insisting that it is static. We therefore begin by
constructing this more general metric. Only after its derivation will we impose
the additional constraint that the metric is static.
The only rotational invariants of the spacelike coordinates xi and their differ-
entials are

x В· x в‰Ў r2 dx В· dx x В· dx

where x в‰Ў x1 x2 x3 and we have defined the coordinate r. Denoting the timelike
coordinate x0 by t, we thus find that the most general form of a spatially isotropic
metric must be

ds2 = A t r dt2 в€’ B t r dt x В· dx в€’ C t r x В· dx 2 в€’ D t r dx2 (9.1)

where A, B, C and D are arbitrary functions of the coordinates t and r.
Let us now transform to the (spherical polar) coordinates t r , defined by

x1 = r sin cos x2 = r sin sin x3 = r cos

In this case, we have

x В· x = r2 x В· dx = r dr dx В· dx = dr 2 + r 2 d + r 2 sin2 d
2 2

and so the general metric (9.1) now takes the form

ds2 = A t r dt2 в€’ B t r r dt dr в€’ C t r r 2 dr 2
в€’D t r dr 2 + r 2 d + r 2 sin2 d
2 2

Collecting together terms and absorbing factors of r into our functions, thereby
redefining A B C D, the metric can be written

ds2 = A t r dt2 в€’ B t r dt dr в€’ C t r dr 2 в€’ D t r d + sin2 d
2 2

If we now define a new radial coordinate by r 2 = D t r and collect together
ВЇ
ВЇ
terms into new arbitrary functions of t and r , thereby again redefining A B C,
we can write the metric as

ds2 = A t r dt2 в€’ B t r dt dВЇ в€’ C t r dВЇ 2 в€’ r 2 d
ВЇ ВЇ ВЇr ВЇ + sin2 d
2 2
(9.2)
r
198 The Schwarzschild geometry

ВЇ
Let us also introduce a new timelike coordinate t defined by the relation

ВЇ
dt = t r A t r dt в€’ 2 B t r dВЇ
ВЇ ВЇ ВЇr
1

where t r is an integrating factor that makes the right-hand side an exact
differential. Squaring, we obtain

ВЇ
dt 2 = A2 dt2 в€’ AB dt dВЇ + 4 B2 dВЇ 2
2
r1 r

from which we find
1 B
ВЇ
A dt2 в€’ B dt dВЇ = dt 2 в€’ dВЇ 2
r r
2 4A
A
ВЇ ВЇ
Thus defining the new functions A = 1/ A and B = C + B/ 4A , our metric
2

(9.2) becomes diagonal and takes the form
ВЇВЇВЇ ВЇ ВЇВЇВЇ r
ds2 = A t r dt2 в€’ B t r dВЇ 2 в€’ r 2 d
ВЇ + sin2 d
2 2

There is no need to retain the bars on the variables, so we can write the metric as

ds2 = A t r dt2 в€’ B t r dr 2 в€’ r 2 d + sin2 d
2 2
(9.3)

Thus, the general isotropic metric is specified by two functions of t and r, namely
A t r and B t r . We will also see that, for surfaces given by t r constant,
the line element (9.3) describes the geometry of 2-spheres, which expresses the
isotropy of the metric. In fact this line element shows that such a surface has a
surface area 4 r 2 . However, because B t r is not necessarily equal to unity we
cannot assume that r is the radial distance.
The final step in obtaining the most general stationary isotropic metric is now
trivial. We require the metric functions g to be independent of the timelike
coordinate, which means simply that A and B must be functions only of r. Thus,
we have

ds2 = A r dt2 в€’ B r dr 2 в€’ r 2 d + sin2 d
2 2
(9.4)

Moreover, we see immediately that ds2 is already invariant under t в†’ в€’t, and
so this is the required form of the metric for a general static spatially isotropic
spacetime.

9.2 Solution of the empty-space field equations
The functions A r and B r in the general static isotropic metric are determined by
solving the Einstein field equations. We are interested in the spacetime geometry
199
9.2 Solution of the empty-space field equations

outside a spherical mass distribution, so we must solve the empty-space field
equations, which simply require the Ricci tensor to vanish:

=0 (9.5)
R

From equation (7.13) we can write the Ricci tensor as

= в€’ + в€’ (9.6)
R

and, in turn, the connection is defined in terms of the metric g by

= 2g + gв€’
1
(9.7)
g g

Thus, we see that the deceptively simple expression (9.5) in fact equates to a rather
complicated set of differential equations for the components of the metric g .
To proceed further, we must calculate the connection coefficients corre-
sponding to our static isotropic metric. This can be done in two ways. The quicker
route (with any metric) is to use the Lagrangian procedure for geodesics discussed
in Section 3.19. This involves writing down the вЂ˜LagrangianвЂ™

L=g x x
Л™Л™

Л™
in which x denotes dx /d , where is some affine parameter along the
geodesic. Subtituting L into the EulerвЂ“Lagrange equations then yields the equa-
tions of an affinely parameterised geodesic, from which the connection coeffi-
cients can be identified. Since later we will discuss the motion of particles in the
Schwarzschild geometry, this procedure would be doubly beneficial.
method, in which the are calculated directly from the metric g using (9.7).
Thus we first need to identify the metric components from the line element (9.4).
The non-zero elements of g and g are

g00 = A r g 00 = 1/A r
g11 = в€’B r g 11 = в€’1/B r
g22 = в€’r 2 g 22 = в€’1/r 2
g33 = в€’r 2 sin2 g 33 = в€’1/ r 2 sin2

where we note that the contravariant components of the metric are simply the
reciprocals of the covariant components, since the metric is diagonal.
Substituting the metric components into the expression (9.7) for the connection,
we find the expressions given in Table 9.1 (with no sums on Latin indices) with
all the other components equalling zero. Thus, summarising these results, we find
200 The Schwarzschild geometry

Table 9.1 The connection coefficients of the general static isotropic metric

=0
0
00

1 dA r
= в€’ 2 gi в‡’ =
1 1
i
g00
00 00
2B r dr
1 dA r
= 2 g0 ig 0 + 0g i в€’ g0i = 2 g 00 i g00 в‡’ 01 =
1 1
0 0
0i
2A r dr
=0
0
ij

1 dB r
= 2 gi ig i + ig i в€’ gii = 2 g ii i gii в‡’ =
1 1 1
i
11
ii
2B r dr
r
= 2 g 11 2 g12 + 2 g12 в€’ 1 g22 в‡’ 22 = в€’
1
1 1
22
Br
r sin2
= в€’ 2 g 11 1 g33 в‡’ 33 = в€’
1
1 1
33
Br
1
= 2 g 22 1 g22 в‡’ 21 =
1
2 2
21
r
= в€’ 2 g 22 2 g33 в‡’ = в€’ sin cos
1
2 2
33 33

1
= 2 g 33 1 g33 в‡’ =
1
3 3
31 31
r
cos
= 2 g 33 2 g33 в‡’ 32 =
1
3 3
32
sin

that only nine of the 40 independent connection coefficients are non-zero; they

= A / 2A = A / 2B = B / 2B
0 1 1
01 00 11

= в€’r/B = в€’ r sin2 = 1/r
1 1 2
/B
22 33 12

= в€’ sin cos = 1/r = cot
2 3 3
33 13 23

We now substitute these connection coefficients into the expression (9.6) in
order to obtain the components R of the Ricci tensor. This requires quite a lot
of tedious (but simple) algebra. Fortunately the off-diagonal components R for
= are identically zero, and we find that the diagonal components are
A A A B A
R00 = в€’ + + в€’ (9.8)
2B 4B A B rB
A A A B B
R11 = в€’ + в€’ (9.9)
2A 4A A B rB
1 r A B
R22 = в€’1+ в€’ (9.10)
2B
B A B
R33 = R22 sin2 (9.11)
201
9.2 Solution of the empty-space field equations

The empty-space field equations (9.5) are thus obtained by setting each of the
expressions (9.8вЂ“9.11) equal to zero. Of these four equations, only the first three
are useful, since the fourth merely repeats the information contained in the third.
Adding B/A times (9.8) to (9.9) and rearranging gives

A B + AB = 0

which implies that AB = constant. Let us denote this constant by . Substituting
B = /A into (9.10) we obtain A + rA = , which can be written as
d rA
=
dr
Integrating this equation gives rA = r + k , where k is another integration
constant. Thus the functions A r and B r are given by
в€’1
k k
Ar = 1+ B r = 1+
and
r r
In solving for A and B we have used only the sum of equations (9.8) and (9.9),
not the separate equations. It is, however, straightforward to check that, with these
forms for A and B, the equations (9.8вЂ“9.11) are satisfied separately.
It can be seen that the integration constant k must in some way represent the
mass of the object producing the gravitational field, as follows. We can identify
k (and ) by considering the weak-field limit, in which we require that
2
Ar
в†’ 1+ 2
c2 c
where is the Newtonian gravitational potential. Moreover, in the weak-field
limit r can be identified as the radial distance, to a very good approximation. For
a spherically symmetric mass M we thus have = в€’GM/r, and so we conclude
that k = в€’2GM/c2 and = c2 . Therefore, the Schwarzschild metric for the empty
spacetime outside a spherical body of mass M is2

в€’1
2GM 2GM
ds = c 1в€’ 2 dt в€’ 1 в€’ 2 dr 2 в€’ r 2 d в€’ r 2 sin2 d
2 2 2 2 2
cr cr

(9.12)

We will use this metric to investigate the physics in the vicinity of a spherical
object of mass M, in particular the trajectories of freely falling massive particles

2
We note that the constant could have been identified earlier by making the additional assumption that
spacetime is asymptotically flat, i.e. that the line element (9.4) tends to the Minkowski line element in the
limit r в†’ . Thus we require that, in this limit, A r в†’ c2 and B r в†’ 1 and so AB = c2 .
202 The Schwarzschild geometry

and photons. The Schwarzschild metric is valid down to the surface of the spheri-
cal object, at which point the empty-space field equations no longer hold. Clearly,
the metric functions are infinite at r = 2 , which is known as the Schwarzschild
radius. As we shall see, if the surface of the massive body contracts within this
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