≈√ (16.57)

™0

k

2k3 k=RH

This is a famous and important result in inflationary theory; it gives the (constant)

value of the amplitude of the plane-wave curvature perturbation having comoving

wavenumber k for modes whose length scale exceeds the horizon.

456 Inflationary cosmology

16.17 Power spectrum of curvature perturbations

From the result (16.57), we can deduce an expression for the power spectrum

k of the primordial curvature perturbations. The precise definition of this

spectrum is a matter of convention, of which there are several, but we will adopt

the most commonly used. In this case, the power spectrum of a given spatially

varying field is defined as the contribution to the total variance of the field per unit

logarithmic interval in k. Thus, we define the curvature spectrum k such that

—

≡ k d ln k (16.58)

x x

0

where · · · denotes a expectation value and the total spatial variation of the

curvature perturbations is

1

x= exp ik · x d3 k (16.59)

k

3/2

2

In these expressions, x refers to comoving coordinates and k = k . Evaluating

x — x , and remembering that d3 k = 4 k2 dk, one finds that (16.58) is

satisfied providing that

22

—

=3 k’k

3

k

kk k

where 3 k ’ k is the three-dimensional delta function. We may therefore write

k = k3 k 2 / 2 2 and, using (16.57), we finally obtain8

2

H2

k= (16.60)

2 ™0 k=RH

In the slow-roll approximation, we know that H is only slowly decreasing

whilst ™ 0 is approximately constant. To a first approximation, therefore, the

power spectrum of the perturbations expected from inflation, as measured by

the contribution to the total fluctuation per unit logarithmic interval, is constant.

Such a spectrum is called scale invariant and was proposed in the late 1960s as

being the most likely to lead to structure appropriately distributed over the scales

we see today. It is also known as a Harrison“Zel™dovich spectrum, after its two

co-proposers. Here we can see it emerging as a prediction of inflation. We can

go further, however, by noting that, during inflation, H is slowly declining, ™ 0

is approximately constant and R is increasing exponentially. Thus modes with

k is often written using the alternative notation 2 k . In addition, it is common

8

We note that the quantity

to define the quantity P k ≡ k 2 , which is also often called the power spectrum and is related to k

k =k P k / 2

3 2

by .

457

16.17 Power spectrum of curvature perturbations

higher k, which leave the horizon later in time, have a slightly lower value of

k since H is lower there. As a result, the spectrum is predicted to be not

exactly constant, but slightly declining, as a function of k. The details of this

depend on the details of the potential V 0 , but we can see from the analysis

here that this is a generic prediction of inflation (assuming that slow-roll is an

accurate model).

Before going on to discuss in the next section the comparison of the prediction

(16.60) with cosmological observations, it is worthwhile re-deriving this result

in a more heuristic (and perhaps enlightening) manner. For a scalar field in an

ordinary Minkowski spacetime, the zero-point uncertainty fluctuation is given by

1 e’ikp t

≈ 1/2 (16.61)

kp

V 2kp

for a mode with physical wavenumber kp , where V is a normalising volume. Here,

instead of kp , we wish to use the comoving wavenumber k, which is related to

the physical wavenumber by k = Rkp . Moreover, an obvious length scale for the

normalising volume is the scale factor R. Thus, in our expanding FRW spacetime,

we assume that

e’ikt/R

k≈ √ (16.62)

R 2k

As explained above, the corresponding power spectrum of the fluctuations k

is obtained by multiplying its squared norm by 4 k3 / 2 3 , which gives

2 2

k H

k= = (16.63)

2R 2

k=RH

As above, we have evaluated the second expression at the ˜horizon crossing™

value of k, RH, since fluctuations on larger length scales are ˜frozen in™ at the

value they reached at this point. To translate this result into the power spectrum

of curvature perturbations , we need to link and . Consider the change t

in time coordinate that would be needed to move between the ˜comoving slicing™,

in which vanishes, to a ˜flat slicing™, in which vanishes. Since , as defined

in (16.39), remains constant in this process, we see that, in this case,

=’™0 t

= =’ =H t =

and (16.64)

/ ™ 0 and hence we recover the result

=H

Eliminating t we find that

2

H2

k= k= (16.65)

2 ™0 k=RH

458 Inflationary cosmology

16.18 Power spectrum of matter-density perturbations

As discussed in Section 16.6, at the end of inflation the scalar field decays

into other particles. Thus, one is left with a spectrum of curvature perturbations,

which from (16.40) is equivalent to the spectrum of fluctuations in the gravita-

tional potential in a comoving slicing. These in turn may be related to the

corresponding fluctuations in the matter density. The full general-relativistic

equations describing the evolution under gravity of these density fluctuations may

be obtained by repeating all the above discussion for a perfect fluid rather than

a scalar field. We will not pursue this calculation here but merely note that the

resulting equations are the same as those obtained using Newtonian theory, except

for a term that is important only on super-horizon scales. Therefore, on sub-

horizon scales, to a good approximation we may take these potential fluctuations

as obeying the perturbed Poisson™s equation in Newtonian gravity,

=4 G

2

where is the fluctuation in the matter density corresponding to that in the

potential. Indeed, we might have expected the Newtonian theory to be a good

approximation on sub-horizon scales since the gravitational field associated with

the perturbations is weak.

It is more common to work instead in terms of the fractional-density fluctuation

≡ / 0 , where 0 is the background matter density. Thus, working in Fourier

space, we have

4 G 0 R2

k=’ k

k2

Using 0 = 3H 2 / 8 G , for the simple spatially flat case that we are considering

we see that

2

2 k

=’ (16.66)

k k

3 RH

2 ∝ k4 2 . Therefore, defining the matter

from which we deduce that k k

power spectrum by P k ≡ k 2 (note that this differs from definition of k

by a factor k3 / 2 2 , as mentioned earlier), we find that P k ∝ k k . Since

k is roughly constant for slow-roll inflation, we thus obtain

P k ∝k (16.67)

In general, the matter power spectrum is parameterised as P k ∝ kn , where n is

known as the primordial spectral index. We therefore see that inflation naturally

predicts n = 1, which is also known as the Harrison“Zel™dovich spectrum.

459

16.19 Comparison of theory and observation

An alternative way of characterising this spectrum is to note from (16.66) that

if we do not define the perturbation spectrum at a single instant of cosmic time

but evaluate it when a given scale re-enters the horizon (k = RH) then

∝

k k

Since the spectrum k , defined as the contribution to the total variance per

unit logarithm interval of k at a single instant of time, is roughly constant then

so too is the matter power spectrum defined in the same way but evaluated at

horizon entry. This is why the Harrison“Zel™dovich spectrum is also known as the

scale-invariant spectrum. The fractional-density perturbations, as they enter the

horizon, make a constant contribution to the total variance per unit logarithmic

interval of k.

Finally, we note from (16.66) that, at a given k, the time evolution of the

fractional-density perturbation k is given by

1

k∝ 2

RH

For a radiation-dominated model we have R ∝ t1/2 and H = 1/ 2t , whereas for

a matter-dominated model R ∝ t2/3 and H = 2/ 3t . Thus, we find

(radiation-dominated)

t

t∝

k

t2/3 (matter-dominated)

which provides a quick derivation of the time dependence of what is known as

the growing mode of the matter-density perturbations. In particular, we note that

the time dependence of this mode is the same as that of the scale factor R in the

matter-dominated case.

16.19 Comparison of theory and observation

The details of the comparison of the inflationary prediction for the perturbation

spectrum with cosmological observations would take us too far afield here. We

thus content ourselves with two brief illustrations. Figure 16.6 shows the prediction

for the power spectrum of anisotropies in the cosmic microwave background

radiation, assuming an early-universe perturbation spectrum that is exactly scale

invariant. The anisotropies in the temperature of the CMB radiation provide a

˜snapshot™ of the (projected) density perturbations at the epoch of recombination

460 Inflationary cosmology

7000

6000

5000

l(l + 1)Cl / (2π) (µK2)

4000

3000

2000

1000

0

0 500 1000 1500 2000 2500

l

Figure 16.6 The predicted power spectrum of CMB temperature anisotropies

(solid line), assuming an early-universe perturbation spectrum that is exactly

scale invariant. The points show the results of recent observations of the CMB

anisotropies by the Wilkinson Microwave Anisotropy Probe (WMAP, circles),

Very Small Array (VSA, squares) and Arcminute Cosmology Bolometer Array

Receiver (ACBAR, triangles) experiments. The vertical error bars indicate the

68 per cent uncertainty in the measured value.

(zrec ≈ 1500). The CMB anisotropies over the sky are usually decomposed in

terms of spherical harmonics as

=

T a mY m

=2 m=’

where the = 0 (constant) and = 1 (dipole) terms are usually ignored, since

the former is unrelated to the anisotropies and the latter is due to the peculiar

velocity of the Earth with respect to the comoving frame of the CMB. The power

in the fluctuations as a function of angular scale is therefore characterised by the

spectrum

1

C= 2

a m

2 + 1 m=’

The characteristic peaks in the predicted CMB power spectrum (solid line) are

a consequence of another feature of inflation that we have already seen in our

461

16.19 Comparison of theory and observation

equations, namely that all modes outside the horizon are frozen and can only start

to oscillate once they re-enter the horizon later in the universe™s evolution. This

means that a ˜phasing up™ is able to occur, in which all modes of interest start

from effectively a ˜zero velocity™ state when they begin the oscillations, during

the epoch of recombination, that lead to the CMB imprints. This is what enables

peaks to be visible in the power spectrum, with modes on different scales able

to complete a different number of oscillations before the end of recombination.

Coherence, leading to peaks, is maintained since each mode has the same starting

conditions. This is only possible if the modes of interest are indeed on super-

horizon scales prior to recombination, and the only known way of achieving

this is via inflation. Thus the peaks visible in the predictions of Figure 16.6

are a powerful means of testing for inflation. The points shown in the figure

are the results of recent observations of the CMB anisotropies by the WMAP

(circles), VSA (squares) and ACBAR (triangles) experiments, which yield a very

impressive confirmation of the peak structure and thereby a direct confirmation

that inflation occurred.

100 000

10 000

Pδ (k)

1000

100

0.01 0.1 1

“1

k (h Mpc)

Figure 16.7 The predicted power spectrum of matter fluctuations (solid line)

assuming an early-universe perturbation spectrum that is exactly scale invariant.

The points show the results derived from the 2dF sample of galaxy redshift

measurments. The horizontal error bars indicate the width of the bin in k-space

over which the measurement is made and the vertical error bars indicate the

68 per cent uncertainty in the measured value.

462 Inflationary cosmology

From the current data it is, however, not possible to tell whether the primordial

spectrum is exactly scale invariant, as assumed in generating the prediction, or

whether it has the slight decrease at larger k, and therefore smaller scales, that we

said was also a generic prediction of inflation. This question should be resolved by

future experimental results, particularly from the CMB on smaller angular scales

and from measurements of the matter distribution on a range of scales. In the

latter case, one may compare the observed distribution of galaxies (both over the

sky and in redshift) with the predicted power spectrum for matter fluctuations in

the universe. The primordial matter power spectrum P k in (16.67) is modified

by the evolution under gravity of the perturbations once they re-enter the horizon.

This effect may be calculated, and the predicted matter power spectrum resulting

from an exactly scale-invariant primordial spectrum from inflation is shown as

the solid line in Figure 16.7. Once again, we see that the predicted spectrum

has oscillations resulting from an mechanism analogous to that which produces

the oscillations in the CMB power spectrum discussed above. The points in the

figure show the measurements derived from the 2dF (2 degree field) sample of

galaxy redshift measurments. Again, a good fit to the data is visible, and time

will tell whether the detailed dynamics of inflation, which can be measured by the

departures from scale invariance, will become accessible from the combination

of data of this type and future CMB experiments.

Exercises

16.1 In the cosmological field equation

™

R2 = R2 ’ k

1

3

show that, if p < ’ 1 , the curvature term becomes negligible as the universe

3

expands.

16.2 Show that the energy“momentum tensor of a scalar field,

= ’g ’V

1

T 2

satisfies the condition T = 0.

16.3 Show that a scalar field acts like a perfect fluid with an energy density and pressure

given by

™ 2 +V

= +2 2

1 1

2

™ 2 ’V

p= ’1 2

1

2 6

Show further that if the field is spatially constant then inflation will occur,

provided that ™ < V . If, in addition, the scalar field does not change with time,

2

show that its equation of state is p = ’ and that it thus acts as an effective

cosmological constant.

463

Exercises

16.4 Show that the equation of motion for a scalar field with potential V is

¨ + 3H ™ + dV = 0

d

, for which V = 2 m2

1 2

Hence find the general solution for a free scalar field , in

the case where H is approximately constant.

16.5 For a potential of the form

= V0 exp ’

V

where is a positive constant, show that the inflation equations can be solved

exactly to give

2

2/

2

t V0

R t = R0 t= 0+

2

ln t

2 6’ 2

t0

√

Hence show that, provided < 2, the solution corresponds to a period of inflation.

Show further that the slow-roll parameters for this model are = 2 = 2 2 , and so

1 1

the inflationary epoch never ends. This model is known as power-law inflation.

16.6 Show that, in general,

¨ ™

R = R H + H2

™

Show that H > 0 only if p < ’ , which is forbidden by the weak energy condition

(see Exercise 8.8). Hence show that, for inflation to occur, one requires

™

H

’ <1

H2

and thus that the first slow-roll parameter must obey < 1.

16.7 In the slow-roll approximation, show that

H = ’2 ™ 2

™ 1

Assuming that ™ varies monotonically with t throughout the period of inflation,

show that

™ = ’2H

where H is now considered as a function of , and hence that we may write the

cosmological field equation as

’ 2 H2 = ’2V

2 3 1

H

This is known as the Hamilton“Jacobi formalism for inflation.

16.8 Repeat Exercise 16.5 using the Hamilton“Jacobi formalism developed in Exer-

cise 16.7.

464 Inflationary cosmology

16.9 In the Hamilton“Jacobi formalism developed in Exercise 16.7, show that the

condition for inflation to occur is

2

H

2 <1

H

16.10 Show that, during an exponential expansion phase of the universe, the proper

distance between any two comoving objects separated by more than H ’1 grows

at a speed exceeding the speed of light. Hence show that an observer in such a

universe can only see processes occurring inside the ˜horizon™ radius H ’1 , and so

the process of inflation in any spatial domain of radius H ’1 (or ˜mini-universe™)

occurs independently of any events outside it.

16.11 A fluctuation in the scalar inflation field leads to a local delay of the end of

inflation by t ∼ / ™ . Assuming that the density of the universe after inflation

decreases as t’2 , show that the fluctuation in the scalar field leads to a relative

density contrast at the end of inflation given by

H

∼

™

∼

Assuming the root mean square (rms) scalar field perturbation to be rms

H/ 2 , show that

H2

∼

2™

rms

16.12 Consider an inflationary domain (or mini-universe in the context of Exercise 16.10)

of initial radius H ’1 , in which the value of the scalar field 1. In a time interval

’1

t = H , show that classically, in the slow-roll approximation, the value of

will change by

2

≈’

Assuming that the typical amplitude of quantum fluctuations in the scalar field is

≈ H/ 2 , show that

1 V

≈

2 3

= 2 m2 2 , show that the decrease in the value of the scalar

1

Hence, for the case V

field due to its classical motion is less than changes due to quantum fluctuations

generated in the same time interval, provided that

6

√

m

is H ’1 ,

Assuming that the typical wavelength of the quantum fluctuation is

show that, after a time interval t = H ’1 , the original domain becomes effectively

divided into e3 ∼ 20 domains of radius H ’1 , each containing a roughly homoge-

neous scalar field + + . Thus, on average, the volume of the universe

465

Exercises

containing a growing -field increases by a factor ∼ 10 after every time interval

t = H ’1 .

Note: This is the mechanism underlying stochastic inflation.

16.13 For the line element

ds2 = 1 + 2 dt2 ’ 1 ’ 2 R2 t dx2 + dy2 + dz2

show that, to first order in , the perturbed parts of the connection coefficients

take the form

=

0

0

= ’R2 ™ + 4H

0

ii

1

=

i

00 i

R2

=’

i

i

where no sum over repeated i indices is implied and and the remaining perturbed

coefficients either follow from symmetry or are zero. Hence show that the

perturbed part of the Einstein tensor is given by

™ +H

G0 = ’2 i

i

’ 3H ™ ’ 3H 2

G0 = ’2 2

0

Gii = 2 ¨ + 4 ™ H + 2H + 3H 2

™

where again no sum over repeated i indices is implied and the remaining entries

either follow from symmetry or are zero.

16.14 For the scalar-field perturbation

t’ t+ tx

0

show that, to first order in , the perturbed parts of the scalar-field energy“

momentum tensor are given by

Ti0 = ™ 0 i

T0 = ’ ™ 2 + ™ 0 ™ + V

0

0

Tii = ™ 2 ’ ™ 0 ™ + V

0

where V = dV/d 0 and the remaining components either follow from symmetry

or are zero.

16.15 Use your answers to Exercises 16.13 and 16.14 to show that the perturbed Einstein

field equations yield only the two equations

1™

™ +H =

20

d

™ 2 +2 = ™2

2

™0

0 0

dt

466 Inflationary cosmology

16.16 Show that the gauge-invariant Fourier curvature perturbation

≡ k +H

k

™0

k

satisfies the equation of motion

™2 2¨0 2

™k + k

¨k + + + 3H =0

0

™0 k

R2

H

= R ™ 0 /H, show

Defining the new variables d = c dt/R and = k, where

k

further that

+ k2 ’ =0

k

k

where a prime denotes d/d and the ˜effective mass™ m2 = ’ / is given by

2 4

2

m= ’ ’ ’

0 0 0 0 0

2

2R2 H 2

2 RH 0

...

™2 ™4 ™0 ¨0 3¨0

= ’2R2 H 2 1 + + + + + 0

0 0

2H ™ 0 2H 2 ™ 0

2H 2 4H 4 H3

16.17 Consider the equation of motion

2

+ k2 ’ =0

end ’

k

k 2

Show that the unique solution (up to a phase factor) that tends to (16.51) for small

is given by

1 i + k end ’

e’ik

k=√

end ’

2k3

17

Linearised general relativity

The gravitational field equations give a quantitative description of how the curva-

ture of spacetime at any event is related to the energy“momentum distribution at

that event. The high degree of non-linearity in these field equations means that

a general solution for an arbitrary matter distribution is analytically intractable.

Consequently, thus far we have concentrated primarily on investigating a number

of special solutions that represent spacetimes with particular symmetries (aside

from our discussion of perturbations in the previous chapter). In this chapter,

we return to a more general investigation of the gravitational field equations and

their solutions. To enable such a study, however, one must make the physical

assumption that the gravitational fields are weak. Mathematically, this assumption

corresponds to linearising the gravitational field equations.

17.1 The weak-field metric

As discussed in Sections 7.6 and 8.6, a weak gravitational field corresponds to a

region of spacetime that is only ˜slightly™ curved. Thus, throughout such a region,

there exist coordinate systems x in which the spacetime metric takes the form

= +h where h 1 (17.1)

g

and the first and higher partial derivatives of h are also small.1 Such coordinates

are often termed quasi-Minkowskian coordinates, since they allow the metric

to be written in a close-to-Minkowski form. Clearly, h must be symmetric

with respect to the swapping of its indices. We also note that, when previously

1

We note that one could equally well consider small perturbations about some other background metric, such

0

that g = g + h . This was the case in our discussion of inflationary perturbations in the previous chapter,

0

in which g was the metric for the background Friedmann“Robertson“Walker spacetime in comoving

Cartesian coordinates.

467

468 Linearised general relativity

considering the weak-field limit, we further assumed that the metric was stationary,

so that 0 g = 0 h = 0 where x0 is the timelike coordinate. In our present

discussion, however, we wish to retain the possibility of describing time-varying

weak gravitational fields, and so we shall not make this additional assumption here.

As we have stressed many times, coordinates are arbitrary and, in principle,

one could develop the description of weak gravitational fields in any coordinate

system. Nevertheless, by adopting quasi-Minkowsian coordinates the mathemati-

cal labour of pursuing our analysis is greatly simplified, as is the interpretation of

the resulting expressions. If one coordinate system exists in which (17.1) holds,

however, then there must be many such coordinate systems. Indeed, two differ-

ent types of coordinate transformation connect quasi-Minkowskian systems to

each other: global Lorentz transformations and infinitesimal general coordinate

transformations, both of which we now discuss.

Global Lorentz transformations

Global Lorentz transformations are of the form

= =

where

x x

and the quantities are constant everywhere. These transform the metric

coefficients as follows:

xx

g= g= +h = + h

x x

Thus, g is also of the form (17.1), with

=

h h

Moreover, we see from this expression that, under a Lorentz transformation, h

itself transforms like the components of a tensor in Minkowski spacetime.

The above property suggests a convenient alternative viewpoint when describ-

ing weak gravitational fields. Instead of considering a slightly curved spacetime

representing the general-relativistic weak field, we can consider h simply as a

symmetric rank-2 tensor field defined on the flat Minkowski background space-

time in Cartesian inertial coordinates. In other words, h is considered as a

special-relativistic gravitational field, in an analogous way to that in which the

4-potential A describes the electromagnetic field in Minkowski spacetime, as

discussed in Chapter 6; we return to this point below. We note, however, that

h does not transform as a tensor under a general coordinate transformation but

only under the restricted class of global Lorentz transformations; for this reason

h and tensors derived from it are sometimes called pseudotensors, although we

will not use this terminology.

469

17.1 The weak-field metric

Infinitesimal general coordinate transformations

Infinitesimal general coordinate transformations take the form

=x + (17.2)

x x

where the x are four arbitrary functions of position of the same order of

smallness as the h . Infinitesimal transformations of this sort make tiny changes

in the forms of all scalar, vector and tensor fields, but these can be ignored in

all quantities except the metric, where tiny deviations from contain all the

information about gravity. From (17.2), we have

x

= +

x

and, working to first order in small quantities, it is straightforward to show that

the inverse transformation is given by2

x

= ’ (17.3)

x

Thus, again working to first order in small quantities, the metric transforms as

follows:

x x

= = ’ ’ +h

g g

x x

= +h ’ ’

=

where we have defined . Hence, we see that g is also of the form

(17.1), the new metric perturbation functions being related to the old ones via

=h ’ ’ (17.4)

h

If we adopt the viewpoint in which h is considered as a tensor field defined

on the flat Minkowski background spacetime, then (17.4) can be considered

as analogous to a gauge transformation in electromagnetism. As discussed in

Chapter 6, if A is a solution of the electromagnetic field equations then another

solution that describes precisely the same physical situation is given by

A new = A +

where is any scalar field. An analogous situation holds in the case of the

gravitational field. From (17.4), it is clear that if h is a solution to the linearised

2

Note that, for the remainder of this chapter, the normal symbol for equality will be used to indicate equality

up to first order in small quantities as well as exact equality.

470 Linearised general relativity

gravitational field equations (see below) then the same physical situation is also

described by

h new = h ’ ’ (17.5)

In this interpretation, however, (17.5) is viewed as a gauge transformation rather

than a coordinate transformation. In other words, we are still working in the same

new

set of coordinates x and have defined a new tensor h whose components in

this basis are given by (17.5).

Now that we have considered the coordinate transformations that preserve the

form of the metric g in (17.1), it is useful to obtain the corresponding form for

the contravariant metric coefficients g . By demanding that g g = , it is

straightforward to verify that, to first order in small quantities, we must have

= ’h

g

where h = h . Moreover, it follows that indices on small quantities

may be respectively raised and lowered using and rather than g and

g . For example, to first order in small quantities, we may write

h =g = ’h =

h h h

17.2 The linearised gravitational field equations

In the weak-field approximation to general relativity, one expands the gravitational

field equations in powers of h , using a coordinate system where (17.1) holds. On

keeping only the linear terms, we thus arrive at the linearised version of general

relativity. The Einstein gravitational field equations were derived in Section 8.4

and read

’ 2g R = ’ T

1

R

To obtain the linearised form of these equations, we thus need to find the linearised

expression for the Riemann tensor R ; the corresponding expressions for the

Ricci tensor R and the Ricci scalar R then follow by the contraction of indices.

To perform this task, we first need the linearised form of the connection

coefficients . To first order in small quantities we have

= + h’ = h+ h’

1 1

(17.6)

h h h

2 2

≡

where we have defined . We may now substitute (17.6) directly into

the expression (7.13) for the Riemann tensor, namely

= ’ + ’ (17.7)

R

471

17.2 The linearised gravitational field equations

The last two terms on the right-hand side are products of connection coefficients

and so will clearly be second order in h ; they will therefore be ignored. Hence,

to first order, we obtain

= h+ h’ ’2 h+ h’

1 1

R h h

2

= h+ ’ ’

1

h h h

2

which is easily shown to be invariant to a gauge transformation of the form (17.5).

The linearised Ricci tensor is obtained by contracting the above expression for

on its first and last indices. This yields

R

= h+ ’ h’

2

1

(17.8)

R h h

2

where we have defined the trace h ≡ h and the d™Alembertian operator ≡

2

. The Ricci scalar is obtained by a further contraction, giving

R=R = = h’

2

(17.9)

R h

Substituting the expressions (17.8) and (17.9) into the gravitational field equa-

tions we obtain the linearised form

h+ ’ h’ h’ h’ = ’2 T

2 2

(17.10)

h h

The number of terms on the left-hand side of the field equations has clearly

increased in the linearisation process. This can be simplified somewhat by defining

the ˜trace reverse™ of h , which is given by

¯ ≡h ’2

1

h h

¯

On contracting indices we find that h = ’h. It is also straightforward to show

¯

¯ ¯ ¯

that h = h , i.e. h = h ’ 2 1

h. On substituting these expressions into

(17.10), the field equations become

2¯ ¯ ¯ ¯

+ ’ h’ h = ’2 T (17.11)

h h

These are the basic field equations of linearised general relativity and are valid

whenever the metric takes the form (17.1). Unless otherwise stated, for the remain-

der of this chapter we will adopt the viewpoint that h is simply a symmetric

tensor field (under global Lorentz transformations) defined in quasi-Cartesian

coordinates on a flat Minkowski background spacetime.

472 Linearised general relativity

17.3 Linearised gravity in the Lorenz gauge

The field equations (17.11) can be simplified further by making use of the gauge

transformation (17.5). Denoting the gauge-transformed field by h for conve-

nience, the components of its trace-reverse transform as

¯ =h ’2

1

h h

=h ’ ’ ’2 h’2

1

¯

=h ’ ’ + (17.12)

and hence we find that

¯ ¯

= ’ 2

h h

Therefore, if we choose the functions x so that they satisfy

¯

=

2

h

¯

then we have h = 0. The importance of this result is that, in this new gauge,

each of the last three terms on the left-hand side of (17.11) vanishes. Thus, the

field equations in the new gauge become

2¯

= ’2 T

h

Let us take stock of the simplification we have just achieved. Dropping primes

and raising indices for convenience, we have found that the linearised field

equations may be written in the simplified form

2¯

= ’2 T (17.13)

h

¯

provided that the h satisfy the gauge condition

¯ =0 (17.14)

h

Moreover, we note that this gauge condition is preserved by any further gauge

satisfy 2 = 0.

transformation of the form (17.5) provided that the functions

The above simplification is entirely analogous to that introduced in electro-

magnetism in Chapter 6. In that case, the electromagnetic field equations were

reduced to the simple form 2 A = 0 j by adoption of the Lorenz gauge condi-

tion A = 0. This condition is preserved by any further gauge transformation

A ’A + if and only if 2 = 0. As a result of the similarities between

the electromagnetic and gravitational cases, (17.14) is often also referred to as the

Lorenz gauge.

473

17.4 General properties of the linearised field equations

17.4 General properties of the linearised field equations

Now that we have arrived at the form of the field equations for linearised general

relativity, it is instructive to consider some general consequences of our lineari-

sation process for the resulting physical theory. The non-linearity of the original

Einstein equations is a direct result of the fact that ˜gravity gravitates™. In other

words, any form of energy“momentum acts as a source for the gravitational field,

including the energy“momentum associated with the gravitational field itself. By

linearising the field equations we have ignored this effect.

One may straightforwardly take steps to address this shortcoming by ˜bootstrap-

ping™ the theory as follows: (i) the energy“momentum carried by the linearised

gravitational field h is calculated; (ii) this energy“momentum acts as a source for

1

corrections h to the field; (iii) the energy“momentum carried by the corrections

1

h is calculated ; (iv) this energy“momentum acts as a source for corrections

2 1

h to the corrections h ; and so on. It is widely stated in the literature3 that,

on completing this bootstrapping process, one arrives back at the original non-

linear field equations of general relativity, although this claim has recently been

brought into question.4 In either case, it is worth noting that this approach allows

the resulting equations to be interpreted simply as a (fully self-consistent) rela-

tivistic theory of gravity in a fixed Minkowski spacetime. This viewpoint brings

gravitation closer in spirit to the field theories describing the other fundamental

forces. Indeed, the remarkable point is that only the field theory of gravitation

has the elegant geometrical interpretation that we have spent so long exploring.

Returning to the linearised theory, one result of ignoring the energy“momentum

carried by the gravitational field is an inconsistency between the linearised field

equations (17.11) and the equations of motion for matter in a gravitational field.

Raising the indices and on (17.11) and operating on both sides of the resulting

equation with , one quickly finds that

=0 (17.15)

T

This should be contrasted with the requirement, derived from the full non-linear

field equations, that T = 0. As was shown in Section 8.8, the latter require-

ment leads directly to the geodesic equation of motion for the worldline x of

a test particle, namely

x+

¨ x x =0

™™ (17.16)

where the dots denote differentiation with respect to the proper time . Performing

a similar calculation for the condition (17.15), however, leads to the equation

3

See, for example, R. P. Feynman, F. B. Morinigo & W. G. Wagner, Feynman Lectures on Gravitation,

Addison“Wesley, 1995.

4

See T. Padmanabhan, From Gravitons to Gravity: Myths and Reality, http://arxiv.org/abs/gr-qc/0409089.

474 Linearised general relativity

of motion x = 0, which means that the gravitational field has no effect on the

¨

motion of the particle. In general, this clearly contradicts the geodesic postulate.

An alternative way to uncover this inconsistency is to note that an immediate

consequence of having linearised the field equations is that solutions can be added.

In other words, if the pairs of tensors h i and T i individually satisfy (17.11)

for i = 1 2 then the quantity i h i is also a solution, corresponding to

the energy“momentum tensor i T i . Thus, for example, two point masses

could remain at a fixed separation from one another indefinitely, the resulting

gravitational field being simply the superposition of their individual radial fields.

Despite this inconsistency, linearised general relativity is still a useful approx-

imation, provided that we are interested only in the far field of sources whose

motion we know a priori and that we are willing to neglect the ˜gravity of grav-

ity™. In such cases, the effect of weak gravitational fields on test particles can

be computed by inserting the form (17.6) for the connection coefficients into the

geodesic equations (17.16). To calculate how the sources themselves move under

their own gravity, however, one would need to re-insert into the field equations

the non-linear terms that the linear theory discards.

17.5 Solution of the linearised field equations in vacuo

In empty space, the linearised field equations in the Lorenz gauge reduce to the

wave equation

2¯

=0 (17.17)

h

with the attendant gauge condition

¯ =0 (17.18)

h

It is straightforward to show that the field equations have plane-wave solutions

of the form

¯

h = A exp ik x (17.19)

where the A are constant (and, in general, complex) components of a symmetric

tensor, and the k are the constant (real) components of a vector. Substituting

the expression (17.19) into the wave equation (17.17) and using the fact that

¯ ¯

h = k h , we find that

2¯ ¯ ¯

= = =0

h h kkh

This can only be satisfied if

k k =k k =0 (17.20)

475

17.6 General solution of the linearised field equations

and hence the vector k must be null. Since the linearised Einstein equations only

take the simple form (17.17) in the Lorenz gauge, we must also take into account

the gauge condition (17.18). On substituting into the latter the plane-wave form

(17.19), we immediately find that the gauge condition is satisfied provided that

one obeys the additional constraint

A k =0 (17.21)

Thus any plane wave of the form (17.19) is a valid solution of the linearised

vacuum field equations in the Lorenz gauge, provided that the vector k satisfies

(17.20) and (17.21). We will discuss plane gravitational waves in detail in the

next chapter.

Since the vacuum field equations are linear (by design), any solution of them

may be written as a superposition of such plane-wave solutions of the form

¯ x= k exp ik x d3 k (17.22)

h A

where k = k0 k and the integral is taken over all values of k. Physical

solutions are obtained by taking the real part of (17.22).

17.6 General solution of the linearised field equations

We now consider the general form of the solution to the linearised field equations

in the presence of some non-zero energy“momentum tensor T . In this case,

the field equations take the form of an inhomogeneous wave equation for each

component,

2¯

h = ’2 T (17.23)

¯

together with the attendant gauge condition h = 0. The general solution to

(17.23) is most easily obtained by using a Green™s function, in a similar manner

to that employed for solving the analogous problem in electromagnetism. We will

now outline this approach.

One begins by considering the solution to the inhomogeneous wave equation

when the source is a -function, i.e. it is located at a definite event in spacetime.

If this event has coordinates y , one is therefore interested in solving an equation

of the form

x ’y = x ’y

2 4

(17.24)

xG

where the subscript on 2 makes explicit that the d™Alembertian operator is with

x

respect to the coordinates x and G x ’ y is the Green™s function for our

problem, which in the absence of boundaries must be a function only of the

difference x ’ y . Since the field equations (17.23) are linear, sources that are

476 Linearised general relativity

more general can be built up by adding further -function sources located at

different events. Thus, the general solution to the linearised field equations can

be written5

¯ ¯

=h0 x ’2 G x ’y T d4 y (17.25)

h x y

where, for completeness, we have made use of the freedom to add any solution

¯

h 0 x of the homogeneous field equations (i.e. the in vacuo field equations). It

may be verified immediately by direct substitution that (17.25) does indeed solve

¯

(17.23). For the discussions in this chapter, however, we will take h 0 x = 0

without loss of generality.

The problem of obtaining a general solution of the linearised field equations has

thus been reduced to solving (17.24) to obtain the appropriate Green™s function.

This may be achieved in a number of ways, and here we shall take a physically

motivated approach. For convenience, we begin by placing the -function source

at the origin of our coordinate system. We will also make the identifications

x = ct x and r = x . With the source at the origin, we may write (17.24) as

= 4

(17.26)

Gx x

We first integrate this equation over a four-dimensional hypervolume V . Since

the spatial spherical symmetry of the problem suggests that the Green™s function

should only depend on ct and r, we choose the hypervolume to be a sphere of

radius r in its spatial dimensions and we integrate in t from ’ to . The

geometry of the bounding surface S of the hypervolume is illustrated by the

vertical cylinder in Figure 17.1, in which the third spatial dimension x3 has been

suppressed. Performing the integration of (17.26) over V we obtain

d4 x = n dS = 1 (17.27)

Gx Gx

V S

where in the first equality we have used the divergence theorem to rewrite the

volume integral as an integral over the bounding surface S with unit normal n .

Since we are working with a metric of signature + ’ ’ ’ , it should be noted

that n is chosen to be outward pointing if it is timelike and inward pointing if

spacelike.

Let us now consider the contributions to this surface integral over S. Since

gravitational field variations travel at speed c, the only points in spacetime that

can be influenced by a -function source at the origin are those lying on the

√

Note that there is no need to include ’g factors in our integral or delta-function definition, since we are

5

¯

considering the problem simply as a tensor field h x defined on a Minkowski spacetime background in a

Cartesian coordinate system.

477

17.6 General solution of the linearised field equations

x0

r

nµ

x2

x1

L

S

Figure 17.1 The geometry of the surface S in spacetime used to evaluate the

Green™s function for the wave equation. The lightcone L emanating from the

origin is also shown. The x3 -direction has been suppressed.

future-pointing part of the lightcone L. Thus G x must be zero at all points in

spacetime except those lying on the future lightcone, and so must be of the form

ct ’ r for ct ≥ 0

fr

= (17.28)

Gx

0 for ct < 0

where f is an arbitrary function of r. The intersection of the future lightcone

with the surface S is a sphere (corresponding to a circle in Figure 17.1) of radius

r lying in the spatial hypersurface ct = r. Thus, the only contribution to the

surface integral in (17.27) is from this sphere (a circle in the figure), for which

the (spacelike) unit normal n points in the inward spatial radial direction (as

illustrated). Rewriting the surface integral using dS = c dt d (where d is an

= ’ r , and performing the integral over the

element of solid angle) and n

spatial sphere, we thus have

Gx

’4 r 2 c dt = 1 (17.29)

r

’

where the only contribution to the integral over t occurs at ct = r. Substituting

(17.28) into (17.29), we find that

df r

ct ’ r c dt ’ 4 r 2 ct ’ r c dt = 1

4 r 2f r (17.30)

dr 0

0

where the prime on the -function denotes differentiation with respect to its

argument. Integration by parts quickly shows the first integral on the left-hand

side of (17.30) to be zero, whereas the second integral equals unity. We therefore

478 Linearised general relativity

require ’4 r 2 df/dr = 1 and so f r = 1/ 4 r , where the constant of integration

vanishes since the Green™s function must tend to zero at spatial infinity. Thus,

re-expressing the result in terms of the coordinates x , the required Green™s

function is

x0 ’ x

= x0

Gx

4x

where the Heaviside function x0 equals unity if x0 ≥ 0 and zero if x0 < 0.

¯

We may now use this form to substitute for G x ’ y in (17.25), with h 0

set to zero, to obtain

x0 ’ y 0 ’ x ’ y

¯ =’ x0 ’ y 0 T d4 y

h x y

x’y

2

Using the delta function to perform the integral over y0 , we finally find that the

general solution to the linearised field equations (17.23) is given by

ct ’ x ’ y y 3

4G T

¯ ct x = ’ (17.31)

h dy

x’y

c4

The interpretation of (17.31) requires some words of explanation. Here x repre-

¯

sents the spatial coordinates of the field point at which h is determined, y

represents the spatial coordinates of a point in the source and x ’ y is the spatial

distance between them. We see that the disturbance in the gravitational field at the

event ct x is the integral over the region of spacetime occupied by the points

of the source at the retarded times tr given by

ctr = ct ’ x ’ y

This region is the intersection of the past lightcone of the field point with the

world tube of the source. An illustration of the geometric meaning of the retarded

time is shown in Figure 17.2.

Although we have shown that (17.31) satisfies the linearised field equations

(17.23), this form of the field equations is only valid in the Lorenz gauge. We must

¯

therefore verify that (17.31) also satisfies the Lorenz gauge condition h = 0.

Before embarking on this we first remind ourselves how to differentiate a function

of retarded time. Setting xr ≡ ctr = x0 ’ x ’ y , for any function f we have

0

0

f y0 y 0

f xr y xr

= (17.32)

y0

x x

r

0

f y0 y f y0 y 0

f xr y xr

= + (17.33)

y0

yi yi yi

r r

479

17.6 General solution of the linearised field equations

ct

(ct, x i )

(ct, y i )

x2

(ctr, yi)

x1

Figure 17.2 The disturbance in the gravitational field at the event ct xi is the

sum of the influences of the energy and momentum sources at the points ctr yi

on the past lightcone.

where r denotes that the expression contained within the brackets is eval-

uated at y0 = xr and where i = 1 2 3. In addition to (17.33), we note that

0

f xr y / y0 = 0.

0

Let us now verify that the solution (17.31) does indeed satisfy the Lorenz gauge

condition. Differentiating, we obtain

¯ T 0 xr y

0 0

T i xr y

x0 x 4G 1

h

=’ 4 +i d3 y

x’y x’y

x0

x c x

0

T xr y

4G 1 1

=’ 4 + T i xr y

0

d3 y

x’y x’y

xi

c x

(17.34)

where we show explicitly that the partial derivatives are with respect to the

coordinates x . Using (17.32), the derivative in the first term of the integrand can

be rewritten as follows:

0

y0 y 0 T 0 y0 y T i y0 y 0

T xr y T xr xr

= = ’

y0 y0 y0 yi

x x

r r r

where in the second equality we have used the fact that xr / xi = ’ xr / yi .

0 0

Returning to (17.34), in a similar manner we may replace / xi by ’ / yi in

the second term of the integrand, which then allows this term to be integrated by

parts, since

0 0

T i xr y T i xr y 3

1 1

d y= ni dS ’

0 3

i

T xr y dy

x’y x’y x’y

yi yi

S

480 Linearised general relativity

where S is the surface of the region of intersection between the past lightcone of

the field point and the world tube of the source and ni is the outward-pointing

0

normal to the surface. Moreover, since T i xr y vanishes on S, the surface

integral is zero.

Combining our results, we may therefore write (17.34) as

¯ T 0 y0 y 0

T i xr y

T i y0 y 0 d3 y

4G

h xr

=’ 4 ’ +

x’y

y0 y0 yi yi

x c r

r

Making use of the result (17.33) to combine the last two terms within the braces,

we thus arrive at the final form

¯ y0 y

4G 1

h T

=’ 4 d3 y (17.35)

x’y

x c y r

As shown in Section 17.4, however, in the linearised theory the energy“momentum

tensor obeys T = 0. Thus the integrand in (17.35) vanishes, and so we have

¯

verified that the solution (17.31) satisfies the Lorenz gauge condition h = 0.

17.7 Multipole expansion of the general solution

In general, the source of the gravitational field may be dynamic and have a spatial

extent that is not small compared with the distance to the point at which one

wishes to calculate the field. In such cases, obtaining a simple expression for the

solution (17.31) is often analytically intractable. In an analogous manner to that

used in electromagnetism, it is often convenient to perform a multipole expansion

of (17.31), which lends itself to the calculation of successive approximations to

the solution. One begins by writing down the Taylor expansion

1 1 1 1 1

= + ’yi + ’yi ’yj +···