ñòð. 22 |

xâ†’ 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

â†’ = +

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

= âˆ’ = âˆ’ =

a a a a

(19.11)

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

âˆ’ d4 x

a

a

where we have integrated by parts (which corresponds simply to rewriting the

integrand using Leibnitzâ€™ 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

Sâ‰¡ d4 x = âˆ’ 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

= âˆ’ =0 (19.13)

a a a

These are the Eulerâ€“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

= âˆ’ + =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 Eulerâ€“Lagrange equations

19.4 Alternative form of the Eulerâ€“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

âˆš

components then the presence of the âˆ’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

âˆš

S= âˆ’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

â†’ = +

a a a

(19.16)

In a similar way to before, we note that the -operator commutes with covariant

a= a . The variations (19.9â€“19.16) lead, in

derivatives, so that

turn, to a variation in the action S â†’ S + S, with

âˆš âˆš

L L

S= L âˆ’g d4 x = + âˆ’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 Leibnitzâ€™

theorem for the covariant differentiation of a product, this term may be written

âˆš âˆš

L L

âˆ’g d4 x = âˆ’g d4 x

a a

a a

aâˆš

L

âˆ’ âˆ’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

reads

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

aâˆš aâˆš

L L L

Sâ‰¡ âˆ’g d4 x = âˆ’ âˆ’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 Eulerâ€“Lagrange equations

L L L

= âˆ’ =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:

â†’ Â¯= + 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

Â¯

S =S+ Q d4 x

a

The variation in this action under the variation in the fields (19.9) is given by

Q

Â¯

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

Â¯

, this surface integral is zero and so S = S. Hence demanding

a vanish on

Â¯

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

Â¯= + 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

Â¯

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

Â¯

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

âˆš

Â¯

S =S+ Q âˆ’g d4 x (19.23)

and its variation is given by

âˆš âˆš

Q

Â¯

S = S+ âˆ’g d4 x = S + âˆ’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

Â¯

, we find that S = S, and so we have obtained the same EL

a vanish on

Â¯

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

Â¯

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 âˆ’V

1

(19.24)

where the first term may be loosely regarded as the â€˜kinetic energyâ€™ of the

field and the second term as its â€˜potential energyâ€™. 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

âˆš

S= âˆ’V âˆ’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

=âˆ’ = 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

âˆ’ âˆ’ =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 â‰¡ =g is the covariant dâ€™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 Kleinâ€“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

= Aâˆ’ A= Aâˆ’ (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 âˆ’1/ 4 0 for later convenience, we shall take the â€˜free-fieldâ€™

part of the Lagrangian to be

1

Lf = âˆ’ Aâˆ’ Aâˆ’

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 â€˜interaction termâ€™ 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 = âˆ’j A . Taking the full

Lagrangian to be L = Lf + Li , the action reads

âˆš

1

S= âˆ’ Aâˆ’ Aâˆ’ âˆ’j A âˆ’g d4 x

gg A A

4 0

(19.29)

As is immediately apparent, however, the interaction term âˆ’j A is not auto-

matically gauge invariant. Under the gauge transformation (19.27) the correspond-

ing term in the action becomes

âˆš âˆš

âˆ’ j A +j âˆ’g d4 x = âˆ’ jA+ âˆ’ âˆ’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 Eulerâ€“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

âˆ’ =0 (19.30)

A A

For the action in (19.29), we have immediately

L

= âˆ’j = âˆ’j (19.31)

A

but evaluation of the second term on the left-hand side of (19.30) requires more

A âˆ’ A = F for conve-

care. Relabelling dummy indices, and writing

nience, we have

1

L

= âˆ’ gg FF

4

A A 0

1

=âˆ’ âˆ’ +F âˆ’

gg F

4 0

1 1

=âˆ’ âˆ’g g âˆ’ âˆ’g g

g g F gg F

4 4

0 0

1 1 1

=âˆ’ âˆ’F âˆ’ âˆ’F =âˆ’

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

(19.30) read

=

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 Einsteinâ€“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 Einsteinâ€“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 Einsteinâ€“Hilbert action

âˆš

SEH = R âˆ’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

âˆš

density EH = R âˆ’g. The resulting EL equations thus take the form (19.13),

which in this case reads

âˆ’ + =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

â†’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

= âˆ’g g

g g

Writing the Ricci scalar as R = g R , the first-order variation in the Einsteinâ€“

Hilbert action (19.32) can be written as

âˆš âˆš âˆš

SEH = âˆ’g d4 x + âˆ’g d4 x + âˆ’g d4 x

gR g R gR

â‰¡ 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

= âˆ’ + âˆ’

R

Let us first consider the variation in the curvature tensor resulting from an arbitrary

variation in the connection coefficients,

â†’ +

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

= âˆ’

R

Moreover, partial derivatives and covariant derivatives coincide at P and so

= âˆ’ (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 Einsteinâ€“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

= âˆ’ (19.36)

R

We may therefore write the second term on the right-hand side of (19.34) as

âˆš

S2 = âˆ’ âˆ’g d4 x

g

âˆš

= âˆ’g âˆ’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

âˆš

âˆ’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 = âˆ’gg

g g

where in the second equality we have used the result (19.33). Thus, we have

âˆš 1âˆš

âˆ’g = âˆ’ 2 âˆ’g âˆ’1/2 g = âˆ’ 2 âˆ’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 Einsteinâ€“Hilbert action

may be written as

âˆš

SEH = âˆ’ 2g R âˆ’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 Einsteinâ€™s field equations in vacuo:

â‰¡R âˆ’ 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 Einsteinâ€™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 Einsteinâ€“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.

âˆš

The Lagrangian density EH = âˆ’gR in the Einsteinâ€“Hilbert action (19.32)

may be written as

âˆš

= âˆ’gg R

EH

âˆš

= âˆ’gg âˆ’ + âˆ’

âˆš

âˆ’Â¯

= âˆ’gg âˆ’ (19.40)

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

Â¯ â‰¡ âˆšâˆ’g g âˆ’ (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

Leibnitzâ€™ rule for the differentiation of products, we can write the first term in

(19.40) as

âˆš âˆš âˆš

âˆ’gg âˆ’ = âˆ’gg âˆ’ âˆ’gg

âˆš âˆš

âˆ’ âˆ’gg + âˆ’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

âˆš âˆš

= 2 âˆ’g âˆ’1/2 g

âˆ’gg g + âˆ’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

âˆš âˆš

âˆ’gg = âˆ’g âˆ’ âˆ’

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

âˆš âˆš âˆš

+2 Â¯

âˆ’gg âˆ’ = âˆ’gg âˆ’ âˆ’gg

Thus, we finally discover that the Einsteinâ€“Hilbert Lagrangian density (19.40)

can be written

âˆš âˆš

Â¯+

EH = âˆ’gg âˆ’ âˆ’gg (19.44)

where Â¯ 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 Â¯ 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

âˆš

Â¯

S= âˆ’ âˆ’g d4 x (19.45)

g

will thus lead to the same field equations as did the Einsteinâ€“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 Einsteinâ€™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

Â¯

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 Einsteinâ€“Hilbert Lagrangian density

âˆš âˆš

EH = âˆ’gg R = âˆ’gg âˆ’ + âˆ’

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

âˆš

SEH = âˆ’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

âˆš

SEH = âˆ’gg d4 x

R

âˆš

= âˆ’gg âˆ’ d4 x (19.46)

where in the second line we have used the contracted version (19.36) of the

Palatini equation. Using Leibnitzâ€™ theorem for the differentiation of products and

relabelling some dummy indices, we may write (19.46) as

âˆš

SEH = âˆ’g âˆ’g d4 x

g

âˆš

âˆ’ âˆ’ âˆ’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

âˆš

SEH = âˆ’ âˆ’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 âˆ’ =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 Einsteinâ€“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 + âˆ’

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 Einsteinâ€“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

â€˜matterâ€™ 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

âˆš

EH

= âˆ’g G

g

where G = R âˆ’ 2 g R is the Einstein tensor. Thus, if we make the bold

1

assertion that the energyâ€“momentum tensor of the non-gravitational fields (or

â€˜matterâ€™) is given by

2 M

T =âˆš (19.50)

âˆ’g g

then we recover the full Einstein equations

=âˆ’ T

G

The definition (19.50) of the â€˜matterâ€™ energyâ€“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 energyâ€“momentum tensor.

19.12 The dynamical energyâ€“momentum tensor

The quantities T defined in (19.50) are clearly the components of a tensor, which

is known more properly as the dynamical energyâ€“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

âˆš

M

SM â‰¡ d4 x = âˆ’g d4 x

1

g T g

2

g

âˆš

= âˆ’2 âˆ’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 energyâ€“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 energyâ€“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= âˆ’ âˆ’ (19.53)

g g x xg x

x x

=g x âˆ’g x âˆ’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 â‰¡g x âˆ’g x=g x âˆ’g x âˆ’g x âˆ’g

g x

=g x âˆ’g xâˆ’ x g x

=g x âˆ’g xâˆ’ x g x

to first order in . Using the expression (19.54) and dropping the explicit

dependence on x, we find that

= âˆ’g âˆ’g âˆ’ =âˆ’ +

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

âˆš

SM = T âˆ’g d4 x = 0

Using Leibnitzâ€™ theorem for the covariant differentiation of a product, we write

âˆš âˆš

SM = âˆ’g d4 x âˆ’ âˆ’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 energyâ€“momentum tensor vanishes, as

required. Thus, we see that the general covariance of the matter action implies

energyâ€“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 energyâ€“momentum tensor, we may calculate the

explicit form of this tensor for some specific â€˜matterâ€™ 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

âˆš

S= âˆ’g

1

g

2

âˆš

+ âˆ’V âˆ’g d4 x

1

2g

âˆš

= âˆ’ 2g âˆ’V âˆ’g d4 x

1 1 1

2g g

2

âˆš

âˆ’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 energyâ€“

momentum tensor for a real scalar field is given by

= âˆ’g âˆ’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 energyâ€“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

âˆš

1

SEM = âˆ’ âˆ’g d4 x

ggFF

4 0

where F = A âˆ’ A and so does not depend on the metric. Varying this

action with respect to the (inverse) metric, we have

âˆš âˆš

1

SEM = âˆ’ âˆ’g + F F âˆ’g d4 x

gg FF

4 0

âˆš

1

=âˆ’ âˆ’ 2g F F âˆ’g d4 x

1

2g F F g

4 0

where in the second equality we have substituted the expression (19.37) for

âˆš

âˆ’g and relabelled some dummy indices. Comparing the above expression with

549

Exercises

(19.51), we find that the energyâ€“momentum tensor for the electromagnetic field

is given by

âˆ’1

=âˆ’ âˆ’ 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

energyâ€“momentum tensor, which is based on Noetherâ€™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 â†’ 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 energyâ€“

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

â‰¡

Ë™a

a

where Ë™ a â‰¡ and x0 is a timelike coordinate. The Hamiltonian density is then

a

0

defined as

Ë™aâˆ’

â‰¡ a

Use the Eulerâ€“Lagrange equations to show that

Ë™a= Ë™a = âˆ’

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 â‰¡ t xi and

using a dot to denote t , show that

dE Ë™a+ Ë™a+

= Ë™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

âˆ’w

=J Â·Â·Â· d

aÂ·Â·Â· cÂ·Â·Â·

bÂ·Â·Â· dÂ·Â·Â·

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

âˆš

tensor of weight w1 + w2 . Show further that âˆ’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 Eulerâ€“Lagrange equations

take the form

= âˆ’ + =0

a a a a

a

provided that the variations and their first derivatives vanish on the boundary

. How do the Eulerâ€“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 âˆ’

a a

x d4 x

a a

x x x

551

Exercises

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

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

x x x

x= x âˆ’ x and x= x âˆ’ 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

âˆ’ + + =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= âˆ’ âˆ’

a a

a a

x = a x âˆ’ a x . This result is known as Noetherâ€™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

= âˆ’

a

S a

which is known as the canonical energyâ€“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 Eulerâ€“Lagrange equations by and summing over a, use

the above result to show directly that S = 0, where S is the canonical

energyâ€“momentum tensor given in Exercise 19.9.

19.11 Consider the â€˜modifiedâ€™ energyâ€“momentum tensor

=S +

where S is the canonical energyâ€“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

âˆš

S= âˆ’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 Eulerâ€“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 energyâ€“momentum tensor S is given by

L

= âˆ’g L

a

S a

19.13 Consider the â€˜modifiedâ€™ energyâ€“momentum tensor

=S +

where S is the canonical energyâ€“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

âˆš

âˆš

âˆ’gv d4 x = âˆ’ 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.

âˆš

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

âˆ— âˆ—

L=g âˆ’m

âˆ—

where the asterisk denotes the complex conjugate. By varying and indepen-

dently, show that

âˆ— âˆ—

+ m2 = 0 + m2 =0

2 2

and

where 2 â‰¡ =g is the covariant dâ€™Alembertian operator.

19.16 In the theory of electromagnetism in arbitrary coordinates, the field tensor is

defined by F = A âˆ’ A . Show directly that

= Aâˆ’ A

F

553

Exercises

and that

F+ + = F+F +

F F F

Hence show that F automatically satisfies the relation

F+ + =0

F F

= Aâˆ’

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= âˆ’ Aâˆ’ âˆ’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 Eulerâ€“Lagrange equations

yield

= = Aâˆ’

and

F 0j F A

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

L = âˆ’4g g Aâˆ’ Aâˆ’ âˆ’ 2 m2 A A

1 1

A A

Show that the field equation for A is given by

Aâˆ’ 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 â‰¡ =g is the covariant dâ€™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 âˆ’ 2F Aâˆ’ âˆ’ 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

Eulerâ€“Lagrange equations yield

+ m2 A = 0 = Aâˆ’

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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