where not all the hij are constant (in any gauge) and i j = 1 2 3. Thus we see

that the passing of a gravitational wave will indeed cause the physical separation

of nearby particles to vary. It is convenient at this point to introduce the quantities

= + 2 hi

1

i i k

(18.12)

k

One then finds straightforwardly that, in terms of these new variables (to first

order in h ),

l2 = ij

ij

which is again valid in any gauge. Thus, the i may be regarded as the components

of a position vector giving the correct physical spatial separation when contracted

with the Euclidean metric tensor ij .

Let us now discuss the particular case of a plane gravitational wave propagating

in the x3 -direction and consider a set of particles initially at rest in the x1 x2 -

plane, i.e. the plane perpendicular to the direction of wave propagation. Thus,

the coordinate separation vector between any two particles has 3 = 0. In the

TT gauge, however, we see from (18.7) that hTT 3 = 0, and so (18.12) implies

k

that 3 = 0 throughout the passage of the wave. Hence the particles remain in

the plane perpendicular to the wave propagation direction; it is only the physical

separations in the transverse directions that vary. Thus the gravitational wave is

transverse not only in its mathematical description hTT but also in its physical

effects.

We first consider the effect of the passage of a gravitational wave with A =

ae1 (i.e. a single polarisation), where we take a to be real and positive for

convenience, and e1 was introduced at the end of Section 18.1. Remembering

¯

that hTT = hTT , we thus have

hTT = ae1 cos k x = ae1 cos k x0 ’ x3

where k = /c, and using (18.12) we quickly find that

= 0 ’ 2 a cos k x0 ’ x3 ’

1 2 1 2

1

i

0

Thus, for two particles initially separated in the x1 -direction 1 = 0 the physical

separation in the x1 -direction will oscillate, and likewise for two particles with an

initial x2 separation. Let us consider a set of particles that, when cos k x0 ’x3 = 0,

form a circle in the x1 x2 -plane with a reference particle at the centre, with

respect to which we refer to the other particles, using the i -vector components.

Then, as the wave passes, the particles remain coplanar and at other times have

spatial separations as illustrated in Figure 18.1.

506 Gravitational waves

ζ2

ζ1

Figure 18.1 The solid dots show the effect of a plane gravitational wave with

A = ae1 on a transverse circle of particles. The initial configuration of parti-

cles is shown by the open dots. From left to right, k x0 ’ x3 is equal to

2n + 2 2n + 1 2n + 2

1 3

2n respectively.

ζ2

ζ1

Figure 18.2 The solid dots show the effect of a plane gravitational wave with

A = be2 on a transverse circle of particles. The initial configuration of

particles is shown by the open dots. From left to right, k x0 ’ x3 is equal to

2n + 2 2n + 1 2n + 2

1 3

2n respectively.

We may straightforwardly repeat our analysis for a gravitational wave with the

other polarisation, i.e. A = be2 again with real and positive b. In this case one

finds that

= 0 ’ 2 b cos k x0 ’ x3

1 2 2 1

1

i

0

and this results in our initial circle of particles having spatial separations as

illustrated in Figure 18.2, which may be obtained from Figure 18.1 by a 45

rotation.

Having determined the relative displacements of test particles induced by the

two separate polarisations of a plane gravitational wave, it is straightforward to

find the effect in the general case in which A = ae1 + be1 , where a and b

507

18.5 The generation of gravitational waves

ζ2

ζ1

Figure 18.3 The solid dots show the effect of a plane gravitational wave with

A = a e1 + ie2 (i.e. right-handed circular polarisation) on a transverse circle

of particles. The initial configuration of particles is shown by the open dots.

From left to right, k x0 ’ x3 is equal to 2n 2n + 2 2n + 1 2n + 2

1 3

respectively.

may, in general, be complex. Of particular interest are the left- and right-handed

circularly polarised modes, for which b = ’ia and b = ia respectively. The effect

of, for example, a right-handed circularly polarised wave would be to distort our

initial circle of particles into an ellipse and to rotate the ellipse in a right-handed

sense, as illustrated in Figure 18.3. Note that the individual particles do not move

around the ring but instead execute small circular ˜epicycles™.

18.5 The generation of gravitational waves

Let us suppose that we have a matter distribution (the source) localised near the

origin O of our coordinate system that we and take our field point x to be a

distance r from O that is large compared with the spatial extent of the source. We

may therefore use the compact-source approximation discussed in Section 17.8.

Without loss of generality, we may take our spatial coordinates xi to correspond

to the ˜centre-of-momentum™ frame of the source particles, in which case from

(17.38) we have

4GM

¯ ¯ ¯

h00 = ’ 2 hi0 = h0i = 0 (18.13)

cr

The remaining (spatial) components of the gravitational field are given by

the integrated stress within the source, which may be written in terms of the

quadrupole formula (17.44) as

2 ij

¯ ij ct x = ’ 2G d I ct (18.14)

h

dt 2

c6 r r

508 Gravitational waves

In this expression r denotes that the expression in the brackets is evaluated at

the retarded time ct = ct ’ r, and the quadrupole-moment tensor of the source is

I ij ct = T 00 ct y yi yj d3 y (18.15)

Thus, we see that, in the compact-source approximation, the far field of the source

falls into two parts: a steady field (18.13) from the total constant ˜mass™ M of the

source and a possibly varying field (18.14) arising from the integrated internal

stresses of the source. It is clearly the latter that will be responsible for any emitted

gravitational radiation.

For slowly moving source particles we have T 00 ≈ c2 , where is the proper

density of the source, and so the integral (18.15) may be written as

I ij ct = c2 ct x xi xj d3 x (18.16)

Thus, the gravitational wave produced by an isolated non-relativistic source is

proportional to the second derivative of the quadrupole moment of the matter-

density distribution. By contrast, the leading contribution to electromagnetic radi-

ation is the first derivative of the dipole moment of the charge density distribution.

This fundamental difference between the two theories may be easily understood

from elementary considerations. Using to denote either the proper mass density

or the proper charge density, the volume integral dV over the source is constant

in time for both electromagnetism and linearised gravitation and so generates no

xi dV , i.e. the dipole moment. For

radiation. Now consider the next moment

electromagnetism, this gives the position of the centre of charge of the source,

which can move with time and hence have a non-zero time derivative; this

provides the dominant contribution in the generation of electromagnetic radiation.

xi dV gives the centre of mass of the source and,

For gravitation, however,

for an isolated system, conservation of momentum means that it cannot change

with time and so cannot contribute to the generation of gravitational waves. Thus,

it is the generally much smaller quadrupole moment, which measures the shape

of the source, that is dominant in generating gravitational waves. This fact, and

the weak coupling of gravitation to matter, means that gravitational radiation

is much weaker than electromagnetic radiation. As a corollary, we note that a

spherically symmetric system has a zero quadrupole moment and thus cannot emit

gravitational radiation.

As an illustration of the generation of gravitational waves, let us consider two

particles A and B of equal mass M moving (non-relativistically) in circular orbits

of radius a about their common centre of mass with an angular speed (see

Figure 18.4). This might represent a simple model of a binary star system, in

509

18.5 The generation of gravitational waves

field

x1 point

A

x

x3

O

x2

B

Figure 18.4 Two particles A, B of equal mass M rotating at angular speed

in circular orbits of radius a about their common centre of mass.

which mutual gravitational attraction keeps the particles (stars) in orbit. In this

case, treating the motion in the Newtonian limit, we require that

1/2

GM

= (18.17)

4a3

Alternatively, in a more terrestrial setting, one might imagine the particles to

be connected by a light rod of length 2a that is spun with constant angular

velocity about its centre point, in which case need not be related to M and a.

For simplicity, we shall assume the particle orbits to lie in the plane x3 = 0, as

illustrated in Figure 18.4.

At any time t, the coordinates of particles A and B may be written

xA = a cos t a sin t 0 xB = ’ a cos t a sin t 0

i i

Thus, the proper density of the source is given by

ct x = M x1 ’ a cos t x2 ’ a sin t

+ x1 + a cos t x2 + a sin t x3

On substituting into (18.16) and making use of the standard trigonometric identi-

ties 2 cos2 t = 1+cos 2 t 2 sin2 t = 1’cos 2 t and 2 sin t cos t = sin 2 t,

one quickly finds the quadrupole-moment tensor,

⎛ ⎞

1 + cos 2 t sin 2 t 0

2 2⎜ ⎟

I ct = Mc a ⎝ sin 2 t 1 ’ cos 2 t 0⎠

ij

(18.18)

0 0 0

510 Gravitational waves

Inserting this expression into the quadrupole formula (18.14) and performing the

necessary differentiations, we finally obtain

⎛ ⎞

cos 2 t ’ r/c sin 2 t ’ r/c 0

8GMa2 2 ⎜ ⎟

¯

h ct x = ⎝ sin 2 t ’ r/c ’ cos 2 t ’ r/c 0⎠

ij

c4 r

0 0 0

We note that, for the physical arrangement illustrated in Figure 18.4, the coordi-

nates xi already correspond to the centre-of-momentum frame of the source and

¯

so the remaining components of h are given by (18.13).

¯

In fact, one is often only interested in the radiative part hrad of the gravitational

field (i.e. the part corresponding only to gravitational radiation). In general, the

¯0 ¯ ij

remaining components of hrad may be found from the spatial components hrad

using the Lorenz gauge condition. For the two-particle system discussed above,

¯0

we see from (18.13) that all the remaining components hrad are zero, and so

⎛ ⎞

0 0 0 0

8GMa2 2 ⎜0 cos 2 t ’ r/c 0⎟

sin 2 t ’ r/c

⎜ ⎟

¯

hrad ct x = ⎜ ⎟

⎝0 sin 2 t ’ r/c ’ cos 2 t ’ r/c 0⎠

4r

c

0 0 0 0

(18.19)

Since the amplitude goes as 1/r, the gravitational perturbation has the form of

a spherical wave rather than a plane wave. Nevertheless, for large r the wave

is well approximated by a plane wave in a small range of angles about any

particular direction. We also note that the angular frequency of the wave is twice

the rotational angular frequency of the two particles.

It is of interest to determine the polarisation of the gravitational waves received

by observers located in different directions relative to the orbiting particles. To

do this, one must transform to the TT gauge appropriate to each observer. Let us

first consider an observer located on the x3 -axis (at some large distance from O).

By comparing with (18.7), we see that (18.19) is already in transverse-traceless

¯

form for a wave travelling in the x3 -direction. Remembering that hTT = hTT and

using the fact that r = x3 , it is straightforward to show that

8GMa2 2

= e1 ’ ie2 t ’ x3 /c

hTT exp 2i (18.20)

rad

c4 r

where e1 and e2 are the linear polarisation tensors introduced at the end of

Section 18.1. Since the amplitude tensor has the form A = ae1 + be2 with

b = ’ia, this corresponds to right-handed circularly polarised radiation, as one

might expect.

511

18.6 Energy flow in gravitational waves

Let us now consider an observer located on the x1 -axis. The form (18.19) is

not in the transverse-traceless gauge for a wave travelling in the x1 -direction. To

transform to the TT gauge, we follow the prescription outlined in Section 18.3.

We first set to zero all non- transverse components, i.e. all entries except those

with i j = 2 2 , (2, 3), (3, 2) and (3, 3). We then subtract one-half of the

resulting trace from the remaining diagonal elements (2, 2) and (3, 3) to make the

¯

final tensor traceless. Remembering that r = x1 in this case, and that hTT = hTT ,

we obtain

⎛ ⎞

0 0 0 0

2⎜ ⎟

4GMa2 ⎜0 ⎟

0 0 0

=

hTT ⎜ ⎟

ct x

⎝0 ⎠

rad

’ cos 2 t ’ r/c

c4 r 0 0

cos 2 t ’ r/c

0 0 0

4GMa2 2

= ’˜ 1 exp 2i t ’ x1 /c (18.21)

e

c4 r

˜

where e1 is a linear polarisation tensor analogous to those used above, but for

propagation in the x1 - direction. Thus, the gravitational waves received by the

observer are linearly polarised in the ˜+™ orientation illustrated in Figure 18.1 “

again as one might have expected.

18.6 Energy flow in gravitational waves

Physically, one would expect gravitational waves to carry energy away from a

radiating source. As discussed in Section 17.11, however, the task of assigning

an energy density to a gravitational field is notoriously difficult. Nevertheless,

bearing in mind the caveats made in Section 17.11, from (17.64) an appropriate

expression for the energy“momentum tensor of the gravitational field in vacuo is

c4 ¯ ¯ ¯ ¯ ¯ ¯

= ’2 ’2

1

t h h h h h h

32 G

where · · · denotes an average over a small region at each point in spacetime.

¯

If we adopt the TT gauge, however, the Lorenz gauge condition hTT = 0 is

¯ ¯

automatically satisfied, and also hTT = 0 and hTT = hTT . Thus in this gauge the

energy“momentum tensor in vacuo reduces to

c4

= hTT

t hTT

32 G

512 Gravitational waves

We will assume further that we are considering only the radiative part of the

gravitational field, in which case we know from the discussion in Section 18.3

0

that hTT = 0, and so

c4 ij

= hTT (18.22)

t hTT

ij

32 G

In particular, from our discussion of energy“momentum tensors in Section 8.1, at

any given time and spatial position the energy flux (i.e. the energy crossing unit

area per unit time) of the gravitational radiation in the unit spatial direction ni is

F n = ’ct0k nk (18.23)

where the minus sign appears as a result of our choice of metric signature, since

then F n = ’c kj t0k nj = kj t0k nj , as required.

As an illustration of these general results, let us calculate the energy flux in the

direction of propagation for a plane gravitational wave of the form

ij ij

hTT = ATT cos k x

ij

where ATT are constants and, for convenience, we have chosen the arbitrary phase

of the wave in such a way that the amplitude matrix is real. Substituting this

= 2 when averaged

expression into (18.22), and using the fact that sin2 k x 1

over several wavelengths, the energy“momentum tensor reads

c4 ij

= k k ATT ATT (18.24)

t ij

64 G

Thus, the flux F in the k-direction is given by

c5 0 l ˆ ij TT c5 0 0 ij TT

0l ˆ

F = ’ct kl = ’ k k kl ATT Aij = k k ATT Aij = ct00 (18.25)

64 G 64 G

ˆ

where in the third equality we have used the fact that k0 = k = ’kl kl , since the

wavevector is null. The final expression is simply the energy density associated

with the plane wave multiplied by its speed, and hence makes good physical sense

as the energy flux carried by the wave in its direction of propagation.

Specialising still further, we may calculate the forms of the expressions (18.24)

and (18.25) explicitly for a wave travelling in the x3 -direction, in which case

513

18.7 Energy loss due to gravitational-wave emission

k = k 0 0 ’k where k = /c and ATT is given by (18.7). Thus, in this case,

the energy“momentum tensor (18.24) can be written as

⎛ ⎞

1 0 0 ’1

⎜000 0⎟

c2 2⎜ ⎟

t= a +b ⎜

22

⎟

⎝000 0⎠

32 G

’1 0 0 1

and the flux in the direction of propagation is

c3

F= a2 + b2

2

(18.26)

32 G

Clearly, similar results hold for a plane gravitational wave travelling along any

of the coordinate axes.

Using the result (18.26) and the expressions (18.20) and (18.21), we find

that, for the two-particle rotating system considered in the previous section, the

gravitational-wave energy flux at a (large) distance r in the x1 - and x3 - directions

respectively is

2 2

c3 4GMa2 3 Ma2 3

2G

F1 = =5

2

2

c4 r

32 G r

c

2 2

c3 8GMa2 3 Ma2 3

16G

F3 = 2 =

2

2

c4 r c5

32 G r

Thus, we see that the energy flux in the x3 -direction is eight times that in the

x1 -direction (or, by symmetry, in any direction in the x3 = 0 plane). Hence the

energy flux due to the gravitational radiation emitted from this system is highly

anisotropic.

18.7 Energy loss due to gravitational-wave emission

Since gravitational waves carry away energy, we expect energy to be lost at a

corresponding rate by the physical system generating the gravitational radiation.

Let us suppose that the source matter distribution is localised near the origin O of

our coordinates. To calculate the rate at which the physical system loses energy,

we equate it to the energy flux of the emitted gravitational radiation evaluated

over a sphere S of large radius r centered on O. Thus, if E is the energy of the

physical system, we have

dE

= ’LGW = ’r 2 (18.27)

F er d

dt 4

514 Gravitational waves

where LGW is the total gravitational-wave luminosity, F er is the gravitational-

wave energy flux at a radius r in the (unit) radial direction er and d is an

element of solid angle.

In general, using (18.22) and (18.23) we may write the gravitational-wave flux

in a unit spatial direction n as

c4 c4

ij ij

F n =’ n =’ n·

TT TT

k

t hij k hTT t hij hTT

32 G 32 G

where we have made the identification x0 ≡ ct and where t ≡ / t. In the second

equality the operator n · returns simply the rate of change of its argument in the

direction n. Thus, taking n to lie in the radial direction and writing r ≡ / r, we

have

c4 ij

F er = ’ TT

(18.28)

t hij r hTT

32 G

To obtain a general formula for (18.27), we must calculate the above energy flux

in terms of properties of the source distribution. From the quadrupole formula

(18.14) we have

2G

¯ ¨

hij = ’ 6 I ij r

cr

where I ij is the quadrupole-moment tensor of the source distribution defined in

(18.16), the dots denote d/dt and r denotes that the expression should be

evaluated at the retarded time ctr = ct ’ r. It will, in fact, be more convenient to

work in terms of the reduced quadrupole-moment tensor of the source distribution,

which is defined by

Jij = Iij ’ 1 (18.29)

ij I

3

j

where I = Ij is the trace of the original tensor. One immediately sees that Jij

is simply the traceless version of Iij . As a result, we may write the transverse-

traceless part of the gravitational field tensor as

2G ij 2G ij

¯ ij ¨ ¨

ij

hTT = hTT = ’ 6 ITT =’ (18.30)

J

c6 r TT

cr r r

ij

where JTT is the transverse-traceless part of (18.29). Since at any point on the

sphere S the direction of gravitational-wave propagation is radial, from (18.10)

we have

ij j

JTT = Pk Pl ’ 2 P ij Pkl J kl

1

i

(18.31)

j

where P ij = ij ’ er er is the spatial projection tensor, which projects tensor

i

components onto the spatial surface orthogonal to the radial direction at any point.

515

18.7 Energy loss due to gravitational-wave emission

Using (18.30) and the expressions (17.32, 17.33) for the derivatives of time-

retarded quantities, the derivatives in the expression (18.28) for the energy flux

are given by

...ij

2G

=’

TT

t hij J TT

c6 r r

2G ...ij 2G ...ij

2G

¨ ij

= 62 JTT + ≈

TT

r hij J TT J TT

c7 r c7 r

cr r r r

where, in the second equation, we have retained only the term in 1/r, which

dominates for large r. Substituting these expressions into (18.28) we obtain

...TT ...ij

G

F er = J ij J TT

8 r 2 c9 r

For convenience, we now use (18.31) to rewrite the product of transverse-traceless

quadrupole moments in terms of products of reduced moments. Denoting the

ˆ

components er of the unit radial vector by xi , this yields

i

ij j

Jij JTT = Jij J ij ’ 2Ji J ik xj xk + 2 J ij J kl xi xj xk xl

ˆˆ ˆˆˆ ˆ

TT 1

where we have made use of the fact that Jij is traceless. Thus, the total

gravitational-wave luminosity is given by

... ...ij ...j ...ik 1 ...ij ...kl

G

LGW = J ij J ’ 2 J i J xj xk + 2 J J xi xj xk xl

ˆˆ ˆˆˆ ˆ d

8 c9 r

4

Since the reduced quadrupole moment Jij is defined as an integral over all space,

it does not depend on the angular coordinates and so may be taken outside the

integral. The three remaining integrals are easily evaluated to give

4

=4 ˆˆ =

d xi xj d ij

3

4 4

4

ˆˆˆ ˆ = ij kl + ik jl + il jk

xi xj xk xl d

15

4

The first result is trivial. The second result may be obtained by noting that

integration over all angles yields zero for i = j, whereas on raising one index and

setting i = j the integrand becomes xi xi = 1 and so the integral equals 4 . Similar

ˆˆ

reasoning leads to the third result. Substituting these three results into (18.27) and

simplifying, one finally obtains

... ...ij

dE G

= ’LGW = ’ 9 (18.32)

J ij J

5c

dt r

516 Gravitational waves

As an illustration, let us apply the general formula (18.32) to the specific exam-

ple of the two-particle rotating system discussed in Section 18.5. The quadrupole-

moment tensor I ij for this system is given in (18.18), from which we quickly find

that the reduced quadrupole-moment tensor (18.29) is given by

⎛ ⎞

3 + cos 2 t

1

sin 2 t 0

2 2⎜ ⎟

J = Mc a ⎝ sin 2 t 3 ’ cos 2 t

ij 1

0⎠

’2

0 0 3

The corresponding third time derivative reads

⎛ ⎞

’ cos 2 t

sin 2 t 0

⎜ ⎟

...ij

J = 8Mc2 a2 3 ⎝’ cos 2 t ’ sin 2 t 0⎠

0 0 0

and so (18.32) becomes

dE G

= ’LGW = ’ 9 8Mc2 a2 3 2 2 sin2 2 t ’ r/c + 2 cos2 2 t ’ r/c

5c

dt

G

= ’ 5 128M 2 a4 6 (18.33)

5c

18.8 Spin-up of binary systems: the binary pulsar PSR B1913 + 16

As discussed in Section 18.5, our simple two-particle rotating system can be used

to model an equal-mass astrophysical binary system, in which case is given by

(18.17). Inserting this expression into (18.33), we find that the total energy E of

the binary system obeys

2 G4 M 5

dE

=’ (18.34)

5 a5

dt

Treating the binary in the Newtonian limit, the total energy is simply

GM 2

1

E = 2Mv ’ 2

2 2a

where v is the orbital speed of either object. Using the radial equation of motion

Mv2 /a = GM 2 / 2a 2 , we may write

GM 2

E=’ = ’Mv2

4a

from which we see that the total energy is negative, since the binary system is

gravitationally bound. Moreover, we note that as E decreases (i.e. becomes more

negative), according to (18.34) the radius a of the orbit must decrease whereas

the orbital speed v must increase. Thus, the emission of gravitational radiation

517

18.9 The detection of gravitational waves

causes the binary system to ˜spin-up™, ending ultimately in the coalescence of the

two objects.

For comparison with observations of binary systems, the most useful way of

characterising the spin-up is by the rate of change of the orbital period P. For our

simple system P = 2 a/v, and so we may write the total energy as

1/3

2 GM 5

P ’2/3

E=’ (18.35)

4

Differentiating this expression with respect to t and inverting, we find that the

rate of change of the orbital period is related to the rate of change of energy by

3P dE

dP

=’ (18.36)

2E dt

dt

Substituting a = ’GM 2 /4E into (18.34) and then substituting for E using (18.35),

we find that (18.36) can be written as follows:

5/3

96 2 GM

dP

= ’ 41/3

5

dt P

This expression gives the rate of change of the orbital period solely in terms

of some constants and P itself, which can be determined straightforwardly from

observations.

The spin-up of a binary system resulting from the emission of gravitational

waves has already been observed in the binary pulsar PSR B1913 + 16. This

system was discovered in 1974 by Hulse and Taylor and consists of a pulsar and

an unseen companion, each with a mass of about 1 4M ; the orbital period is

7.75 hours. The pulsar provides a very accurate clock, so that the change in the

orbital period as the system loses energy can be measured. In practice, our results

above have to be modified slightly to allow for the considerable eccentricity of

the orbit e = 0 617 , but this is relatively straightforward. Timing measurements

made by Taylor and colleagues over several decades show that the decrease in

orbital period as a function of time is in agreement with that predicted from

the emission of gravitational radiation, to within one-third of one per cent. This

constitutes an additional, and highly accurate, experimental verification of general

relativity (albeit in the weak-field regime), for which Hulse and Taylor received

the Nobel Prize in Physics in 1993.

18.9 The detection of gravitational waves

Although the measurement of the spin-up of the binary pulsar PSR B1913 + 16

provides indirect evidence of the existence of gravitational radiation, a major goal

518 Gravitational waves

of modern experimental astrophysics is to make a direct detection of gravitational

waves by measuring their influence on some test bodies.

There are two distinct approaches to gravitational-wave detection, ˜free-particle™

and ˜resonant™ detection. In our discussion in Section 18.4, we found that the

effect of a gravitational wave on a cloud of free test particles is a variation in

their relative separations. Thus one may attempt to detect gravitational waves by

measuring the separations of a set of free test particles as a function of time,

which is the basis of free-particle detection experiments. Alternatively, if the

particles are not free, but are instead the constituent particles of some elastic

body, then tidal forces on the particles induced by a gravitational wave will give

rise to vibrations in the body, which one can attempt to measure. In particular, if

the incident gravitational radiation were in the form of a plane wave of a given

frequency then the amplitude of the induced vibrations would be enhanced if

the elastic body were designed to have a resonant frequency close to that of the

incident wave. This is the basis of resonant detection.

Resonant detectors are the older type of realistic gravitational-wave detector,

having been pioneered by Weber in the early 1960s and refined by him and

others over several decades. We will concentrate our discussion, however, on

free-particle gravitational-wave detectors, which have gained in popularity over

recent years and are also very much easier to analyse. In our discussion of the

motion of free test particles in the presence of a passing gravitational wave, we

showed in Section 18.4 that the relative physical separation l of two free particles

varies as

l2 = ’ hij ij

ij

where i is the separation vector between the two particles. In the absence of a

gravitational wave, the undisturbed distance l0 between the particles is given by

l0 = ij i j . To first order in hij , the fractional change in the physical separation

of the particles is therefore given by

l

= ’ 2 hij ni nj

1

l0

where ni is a unit vector in the direction of separation of the two particles. Thus,

we see that the passing of a gravitational wave produces a linear strain, i.e. the

change in the relative separation of the particles is proportional to their original

undisturbed separation. For typical astrophysical sources, the largest strain one

might reasonably expect to receive at the Earth is of order

l

∼ 10’21

l

519

18.9 The detection of gravitational waves

Thus, even if the two test masses were separated by a distance l0 = 1 km, the

change l in this distance is of order 10’16 cm, which corresponds to ∼10’6 of

the size of the atoms that comprise the test masses!

Fortunately, laser Michelson interferometers provide a means of measuring

such tiny changes in the separation of the test masses. The principle of operation

of such an experiment is quite straightforward and is illustrated in Figure 18.5.

The basic system of made up of three test masses. Two have mirrors M attached to

them, and to the third is attached a beamsplitter B. Each mass is suspended from

a support that isolates the mass from external vibrations but allows it to swing

freely in the horizontal direction. A laser L (with typical wavelength ∼10’4 cm)

is aimed at B, which splits the laser light into two beams directed down the arms

of the interferometer. The beams are reflected by the mirrors at the end of each

arm and then recombined in B before being detected in the detector D. When

the beams are recombined they will interfere constructively if the lengths of the

two arms L1 and L2 differ by an amount L = n and will interfere destruc-

tively if L = n + 2 , where n is an integer. The system is arranged so that

1

the beams interfere destructively if all three masses are perfectly stationary. In

practice, the experimental set-up is more sophisticated than the simple Michelson

M

P

L B P M

D

Figure 18.5 A schematic representation of a laser Michelson interferometer

designed to detect gravitational waves (see the main text for details).

520 Gravitational waves

interferometer we have discussed. The most important improvement is the intro-

duction of an additional test mass with a partially reflecting mirror P in each arm

of the interferometer, thereby forming a ˜cavity™, as illustrated in Figure 18.5.

A typical photon may travel up and down this cavity many times before eventually

arriving at the beamsplitter, thereby greatly increasing the effective arm length of

the interferometer. The use of large laser Michelson interferometers as a means

for attempting to detect gravitational waves is currently being actively pursued

by a number of laboratories around the world.

Exercises

¯

18.1 For a plane gravitational wave of the form h = A exp ik x , show that, under

the gauge transformation (17.5) with = exp ik x , the amplitude tensor trans-

forms as

= A ’i k ’i k +i

A k

18.2 The trace-reverse gravitational-field tensor transforms as

¯ ¯

=h ’ ’ +

h

¯ ¯

Since the components h also satisfy the in vacuo wave equation 2 h = 0, show

that this gauge transformation may be used to set any four linear combinations of

¯

the h to zero.

18.3 The transverse-traceless (TT) gauge is defined by choosing

¯ TT ¯

h0i = 0 hTT = 0

and

Hence show that

¯ 00 ¯ ij

=0 =0

and

0 hTT i hTT

¯ = A exp ik x , show that the four

18.4 For a plane gravitational wave of the form h

conditions in Exercise 18.3 become

A0i = 0 =0 A00 = 0 Aij kj = 0

ATT TT

TT TT

18.5 Show that the spatial projection tensor Pij ≡ ’ ni nj , where ni is a unit vector,

ij

satisfies the relations

ni Pji vj = 0 Pk Pjk vj = Pji vj

i

and

and interpret these relations geometrically.

18.6 The quantities Aij are the spatial components of the amplitude tensor for a plane

gravitational wave with spatial wavevector ki . Consider the tensor

Aij = Pk Plj ’ 2 P ij Pkl Akl

1

i

TT

521

Exercises

ˆˆ

where Pij = ij ’ ki kj . Show that Aij is both transverse, so that Aij kj = 0, and

TT TT

traceless.

18.7 Use your answer to Exercise 18.6 to show that, for a plane gravitational wave

propagating in the x1 -direction,

⎛ ⎞

00 0 0

⎜0 0 ⎟

⎜ ⎟

0 0

ATT = ⎜ ⎟

⎝0 0 2 A22 ’ A33 ⎠

1

A23

A33 ’ A22

1

A23

00 2

18.8 In the TT gauge show that, to first order in h ,

=0 = 1

and hTT

00 0 20

18.9 Consider two nearby particles, initially at rest in our chosen coordinate system x ,

which have a coordinate separation given by a small spacelike connecting vector

= 0 1 2 3 . During the passage of a gravitational wave show that, to

first order in h in the TT gauge, the equation of geodesic deviation may be

written as

D2

= c2 R 00 = 2 c2 0 0 h

1

2

D

Show further that, to the same order of approximation, in the TT gauge one has

D2 d2

= + 2 c2

1

0 0h

2 2

D d

= constant is a solution of the geodesic equation, and so the

Hence show that

coordinate separation of the two particles remains unaltered during the passage of

the gravitational wave.

18.10 If i is the spatial coordinate separation vector of two nearby particles, show that

the square of their physical separation is given by

l2 = ij

ij

where i = i + 2 hik k . Show that, during the passage of a gravitational wave with

1

A = be2 that is travelling in the x3 -direction,

= 0 ’ 2 b cos k x0 ’ x3

1 2 2 1

1

i

0

18.11 For two test particles reacting to the passage of a circularly polarised gravitational

wave, show that one particle moves in a circle with respect to the other.

18.12 For the two-particle system considered in Section 18.5, verify that

⎛ ⎞

0 0 0 0

2 2⎜

0⎟

⎜0 cos 2 t ’ r/c sin 2 t ’ r/c ⎟

¯ rad ct x = 8GMa ⎜ ⎟

h

⎝0 sin 2 t ’ r/c 0⎠

’ cos 2 t ’ r/c

4r

c

0 0 0 0

522 Gravitational waves

and hence show that an observer on the x3 -axis measures a right-handed circularly

polarised gravitational wave of the form

8GMa2 2

= e1 ’ ie2 exp 2i t ’ x3 /c

hTT

rad 4r

c

18.13 Consider a system of four equal masses attached to the ends of a cross formed

from massless rods of equal length, set at 90 . If the system rotates freely about

an axis through the centre of the cross and perpendicular to its plane, show that

in the far field there is no quadrupole gravitational radiation.

Hint: Consider the system as the superposition of two systems, each like that in

Exercise 18.12 but 90 out of phase.

18.14 For a plane gravitational wave of the form

hij = Aij cos k x

TT TT

travelling in the x3 -direction, verify that the energy“momentum tensor of the

linearised gravitational field is given by

⎛ ⎞

1 0 0 ’1

⎜000 0⎟

c2 2⎜ ⎟

t= a +b ⎜ ⎟

2 2

⎝000 0⎠

32 G

’1 0 0 1

and that the flux in the direction of propagation is

c3

F= a2 + b 2

2

32 G

18.15 For the two-particle system considered in Section 18.5, verify that the gravitational-

wave energy flux at a (large) distance r is, in the x1 - and x3 -directions respectively,

2 2

c3 4GMa2 3

Ma2 3

2G

F1 = =

2

2

c4 r c5

32 G r

2 2

c3 8GMa2 3

Ma2 3

16G

F3 = 2 =

2

2

c4 r c5

32 G r

18.16 If JTT = Pk Plj ’ 2 P ij Pkl J kl and P ij = ’ xi xj , show that

ˆˆ

ij 1

i ij

1

Jij JTT = Jij J ij ’ 2Jij J ik xj xk + J ij J kl xi xj xk xl

ˆˆ ˆˆˆ ˆ

TT ij

2

ˆ

18.17 If xi is a unit radial vector, show that

4 4

ˆˆ = ˆˆˆ ˆ = ij kl + ik jl +

xi xj d xi x j xk x l d

ij il jk

3 15

4 4

18.18 For the two-particle system considered in Section 18.5, verify that gravitational-

wave emission causes the the total energy E of the system to decrease according to

dE G

= ’ 5 128M 2 a4 6

5c

dt

523

Exercises

18.19 For a binary star system containing two stars of mass M and separation 2a, show

that the orbital angular speed is

1/2

GM

=

4a3

Hence show that gravitational-wave emission causes the the total energy E of the

system to decrease according to

2 G4 M 5

dE

=’

5 a5

dt

Thus show that the orbital period P decreases according to

5/3

96 2 GM

dP

= ’ 41/3

5

dt P

18.20 Show that, to first order in hij , the fractional change in the physical separation of

the particles during the passage of a gravitational wave is

l

= ’ 2 hij ni nj

1

l0

where ni is a unit vector in the direction of separation of the two particles.

18.21 Consider a line element of the form

ds2 = c2 dt2 ’ dx2 ’ f 2 u dy2 ’ g 2 u dz2

where f u and g u are functions of u = ct ’ x. Calculate the connection coeffi-

cients and hence the Ricci tensor for this line element. Hence show that the line

element is a solution to the full empty-space field equations R = 0, provided that

f g

+ =0

f g

where a prime denotes d/du. Show that this solution may be interpreted, with no

approximation, as a linearly polarised plane gravitational wave travelling in the

x-direction.

19

A variational approach to general relativity

Most of classical and quantum physics can be expressed in terms of variational

principles, and it is often when written in this form that the physical meaning

is most clearly understood. Moreover, once a physical theory has been writ-

ten as a variational principle it is usually straightforward to identify conserved

quantities, or symmetries of the system of interest, that otherwise might have

been found only with considerable effort. Conversely, by demanding that the

variational principle be invariant under some symmetry, one ensures that the

equations of motion derived from it also respect that symmetry. In this final

chapter, we therefore present an introductory account of variational principles

and the Lagrangian formalism. Our ultimate aim will be to derive afresh the field

equations of general relativity from this new perspective. This will require us to

consider some general aspects of classical field theory in flat and curved space-

times. As a result, this chapter lies somewhat outside the mainstream discussion

presented in preceding chapters and may be omitted on a first reading. Never-

theless the variational approach that we shall outline is extremely powerful and

provides the basis for most current research into the formulation of classical (and

quantum) field theories, including general relativity and other candidate theories

of gravitation.

19.1 Hamilton™s principle in Newtonian mechanics

To begin, let us remind ourselves of a familiar example of a physical varia-

tional principle, namely Hamilton™s principle in Newtonian mechanics. Consider

a mechanical system whose configuration can be defined uniquely by a number

of generalised coordinates q a , a = 1 2 n (usually distances and angles),

together with time t, and which experiences only forces derivable from a potential.

Hamilton™s principle states that in moving from one configuration at time t1 to

524

525

19.1 Hamilton™s principle in Newtonian mechanics

another at time t2 the motion of such a system is such as to make stationary the

action

t2

S= ™

L q a q a t dt (19.1)

t1

The Lagrangian L is defined, in terms of the kinetic energy T and the potential

energy V (with respect to some reference situation), by L = T ’ V . Here V

™

is a function of the q a (and possibly t) only, but not of the q a . As discussed

in Section 3.19, the coordinates define a configuration space with line element

ds2 = gab dq a dq b . For example, the Lagrangian for a particle of mass m can be

written as

L = T ’ V = 2 mgab q a q b ’ V

™™

1

(19.2)

Returning to the general expression (19.1), let us consider an arbitrary variation

a

qa t ’ q t = qa t + qa t

in the trajectory in configuration space and demand that the corresponding varia-

tion S in the action vanishes. Assuming that q a t = 0 at the endpoints t1 and

t2 , we know from our discussion of the calculus of variations in Appendix 3C at

the end of Chapter 3 that the Lagrangian L must satisfy the Euler“Lagrange (EL)

equations

L d L

’ =0 a=1 2 n

™

q a dt qa

For example, as shown in Section 3.19, the EL equations for the Lagrangian

(19.2) are

m qa +

¨ ™ ™ = ’g ab b V

a bc

bc q q

which corresponds to Newton™s second law in an arbitrary coordinate system.

If the q a t are taken to be the Cartesian coordinates xa t of the particle, we

immediately recover the more familiar form m¨ a = ’ ab b V .

x

Hamilton™s principle is easily extended from the notion of discrete particles

to continuous systems. As an example, let us consider a flexible string stretched

between two fixed points at x = 0 and x = l. In this case, we again have one inde-

pendent time coordinate t, but now in the context of a continuum in which the q a t

become the continuous variable t x describing the transverse displacement

of the string as a function of position and time (see Figure 19.1). Consequently,

526 A variational approach to general relativity

t

φ φ (t2, x)

t2

φ (t1, x)

t1

x

0 l

Figure 19.1 The transverse displacement t x of a taut string fixed at two

points a distance l apart, viewed as a function in the t x -plane.

the expressions for T and V become integrals over x rather than sums over the

label a. If x and x are the local line density and tension of the string then the

kinetic and potential energies of the string for small displacements are given by

2 2

l l

T= V=

1 1

and

dx dx

2 2

t x

0 0

Thus, the action (19.1) becomes

t2 l t2 l

S≡ dx dt = ’

2 2

1

(19.3)

dx dt

t x

2

0 0

t1 t1

where in the first equality we have defined the Lagrangian density and in the

final expression we have adopted the shorthand t = / t and x = / x. Let us

now consider an arbitrary variation in the function of the form

tx’ tx= tx+ (19.4)

tx

This leads to a variation in the action (19.1) given by

t2 l

S= + (19.5)

dx dt

t x

0

t1 t x

=t =x

From (19.4), one immediately notes that and .

t x

Substituting these expressions in (19.5) and using Leibnitz™ rule for the differen-

tiation of a product, we may write

t2 l

S = Sb ’ + (19.6)

dx dt

t x

0

t1 t x

527

19.2 Classical field theory and the action

where the ˜boundary™ (or ˜surface™) term is given by

t2 l

Sb = + dx dt

t x

0

t1 t x

t=t2 x=l

l t2

= dx + dt

0 t1

t x

t=t1 x=0

If we assume that the variation is such that

t1 x = 0 = t 0 =0=

and

t2 x tl

then it vanishes on the entire ˜boundary™ of the region of interest in the t x -

plane, and we have Sb = 0. Thus, in this case, by demanding that the total

variation (19.6) in the action vanishes ( S = 0) and using the fact that is

arbitrary, we obtain

+ = ’ =0

t x t t x x

t x

where, in the first equality, we have evaluated the derivatives of with respect

to t and x using (19.3). If, in addition, and do not depend on x or t then

2 12

=2 2

x2 ct

where c2 = / . This is the wave equation for small transverse oscillations of a

taut uniform string.

19.2 Classical field theory and the action

In the above discussion, the function t x may be regarded as a ˜field™ defined

on a two-dimensional space (or manifold) parameterised by the coordinates

x and t. To extend the idea of a variational principle to a field theory in spacetime,

t x by a (finite) set of fields a x

one therefore needs only to replace

defined on a four-dimensional spacetime parameterised in terms of some (in

general) arbitrary set of continuous coordinates x . Alternatively, one could even

consider each member of the (finite) set of generalised coordinates q a t in (19.1)

as a ˜field™ defined on a one-dimensional manifold parameterised by the continu-

ous coordinate t, and simply replace the q a t by the set of fields a x . In either

case, the index a acts merely as a label for the individual fields in the theory.

This last point is worth clarifying. If, for example, one were considering a field

theory containing a set of M scalar fields 1 2 M then the set of fields

would be simply a = 1 2 M . Alternatively, one might be interested

in a field theory containing a vector field (such as electromagnetism). In this

528 A variational approach to general relativity

case, the label a would run over the four components of the vector field in the

a = A0 A1 A2 A3 = A

chosen coordinate system, i.e. we would write

and so a would then be a spacetime index. Similar considerations apply to the

components of tensor fields. Use of the index a may also be trivially extended to

label the components of two or more vector or tensor fields involved in the theory.

Indeed, when considering field theories defined on some arbitrary manifold and

in arbitrary coordinates, one must always include the metric tensor components

in the set of fields. For example, in electromagnetism on an arbitrary manifold,

a=A g

the full set of fields is in fact .

By analogy with (19.3), the action S for a set of fields defined on some

general four-dimensional spacetime manifold should take the form of an integral

of some function , called the Lagrangian density, of the fields a and their

first (and possibly higher) derivatives over some four-dimensional region of

the spacetime. Thus, we take the action integral to be

S= d4 x

a a a

(19.7)

where d4 x denotes the product of coordinate differentials dx0 dx1 dx2 dx3 . It is

believed that physical theories should be generally covariant and so this symmetry

must be reflected in the action S, which therefore has to be a scalar under

general coordinate transformations. From the discussion in Section 2.14, we know

that in any arbitrary coordinate system x the invariant volume element (which

√

transforms as a scalar field) is d4 V = ’g d4 x, where g is the determinant of the

metric tensor in that coordinate system (and is negative for the signature of the

metric used in this book). It is therefore convenient to write the action (19.7) in

the form

√

S = L ’g d4 x

where we have introduced the field Lagrangian L, which is clearly related to the

Lagrangian density by1

√

= L ’g (19.8)

√

For the action S to be a scalar, the quantity L ’g d4 x must be a scalar field at

√

each point in . Since the invariant volume element ’g d4 x is already a scalar

field, then so too must be the Lagrangian L. Taking L to be in general a function

of the fields a and their first (and possibly higher) derivatives, the action for a

1

Although most authors agree that is called the Lagrangian density, it is common in field theory for the

term Lagrangian (and the symbol L) to mean the integral of over some three-dimensional spacelike

hypersurface, rather than the relationship given in (19.8). We will adopt the convention (19.8) throughout

this chapter.

529

19.3 Euler“Lagrange equations

set of classical fields defined on some 4-dimensional spacetime manifold may be

written as

√

S= ’g d4 x

a a a

L

where L is a scalar function of spacetime position. We note finally that the

Lagrangian density in (19.8) will not transform as a scalar field under coor-

dinate transformations; in fact, it is what is known as a scalar density of weight

unity, although we need not concern ourselves here with the definition of such

objects.

19.3 Euler“Lagrange equations

We now derive the form of the field equations for (some subset of) the fields a

by demanding that the action is stationary, or invariant, under small variations in

(the same subset of) the fields of the form