Chapter 9. Gravitational waves

9.1 The speed of gravity

In Newtonian gravity, gravitational effects are assumed to propagate at infinite speed, so that for example the lunar tides correspond at any time to the position of the moon at the same instant. This clearly can't be true in relativity, since simultaneity isn't something that different observers even agree on. Not only should the “speed of gravity” be finite, but it seems implausible that it would be greater than c; in section 2.2 (p. 46), we argued based on empirically well established principles that there must be a maximum speed of cause and effect. Although the argument was only applicable to special relativity, i.e., to a flat spacetime, it seems likely to apply to general relativity as well, at least for low-amplitude waves on a flat background. As early as 1913, before Einstein had even developed the full theory of general relativity, he had carried out calculations in the weak-field limit showing that gravitational effects should propagate at c. This seems eminently reasonable, since (a) it is likely to be consistent with causality, and (b) G and c are the only constants with units that appear in the field equations (obscured by our choice of units, in which G=1 and c=1), and the only velocity-scale that can be constructed from these two constants is c itself.¹

Although extremely well founded theoretically, this turns out to be extremely difficult to test empirically. In a 2003 experiment,² Fomalont and Kopeikin used a world-wide array of radio telescopes to observe a conjunction in which Jupiter passed within 3.7' of a quasar, so that the quasar's radio waves came within about 3 light-seconds of the planet on their way to the earth. Since Jupiter moves with v=4×10^-5, one expects naively that the radio waves passing by it should be deflected by the field produced by Jupiter at the position it had 3 seconds earlier. This position differs from its present position by about 10^-4 light-seconds, and the result should be a difference in propagation time, which should be different when observed from different locations on earth. Fomalont and Kopeikin measured these phase differences with picosecond precision, and found them to be in good agreement with the predictions of general relativity. The real excitement started when they published their result with the interpretation that they had measured, for the first time, the speed of gravity, and found it to be within 20% error bars of c. Samuel³ and Will⁴ published refutations, arguing that Kopeikin's calculations contained mistakes, and that what had really been measured was the speed of light, not the speed of gravity.

The reason that the interpretation of this type of experiment is likely to be controversial is that although we do have theories of gravity that are viable alternatives to general relativity (e.g., the Brans-Dicke theory, in which the gravitational constant is a dynamically changing variable), such theories have generally been carefully designed to agree with general relativity in the weak-field limit, and in particular every such theory (or at least every theory that remains viable given current experimental data) predicts that gravitational effects propagate at c in the weak-field limit. Without an alternative theory to act as a framework --- one that disagrees with relativity about the speed of gravity --- it is difficult to know whether an observation that agrees with relativity is a test of this specific aspect of relativity.

9.2 Gravitational radiation

9.2.1 Empirical evidence

So we still don't know, a century after Einstein found the field equations, whether gravitational “ripples” travel at c. Nevertheless, we do have strong empirical evidence that such ripples exist. The Hulse-Taylor system (page 204) contains two neutron stars orbiting around their common center of mass, and the period of the orbit is observed to be decreasing gradually over time (figure a). This is interpreted as evidence that the stars are losing energy to radiation of gravitational waves.⁵ As we'll see in section 9.2.5, the rate of energy loss is in excellent agreement with the predictions of general relativity.

a / The Hulse-Taylor pulsar's orbital motion is gradually losing energy due to the emission of gravitational waves. The linear decrease of the period is integrated on this plot, resulting in a parabola. From Weisberg and Taylor, http://arxiv.org/abs/astro-ph/0211217.

An even more dramatic, if less clearcut, piece of evidence is Komossa, Zhou, and Lu's observation⁶ of a supermassive black hole that appears to be recoiling from its parent galaxy at a velocity of 2650 km/s (projected along the line of sight). They interpret this as evidence for the following scenario. In the early universe, galaxies form with supermassive black holes at their centers. When two such galaxies collide, the black holes can merge. The merger is a violent process in which intense gravitational waves are emitted, and these waves carry a large amount of momentum, causing the black holes to recoil at a velocity greater than the escape velocity of the merged galaxy.

Although the energy loss from systems such as the Hulse-Taylor binary provide strong evidence that gravitational waves exist and carry enery, we would also like to detect them directly. This would not only be a definitive test of a century-old prediction of general relativity, it would also open a window into a completely new method of astronomical observation. A series of attempts is under way to observe gravitational waves directly using interferometers, which detect oscillations in the lengths of their own arms. These can be either ground-based or space-based. The early iterations of the ground-based LIGO interferometer did not detect gravitational waves, but a new version, Advanced LIGO, is under construction. A space-based system, LISA, has been proposed for launch in 2020, but its funding is uncertain. The two devices would operate in complementary frequency ranges (figure b). A selling point of LISA is that if it is launched, there are a number of known sources in the sky that are known to be easily within its range of sensitivity.⁷ One excellent candidate is HM Cancri, a pair of white dwarfs with an orbital period of 5.4 minutes, shorter than that of any other known binary star.⁸

b / Predicted sensitivities of LISA and LIGO to gravitational waves of various frequences.

9.2.2 Energy content

Even without performing the calculations for a system like the Hulse-Taylor binary, it is easy to show that if such waves exist, they must be capable of carrying away energy. Consider two equal masses in highly elliptical orbits about their common center of mass, figure c. The motion is nearly one-dimensional. As the masses recede from one another, they feel a delayed version of the gravitational force originating from a time when they were closer together and the force was stronger. The result is that in the near-Newtonian limit, they lose more kinetic and gravitational energy than they would have lost in the purely Newtonian theory. Now they come back inward in their orbits. As they approach one another, the time-delayed force is anomalously weak, so they gain less mechanical energy than expected. The result is that with each cycle, mechanical energy is lost. We expect that this energy is carried by the waves, in the same way that radio waves carry the energy lost by a transmitting antenna.⁹

Not only can these waves remove mechanical energy from a system, they can also deposit energy in a detector, as shown by the nonmathematical “sticky bead argument” (figure d), which was originated by Feynman in 1957 and later popularized by Bondi.

Now strictly speaking, we have only shown that gravitational waves can extract or donate mechanical energy, but not that the waves themselves transmit this energy. The distinction isn't one that normally occurs to us, since we are trained to believe that energy is always conserved. But we know that, for fundamental reasons, general relativity doesn't have global conservation laws that apply to all spacetimes (p. 133). Perhaps the energy lost by the Hulse-Taylor system is simply gone, never to reappear, and the energy imparted to the sticky bead is simply generated out of nowhere. On the other hand, general relativity does have global conservation laws for certain specific classes of spacetimes, including, for example, a conserved scalar mass-energy in the case of a stationary spacetime (p. 225). Spacetimes containing gravitational waves are not stationary, but perhaps there is something similar we can do in some appropriate special case.

Suppose we want an expression for the energy of a gravitational wave in terms of its amplitude. This seems like it ought to be straightforward. We have such expressions in other classical field theories. In electromagnetism, we have energy densities and associated with the electric and magnetic fields. In Newtonian gravity, we can assign an energy density to the gravitational field g; the minus sign indicates that when masses glom onto each other, they produce a greater field, and energy is released.

In general relativity, however, the equivalence principle tells us that for any gravitational field measured by one observer, we can find another observer, one who is free-falling, who says that the local field is zero. It follows that we cannot associate an energy with the curvature of a particular region of spacetime in any exact way. The best we can do is to find expressions that give the energy density (1) in the limit of weak fields, and (2) when averaged over a region of space that is large compared to the wavelength. These expressions are not unique. There are a number of ways to write them in terms of the metric and its derivatives, and they all give the same result in the appropriate limit. The reader who is interested in seeing the subject developed in detail is referred to Carroll's Lecture Notes on General Relativity, http://arxiv.org/abs/gr-qc/?9712019. Although this sort of thing is technically messy, we can accomplish quite a bit simply by knowing that such results do exist, and that although they are non-unique in general, they are uniquely well defined in certain cases. Specifically, when one wants to discuss gravitational waves, it is usually possible to assume an asymptotically flat spacetime. In an asymptotically flat spacetime, there is a scalar mass-energy, called the ADM mass, that is conserved. In this restricted sense, we are assured that the books balance, and that the emission and absorption of gravitational waves really does mean the transmission of a fixed amount of energy.

9.2.3 Expected properties

To see what properties we should expect for gravitational radiation, first consider the reasoning that led to the construction of the Ricci and Einstein tensors. If a certain volume of space is filled with test particles, then the Ricci and Einstein tensors measure the tendency for this volume to “accelerate;” i.e., -d² V/dt² is a measure of the attraction of any mass lying inside the volume. A distant mass, however, will exert only tidal forces, which distort a region without changing its volume. This suggests that as a gravitational wave passes through a certain region of space, it should distort the shape of a given region, without changing its volume.

When the idea of gravitational waves was first discussed, there was some skepticism about whether they represented an effect that was observable, even in principle. The most naive such doubt is of the same flavor as the one discussed in section 8.2.6 about the observability of the universe's expansion: if everything distorts, then don't our meter-sticks distort as well, making it impossible to measure the effect? The answer is the same as before in section 8.2.6; systems that are gravitationally or electromagnetically bound do not have their scales distorted by an amount equal to the change in the elements of the metric.

A less naive reason to be skeptical about gravitational waves is that just because a metric looks oscillatory, that doesn't mean its oscillatory behavior is observable. Consider the following example.

The Christoffel symbols depend on derivatives of the form ∂_a g_bc, so here the only nonvanishing Christoffel symbol is . It is then straightforward to check that the Riemann tensor vanishes by symmetry. Therefore this metric must really just be a flat-spacetime metric that has been subjected to a silly change of coordinates.

Self-check: R vanishes, but Γ doesn't. Is there a reason for paying more attention to one or the other?

To keep the curvature from vanishing, it looks like we need a metric in which the oscillation is not restricted to a single variable. For example, the metric

does have nonvanishing curvature. In other words, it seems like we should be looking for transverse waves rather than longitudinal ones.¹⁰ On the other hand, this metric cannot be a solution to the vacuum field equations, since it doesn't preserve volume. It also stands still, whereas we expect that solutions to the field equations should propagate at the velocity of light, at least for small amplitudes. These conclusions are self-consistent, because a wave's polarization can only be constrained if it propagates at c (see p. 115).

Based on what we've found out, the following seems like a metric that might have a fighting chance of representing a real gravitational wave:

It is transverse, it propagates at c(=1), and the fact that g_xx is the reciprocal of g_yy makes it volume-conserving. The following Maxima program calculates its Einstein tensor:

load(ctensor);
ct_coords:[t,x,y,z];
lg:matrix([1,0,0,0],
          [0,-(1+A*sin(z-t)),0,0],
          [0,0,-1/(1+A*sin(z-t)),0],
          [0,0,0,-1]);
cmetric();
einstein(true);

For a representative component of the Einstein tensor, we find

For small values of A, we have |G_tt| ≤sssim A²/2. The vacuum field equations require G_tt=0, so this isn't an exact solution. But all the components of G, not just G_tt, are of order A², so this is an approximate solution to the equations.

It is also straightforward to check that propagation at approximately c was a necessary feature. For example, if we replace the factors of sin(z-t) in the metric with sin(z-2t), we get a G_xx that is of order unity, not of order A².

To prove that gravitational waves are an observable effect, we would like to be able to display a metric that (1) is an exact solution of the vacuum field equations; (2) is not merely a coordinate wave; and (3) carries momentum and energy. As late as 1936, Einstein and Rosen published a paper claiming that gravitational waves were a mathematical artifact, and did not actually exist.¹¹

9.2.4 Some exact solutions

In this section we study several examples of exact solutions to the field equations. Each of these can readily be shown not to be a mere coordinate wave, since in each case the Riemann tensor has nonzero elements.

Example 1: An exact solution

We've already seen, e.g., in the derivation of the Schwarzschild metric in section 6.2.4, that once we have an approximate solution to the equations of general relativity, we may be able to find a series solution. Historically this approach was only used as a last resort, because the lack of computers made the calculations too complex to handle, and the tendency was to look for tricks that would make a closed-form solution possible. But today the series method has the advantage that any mere mortal can have some reasonable hope of success with it --- and there is nothing more boring (or demoralizing) than laboriously learning someone else's special trick that only works for a specific problem. In this example, we'll see that such an approach comes tantalizingly close to providing an exact, oscillatory plane wave solution to the field equations.

Our best solution so far was of the form

where f=Asin (z-t). This doesn't seem likely to be an exact solution for large amplitudes, since the x and y coordinates are treated asymmetrically. In the extreme case of , there would be singularities in g_yy, but not in g_xx. Clearly the metric will have to have some kind of nonlinear dependence on f, but we just haven't found quite the right nonlinear dependence. Suppose we try something of this form:

ds² = dt² - (1+f+cf²)dx² - (1-f+df²)dy² - dz²

This approximately conserves volume, since (1+f+…)(1-f+…) equals unity, up to terms of order f². The following program tests this form.

load(ctensor);
ct_coords:[t,x,y,z];
f : A*exp(%i*k*(z-t)); 
lg:matrix([1,0,0,0],
          [0,-(1+f+c*f^2),0,0],      
          [0,0,-(1-f+d*f^2),0],     
          [0,0,0,-1]);
cmetric();
einstein(true);

In line 3, the motivation for using the complex exponential rather than a sine wave in f is the usual one of obtaining simpler expressions; as we'll see, this ends up causing problems. In lines 5 and 6, the symbols c and d have not been defined, and have not been declared as depending on other variables, so Maxima treats them as unknown constants. The result is G_tt ∼ (4 d + 4 c - 3)A² for small A, so we can make the A² term disappear by an appropriate choice of d and c. For symmetry, we choose c=d=3/8. With these values of the constants, the result for G_tt is of order A⁴. This technique can be extended to higher and higher orders of approximation, resulting in an exact series solution to the field equations.

Unfortunately, the whole story ends up being too good to be true. The resulting metric has complex-valued elements. If general relativity were a linear field theory, then we could apply the usual technique of forming linear combinations of expressions of the form e^+i… and e^-i…, so as to give a real result. Unfortunately the field equations of general relativity are nonlinear, so the resulting linear combination is no longer a solution. The best we can do is to make a non-oscillatory real exponential solution (problem 2).

load(ctensor);
ct_coords:[t,x,y,z];
depends(p,[z,t]);
depends(q,[z,t]);
lg:matrix([1,0,0,0],
          [0,-p^2,0,0],
          [0,0,-q^2,0],
          [0,0,0,-1]);
cmetric();
einstein(true);

9.2.5 Rate of radiation

How can we find the rate of gravitational radiation from a system such as the Hulse-Taylor pulsar?

Let's proceed by analogy. The simplest source of sound waves is something like the cone of a stereo speaker. Since typical sound waves have wavelengths measured in meters, the entire speaker is generally small compared to the wavelength. The speaker cone is a surface of oscillating displacement x=x_osinω t. Idealizing such a source to a radially pulsating spherical surface, we have an oscillating monopole that radiates sound waves uniformly in all directions. To find the power radiated, we note that the velocity of the source-surface is proportional to x_oω, so the kinetic energy of the air immediately in contact with it is proportional to ω² x_o². The power radiated is therefore proportional to ω² x_o².

In electromagnetism, conservation of charge forbids the existence of an oscillating electric monopole. The simplest radiating source is therefore an oscillating electric dipole D=D_osinω t. If the dipole's physical size is small compared to a wavelength of the radiation, then the radiation is an inefficient process; at any point in space, there is only a small difference in path length between the positive and negative portions of the dipole, so there tends to be strong cancellation of their contributions, which were emitted with opposite phases. The result is that the wave's electromagnetic potential four-vector (section 4.2.5) is proportional to D_oω, the fields to D_oω², and the radiated power to D_o²ω⁴. The factor of ω⁴ can be broken down into (ω²)(ω²), where the first factor of ω² occurs for reasons similar to the ones that explain the ω² factor for the monopole radiation of sound, while the second ω² arises because the smaller ω is, the longer the wavelength, and the greater the inefficiency in radiation caused by the small size of the source compared to the wavelength.

Since our universe doesn't seem to have particles with negative mass, we can't form a gravitational dipole by putting positive and negative masses on opposite ends of a stick --- and furthermore, such a stick will not spin freely about its center, because its center of mass does not lie at its center! In a more realistic system, such as the Hulse-Taylor pulsar, we have two unequal masses orbiting about their common center of mass. By conservation of momentum, the mass dipole moment of such a system is constant, so we cannot have an oscillating mass dipole. The simplest source of gravitational radiation is therefore an oscillating mass quadrupole, Q=Q_osinω t. As in the case of the oscillating electric dipole, the radiation is suppressed if, as is usually the case, the source is small compared to the wavelength. The suppression is even stronger in the case of a quadrupole, and the result is that the radiated power is proportional to Q_o²ω⁶.

This result has the interesting property of being invariant under a rescaling of coordinates. In geometrized units, mass, distance, and time all have the same units, so that Q_o² has units of (length³)² while ω⁶ has units of (length)^-6. This is exactly what is required, because in geometrized units, power is unitless, energy/time=length/length=1.

We can also tie the ω⁶ dependence to our earlier argument, on p. 298, for the dissipation of energy by gravitational waves. The argument was that gravitating bodies are subject to time-delayed gravitational forces, with the result that orbits tend to decay. This argument only works if the forces are time-varying; if the forces are constant over time, then the time delay has no effect. For example, in the semi-Newtonian limit the field of a sheet of mass is independent of distance from the sheet. (The electrical analog of this fact is easily proved using Gauss's law.) If two parallel sheets fall toward one another, then neither is subject to a time-varying force, so there will be no dissipation of energy. In general, we expect that there will be no gravitational radiation from a particle unless the third derivative of its position d³ x/dt³ is nonzero. (The same is true for electric quadrupole radiation.) In the special case where the position oscillates sinusoidally, the chain rule tells us that taking the third derivative is equivalent to bringing out a factor of ω³. Since the amplitude of gravitational waves is proportional to d³ x/dt³, their energy varies as (d³ x/dt³)², or ω⁶.

The general pattern we have observed is that for multipole radiation of order m (0=monopole, 1=dipole, 2=quadrupole), the radiated power depends on ω^2(m+1). Since gravitational radiation must always have m=2 or higher, we have the very steep ω⁶ dependence of power on frequency. This demonstrates that if we want to see strong gravitational radiation, we need to look at systems that are oscillating extremely rapidly. For a binary system with unequal masses of order m, with orbits having radii of order r, we have Q_o∼ mr². Newton's laws give ω∼ m^1/2 r^-3/2, which is essentially Kepler's law of periods. The result is that the radiated power should depend on (m/r)⁵. Reinserting the proper constants to give an equation that allows practical calculation in SI units, we have

where k is a unitless constant of order unity.

For the Hulse-Taylor pulsar,¹² we have m ∼ 3× 10³⁰ kg (about one and a half solar masses) and r ∼ 10⁹ m. The binary pulsar is made to order our purposes, since m/r is extremely large compared to what one sees in almost any other astronomical system. The resulting estimate for the power is about 10²⁴ watts.

The pulsar's period is observed to be steadily lengthening at a rate of α=2.418× 10^-12 seconds per second. To compare this with our crude theoretical estimate, we take the Newtonian energy of the system Gm²/r and multiply by ωα, giving 10²⁵ W, which checks to within an order of magnitude. A full general-relativistic calculation reproduces the observed value of α to within the 0.1% error bars of the data.

Homework Problems

2. Show that the metric ds² = dt² -Adx² -Bdy² - dz² with

is an approximate solution to the vacuum field equations, provided that k is real --- which prevents this from being a physically realistic, oscillating wave. Find the next nonvanishing term in each series.