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a
xв†’ x= x+
a a a
(19.9)
x

It is important to note that we are not performing any coordinate transformation
here; we are considering only variations in the functional forms of the fields a in
a fixed coordinate system. For simplicity, we shall perform our derivation of the
field equations under the assumption that the field theory is local, which means
that second- or higher-order derivatives of the fields do not appear in the action.
Thus, we need only consider the consequent variation in the first derivatives of
the fields, which, from (19.9), is given by
a
в†’ = +
a a a
(19.10)

We also note for later use that, from its definition (19.9), the -operator commutes
with derivatives since
a a
= в€’ = в€’ =
a a a a
(19.11)

The variations (19.9, 19.10) lead to a variation in the action S в†’ S + S, with

S= d4 x = + d4 x
a a
(19.12)
a a

where, for the time being, it is convenient to work in terms of the Lagrangian
density defined in (19.8). To derive the field equations, we wish to factor out
530 A variational approach to general relativity
a in the second term of the integrand. Using (19.11), this second
the variation
term may be written

d4 x = d4 x
a a
a a

в€’ d4 x
a
a

where we have integrated by parts (which corresponds simply to rewriting the
integrand using LeibnitzвЂ™ theorem for the derivative of a product). The first
integral on the right-hand side is a total derivative and can therefore be converted
into an integral over the bounding surface of the region , by straightforward
a to those that vanish on the
calculus. If we restrict the permissible variations
, this integral will also vanish and so (19.12) becomes2
boundary

Sв‰Ў d4 x = в€’ d4 x
a a
a a a

/ a of the
where, in the first equality, we define the variational derivative
Lagrangian density with respect to the field a . If we demand that the action is
stationary, so that S = 0, under the arbitrary variations a we thus require that

= в€’ =0 (19.13)
a a a

These are the EulerвЂ“Lagrange (EL) equations, which correspond to the field
equations of the (local) field theory defined by the action S = d4 x. If, in
addition, the Lagrangian density depends on second- or higher-order derivatives of
the fields then the above derivation is straightforwardly generalised. For example,
if second-order derivatives also appear then one obtains

= в€’ + =0 (19.14)
a a a a

a
provided that the variations and their first derivatives vanish on the bound-
ary .

2 a
The restriction that the variation vanishes on the boundary is generally allowable, except when
discussing topological objects in field theory such as instantons, which are beyond the scope of our discussion.
531
19.4 Alternative form of the EulerвЂ“Lagrange equations

19.4 Alternative form of the EulerвЂ“Lagrange equations
The EL equations in the form (19.13), or generalised to higher-order derivatives
of the fields, provide a straightforward means of determing the field equations
corresponding to a given action. In particular, these equations still hold if (some
of) the fields a being varied are the components of the metric tensor g (or
functions thereof), as will be the case when we derive the Einstein equations from
the gravitational action in Section 19.8.
Nevertheless, if the fields a being varied are not functions of the metric tensor
в€љ
components then the presence of the в€’g factor in the Lagrangian density (19.8)
makes evaluation of the derivative terms in the EL equations (19.13) unnecessarily
cumbersome, although one will nevertheless arrive at the correct field equations.
In such cases, however, the Lagrangian L can often be written in terms of the
fields a and their first (and possibly higher-order) covariant derivatives a,

as opposed to partial derivatives. Indeed, recalling that L should be a scalar
function of spacetime position, one might expect this to be the case since scalars
are most easily obtained by contracting tensor indices. Let us therefore repeat our
derivation of the form of the EL equations, working instead with an action of the
form

в€љ
S= в€’g d4 x
a a a
(19.15)
L g g

where the fields a being varied are independent of the metric tensor g but
L might still contain g , to raise or lower indices, for example (L might also
contain the partial derivatives of g ; recall that the covariant derivatives of the
metric vanish identically).
For simplicity, let us again assume that no second- or higher-order covari-
ant derivatives appear in L. The variation (19.9) leads to variations in the first
covariant derivatives of the fields given by
a
в†’ = +
a a a
(19.16)

In a similar way to before, we note that the -operator commutes with covariant
a= a . The variations (19.9вЂ“19.16) lead, in
derivatives, so that
turn, to a variation in the action S в†’ S + S, with

в€љ в€љ
L L
S= L в€’g d4 x = + в€’g d4 x
a a
(19.17)
a a

where we are now working in terms of the Lagrangian L (as opposed to the
Lagrangian density ). The partial derivative appearing in the first term of
the integrand on the right-hand side deserves some comment. In (19.17), we
532 A variational approach to general relativity

are treating a and a as independent variables. In general, however, the
a will contain terms involving the fields a multiplied
covariant derivatives
by some connection coefficient. If, for some reason, these terms are written out
explicitly in the Lagrangian, they must not be included when calculating the
partial derivative of L with respect to the fields a .
As in our previous derivation of the EL equations, we must now factor out the
a in the second term of the integrand in (19.17). Using the fact that
variation
the -operator commutes with the covariant derivative and employing LeibnitzвЂ™
theorem for the covariant differentiation of a product, this term may be written

в€љ в€љ
L L
в€’g d4 x = в€’g d4 x
a a
a a

aв€љ
L
в€’ в€’g d4 x (19.18)
a

We may now use the divergence theorem to convert the first integral on the
right-hand side to an integral over the boundary . The divergence theorem

g d4 x = d3 y (19.19)
V nV

where V is an arbitrary vector field, is the determinant of the induced metric
on the boundary in the coordinates yi (see Section 2.14) and n is a unit normal
to the boundary. Applying this theorem to the first integral on the right-hand side
a to vanish on
of (19.18) and restricting the allowed variations , we see
that this integral is zero. Thus (19.18) becomes

aв€љ aв€љ
L L L
Sв‰Ў в€’g d4 x = в€’ в€’g d4 x
a a a

where, in the first equality, we define the variational derivative L/ a of the
Lagrangian with respect to the field a . Thus, demanding stationarity of the
action, S = 0, we obtain the alternative form for the EulerвЂ“Lagrange equations

L L L
= в€’ =0 (19.20)
a a a

We shall make use of this form for the EL equations when we consider the field
theories of a real scalar field in Section 19.6 and electromagnetism in Section 19.7.
533
19.5 Equivalent actions

19.5 Equivalent actions
From the derivation of the EL equations, (19.13), the alert reader will have noticed
that there exists an ambigiuity in the definition of the action. This derives from
the fact that one can always convert the integral of a total derivative over some
region into an integral over the bounding surface . Let us therefore consider
the following modification of the Lagrangian density:

в†’ ВЇ= + a
(19.21)
Q

where the Q may, in general, be four arbitrary functions of the fields (but not
of their derivatives). The corresponding action thus reads
ВЇ
S =S+ Q d4 x
a
The variation in this action under the variation in the fields (19.9) is given by
Q
ВЇ
S = S+ d4 x = S + d4 x
a
Q a

where S is the variation in the original action given by the equation before (19.13)
and we have used the fact that the -operator commutes with derivatives. Since
the last integral on the right-hand side is a total derivative, it can be converted to
a surface integral over the boundary . Assuming once again that the variations
ВЇ
, this surface integral is zero and so S = S. Hence demanding
a vanish on
ВЇ
that S = 0 yields the same EL equations as demanding that S = 0, and the two
actions are said to be equivalent. In other words, any two Lagrangian densities
related by an expression of the form (19.21) lead to the same EL equations. The
above argument is easily extended to the case in which contains second- or
higher-order derivatives of the fields. For example, if second-order derivatives
also appear in then the same EL equations (19.14) will be obtained from any
Lagrangian density of the form
ВЇ= + a a
(19.22)
Q
a and their first derivatives vanish on the
provided that the variations
boundary .
Despite the appealing features of the above mathematical manoeuvre, the very
general nature of the allowed transformation (19.21) can lead to problems of
principle. In particular, we have not constrained in any way the transformation
properties of the four quantities Q . Thus, we have not ensured that the quantity
Q d4 x is a scalar function under coordinate transformations. Strictly speaking,
one should ensure that this is true in order that the second term on the right-
hand side of (19.21) is a scalar quantity. Without this criterion, the value of this
534 A variational approach to general relativity

ВЇ
integral (and hence the action S) is not a scalar, i.e. its value changes depending
on the choice of coordinates. We shall see in Section 19.9, however, that the
necessary requirements on the quantities Q are not always imposed. A partial
defence of such practices is that, as stated earlier, in the variation (19.9) we are
not performing any coordinate transformation; we are considering only variations
in the functional forms of the fields a in a fixed coordinate system. One might
therefore be persuaded that the variational formalism outlined above would survive
the introduction of terms in the action that are not scalars under general coordinate
transformations. In principle, however, such sleight of hand is best avoided, and
one should always aim to construct an action that is a true covariant scalar.
We may also construct equivalent actions when the original action takes the
form (19.15), remembering that in this case we are assuming that the fields of
interest a are independent of the components of the metric tensor g . Suppose,
for example, that no second- or higher-order covariant derivatives of the fields
appear in L, and consider the new Lagrangian

ВЇ
L = L+ a
Q

where the functions Q depend only on the fields and not on their first covariant
derivatives. The corresponding action then reads
в€љ
ВЇ
S =S+ Q в€’g d4 x (19.23)

and its variation is given by
в€љ в€љ
Q
ВЇ
S = S+ в€’g d4 x = S + в€’g d4 x
a
Q a

where S is the variation in the original action and again we have used the result
that the -operator commutes with covariant derivatives. Using the divergence
theorem (19.19), the last integral on the right-hand side can be converted to a
surface integral over the boundary . Assuming once again that the variations
ВЇ
, we find that S = S, and so we have obtained the same EL
a vanish on
ВЇ
equations (19.20) by demanding that S = 0 as we did by demanding that S = 0.
We note that, by using the divergence theorem to obtain a surface integral, in
the present case we require the Q to be the components of a vector. This also
ensures that Q is a scalar field, and so the second term on the right-hand side
ВЇ
of (19.23) (and hence the total action S) is a scalar integral.

19.6 Field theory of a real scalar field
The simplest example of a field theory is that of a single real scalar field x
defined on the spacetime. We will also restrict our considerations to a local field
535
19.6 Field theory of a real scalar field

theory, so that no second- or higher-order derivatives of the field appear in the
Lagrangian L.
As a starting point, we take as inspiration the Lagrangian (19.2) for the classical
motion of a mechanical system in Newtonian mechanics. This Lagrangian is
expressed in terms of the derivatives of the generalised coordinates q a t with
respect to the time parameter t, the metric gab of the configuration space of the
system and a potential V q a . Replacing the generalised coordinates by the field
x and time derivatives by derivatives with respect to spacetime position, a
reasonable choice of Lagrangian is given by

L = 2g в€’V
1
(19.24)

where the first term may be loosely regarded as the вЂ˜kinetic energyвЂ™ of the
field and the second term as its вЂ˜potential energyвЂ™. In the expression (19.24), we
have used covariant derivatives rather than partial derivatives since, as stated in
Section 19.2, L must itself be a scalar function of spacetime position. However,
since the covariant derivative of a scalar quantity reduces to a partial derivative,
in this case the latter could be used. Nevertheless, it is usually wiser to retain the
manifestly covariant notation in (19.24). In particular, we see immediately that
the corresponding action is given by
в€љ
S= в€’V в€’g d4 x
1
(19.25)
2g

which is of the general form given in (19.15). Varying this action with respect to
, we may therefore use the convenient form of the EL equations given in (19.20).
For the form of Lagrangian (19.24) we have
L dV L
=в€’ = 1
and 2g
d
where in the second equation we have relabelled the dummy indices in order
to make the differentiation more transparent. Evaluating this derivative explicitly
gives3
L
= 2g + = +g =g
1 1
g
2

and so the EL equations (19.20) become
dV
в€’ в€’ =0
g
d

3
With a little practice, derivatives of this sort can in fact be evaluated very quickly, without needing to employ
the explicit relabelling step used above.
536 A variational approach to general relativity

Remembering that the covariant derivative of the metric tensor is zero, and
rearranging, we thus find that the dynamical field equation satisfied by is

dV
+ =0
2
(19.26)
d

where 2 в‰Ў =g is the covariant dвЂ™Alembertian operator.
A common choice for the potential is V = 2 m2 2 , where m is a constant
1

parameter that characterises the dynamics of the scalar field. The field equation
(19.26) then becomes
+ m2 = 0
2

which is known as the KleinвЂ“Gordon equation. Upon quantisation (which is
beyond the scope of our discussion), this field theory describes collections of
neutral spinless particles of mass m that do not interact with each other except
through their mutual gravitational attraction.

19.7 Electromagnetism from a variational principle
As discussed in Chapter 6, electromagnetism may be described in terms of the
vector field A . Thus, using the general description given in Section 19.2, the
fields a a = 1 4 being varied are the components of this vector field and
so a is a spacetime index. To describe the dynamics of the electromagnetic field
in terms of the variational principle, again we begin by constructing a Lagrangian
L which is a function of A and its first derivatives and which behaves as a scalar
field under general coordinate transformations. We will work from the outset
assuming arbitrary coordinates.
In the case of electromagnetism, however, we saw in Chapter 6 that the theory
also possesses a gauge invariance. If A describes the electromagnetic field in
some physical situation then the same situation is also described by any other
field of the form
=A + =A + (19.27)
A

where is any scalar field (the last equality holds because the covariant derivative
of the scalar is simply its partial derivative). As discussed earlier, by demanding
that the action be invariant under some symmetry one ensures that the resulting
equations of motion also respect that symmetry. We must therefore make sure
that the action is invariant under the gauge transformation (19.27). This precludes
us from forming scalars depending on A A , since it is easy to show that this
537
19.7 Electromagnetism from a variational principle

expression is not gauge invariant. Nevertheless, the electromagnetic field-strength
tensor
= Aв€’ A= Aв€’ (19.28)
F A
is easily shown to be gauge invariant; the second equality in (19.28) holds since
a convenient cancellation occurs between the terms containing connection coef-
ficients arising from the two covariant derivatives. The most obvious scalar to
be constructed from the field-strength tensor is simply F F = g g F F .
Including a factor of в€’1/ 4 0 for later convenience, we shall take the вЂ˜free-fieldвЂ™
part of the Lagrangian to be
1
Lf = в€’ Aв€’ Aв€’
gg A A
4 0
where, again for later convenience, we have written the expression in terms of
covariant derivatives rather than partial derivatives.
So far we have not taken into account that the source of the electromagnetic
field is the 4-current density j of any charged matter present. To describe this,
we must include an вЂ˜interaction termвЂ™ in the Lagrangian. The most straightforward
scalar we may construct from the electromagnetic field and the current density is
j A , and we will take the interaction term to be Li = в€’j A . Taking the full
Lagrangian to be L = Lf + Li , the action reads

в€љ
1
S= в€’ Aв€’ Aв€’ в€’j A в€’g d4 x
gg A A
4 0

(19.29)
As is immediately apparent, however, the interaction term в€’j A is not auto-
matically gauge invariant. Under the gauge transformation (19.27) the correspond-
ing term in the action becomes
в€љ в€љ
в€’ j A +j в€’g d4 x = в€’ jA+ в€’ в€’g d4 x
j j

Using the divergence theorem (19.19), we may write the second term in the
integrand on the right-hand side as a surface integral over the boundary .
Taking the source j to vanish on (by, for example, taking the boundary to
be at spatial infinity), the surface integral is zero. Thus, we see that the part of
the action arising from the interaction term is, in fact, gauge invariant, provided
that the source j satisfies the covariant continuity equation
j =0
and so the requirement of gauge invariance implies the conservation of charge.
538 A variational approach to general relativity

Thus, under the appropriate conditions, the action (19.29) is invariant under
the gauge transformation (19.27) and, by construction, is a scalar under general
coordinate transformations. Let us now determine the EulerвЂ“Lagrange equations
resulting from varying the fields A in this action (while keeping the source
j fixed). From (19.29), we see that the action has the general form (19.15).
Therefore we may once again use the form of the EL equations given in (19.20),
which in this case read
L L
в€’ =0 (19.30)
A A

For the action in (19.29), we have immediately
L
= в€’j = в€’j (19.31)
A
but evaluation of the second term on the left-hand side of (19.30) requires more
A в€’ A = F for conve-
care. Relabelling dummy indices, and writing
nience, we have
1
L
= в€’ gg FF
4
A A 0

1
=в€’ в€’ +F в€’
gg F
4 0
1 1
=в€’ в€’g g в€’ в€’g g
g g F gg F
4 4
0 0
1 1 1
=в€’ в€’F в€’ в€’F =в€’
F F F
4 4
0 0 0

where in the last equality we have used the antisymmetry of the field-strength
tensor (19.28). Combining this result with (19.31), the EL Lagrange equations

=
F 0j

which is the same expression as that for the inhomogeneous Maxwell equations in
an arbitrary coordinate system, given in Section 7.7. The remaining homogeneous
Maxwell equations are in fact automatically satisfied from the definition (19.28)
of the field-strength tensor, since
+ + = + + =0
F F F F F F
Of course, one may object to the fact that we carefully constructed the action
(19.29) (by, for example, including specific factors in Lf and Li ) in such a way that
539
19.8 The EinsteinвЂ“Hilbert action and general relativity in vacuo

its variation with respect to A led to the field equations for electromagnetism.
Nevertheless, the derivation above illustrates the natural way in which the action
approach constrains the possible forms for the theory and allows any symmetries
in the theory to be made manifest.

19.8 The EinsteinвЂ“Hilbert action and general relativity in vacuo
We now use our experience in expressing scalar field theory and electromagnetism
as variational principles to construct an action for gravitation from which the
Einstein field equations of general relativity can be derived. For the time being,
we will restrict our attention to general relativity in vacuo.
To construct an action for general relativity, we must define a Lagrangian L
which is a scalar under general coordinate transformations and which depends on
the components g of the metric tensor (these are now the dynamical fields), and
their first- and possibly higher-order derivatives. The simplest non-trivial scalar
that can be constructed from the metric and its derivatives is the Ricci scalar R,
which depends on g and its first- and second-order derivatives. In fact, R is
the only scalar derivable from the metric tensor that depends on derivatives no
higher than second order. From our knowledge of gravitation as a manifestation
of spacetime curvature, we might also expect L to be derived from the curvature
tensor. Thus, in searching for the simplest plausible variational principle for
gravitation, one is immediately led to the EinsteinвЂ“Hilbert action

в€љ
SEH = R в€’g d4 x (19.32)

Since the corresponding Lagrangian LEH = R now depends on the elements
of the metric tensor, it is more convenient to work in terms of the Lagrangian
в€љ
density EH = R в€’g. The resulting EL equations thus take the form (19.13),
which in this case reads

в€’ + =0
g g g

Unfortunately, the task of evaluating each term in the above equation involves
a formidable amount of algebra, albeit straightforward. We shall therefore not
pursue this approach any further. Instead, we shall derive the corresponding field
equations by considering directly the variation in the action resulting from a
variation in the metric tensor.
Let us therefore consider a variation in the metric tensor given by

в†’g +g
g
540 A variational approach to general relativity

where g and its first derivative vanish on the boundary of the region . It
will prove useful also to determine the corresponding variation g in the inverse
metric components. This is most easily achieved by noting that g g = and
using the fact that the constant tensor does not change under a variation. To
first order in the variation, one may therefore write
g g +g =0 (19.33)
g
Multiplying through by g , relabelling indices and rearranging, one obtains
= в€’g g
g g
Writing the Ricci scalar as R = g R , the first-order variation in the EinsteinвЂ“
Hilbert action (19.32) can be written as
в€љ в€љ в€љ
SEH = в€’g d4 x + в€’g d4 x + в€’g d4 x
gR g R gR

в‰Ў S1 + S2 + S3 (19.34)
To derive the field equations, we need to factor out the variation g in the
second and third integrals. Let us first focus on the second term and write the
variation R in terms of the variation g in the metric tensor. It is in fact
more illuminating, and no more work, to determine the variation R in the
full curvature tensor, from which the corresponding variation in the Ricci tensor
can be obtained immediately by contraction. The curvature tensor is given by
= в€’ + в€’
R
Let us first consider the variation in the curvature tensor resulting from an arbitrary
variation in the connection coefficients,
в†’ +
It is worth noting that the variation is the difference of two connections
and is therefore a tensor. As is often the case in proving tensor identities, it is
easiest to work in local geodesic coordinates at some arbitrary point P. In such a
P = 0, and so at the point P we have
coordinate system
= в€’
R
Moreover, partial derivatives and covariant derivatives coincide at P and so
= в€’ (19.35)
R
We now see, however, that the quantities on the right-hand side are tensors, and
therefore (19.35) holds not only in geodesic coordinates at P but in any arbitrary
coordinate system. Since the point P was chosen arbitrarily, the result (19.35)
541
19.8 The EinsteinвЂ“Hilbert action and general relativity in vacuo

thus holds generally and is known as the Palatini equation. The corresponding
variation in the Ricci tensor is obtained by contracting on and in (19.35)
to give
= в€’ (19.36)
R

We may therefore write the second term on the right-hand side of (19.34) as
в€љ
S2 = в€’ в€’g d4 x
g
в€љ
= в€’g в€’g d4 x
g

where in the last line we have used the fact that the covariant derivative of the
metric vanishes and we have relabelled indices in the second term of the integrand.
Using the divergence theorem (19.19), however, we may write S2 as a surface
integral over the boundary , which vanishes provided that the variation in the
connection vanishes on the boundary. This means that variations in the metric
tensor and in its first derivatives vanish on .
Let us now turn our attention to the third term S3 in (19.34), in which we
в€љ
в€’g in terms of the variation g . Recalling that g = det g ,
must express
we note that the cofactor of the element g in this determinant is gg . It follows
that
g = gg = в€’gg
g g

where in the second equality we have used the result (19.33). Thus, we have
в€љ 1в€љ
в€’g = в€’ 2 в€’g в€’1/2 g = в€’ 2 в€’gg g
1
(19.37)

Substituting this expression into the third term S3 in (19.34) and remembering
that S2 = 0, we finally discover that the variation in the EinsteinвЂ“Hilbert action
may be written as
в€љ
SEH = в€’ 2g R в€’g d4 x
1
(19.38)
R g

By demanding that SEH = 0 and using the fact that the variation g is arbitrary,
we thus recover EinsteinвЂ™s field equations in vacuo:

в‰ЎR в€’ 2g R = 0
1
(19.39)
G

This is an impressive result, since we have obtained the field equations of general
relativity by varying an action (19.32) to which we were led very naturally
on the grounds of symmetry and simplicity. This illustrates the power of the
variational approach and should be contrasted with the more heuristic approach
542 A variational approach to general relativity

we had to employ in Section 8.4. Moreover, if one were willing to consider more
complicated actions, the variational formalism suggests how EinsteinвЂ™s theory
might be modified by adding to the Lagrangian terms proportional to R2 R3 , etc.
The formalism also provides a means for investigating alternative gravitational
Lagrangians. For example, the choice L = R leads to an alternative
R
self-consistent theory of gravity considered by Eddington.

19.9 An equivalent action for general relativity in vacuo
The EinsteinвЂ“Hilbert action (19.32) differs from the action (19.25) for scalar field
theory and the action (19.29) for electromagnetism in that it depends on second-
order derivatives of the dynamical fields. It is therefore of interest to consider
whether the empty-space gravitational field equations can be derived from an
action that depends only on the metric tensor and its first derivatives. As stated
in the previous section, however, R is the only scalar derivable from the metric
tensor that depends on derivatives no higher than second order, so at first our goal
appears unattainable. Nevertheless, as we will show, we may use the notion of
equivalent actions discussed in Section 19.5 to circumvent this difficulty, albeit
in a way that results in a new action that is not a scalar under general coordinate
transformations.
в€љ
The Lagrangian density EH = в€’gR in the EinsteinвЂ“Hilbert action (19.32)
may be written as
в€љ
= в€’gg R
EH
в€љ
= в€’gg в€’ + в€’
в€љ
в€’ВЇ
= в€’gg в€’ (19.40)

where in the last line we have defined a new Lagrangian density

ВЇ в‰Ў в€љв€’g g в€’ (19.41)

which clearly depends only on the metric and its first derivatives. (Note that the
minus sign in (19.40) is for later convenience.) By relabelling indices and using
LeibnitzвЂ™ rule for the differentiation of products, we can write the first term in
(19.40) as
в€љ в€љ в€љ
в€’gg в€’ = в€’gg в€’ в€’gg
в€љ в€љ
в€’ в€’gg + в€’gg
(19.42)
543
19.10 The Palatini approach for general relativity in vacuo

To evaluate the last two terms on the right-hand side, we note that
в€љ в€љ
= 2 в€’g в€’1/2 g
в€’gg g + в€’g g
1
(19.43)
Using the result (3.24) derived in Section 3.10, we have g = 2g and, since
the covariant derivative of the metric (or its inverse) is zero,
= + + =0
g g g g
Thus, we may write (19.43) as
в€љ в€љ
в€’gg = в€’g в€’ в€’
g g g
Substituting this result into the last two terms on the right-hand side of (19.42)
(contracting on and for the first of these terms), relabelling indices and
simplifying, one finds that
в€љ в€љ в€љ
+2 ВЇ
в€’gg в€’ = в€’gg в€’ в€’gg
Thus, we finally discover that the EinsteinвЂ“Hilbert Lagrangian density (19.40)
can be written
в€љ в€љ
ВЇ+
EH = в€’gg в€’ в€’gg (19.44)

where ВЇ is given by (19.41).
We see immediately, however, that the second term in (19.44) is a total deriva-
tive, and so EH and ВЇ are related by an expression of the form (19.22). The
two Lagrangian densities are therefore equivalent. As discussed in Section 19.5,
variation of the new action
в€љ
ВЇ
S= в€’ в€’g d4 x (19.45)
g

will thus lead to the same field equations as did the EinsteinвЂ“Hilbert action SEH ,
provided that the variation in the metric and its first derivative vanish on the
boundary . Thus, the variation of (19.45) will again yield EinsteinвЂ™s field
equations in vacuo (which may be checked directly), but the action depends only
on the metric and its first derivatives. There is, however, a price to pay in adopting
ВЇ
the above result, since the new action S is easily shown not to be a scalar with
respect to general coordinate transformations (see the discussion in Section 19.5).

19.10 The Palatini approach for general relativity in vacuo
A more elegant and illuminating method for obtaining the Einstein field equations
from an action depending only on dynamical fields and their first derivatives is
544 A variational approach to general relativity

provided by the Palatini approach, which we now discuss. In this formalism one
treats the metric g and the connection as independent fields. In other
words, one does not assume any explicit relationship between the metric and the
connection.
We begin again with the EinsteinвЂ“Hilbert Lagrangian density
в€љ в€љ
EH = в€’gg R = в€’gg в€’ + в€’

which we now consider as a function of the metric, the connection and first
derivatives of the connection, i.e. EH = EH g . Let us first
consider the variation in the action resulting from a variation in the metric alone.
This may be written as
в€љ
SEH = в€’gg d4 x
R

Demanding that SEH = 0 for an arbitrary variation in the metric, we immediately
find that
=0
R

which gives the Einstein field equations in vacuo.
Let us now consider varying the action with respect to the connection, which
yields
в€љ
SEH = в€’gg d4 x
R
в€љ
= в€’gg в€’ d4 x (19.46)

where in the second line we have used the contracted version (19.36) of the
Palatini equation. Using LeibnitzвЂ™ theorem for the differentiation of products and
relabelling some dummy indices, we may write (19.46) as
в€љ
SEH = в€’g в€’g d4 x
g
в€љ
в€’ в€’ в€’g d4 x (19.47)
g g

where we note that we have not assumed that the covariant derivative of the
metric vanishes, since we have not (yet) specified any relationship between the
connection and the metric. Using the divergence theorem (19.19), we may write
the first integral on the right-hand side of (19.47) as a surface integral over the
boundary , which vanishes if we assume that the variation in the connection
545
19.11 General relativity in the presence of matter

vanishes on the boundary. Relabelling some dummy indices in the second integral
on the right-hand side of (19.47), we thus find
в€љ
SEH = в€’ в€’g d4 x (19.48)
g g

Since we are assuming that the manifold is torsionless, the variation in the
connection, although arbitrary, must be symmetric in its lower two indices. As a
result, demanding that SEH = 0 only requires the symmetric part of the term in
parentheses in (19.48) to vanish; when contracted with , the antisymmetric
part will automatically equal zero. Thus, stationarity of the action requires that

+2 в€’ =0
1 1
g g g
2

We thus deduce that g = 0, which in turn implies that g = 0. Hence by
demanding stationarity of the EinsteinвЂ“Hilbert action with respect to variations in
the (symmetric) connection, we have derived that the covariant derivative of the
metric must vanish. We may thus write

= +
g g g

Cyclically permuting the free indices to obtain similiar expressions for and
g
g , combining the results and contracting with g one finds that

= 2g + в€’
1
g g g

and hence the connection must be the metric connection.

19.11 General relativity in the presence of matter
So far we have confined our attention to deriving the gravitational field equations
in vacuo. We now consider how the full Einstein equations, in the presence of
other (non-gravitational) fields, may be obtained by a variational principle. In
order to accommodate this generalisation, one simply needs to add an extra term
to the action to give
1 1
S= S + SM = EH + d4 x (19.49)
2 EH M
2
where the EinsteinвЂ“Hilbert action SEH is considered as a function of the metric
and of its first- and second-order derivatives (as in Section 19.8). SM is the
вЂ˜matterвЂ™ action for any non-gravitational fields present, and = 8 G/c4 . The
factor 1/ 2 in (19.49) is chosen for later convenience.
546 A variational approach to general relativity

Let us now consider varying the action with respect to the (inverse) metric, to
obtain
1 EH M
+ =0
2g g
From (19.38), we see that
в€љ
EH
= в€’g G
g
where G = R в€’ 2 g R is the Einstein tensor. Thus, if we make the bold
1

assertion that the energyвЂ“momentum tensor of the non-gravitational fields (or
вЂ˜matterвЂ™) is given by
2 M
T =в€љ (19.50)
в€’g g
then we recover the full Einstein equations

=в€’ T
G

The definition (19.50) of the вЂ˜matterвЂ™ energyвЂ“momentum tensor may appear to
be somewhat arbitrary. Nevertheless, as we show in the next section, this tensor
has all the properties required of an energyвЂ“momentum tensor.

19.12 The dynamical energyвЂ“momentum tensor
The quantities T defined in (19.50) are clearly the components of a tensor, which
is known more properly as the dynamical energyвЂ“momentum tensor. From the
definition we also see immediately that T is a symmetric tensor, as is required
by the full Einstein equations (19.39). Most importantly, however, we now show
that it obeys the conservation equation T = 0.
From the definition (19.50), the variation in the matter action resulting from a
variation in the metric is given by
в€љ
M
SM в‰Ў d4 x = в€’g d4 x
1
g T g
2
g
в€љ
= в€’2 в€’g d4 x
1
(19.51)
T g

where, in the last equality, we have written SM in terms of the contravariant
components T of the energyвЂ“momentum tensor for later convenience, using the
result (19.33). Let us now consider making an infinitesimal general coordinate
transformation
=x + (19.52)
x x
547
19.12 The dynamical energyвЂ“momentum tensor

where x is an infinitesimal smooth vector field. Since the action SM is, by
construction, a covariant scalar, then we must have SM = 0 under the coordinate
transformation. We know, however, that the metric coefficients must transform as
x x
x= x= в€’ в€’ (19.53)
g g x xg x
x x
=g x в€’g x в€’g
x x x
(19.54)
to first order in , where we have used the expression (17.3) for the trans-
formation matrix corresponding to the infinitesimal coordinate transformation
(19.52). We have explicitly included the dependence on x and x in (19.54), since
it is crucial to determining the corresponding variation g . As mentioned in
Section 19.3, this variation is only of the functional form of the fields g . Thus,
we have
x в‰Ўg x в€’g x=g x в€’g x в€’g x в€’g
g x
=g x в€’g xв€’ x g x
=g x в€’g xв€’ x g x
to first order in . Using the expression (19.54) and dropping the explicit
dependence on x, we find that
= в€’g в€’g в€’ =в€’ +
g g
where, in the second equality, we have rewritten the partial derivatives in terms of
covariant derivatives, cancelled matching terms involving connection coefficients
and used the fact that g = 0.
Substituting this result into (19.51) and remembering that SM = 0 under a
coordinate transformation and that T is symmetric, we have
в€љ
SM = T в€’g d4 x = 0

Using LeibnitzвЂ™ theorem for the covariant differentiation of a product, we write
в€љ в€љ
SM = в€’g d4 x в€’ в€’g d4 x = 0 (19.55)
T T

We may use the divergence theorem (19.19) to write the first integral as a surface
integral over the boundary in the usual manner. Assuming that the functions
x vanish on the boundary this surface integral vanishes, leaving only the
second integral in (19.55). Since the x are arbitrary, however, one immediately
finds that
=0
T
548 A variational approach to general relativity

and so the covariant divergence of the energyвЂ“momentum tensor vanishes, as
required. Thus, we see that the general covariance of the matter action implies
energyвЂ“momentum conservation in the same way as the gauge invariance
of the action (19.29) for electromagnetism implies charge conservation (see
Section 19.7).
Now that we have shown that the tensor T defined by (19.50) has the
appropriate properties of an energyвЂ“momentum tensor, we may calculate the
explicit form of this tensor for some specific вЂ˜matterвЂ™ actions. Let us begin by
considering the action (19.25) for a real scalar field . Varying this action now
with respect to the (inverse) metric, rather than the field , we obtain
в€љ
S= в€’g
1
g
2
в€љ
+ в€’V в€’g d4 x
1
2g
в€љ
= в€’ 2g в€’V в€’g d4 x
1 1 1
2g g
2

в€љ
в€’g. Comparing
where in the last line we have used the expression (19.37) for
the above expression with that in (19.51), we immediately see that the energyвЂ“
momentum tensor for a real scalar field is given by

= в€’g в€’V
1
T 2

which agrees with the expression (16.7) adopted in our discussion of inflation in
Section 16.1.
We may also obtain the energyвЂ“momentum tensor for the electromagnetic field
in a similar manner. From (19.29) and (19.28), in the absence of sources we may
write the action for electromagnetism as
в€љ
1
SEM = в€’ в€’g d4 x
ggFF
4 0

where F = A в€’ A and so does not depend on the metric. Varying this
action with respect to the (inverse) metric, we have
в€љ в€љ
1
SEM = в€’ в€’g + F F в€’g d4 x
gg FF
4 0
в€љ
1
=в€’ в€’ 2g F F в€’g d4 x
1
2g F F g
4 0

where in the second equality we have substituted the expression (19.37) for
в€љ
в€’g and relabelled some dummy indices. Comparing the above expression with
549
Exercises

(19.51), we find that the energyвЂ“momentum tensor for the electromagnetic field
is given by
в€’1
=в€’ в€’ 4g F F
EM 1
T FF
0

which agrees with the expression derived in Exercise 8.3.
Finally, we note that in field theory it is common to define also a canonical
energyвЂ“momentum tensor, which is based on NoetherвЂ™s theorem.4 This states that
for every symmetry of the action there exists a corresponding conserved quantity.
In particular, if an action is invariant under a spacetime translation, characterised
by a coordinate transformation of the form x в†’ x + a in which the vector
a does not depend on spacetime position, then one can define a tensor S that
S = 0. It is this tensor that is usually called the canonical energyвЂ“
obeys
momentum tensor. Unfortunately, there are some drawbacks in using it, since it
is not necessarily symmetric (although it can be made so) or gauge invariant.

Exercises
19.1 If x and x are the local line density and tension of a string, show that the
kinetic and potential energies of the string for small displacements t x are given
by
2 2
l l
T= V=
1 1
and
dx dx
2 2
t x
0 0

19.2 In classical field theory, the conjugate field momenta are defined in terms of the
Lagrangian density by

в‰Ў
Л™a
a

where Л™ a в‰Ў and x0 is a timelike coordinate. The Hamiltonian density is then
a
0
defined as
Л™aв€’
в‰Ў a

Use the EulerвЂ“Lagrange equations to show that

Л™a= Л™a = в€’
and a
a

19.3 Consider the quantity

E= d3 x
S

4
See, for example, L. H. Ryder, Quantum Field Theory, Cambridge University Press, 1985.
550 A variational approach to general relativity

where is the Hamiltonian density in Exercise 19.2 and the integral extends over
some spacelike hypersurface S for which x0 = constant. Setting x в‰Ў t xi and
using a dot to denote t , show that

dE Л™a+ Л™a+
= Л™a + d3 x
i
a a
dt t
S a i

By integrating the third term in the integrand by parts, show that dE/dt = 0
provided that does not depend explicitly on t.
19.4 Obtain an expression for the Hamiltonian density for the string in Exercise 19.1.
Hence show that the total energy E of the string is given by
l
E= dx
0

and show explicitly that it is a constant of the motion.
19.5 A relative tensor of weight w transforms under a coordinate transformation as

xa xb
в€’w
=J В·В·В· d
aВ·В·В· cВ·В·В·
bВ·В·В· dВ·В·В·
xc x
where J is the Jacobian of the transformation and is given by
xa
J = det
xb
Show that the product of two relative tensors of weights w1 and w2 is a relative
в€љ
tensor of weight w1 + w2 . Show further that в€’g is a relative scalar of weight
w = 1 (called a scalar density).
For a field theory defined by the action S = d4 x show that, if
19.6 depends
on first- and second-order derivatives of the fields, the EulerвЂ“Lagrange equations
take the form

= в€’ + =0
a a a a

a
provided that the variations and their first derivatives vanish on the boundary
. How do the EulerвЂ“Lagrange equations generalise when depends on higher-
order derivatives of the fields? What assumptions are required regarding the value
a
of the variation and its derivatives on R?
19.7 Consider a local field theory for which the action has the form

S= x d4 x
a a
x

=x +
Under an infinitesimal general coordinate transformation x x , the
variation in the action is given by

S= d4 x в€’
a a
x d4 x
a a
x x x
551
Exercises

Adopting the shorthand notation S = x d4 x в€’ x d4 x, show that

S= x+ x d4 x = x+ d4 x
x x x

x= x в€’ x and x= x в€’ x.
where
19.8 Suppose that the action in Exercise 19.7 is invariant under the given coordinate
transformation, so that S = 0. Since the range of integration can be chosen
arbitrarily, show by writing

= +
a a
a a

or otherwise, that

в€’ + + =0
a a
a a a

Hence show that the invariance of the action under the given coordinate transfor-
mation implies that j = 0, where

j= в€’ в€’
a a
a a

x = a x в€’ a x . This result is known as NoetherвЂ™s theorem.
a
in which
19.9 Use your answer to Exercise 19.8 to show that invariance of the action under the
infinitesimal translation x = x + implies that S = 0, where

= в€’
a
S a

which is known as the canonical energyвЂ“momentum tensor of the fields a . Is
S necessarily symmetric in and ?
19.10 For the field theory considered in Exercise 19.7, use the fact that does not
depend explicitly on the coordinates x to write

= +
a a
a a

a
By multiplying the EulerвЂ“Lagrange equations by and summing over a, use
the above result to show directly that S = 0, where S is the canonical
energyвЂ“momentum tensor given in Exercise 19.9.
19.11 Consider the вЂ˜modifiedвЂ™ energyвЂ“momentum tensor

=S +

where S is the canonical energyвЂ“momentum tensor given in Exercise 19.9 and
= 0 and
is any tensor that is antisymmetric in and . Show that
that one can always arrange for to be symmetric in and .
552 A variational approach to general relativity

19.12 Consider a local field theory defined on Minkowski spacetime in an arbitrary
coordinate system x with metric g . The action has the form
в€љ
S= в€’g d4 x
a a
L x xg x

where the fields a are independent of the metric g and L is a scalar under
general coordinate transformations. Use the fact that L does not depend explicitly
on x to write
L L
L= +
a a
a a

a
By multiplying the appropriate form of the EulerвЂ“Lagrange equations by ,
summing over a and noting that covariant derivatives commute in Minkowski
S = 0, where the covariant
spacetime, use the above result to show that
canonical energyвЂ“momentum tensor S is given by
L
= в€’g L
a
S a

19.13 Consider the вЂ˜modifiedвЂ™ energyвЂ“momentum tensor

=S +

where S is the canonical energyвЂ“momentum tensor given in Exercise 19.12 and
is any tensor that is antisymmetric in and . Show that, in a flat spacetime,
= 0 and that one can always arrange for to be symmetric in and .
19.14 In a four-dimensional spacetime, use the divergence theorem to show that
в€љ
в€љ
в€’gv d4 x = в€’ d3 y
nv

where v is an arbitrary vector field, is the determinant of the induced metric
on the boundary in the coordinates yi and n is a unit normal to the boundary.
в€љ
19.15 Consider complex scalar field = 1 + i 2 / 2, where i i = 1 2 are real
scalar fields with potentials of the form V = 2 m2 2 . Show that the Lagrangian for
1
i
may be written as
в€— в€—
L=g в€’m
в€—
where the asterisk denotes the complex conjugate. By varying and indepen-
dently, show that
в€— в€—
+ m2 = 0 + m2 =0
2 2
and

where 2 в‰Ў =g is the covariant dвЂ™Alembertian operator.
19.16 In the theory of electromagnetism in arbitrary coordinates, the field tensor is
defined by F = A в€’ A . Show directly that

= Aв€’ A
F
553
Exercises

and that
F+ + = F+F +
F F F

Hence show that F automatically satisfies the relation

F+ + =0
F F

= Aв€’
19.17 If F A , show that

F+ + =2 R +R +R
F F A

where R is the Riemann tensor. Hence use the cyclic identity (??) to show
that the above expression is zero.
19.18 An alternative Lagrangian for electromagnetism is given by
1 1
L= в€’ Aв€’ в€’j A
FF F A
4 2
0 0

where F and A are considered as independent quantities (i.e. no functional
relationship between them is assumed). By varying the corresponding action with
respect to F and A independently, show that the EulerвЂ“Lagrange equations
yield
= = Aв€’
and
F 0j F A

19.19 The Lagrangian for a free massive vector field A of mass m is

L = в€’4g g Aв€’ Aв€’ в€’ 2 m2 A A
1 1
A A

Show that the field equation for A is given by

Aв€’ A + m2 A = 0

By making use of the fact that covariant derivatives commute in Minkowski
spacetime, show that in this case A = 0 and hence that the field equation can
be written
A + m2 A = 0
2

where 2 в‰Ў =g is the covariant dвЂ™Alembertian operator. These are
called the second-order Proca equations.
19.20 An alternative Lagrangian for a free massive vector field A of mass m, is

L = 4F F в€’ 2F Aв€’ в€’ 2 m2 A A
1 1 1
A

where F and A are considered as independent quantities. By varying the
corresponding action with respect to F and A independently, show that the
EulerвЂ“Lagrange equations yield

+ m2 A = 0 = Aв€’
and
F F A

which are called the first-order Proca equations.
554 A variational approach to general relativity

19.21 The simplest scalar action for gravity in vacuo that one can construct from the
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