Schutz B. A first course in general relativity

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Gravitational radiation

9.1 The propagation of gravitational waves

It may happen that in a region of spacetime the gravitational ﬁeld is weak but not stationary.

This can happen far from a fully relativistic source undergoing rapid changes that took

place long enough ago for the disturbances produced by the changes to reach the distant

region under consideration. We shall study this problem by using the weak-ﬁeld equations

developed in the last chapter, but ﬁrst we study the solutions of the homogeneous system of

equations that we excluded from the Newtonian treatment in § 8.4. The Einstein equations

Eq. (8.42), in vacuum (T

μν

= 0) far outside the source of the ﬁeld, are



−

∂

∂t

+∇



αβ

= 0. (9.1)

In this chapter we do not neglect ∂

/∂t

.Eq.(9.1) is called the three-dimensional wave

equation. We shall show that it has a (complex) solution of the form

αβ

= A

αβ

exp (ik

), (9.2)

where {k

} are the (real) constant components of some one-form and {A

αβ

} the (complex)

constant components of some tensor. (In the end we shall take the real part of any complex

solutions.) Eq. (9.1) can be written as

μν

αβ

,μν

= 0, (9.3)

and, from Eq. (9.2), we have

αβ

,μ

= ik

αβ

. (9.4)

Therefore, Eq. (9.3) becomes

μν

αβ

,μν

=−η

μν

αβ

= 0.

This can vanish only if

μν

= k

= 0. (9.5)

So Eq. (9.2) gives a solution to Eq. (9.1)ifk

is a null one-form or, equivalently, if the

associated four-vector k

is null, i.e. tangent to the world line of a photon. (Recall that

we raise and lower indices with the ﬂat-space metric tensor η

μν

,sok

is a Minkowski

204 Gravitational radiation

null vector.) Eq. (9.2) describes a wavelike solution. The value of

αβ

is constant on a

hypersurface on which k

is constant:

= k

t +k · x = const., (9.6)

where k refers to {k

}. It is conventional to refer to k

as ω, which is called the frequency

of the wave:



k → (ω, k) (9.7)

is the time–space decomposition of



k. Imagine a photon moving in the direction of the null

vector



k.Ittravelsonacurve

(λ) = k

λ + l

, (9.8)

where λ is a parameter and l

is a constant vector (the photon’s position at λ = 0). From

Eqs. (9.8) and (9.5), we ﬁnd

(λ) = k

= const. (9.9)

Comparing this with Eq. (9.6), we see that the photon travels with the gravitational wave,

staying forever at the same phase. We express this by saying that the wave itself travels at

the speed of light, and



k is its direction of travel. The nullity of



k implies

=|k|

, (9.10)

which is referred to as the dispersion relation for the wave. Readers familiar with wave

theory will immediately see from Eq. (9.10) that the wave’s phase velocity is one, as is its

group velocity, and that there is no dispersion.

The Einstein equations only assume the simple form, Eq. (9.1), if we impose the gauge

condition

αβ

,β

= 0, (9.11)

the consequences of which we must therefore consider. From Eq. (9.4), we ﬁnd

αβ

= 0, (9.12)

which is a restriction on A

αβ

: it must be orthogonal to



The solution A

αβ

exp(ik

)iscalledaplane wave. (Of course, in physical applications,

we use only the real part of this expression, allowing A

αβ

to be complex.) By the theorems

of Fourier analysis, any solution of Eqs. (9.1) and (9.11) is a superposition of plane wave

solutions (see Exer. 3, § 9.7).

The transverse-traceless gauge

We so far have only one constraint, Eq. (9.12), on the amplitude A

αβ

, but we can use our

gauge freedom to restrict it further. Recall from Eq. (8.38) that we can change the gauge

while remaining within the Lorentz class of gauges using any vector solving



−

∂

∂t

+∇



= 0. (9.13)

205 9.1 The propagation of gravitational waves

Let us choose a solution

= B

exp(ik

), (9.14)

where B

is a constant and k

is the same null vector as for our wave solution. This

produces a change in h

αβ

, given by Eq. (8.24),

(NEW)

αβ

= h

(OLD)

αβ

− ξ

α,β

− ξ

β,α

(9.15)

and a consequent change in

αβ

, given by Eq. (8.34),

(NEW)

αβ

(OLD)

αβ

− ξ

α,β

− ξ

β,α

+ η

αβ

,μ

. (9.16)

Using Eq. (9.14) and dividing out the exponential factor common to all terms gives

(NEW)

αβ

= A

(OLD)

αβ

− i B

+ i η

αβ

. (9.17)

In Exer. 5, § 9.7, it is shown that B

can be chosen to impose two further restrictions

on A

(NEW)

αβ

= 0 (9.18)

and

αβ

= 0, (9.19)

where



U is some ﬁxed four-velocity, i.e. any constant timelike unit vector we wish to

choose. Eqs. (9.12), (9.18), and (9.19) together are called the transverse–traceless (TT)

gauge conditions. (The word ‘traceless’ refers to Eq. (9.18); ‘transverse’ will be explained

below.) We have now used up all our gauge freedom, so any remaining independent com-

ponents of A

αβ

must be physically important. Notice, by the way, that the trace condition,

Eq. (9.18), implies (see Eq. (8.29))

αβ

= h

αβ

. (9.20)

Let us go to a Lorentz frame for the background Minkowski spacetime (i.e. make a

background Lorentz transformation), in which the vector



U upon which we have based

the TT gauge is the time basis vector U

= δ

. Then Eq. (9.19) implies A

α0

= 0for

all α. In this frame, let us orient our spatial coordinate axes so that the wave is traveling

in the z direction,



k → (ω,0,0,ω). Then, with Eq. (9.19), Eq. (9.12) implies A

αz

= 0for

all α. (This is the origin of the adjective ‘transverse’ for the gauge: A

μν

is ‘across’ the

direction of propagation e

.) These two restrictions mean that only A

, A

, and A

= A

are nonzero. Moreover, the trace condition, Eq. (9.18), implies A

=−A

. In matrix form,

we therefore have in this specially chosen frame

206 Gravitational radiation

αβ

) =

⎛

⎜

⎝

00 0 0

0 A

−Axx 0

00 0 0

⎞

⎟

⎠

. (9.21)

There are only two independent constants, A

and A

. What is their physical

signiﬁcance?

The effect of waves on free particles

As we remarked earlier, any wave is a superposition of plane waves; if the wave travels in

the z direction, we can put all the plane waves in the form of Eq. (9.21), so that any wave

has only the two independent components h

and h

. Consider a situation in which

a particle initially in a wave-free region of spacetime encounters a gravitational wave.

Choose a background Lorentz frame in which the particle is initially at rest, and choose

the TT gauge referred to in this frame (i.e. the four-velocity U

in Eq. (9.19) is the initial

four-velocity of the particle). A free particle obeys the geodesic equation, Eq. (7.9),

dτ

+ 

μν

= 0. (9.22)

Since the particle is initially at rest, the initial value of its acceleration is

(dU

/dτ )

=−

=−

αβ

β0,0

+ h

0β,0

− h

00,β

). (9.23)

But by Eq. (9.21), h

β0

vanishes, so initially the acceleration vanishes. This means the parti-

cle will still be at rest a moment later, and then, by the same argument, the acceleration will

still be zero a moment later. The result is that the particle remains at rest forever, regardless

of the wave! However, being ‘at rest’ simply means remaining at a constant coordinate

position, so we should not be too hasty in its interpretation. All we have discovered is that

by choosing the TT gauge – which means making a particular adjustment in the ‘wiggles’

of our coordinates – we have found a coordinate system that stays attached to individual

particles. This in itself has no invariant geometrical meaning.

To get a better measure of the effect of the wave, let us consider two nearby particles,

one at the origin and another at x = ε, y = z = 0, both beginning at rest. Both then remain

at these coordinate positions, and the proper distance between them is

 l ≡

|ds

1/2

αβ

1/2

dx ≈|g

(x = 0)|

1/2

≈ [1 +

(x = 0)]ε. (9.24)

Now, since h

is not generally zero, the proper distance (as opposed to the coordinate dis-

tance) does change with time. This is an illustration of the difference between computing

a coordinate-dependent number (the position of a particle) and a coordinate-independent

207 9.1 The propagation of gravitational waves

number (the proper distance between two particles). The effect of the wave is unam-

biguously seen in the coordinate-independent number. The proper distance between two

particles can be measured: we will discuss two ways of measuring it in the paragraph

on ‘Measuring the stretching of space’ below. The physical effects of gravitational ﬁelds

always show up in measurables.

Equation (9.24) tells us a lot. First, the change in the distance between two particles

is proportional to their initial separation ε. Gravitational waves create a bigger distance

change if the original distance is bigger. This is the reason that modern gravitational

wave detectors, which we discuss below, are designed and built on huge scales, measuring

changes in separations over many kilometers (for ground-based detectors) or millions of

kilometers (in space). The second thing we learn from Eq. (9.24) is that the effect is small,

proportional to h

. We will see when we study the generation of waves below that these

dimensionless components are typically 10

−21

or smaller. So gravitational wave detectors

have to sense relative distance changes of order one part in 10

. This is the experimental

challenge that was achieved for the ﬁrst time in 2005, and improvements in sensitivity are

continually being made.

Tidal accelerations: gravitational wave forces

Another approach to the same question of how gravitational waves affect free particles

involves the equation of geodesic deviation, Eq. (6.87). This will lead us, in the following

paragraph, to a way of understanding the action of gravitational waves as a tidal force on

particles, whether they are free or not.

Consider again two freely falling particles, and set up the connecting vector ξ

between

them. If we were to work in a TT-coordinate system, as in the previous paragraph, then

the fact that the particles remain at rest in the coordinates means that the components of



ξ would remain constant; although correct, this would not be a helpful result since we

have not associated the components of



ξ in TT-coordinates with the result of any measure-

ment. Instead we shall work in a coordinate system closely associated with measurements,

the local inertial frame at the point of the ﬁrst geodesic where



ξ originates. In this frame,

coordinate distances are proper distances, as long as we can neglect quadratic terms in

the coordinates. That means that in these coordinates the components of



ξ do indeed

correspond to measurable proper distances if the geodesics are near enough to one another.

What is more, in this frame the second covariant derivative in Eq. (6.87) simpliﬁes. It

starts out as ∇

∇

, where we are calling the tangent to the geodesic



U here instead of



V. Now, the ﬁrst derivative acting on



ξ just gives d ξ

/d τ . But the second derivative is a

covariant one, and should contain not just d/d τ but also a term with a Christoffel symbol.

But in this local inertial frame the Christoffel symbols all vanish at this point, so the second

derivative is just an ordinary second derivative with respect to tau. The result is, again in

the locally inertial frame,

dτ

= R

μνβ

, (9.25)

208 Gravitational radiation

where



U = dx/dτ is the four-velocity of the two particles. In these coordinates the compo-

nents of



U are needed only to lowest (i.e. ﬂat-space) order, since any corrections to U

that

depend on h

μν

will give terms second order in h

μν

in the above equation (because R

μνβ

already ﬁrst order in h

μν

). Therefore,



U → (1, 0, 0, 0) and, initially,



ξ → (0, ε, 0, 0). Then,

to ﬁrst order in h

μν

,Eq.(9.25) reduces to

dτ

∂

∂t

= εR

00x

=−εR

0x0

. (9.26)

This is the fundamental result, which shows that the Riemann tensor is locally measurable

by simply watching the proper distance changes between nearby geodesics.

Now, the Riemann tensor is itself gauge invariant, so its components do not depend

on the choice we made between a local inertial frame and the TT coordinates. It follows

also that the left-hand side of Eq. (9.26) must have an interpretation independent of the

coordinate gauge. We identify ξ

as the proper lengths of the components of the connecting

vector



ξ, in other words the proper distances along the four coordinate directions over the

coordinate intervals spanned by the vector. With this interpretation, we free ourselves from

the choice of gauge and arrive at a gauge-invariant interpretation of the whole of Eq. (9.26).

Just to emphasize that we have restored gauge freedom to this equation, let us write the

Riemann tensor components in terms of the components of the metric in TT gauge. This

is possible, since the Riemann components are gauge-invariant. And it is desirable, since

these components are particularly simple in the TT gauge. It is not hard to use Eq. (8.25)

to show that, for a wave traveling in the z-direction, the components are

0x0

= R

x0x0

=−

xx,00,

0x0

= R

y0x0

=−

xy,00,

0y0

= R

y0y0

=−

yy,00,

=−R

0x0,

⎫

⎪

⎬

⎪

⎭

(9.27)

with all other independent components vanishing. This means, for example, that two par-

ticles initially separated in the x direction have a separation vector



ξ whose components’

proper lengths obey

∂

∂t

∂

∂t

∂

∂t

∂

∂t

. (9.28a)

This is clearly consistent with Eq. (9.24). Similarly, two particles initially separated by ε

in the y direction obey

∂

∂t

∂

∂t

=−

∂

∂t

∂

∂t

∂

∂t

. (9.28b)

Remember, from Eq. (9.21), that h

=−h

209 9.1 The propagation of gravitational waves

Measuring the stretching of space

The action of gravitational waves is sometimes characterized as a stretching of space.

Eq. (9.24) makes it clear what this means: as the wave passes through, the proper sepa-

rations of free objects that are simply sitting at rest change with time. However, students of

general relativity sometimes ﬁnd this concept confusing. A frequent question is, if space

is stretched, why is a ruler (which consists, after all, mostly of empty space with a few

electrons and nuclei scattered through it) not also stretched, so that the stretching is not

measurable by the ruler? The answer to this question lies not in Eq. (9.24) but in the

geodesic deviation equation, Eq. (9.26).

Although the two formulations of the action of a gravitational wave, Eqs. (9.24)

and (9.26), are essentially equivalent for free particles, the second one is far more use-

ful and instructive when we consider the behavior of particles that have other forces acting

on them as well. The ﬁrst formulation is the complete solution for the relative motion of

particles that are freely falling, but it gives no way of including other forces. The second

formulation is not a solution but a differential equation. It shows the acceleration of one

particle (let’s call it B), induced by the wave, as measured in a freely falling local inertial

frame that initially coincides with the motion of the other particle (let’s call it A). It says

that, in this local frame, the wave acts just like a force pushing on B. This is called the tidal

force associated with the wave. This force depends on the separation



ξ.FromEq.(9.25), it

is clear that the acceleration resulting from this effective force is

∂

∂t

=−R

0j0

. (9.29)

Now, if particle B has another force on it as well, say



, then to get the complete motion of

B we must simply solve Newton’s second law with all forces included. This means solving

the differential equation

∂

∂t

=−R

0j0

, (9.30)

where m

is the mass of particle B. Indeed, if particle A also has a force



on it, then it

will not remain at rest in the local inertial frame. But we can still make measurements in

that frame, and in this case the separation of the two particles will obey

∂

∂t

=−R

0j0

−

. (9.31)

This equation allows us to treat material systems acted on by gravitational waves. The

continuum version of it can be used to understand how a ‘bar’ detector of gravitational

waves, which we will encounter below, reacts to an incident wave. And it allows us to

answer the question of what happens to a ruler when the wave hits. Since the atoms in

the ruler are not free, but instead are acted upon by electric forces from nearby atoms, the

ruler will stretch by an amount that depends on how strong the tidal gravitational forces are

compared to the internal binding forces. Now, gravitational forces are very weak compared

to electric forces, so in practice the ruler does not stretch at all. In this way the ruler can be

210 Gravitational radiation

(a) (b) (c)

t = t

Figure 9.1

(a) A circle of free particles before a wave traveling in the z direction reaches them. (b)

Distortions of the circle produced by a wave with the ‘+’ polarization. The two pictures represent

the same wave at phases separated by 180

◦

. Particles are positioned according to their proper

distances from one another. (c) As (b) for the ‘ ×’ polarization.

used to measure the tidal displacement of nearby free particles, in other words to measure

the ‘stretching of space’.

There are other ways of measuring the stretching. One of the most important in practice

is to send photons back and forth between the free particles and measure the changes

in the light–travel time between them. This is the principle of the laser interferometer

gravitational wave detector, and we will discuss it in some detail below.

Polarization of gravitational waves

Eqs. (9.28a) and (9.28b) form the foundation of the deﬁnition of the polarization of a

gravitational wave. Consider a ring of particles initially at rest in the x − y plane, as in

Fig. 9.1(a). Suppose a wave has h

= 0, h

= 0. Then the particles will be moved (in

terms of proper distance relative to the one in the center) in the way shown in Fig. 9.1(b),

as the wave oscillates and h

=−h

changes sign. If, instead, the wave had h

= 0but

= h

= 0, then the picture would distort as in Fig. 9.1(c). Since h

and h

are inde-

pendent, (b) and (c) provide a pictorial representation for two different linear polarizations.

Notice that the two states are simply rotated 45

◦

relative to one another. This contrasts

with the two polarization states of an electromagnetic wave, which are 90

◦

to each other.

As Exer. 16, § 9.7, shows, this pattern of polarization is due to the fact that gravity is

represented by the second-rank symmetric tensor h

μν

. (By contrast, electromagnetism is

represented by the vector potential A

of Exer. 11, § 8.6.)

An exact plane wave

Although all waves that we can expect to detect on Earth are so weak that linearized theory

ought to describe them very accurately, it is interesting to see if the linear plane wave

211 9.1 The propagation of gravitational waves

corresponds to some exact solution of the nonlinear equations that has similar properties.

We shall brieﬂy derive such a solution.

Suppose the wave is to travel in the z direction. By analogy with Eq. (9.2), we might

hope to ﬁnd a solution that depends only on

u := t −z.

This suggests using u as a coordinate, as we did in Exer. 34, § 3.10 (with x replaced by z).

In ﬂat space it is natural, then, to deﬁne a complementary null coordinate

v = t + z,

so that the line element of ﬂat spacetime becomes

=−du dv + dx

+ dy

Now, we have seen that the linear wave affects only proper distances perpendicular to its

motion, i.e. in the x −y coordinate plane. So let us look for a nonlinear generalization of

this, i.e. for a solution with the metric

=−du dv + f

(u)dx

+ g

(u)dy

where f and g are functions to be determined by Einstein’s equations. It is a straightforward

calculation to discover that the only nonvanishing Christoffel symbols and Riemann tensor

components are



f /f , 

=˙g/g,



= 2

f /f , 

= 2˙g/g,

uxu

=−

f /f , R

uyu

=−¨g/g,

and others obtainable by symmetries. Here, dots denote derivatives with respect to u.The

only vacuum ﬁeld equation then becomes

f /f +¨g/g = 0. (9.32)

We can therefore prescribe an arbitrary function g(u) and solve this equation for f (u). This

is the same freedom as we had in the linear case, where Eq. (9.2) can be multiplied by

an arbitrary f (k

) and integrated over k

to give the Fourier representation of an arbitrary

function of (z − t). In fact, if g is nearly 1,

g ≈ 1 +ε(u),

so we are near the linear case, then Eq. (9.32) has a solution

f ≈ 1 − ε(a).

This is just the linear wave in Eq. (9.21), with plane polarization with h

= 0, i.e. the

polarization shown in Fig. 9.1(b). Moreover, it is easy to see that the geodesic equation

212 Gravitational radiation

implies, in the nonlinear case, that a particle initially at rest on our coordinates remains

at rest. We have, therefore, a simple nonlinear solution corresponding to our approximate

linear one.

This solution is one of a class called plane-fronted waves with parallel rays. See Ehlers

and Kundt (1962), § 2.5, and Stephani, et al. (2003), § 24.5.

Geometrical optics: waves in a curved spacetime

In this chapter we have made the simplifying assumption that our gravitational waves are

the only gravitational ﬁeld, that they are perturbations of ﬂat spacetime, starting from

Eq. (8.12). We found that they behave like a wave ﬁeld moving at the speed of light in

special relativity. But the real universe contains other gravitational ﬁelds, and gravitational

waves have to make their way to our detectors through the ﬁelds of stars, galaxies, and the

universe as a whole. How do they move?

The answer comes from studying waves as perturbations of a curved metric, of the form

αβ

+ h

αβ

, where g could be the metric created by any combination of sources of gravity.

The computation of the dynamical equation governing h is very similar to the one we went

through at the beginning of this chapter, but the mathematics of curved spacetime must be

used. We won’t go into the details here, but it is important to understand qualitatively that

the results are very similar to the results we would get for electromagnetic waves traveling

through complicated media.

If the waves have short wavelength, then they basically follow a null geodesic, and they

parallel-transport their polarization tensor. This is exactly the same as for electromagnetic

waves, so that photons and gravitational waves leaving the same source at the same time

will continue to travel together through the universe, provided they move through vacuum.

For this geometrical optics approximation to hold, the wavelength has to be short in two

ways. It must be short compared to the typical curvature scale, so that the wave is merely a

ripple on a smoothly curved background spacetime; and its period must be short compared

to the timescale on which the background gravitational ﬁelds might change. If nearby null

geodesics converge, then gravitational and electromagnetic waves traveling on them will

be focussed and become stronger. This is called gravitational lensing, and we will see an

example of it in Ch. 11.

Gravitational waves do not always keep step with their electromagnetic counterparts.

Electromagnetic waves are strongly affected by ordinary matter, so that if their null

geodesic passes through matter, they can suffer additional lensing, scattering, or absorp-

tion. Gravitational waves are hardly disturbed by matter at all. They follow the null

geodesics even through matter. The reason is the weakness of their interaction with matter,

as we saw in Eq. (9.24). If the wave amplitudes h

are small, then their effect on any

matter they pass through is also small, and the back-effect of the matter on them will be

of the same order of smallness. Gravitational waves are therefore highly prized carriers of

information from distant regions of the universe: we can in principle use them to ‘see’ into

the centers of supernova explosions, through obscuring dust clouds, or right back to the

ﬁrst fractions of a second after the Big Bang.