cosmological background. This is a general problem, and it may be that only at frequencies above about 0.1 Hz will the universe be quiet enough to allow us to listen directly to the hiss of gravitational waves from the Big Bang.

Theoretical predictions of the radiation to be expected vary hugely, from below 10−15 up to 10−8 and higher. This reflects the uncertainty in theoretical models of the physical conditions and indeed of the laws of physics themselves during the early Big Bang, and demonstrates the importance that detecting a background would play in constraining these models. An observation of the random (stochastic) background of gravitational waves is possibly the most important observation that gravitational wave detectors can make.

9.6 F u r t h e r re a d i n g

Joseph Weber’s early thinking about detectors is in Weber (1961). One of the most interesting theoretical developments stimulated by research into gravitational wave detection has been the design of so-called ‘quantum nondemolition’ detectors: methods of measuring aspects of the excitation of a vibrating bar to arbitrary precision without disturbing the quantity being measured, even when the bar is excited only at the oneor two-quantum (phonon) level. See early work by Thorne et al. (1979) and Caves et al. (1980).

A full discussion of the wave equation is beyond our scope here, but is amply treated in many texts on electromagnetism, such as Jackson (1975). A simplified discussion of gravitational waves is in Schutz (1984). See also Schutz and Ricci (2001).

The detection of gravitational waves is a rapidly evolving field, so the student who wants the latest picture should consult the literature, starting with the various articles that we have cited from the open-access electronic journal Living Reviews in Relativity, whose review articles are kept up-to-date: Armstrong (2006), Blanchet (2006), Futamase and Itoh (2007), Hough and Rowan (2000), Will (2006). More popular-style articles about gravitational waves and other applications of general relativity can be found on the Einstein Online website: http://www.einstein-online.info/en/.

The websites of the detectors LIGO (http://www.ligo.caltech.edu/), GEO (http://geo600.aei.mpg.de/), LSC (http://www.ligo.org/), VIRGO (http://wwwcascina.virgo.infn.it/), and LISA (http://www.lisascience.org/) are also good sources of current information. The bar detectors of the Rome group (http://www.roma1.infn.it/rog/) and the Auriga detector (http://www.auriga.lnl.infn.it/) are the last operating resonant-mass detectors.

Readers who wish to assist with the compute-intensive analysis of data from the big interferometers may download a screen-saver called Einstein@Home, which uses the idle time on a computer to perform parts of the data analysis. Hundreds of thousands of computers have so far joined this activity. See the website http://einstein.phys.uwm.edu/.

9.7E xe rc i s e s

1A function f (s) has derivative f (s) = df /ds. Prove that ∂f (kμxμ)/ ∂xν = kν f (kμxμ). Use this to prove Eq. (9.4) and the one following it.

2Show that the real and imaginary parts of Eq. (9.2) at a fixed spatial position {xi} oscillate sinusoidally in time with frequency ω = k0.

superposition of plane waves.

4Derive Eqs. (9.16) and (9.17).

5(a) Show that A(NEW)αβ , given by Eq. (9.17), satisfies the gauge condition Aαβ kβ = 0 if

A(OLD)αβ does.

(b)Use Eq. (9.18) for A(NEW)αβ to constrain Bμ.

(c)Show that Eq. (9.19) for A(NEW)αβ imposes only three constraints on Bμ, not the four that we might expect from the fact that the free index α can take any values from 0

to 3. Do this by showing that the particular linear combination kα (Aαβ Uβ ) vanishes for any Bμ.

(d)Using (b) and (c), solve for Bμ as a function of kμ, A(OLD)αβ , and Uμ. These determine Bμ: there is no further gauge freedom.

(e)Show that it is possible to choose ξ β in Eq. (9.15) to make any superposition of plane waves satisfy Eqs. (9.18) and (9.19), so that these are generally applicable to gravitational waves of any sort.

(f)Show that we cannot achieve Eqs. (9.18) and (9.19) for a static solution, i.e. one

for which ω = 0.

6Fill in all the algebra implicit in the paragraph leading to Eq. (9.21).

7Give a more rigorous proof that Eqs. (9.22) and (9.23) imply that a free particle initially at rest in the TT gauge remains at rest.

8Does the free particle of the discussion following Eq. (9.23) feel any acceleration? For example, if the particle is a bowl of soup (whose diameter is much less than a wavelength), does the soup slosh about in the bowl as the wave passes?

9Does the free particle of the discussion following Eq. (9.23) see any acceleration? To answer this, consider the two particles whose relative proper distance is calculated in Eq. (9.24). Let the one at the origin send a beam of light towards the other, and let it be reflected by the other and received back at the origin. Calculate the amount of proper time elapsed at the origin between the emission and reception of the light (you may assume that the particles’ separation is much less than a wavelength of the

gravitational wave). By monitoring changes in this time, the particle at the origin can ‘see’ the relative acceleration of the two particles.

10 (a) We have seen that

hyz = A sin ω(t − x), all other hμν = 0,

with A and ω constants, |A| 1, is a solution to Eqs. (9.1) and (9.11). For this metric tensor, compute all the components of Rαβμν and show that some are not zero, so that the spacetime is not flat.

(b) Another metric is given by

hyz = A sin ω(t − x), htt = 2B(x − t), htx = −B(x − t), all other hμν = 0.

Show that this also satisfies the field equations and the gauge conditions.

(c)For the metric in (b), compute Rαβμν . Show that it is the same as in (a).

(d)From (c) we conclude that the geometries are identical, and that the difference in the metrics is due to a small coordinate change. Find a ξ μ such that

hμν (part a) − hμν (part b) = −ξμ,ν − ξν,μ.

11(a) Derive Eq. (9.27).

(b)Solve Eqs. (9.28a) and (9.28b) for the motion of the test particles in the polarization rings shown in Fig. 9.1.

12Do calculations analogous to those leading to Eqs. (9.28) and (9.32) to show that the separation of particles in the z direction (the direction of travel of the wave) is unaffected.

13One kind of background Lorentz transformation is a simple 45◦ rotation of the x and y axes in the x − y plane. Show that under such a rotation from (x, y) to (x , y ), we have hTTx y = hTTxx , hTTx x = −hTTxy . This is consistent with Fig. 9.1.

14(a) Show that a plane wave with Axy = 0 in Eq. (9.21) has the metric

where h+ = Axx sin[ω(t − z)].

(b)Show that this wave does not change proper separations of free particles if they are aligned along a line bisecting the angle between the x- and y-axes.

(c)Show that a plane wave with Axx = 0 in Eq. (9.21) has the metric

where h× = Axy sin[ω(t − z)].

(d)Show that the wave in (c) does not change proper separations of free particles if they are aligned along the coordinate axes.

(e)Show that the wave in (c) produces an elliptical distortion of the circle that is rotated by 45◦ to that of the wave in (a).

15 (a) A wave is said to be circularly polarized in the x − y plane if hTTyy = −hTTxx and hTTxy = ±ihTTxx . Show that for such a wave, the ellipse in Fig. 9.1 rotates without changing shape.

(b)A wave is said to be elliptically polarized with principal axes x and y if hTTxy =

±iahTTxx , where a is some real number, and hTTyy = −hTTxx . Show that if hTTxy = αhTTxx , where α is a complex number (the general case for a plane wave), new axes x and y can be found for which the wave is elliptically polarized with principal axes x and y . Show that circular and linear polarization are special cases of elliptical.

16Two plane waves with TT amplitudes, Aμν and Bμν , are said to have orthogonal polar-

izations if (Aμν ) Bμν = 0, where (Aμν ) is the complex conjugate of Aμν . Show that if Aμν and Bμν are orthogonal polarizations, a 45◦ rotation of Bμν makes it proportional to Aμν .

17Find the transformation from the coordinates (t, x, y, z) of Eqs. (9.33)–(9.36) to the local inertial frame of Eq. (9.37). Use this to verify Eq. (9.38).

18Prove Eq. (9.39).

19Use the sum of Eqs. (9.40) and (9.41) to show that the center of mass of the spring remains at rest as the wave passes.

20Derive Eq. (9.44) from Eq. (9.43), and then prove Eq. (9.45).

21Generalize Eq. (9.45) to the case of a plane wave with arbitrary elliptical polarization (Exer. 15) traveling in an arbitrary direction relative to the separation of the masses.

22Consider the equation of geodesic deviation, Eq. (6.87), from the point of view of the geodesic at the center of mass of the detector of Eq. (9.45). Show that the vector ξ as we have defined it in Eq. (9.42) is twice the connecting vector from the center of mass to one of the masses, as defined in Eq. (6.83). Show that the tidal force as measured by the center of mass leads directly to Eq. (9.45).

23Derive Eqs. (9.48) and (9.49), and derive the general solution of Eq. (9.45) for arbitrary initial data at t = 0, given Eq. (9.46).

24Prove Eq. (9.53).

25Derive Eq. (9.56) from the given definition of Q.

26(a) Use the metric for a plane wave with ‘+’ polarization, Eq. (9.58), to show that the square of the coordinate speed (in the TT coordinate system) of a photon moving

This is not identically one. Does this violate relativity? Why or why not?

(b)Imagine that an experimenter at the center of the circle of particles in Fig. 9.1 sends a photon to the particle on the circle at coordinate location x = L on the

positive-x axis, and that the photon is reflected when it reaches the particle and returns to the experimenter. Suppose further that this takes such a short time that h+ does not change significantly during the experiment. To first order in h+, show that the experimenter’s proper time that elapses between sending out the photon and receiving it back is (2 + h+)L.

(c)The experimenter says that this proves that the proper distance between herself and the particle is (1 + h+/2)L. Is this a correct interpretation of her experiment? If the experimenter uses an alternative measuring process for proper distance, such as laying out a number of standard meter sticks between her location and the particle, would that produce the same answer? Why or why not?

(d)Show that if the experimenter simultaneously does the same experiment with a particle on the y-axis at y = L, that photon will return after a proper time of (2 − h+)L.

(e)The difference in these return times is 2h+L and can be used to measure the wave’s amplitude. Does this result depend on our use of TT gauge, i.e. would we have obtained the same answer had we used a different coordinate system?

27(a) Derive the full three-term return relation, Eq. (9.63), for the rate of change of the

return time for a beam traveling through a plane wave h+ along the x-direction, when the wave is moving at an angle θ to the z-axis in the x − z plane.

(b)Show that, in the limit where L is small compared to a wavelength of the gravi-

tational wave, the derivative of the return time is the derivative of t + δL, where δL = L cos2 θ h(t) is the excess proper distance for small L. Explain where the factor of cos2 θ comes from.

(c)Examine the limit of the three-term formula in (a) when the gravitational wave is traveling along the x-axis too (θ = ±π/2): what happens to light going parallel to

(a)Show that fl(r) satisfies the equation

%&

(b)Show that the most general spherically symmetric solution is given by Eq. (9.68).

(c)Substitute the variable s = r to show that fl satisfies the equation

This is known as Bessel’s equation, whose solutions are called Bessel functions

(e)Use Eqs. (9.152) and (9.153) to show that for s l, the dominant behavior of jl and nl is

(h)Repeat the calculation of Eqs. (9.69)–(9.74), only this time multiply Eq. (9.67) by jl(r )Ylm(θ , φ) before performing the integrals. Show that the left-hand side of

Eq. (9.67) becomes, when so integrated, exactly

and that when ε l this becomes (with the help of Eqs. (9.159) and (9.156)–(9.157) above, since we assume r = ε is outside the source) simply iAlmμν / . Similarly, show that the right-hand side of Eq. (9.67) integrates to −16π l Tμν rlYlm(θ , φ)d3x/(2l + 1)! ! in the same approximation.

(i) Show, then, that the solution is Eq. (9.159), with

(j)Let l = 0 and deduce Eq. (9.73) and (9.74).

(k)Show that if Jμνlm =0 for some l, then the terms neglected in Eq. (9.161), because of the approximation ε 1, are of the same order as the dominant terms in Eq. (9.161) for l + 1. In particular, this means that if Jμν =0 in Eq. (9.72), any attempt to get a more accurate answer than Eq. (9.74) must take into account not

only the terms for l > 0 but also neglected terms in the derivation of Eq. (9.74),

such as Eq. (9.69).

30 Re-write Eq. (9.82a) for a set of N discrete point particles, where the masses are {m(A), A = 1, . . . , N} and the positions are {x(iA)}.

31Calculate the quadrupole tensor Ijk and its traceless counterpart –I jk (Eq. (9.87)) for the following mass distributions.

(a)A spherical star where density is ρ(r, t). Take the origin of the coordinates in Eq. (9.82) to be the center of the star.

(b)The star in (a), but with the origin of the coordinates at an arbitrary point.

(c)An ellipsoid of uniform density ρ and semiaxes of length a, b, c oriented along the x, y, and z axes respectively. Take the origin to be at the center of the ellipsoid.

(d)The ellipsoid in (c), but rotating about the z axis with angular velocity ω.

(e) Four masses m located respectively at the points (a, 0, 0), (0, a, 0), (−a, 0, 0), (0, −a, 0).

(f)The masses as in (e), but all moving counter-clockwise about the z axis on a circle of radius a with angular velocity ω.

(g)Two masses m connected by a massless spring, each oscillating on the x axis with angular frequency ω and amplitude A about mean equilibrium positions a distance l0 apart, keeping their center of mass fixed.

(h)Unequal masses m and M connected by a spring of spring constant k and equilib-

rium length l0, oscillating (with their center of mass fixed) at the natural frequency of the system, with amplitude 2A (this is the total stretching of the spring). Their separation is along the x axis.

32This exercise develops the TT gauge for spherical waves.

(a)In order to transform Eq. (9.83) to the TT gauge, use a gauge transformation gen-

erated by a vector ξ α = Bα (xμ)ei (r−t)/r, where Bα is a slowly varying function of xμ. Find the general transformation law to order 1/r.

it is possible to find functions Bα , which accomplish such a transformation and which satisfy ξ α = 0 to order 1/r.

(c)Show that Eqs. (9.84)–(9.87) hold in the TT gauge.

(d)By expanding R in Eq. (9.103) but discarding r−1 terms, show that the higher-order

where nj points in the z direction.

34Show that –I jk is trace free, i.e. –I l l = 0.

35For the systems described in Exer. 31, calculate the transverse-traceless quadrupole radiation field, Eqs. (9.85)–(9.86) or (9.163), along the x, y, and z axes. In Eqs. (9.85)– (9.86) be sure to change the indices appropriately when doing the calculation on the x and y axes, as in the discussion leading to Eq. (9.91).

36Use Eq. (9.163) or a rotation of the axes in Eqs. (9.85)–(9.86) to calculate the amplitude and orientation of the polarization ellipse of the radiation from the simple oscillator, Eq. (9.88), traveling at an angle θ to the x axis.

37The ω and 2ω terms in Eq. (9.93) are qualitatively different, in that the 2ω term depends only on the amplitude of oscillator A, while the ω term depends on both A and the separation of the masses l0. Why should l0 be involved – the masses don’t move over that distance? The answer is that stresses are transmitted over that distance by the spring, and stresses cause the radiation. To see this, do an analogous calculation for a similar system, in which stresses are not passed over large distances. Consider a

system consisting of two pairs of masses. Each pair has one particle of mass m and another of mass M m. The masses within each pair are connected by a short spring

whose natural frequency is ω. The pairs’ centers of mass are at rest relative to one another. The springs oscillate with equal amplitude in such a way that each mass m

oscillates sinusoidally with amplitude A, and the centers of oscillation of the masses are separated by l0 A. The masses oscillate out of phase. Use the calculation of Exer. 31(h) to show that the radiation field of the system is Eq. (9.93) without the ω term. The difference between this system and that in Eq. (9.93) may be thought to be the origin of the stresses to maintain the motion of the masses m.

38Do the same as Exer. 36 for the binary system, Eqs. (9.98)–(9.99), but instead of finding the orientation of the linear polarization, find the orientation of the ellipse of elliptical polarization.

39Let two spherical stars of mass m and M be in elliptical orbit about one another in the x − y plane. Let the orbit be characterized by its total energy E and its angular

momentum L.

(a)Use Newtonian gravity to calculate the equation of the orbits of both masses about their center of mass. Express the orbital period P, minimum separation a, and eccentricity e as functions of E and L.

(b)Calculate –I kj for this system.

(c)Calculate from Eq. (9.106) the TT radiation field along the x and z axes. Show that your result reduces to Eqs. (9.98)–(9.99) when m = M and the orbits are circular.

40Show from Eq. (9.101) that spherically symmetric motions produce no gravitational radiation.

41Derive Eq. (9.115) from Eq. (9.114) in the manner suggested in the text.

42(a) Derive Eq. (9.116).

(b)Derive Eqs. (9.117) and (9.118) by superposing Eqs. (9.107) and (9.116) and assuming R is small.

(c)Derive Eq. (9.120) in the indicated manner.

43Show that if we define an averaged stress–energy tensor for the waves

(where denotes an average over both one period of oscillation in time and one wavelength of distance in all spatial directions), then the flux F of Eq. (9.122) is the component T0z for that wave. A more detailed argument shows that Eq. (9.164) can in fact be regarded as the stress–energy tensor of any wave packet, provided the averages

are defined suitably. This is called the Isaacson stress–energy tensor. See Misner et al. (1973) for details.

44(a) Derive Eq. (9.125) from Eq. (9.123).

(b)Justify Eq. (9.127) from Eq. (9.125).

(c)Derive Eq. (9.127) from Eq. (9.122) using Exer. 33(b).

45(a) Consider the integral in Eq. (9.128). We shall do it by the following method.

(i) Argue on grounds of symmetry that njnk sin θ dθ dφ must be proportional

to δjk. (ii) Evaluate the constant of proportionality by explicitly doing the case j = k = z.

(b)Follow the same method for Eq. (9.129). In (i) argue that the integral can depend only on δij, and show that the given tensor is the only one constructed purely from δij that has the symmetry of being unchanged when the values of any two of its

indices are exchanged.

46Derive Eqs. (9.130) and (9.131), remembering Eq. (9.124) and the fact that –I ij is symmetric.

47(a) Recall that the angular momentum of a particle is pφ . It follows that the angular momentum flux of a continuous system across a surface xi = const. is Tiφ . Use this and Exer. 43 to show that the total z component of angular momentum radiated by

a source of gravitational waves (which is the integral over a sphere of large radius

total energy radiated to the total angular momentum radiated is /m.

48Calculate Eq. (9.135).

49For the arbitrary binary system of Exer. 39:

(a)Show that the average energy loss rate over one orbit is

(c) Verify Eq. (9.144).

(Do parts (b) and (c) even if you can’t do (a).) These were originally derived by Peters (1964).