9	Gravitational radiation

9.1 T h e p ro p a g a t i o n o f g ra v i t a t i o n a l wa v e s

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

− ∂t2 +			¯	=
	∂2	2	hαβ		0.	(9.1)

In this chapter we do not neglect ∂2/∂t2. Eq. (9.1) is called the three-dimensional wave equation. We shall show that it has a (complex) solution of the form

¯	=		α
hαβ		Aαβ exp (ik		xα ),	(9.2)

where {kα } are the (real) constant components of some one-form and {Aαβ } 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

η

μν hαβ

=

0,

¯

,μν

and, from Eq. (9.2), we have

¯

=

μ ¯

,μ

hαβ

ik

hαβ .

Therefore, Eq. (9.3) becomes

η

μν hαβ

ημν k

k

ν

hαβ

=

0.

¯

,μν

= −

μ

¯

(9.3)

(9.4)

This can vanish only if

ημν kμkν = kν kν = 0.

(9.5)

So Eq. (9.2) gives a solution to Eq. (9.1) if kα is a null one-form or, equivalently, if the associated four-vector kα 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 ημν , so kα is a Minkowski

204	Gravitational radiation

¯αβ

null vector.) Eq. (9.2) describes a wavelike solution. The value of h hypersurface on which kα xα is constant:

kα xα = k0t + k · x = const.,

is constant on a

(9.6)

where k refers to {ki}. It is conventional to refer to k0 as ω, which is called the frequency of the wave:

→	(ω, k)	(9.7)
k
k. Imagine a photon moving in the direction of the null
is the time–space decomposition of
k. It travels on a curve
vector
xμ(λ) = kμλ + lμ,		(9.8)

where λ is a parameter and lμ is a constant vector (the photon’s position at λ = 0). From Eqs. (9.8) and (9.5), we find

kμxμ(λ) = kμlμ = 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

k is its direction of travel. The nullity of k implies
the speed of light, and
ω2 = |k|2,	(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

hαβ		=	0,	(9.11)
¯	,β
the consequences of which we must therefore consider. From Eq. (9.4), we find
Aαβ kβ = 0,				(9.12)
αβ

which is a restriction on A : it must be orthogonal to k.

The solution Aαβ exp(ikμxμ) is called a plane wave. (Of course, in physical applications, we use only the real part of this expression, allowing Aαβ 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 t ransverse -traceless gauge

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

− ∂t2 + 2			ξα = 0.	(9.13)
	∂2

205

9.1 The propagation of gravitational waves

Let us choose a solution

ξα = Bα exp(ikμxμ),

(9.14)

where Bα is a constant and kμ is the same null vector as for our wave solution. This

produces a change in hαβ , given by Eq. (8.24),

hαβ(NEW) = hαβ(OLD) − ξα,β − ξβ,α

(9.15)

h

and a consequent change in ¯αβ , given by Eq. (8.34),

h(NEW)

h(OLD)

−

ξ

α,β −

ξ

β,α +

η

αβ

ξ μ

.

(9.16)

¯αβ

= ¯αβ

,μ

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

Aαβ(NEW) = Aαβ(OLD) − i Bα kβ − i Bβ kα + i ηαβ Bμkμ.

(9.17)

In Exer. 5, § 9.7, it is shown that Bα can be chosen to impose two further restrictions

on A(NEW)	:
αβ
	Aα α = 0	(9.18)

and

Aαβ Uβ = 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 components of Aαβ must be physically important. Notice, by the way, that the trace condition, Eq. (9.18), implies (see Eq. (8.29))

hTT	=	hTT.	(9.20)
¯αβ		αβ

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 Uβ = δβ 0. Then Eq. (9.19) implies Aα0 = 0 for all α. In this frame, let us orient our spatial coordinate axes so that the wave is traveling

k	(ω, 0, 0, ω). Then, with Eq. (9.19), Eq. (9.12) implies A	αz =	0 for
in the z direction, →

all α. (This is the origin of the adjective ‘transverse’ for the gauge: Aμν is ‘across’ the direction of propagation ez.) These two restrictions mean that only Axx, Ayy, and Axy = Ayx are nonzero. Moreover, the trace condition, Eq. (9.18), implies Axx = −Ayy. In matrix form, we therefore have in this specially chosen frame

206			Gravitational radiation
			=		0	0	0	0
		αβ			0	Axy	−Axx	0
		(ATT)			0	Axx	Axy	0	.		(9.21)
					0	0	0	0
	There are only two independent			constants,			ATT and	ATT		. What	is their physical
							xx		xy

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 hTTxx and hTTxy . 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 Uα in Eq. (9.19) is the initial four-velocity of the particle). A free particle obeys the geodesic equation, Eq. (7.9),

	d
		Uα + α μν UμUν = 0.	(9.22)
	dτ
Since the particle is initially at rest, the initial value of its acceleration is
(dUα /dτ )0 = − α 00 = − 21 ηαβ (hβ0,0 + h0β,0 − h00,β ).			(9.23)

But by Eq. (9.21), hTTβ0 vanishes, so initially the acceleration vanishes. This means the particle 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 ≡ |ds2|1/2 = |gαβ dxα dxβ |1/2

' ε

= |gxx|1/2 dx ≈ |gxx(x = 0)|1/2ε

0
≈ [1 + 21 hxxTT(x = 0)]ε.	(9.24)

Now, since hTTxx is not generally zero, the proper distance (as opposed to the coordinate distance) 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 unambiguously 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 hTTij . 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 1021. 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 α 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

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

ξ

have not associated the components of in TT-coordinates with the result of any measure-

ξ

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 originates. In this frame,

ξ

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 do indeed

ξ

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		U		U ξ α , where we are calling the tangent to the geodesic U here instead of
				ξ just gives d ξ α /d τ . But the second derivative is a

V. Now, the first derivative acting on

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,

d2
dτ 2 ξ α = Rα μνβ UμUν ξ β ,	(9.25)

208						Gravitational radiation
	where U		=
				dx/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 Uα that
	depend on hμν will give terms second order in hμν in the above equation (because Rα μνβ is
								ξ	(0, ε, 0, 0). Then,
	already first order in hμν ). Therefore, U → (1, 0, 0, 0) and, initially, →
	to first order in hμν , Eq. (9.25) reduces to
					d2		∂2
						ξ α =		ξ α = εRα 00x = −εRα 0x0.	(9.26)
					dτ 2		∂t2

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 α as the proper lengths of the components of the connecting

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

ξ

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

Rx0x0 Ry0x0 Ry0y0

= Rx0x0		= − 21 h xxTT,00,
=	Ry0y0	= −	1	h	TT	x
						R 0x0,		(9.27)
=	Ry0x0		21 h xyTT,00,
		= − 2			yy,00, = −

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

ticles initially separated in the x direction have a separation vector whose components’

ξ

proper lengths obey

∂2

∂2

∂2

∂2

ξ x =

21

ε

hxxTT,

ξ y =

21

ε

hxyTT.

(9.28a)

∂t2

∂t2

∂t2

∂t2

This is clearly consistent with Eq. (9.24). Similarly, two particles initially separated by ε in the y direction obey

	∂2					∂2				∂2
			ξ y =	21	ε		hyyTT	= − 21	ε		hxxTT,
	∂t2					∂t2				∂t2
	∂2					∂2
		ξ x =		21	ε		hxyTT.				(9.28b)
	∂t2					∂t2

Remember, from Eq. (9.21), that hTTyy = −hTTxx .