Communicating with 1 W lasers, LISA achieves a remarkable sensitivity, and will be able to see the strongest sources in its band anywhere in the universe. We will return to the kinds of sources that might be detected by LISA and the ground-based detectors after studying the way that gravitational waves are generated in general relativity.

9.3 T h e g e n e ra t i o n o f g ra v i t a t i o n a l wa v e s

Simple estimates

It is easy to see that the amplitude of any gravitational waves incident on Earth should be small. A ‘strong’ gravitational wave would have hμν = 0(1), and we should expect amplitudes like this only very near a highly relativistic source, where the Newtonian potential (if it had any meaning) would be of order one. For a source of mass M, this would be at distances of order M from it. As with all radiation fields, the amplitude of the gravitational waves falls off as r−1 far from the source. (Readers who are not familiar with solutions of the wave equation will find demonstrations of this in the next sections.) So if Earth is a distance R from a source of mass M, the largest amplitude waves we should expect are of order M/R. For the formation of a 10 M black hole in a supernova explosion in a nearby galaxy 1023 m away, this is about 10−17. This is in fact an upper limit in this case, and less-violent events will lead to very much smaller amplitudes.

Slow motion wave generation

We will find the exact solution in a later section. Here we will make some simplifying – but realistic – assumptions. We assume that the time-dependent part of Tμν is in sinusoidal oscillation with frequency , i.e. that it is the real part of

and that the region of space in which Sμν =0 is small compared with 2π/ , the wavelength of a gravitational wave of frequency . The first assumption is not much of a restriction, since a general time dependence can be reduced to a sum over sinusoidal motions by Fourier analysis. Besides, many interesting astrophysical sources are roughly periodic: pulsating stars, pulsars, binary systems. The second assumption is called the slow-motion assumption, since it implies that the typical velocity inside the source region, which is times the size of that region, should be much less than one. All but the most powerful sources of gravitational waves probably satisfy this condition.

¯

Let us look for a solution for hμν of the form

(In the end we must take the real part of this for our answer.) Putting this and Eq. (9.65) into Eq. (9.64) gives

¯

It is important to bear in mind as we proceed that the indices on hμν in Eq. (9.64) play

¯

almost no role. We shall regard each component hμν as simply a function on Minkowski space, satisfying the wave equation. All our steps will be the same as for solving the scalar equation (−∂2/∂t2 + 2)f = g, until we come to Eq. (9.75).

Outside the source (i.e. where Sμν = 0) we want a solution Bμν of Eq. (9.67) that represents outgoing radiation far away; and of all such possible solutions we want the one that dominates in the slow-motion limit. Let us define r to be the spherical polar radial coordinate, where the origin is chosen inside the source. We show in Exer. 29, § 9.7 that the solution we seek is spherical outside the source,

where Aμν and Zμν are constants. The term in e−i r represents a wave traveling toward the origin r = 0 (called an ingoing wave), while the other term is outgoing (see Exer. 28, § 9.7). We want waves emitted by the source, so we choose Zμν = 0.

Our problem is to determine Aμν in terms of the source. Here we make our approxima-

where |Bμν |max is the maximum value Bμν takes inside the source. We will see that this term is negligible. The other term from integrating the left-hand side of Eq. (9.67) is

by Gauss’ theorem. But the surface integral is outside the source, where only the spherical part of Bμν given by Eq. (9.68) survives:

the last step being the application of Gauss’ theorem to any volume completely containing the source. This means that Tμν = 0 on the surface bounding this volume, so that the right-hand side of Eq. (9.79) vanishes. This means that if =0, we have

These conditions basically embody the laws of conservation of total energy and momen-

¯μ0

tum for the oscillating source. The neglected higher-order parts of h are gauge terms (Exer. 32, § 9.7).

The expression for Jij can also be rewritten in an instructive way by using the result of Exer. 23, § 4.10.

For a source in slow motion, we have seen in Ch. 7 that T00 ≈ ρ, the Newtonian mass density. It follows that the integral on the left-hand side of Eq. (9.81) is what is often referred to as the quadrupole moment tensor of the mass distribution,

(Conventions for defining the quadrupole moment vary from one text to another. We follow Misner et al. (1973).) In terms of this we have

It is important to remember that Eq. (9.83) is an approximation which neglects not merely all terms of order r−2 but also r−1 terms that are not dominant in the slow-motion

Because of Eq. (9.83), this approximation is often called the quadrupole approximation for gravitational radiation.

As for the plane waves we studied earlier, we have here the freedom to make a further restriction of the gauge. The obvious choice is to try to find a TT gauge, transverse to the direction of motion of the wave (the radial direction), which has the unit vector nj = xj/r. Exer. 32, § 9.7, shows that this is possible, so that in the TT gauge we have the simplest form of the wave. If we choose our axes so that at the point where we measure the wave it is traveling in the z direction, then we can supplement Eq. (9.80) by

is called the trace-free or reduced quadrupole moment tensor.

Examples

Let us consider the waves emitted by a simple oscillator like the one we used as a detector in § 9.2. If both masses oscillate with angular frequency ω and amplitude A about mean equilibrium positions a distance l0, apart, then, by Exer. 30, § 9.7, the quadrupole tensor has only one nonzero component,

Recall that only the sinusoidal part of Ixx should be used in the formulae developed in the previous paragraph. In this case, there are two such pieces, with frequencies ω and 2ω. Since the wave equation Eq. (9.64) is linear, we shall treat each term separately and simply add the results later. The ω term in Ixx is the real part of 2ml0A exp(−iωt). The trace-free quadrupole tensor has components

all off-diagonal components vanishing. If we consider the radiation traveling in the z direction, we get, from Eqs. (9.84)–(9.86),

The radiation is linearly polarized, with an orientation such that the ellipse in Fig. 9.1 is aligned with the line joining the two masses. The same is true for the radiation going in the