Chapter 21 Gravitational Waves - University of Tennesseeeagle.phys.utk.edu/guidry/astro616/lectures/lecture_ch21.pdf · Chapter 21 Gravitational Waves Just as • Maxwell’s theory

Chapter 21

Gravitational Waves

Just as

• Maxwell’s theory permits the existence of propagat-ing waves in the electromagnetic field,

• General relativity predicts that fluctuations in themetric of spacetime can propagate asgravitationalwaves.

Gravitational waves are of considerable current interestfor three basic reasons:

• The existence of gravitational waves is the last fun-damental prediction of general relativity that has notbeen confirmed directly.

• Gravitational waves are capable of probing the Uni-verse back to the Planck Scale.

• Present technology may now be capable of detectinggravitational waves.

1181

1182 CHAPTER 21. GRAVITATIONAL WAVES

The Last Untested Prediction of General Relativity

Gravitational waves represent the last prediction of the general theoryof relativity that has not been tested directly.

• Although the Binary Pulsar provides strongindirect evidenceforthe emission of gravitational waves from that system, no gravi-tational wave has yet been detected directly.

• This is not because gravitational waves are expected to be rare;

• it is because they are expected to interact so weakly with matter.

• General relativity is tested by the mere existence of gravitationalwaves and by their detailed properties.

• For example, general relativity predicts that gravitational wavescan have only two states of polarization and must travel at lightvelocity.

Some alternative theories of gravity predict gravitationalwaves with as many as six polarization states and withspeeds that can be less thanc.

1183

Last Photon

Scattering

Surface

Now

Last Neutrino

Scattering

Surface

Photons

Neutrinos

Gravity Waves

Planck

Scale

100,000

Years

1 Second

10-43

Seconds

Big

Bang S

ingula

rity

Photons

Strongly

CoupledPhotons and

Neutrinos

Strongly

Coupled

Figure 21.1:Probes of the early Universe. More weakly interacting particles canbe seen from earlier epochs. In principle, gravity waves from times close to thePlanck scale might be visible in the current Universe.

The Deepest Probe

Weakness of gravitational waves makes them difficult to detect but

• That same weakness means that gravitational waves can in prin-ciple be seen from earlier epoches than can be probed directlyby observing other forms of radiation (Fig. 21.1).

• Photons decouple∼ 105 y after the big bang, so we can’t seedirect photons from earlier periods.

• Neutrinos couple more weakly and fell out of equilibrium at∼1 safter the big bang, so neutrinos could give direct informationonly back to a time only∼ 1 safter the big bang.

• In principle, gravitational waves could go back to the inflation-ary or even Planck scales.


Technology May Now Exist to Detect Gravitational Waves

A current and proposed generation of gravity wave detec-tors may have sufficient sensitivity to make the first directmeasurements of gravitational waves within the comingdecade. Thus, there is a real chance that the currently pri-marily theoretical topic of gravity waves will soon becomean experimental science.

21.1. LINEARIZED GRAVITY 1185

21.1 Linearized Gravity

The Einstein equation may be expressed in the form

Rµν = −8πG(Tµν −12gµνTλ

λ ).

• Because this equation is non-linear, general solutions that corre-spond to gravitational waves are difficult to obtain.

• In many instances we may assume that the gravitational wavesare weak, which allows the metric to be expressed in the form

gµν(x) = ηµν +hµν(x),

whereηµν is the metric of (flat) Minkowski space andhµν issmall.

• The linearized vacuum Einstein equationthen results from

– inserting the approximate metric into the vacuum Einsteinequation,

Rµν = 0

obtained by settingTµν = 0, and

– expansion of the resulting equations to first order inhµν .


21.1.1 Linearized Curvature Tensor

The Ricci curvature tensorRµν is given by

Rµν = Γλµν ,λ −Γλ

µλ ,ν +ΓλµνΓσ

λσ −Γσµλ Γλ

νσ ,

where the Christoffel symbolsΓσλ µ are related to the metric tensor

gµν by

Γσλ µ = 1

2gνσ(

∂gµν

∂xλ +∂gλν∂xµ −

∂gµλ∂xν

)

To zeroth order inhµν , the Christoffel coefficients vanish and so doesRµν , since∂g/∂x = 0 for ηµν . To first order inhµν ,

δΓγµν =

12

ηγδ(∂hδ µ

∂xν +∂hδν∂xµ −

∂hµν

∂xδ

)

.

The last two terms in the above equation forRµν are quadratic in∂hand may be discarded to first order inh, giving

δRµν =∂ (δΓγ

µν)

∂xγ −∂ (δΓγ

µγ)

∂xν +O

(

h2)

.


21.1.2 Wave Equation

Substitution and some algebra yields (Exercise)

δRµν = 12(−2hµν +∂µVν +∂νVµ),

where the 4-dimensional Laplacian (d’Alembertian operator) is de-fined by

2 ≡ ηαβ ∂α∂β = −∂ 2

∂ t2 + ∇∇∇2

with the definitions

∂µ ≡ ∂/∂xµ ∂ µ ≡ ∂/∂xµ

and where theVν are defined through

Vν ≡ ∂γhγν −

12∂νhγ

γ

= ∂γηγδ hδν −12∂νηγδ hδγ .

Note that raising and lowering of indices in linearized gravity is gen-erally accomplished through contraction with the flat-space metricηµν ,

hγν ≡ ηγδ hδν

rather than through contraction withgµν = ηµν +hµν .


Thus the vacuum Einstein equation to this order yields the wave equa-tion

2hµν −∂µVν −∂νVµ = 0.

• Sincehµν is symmetric, this constitutes a set of 10linear partialdifferential equations for the metric perturbationhµν .

• One refers to these as thelinearized vacuum Einstein equationsand to the resulting theory aslinearized gravity.

As is clear from the derivation, we may expect this to be avalid approximation to the full gravitational theory whenthe metric departs only slightly from that of flat spacetime.


21.1.3 Degrees of Freedom and Gauge Transformations

The equation


cannot yield unique solutions in its present form because ofthe free-dom of coordinate transformations:

• Given one solution (metric), we may generate another by a co-ordinate transformation.

• This ambiguity is related to a similar ambiguity in electromag-netism that is associated withfreedom to make gauge transfor-mations without altering the fields.

• The symmetric tensorhµν has10 independent componentsbutthey are related byfour Bianchi identities:

Gµν ;µ = 0.

• This leaves10−4= 6 independent equations in the 10 unknownshµν , implying 4 degrees of freedom.

• These are the coordinate transformations: ifhµν solves the lin-earized Einstein equations, so doesh′µν whereh′µν is related tohµν through a general coordinate transformationx→ x′.

• Such a transformation involves four arbitrary functionsx′µ(x).


This is analogous to the gauge ambiguity in electromag-netism,which is removed byfixing a gauge. Here wecan remove the ambiguity byfixing the coordinate system.Four coordinate conditions added to the six independentequations then permit unique solutions to


Let us examine this in a little more detail.


21.1.4 Choice of Gauge

In electromagnetism it is possible to make different choices of the(vector and scalar) potentials that give the same electric and magneticfields and thus the same classical observables.

• This freedom is associated with gauge invariance.

• Something similar is possible in linearized gravity.

• Small changes can be made in the coordinates that leaveηµνunchanged in

gµν(x) = ηµν +hµν(x),

but that alter the functional form ofhµν .

• Under small changes in the coordinates

xµ → x′µ = xµ + εµ(x),

whereεµ(x) is similar in size tohµν .

• Then,

gµν(x) = ηµν +hνµ(x)→ ηµν +h′µν(x)

= ηµν +hµν(x)−∂µεν −∂νεµ .

• The transformationhµν → h′µν , with

h′µν = hµν(x)−∂µεν −∂νεµ ,

is termed agauge transformation,by analogy with electromag-netism. Ifhµν is a solution of


so ish′µν .


Table 21.1: Gauge invariance in linearized gravity and electromagnetism

Linearized Gravity Electromagnetism†

Potentials hµν Vector potential:AAA(t,xxx)Scalar potential :Φ(t,xxx)

Fields Linearized Riemann Electric field:EEE(t,xxx)curvature:Rαβδγ(x) Magnetic field:BBB(t,xxx)

Gauge transformation hµν → hµν −∂µεν −∂νεµ Aµ → Aµ −∂ µ χ

Example of a gauge ∂νhνµ − 1

2∂µhνν = 0 ∂ µAµ = 0

condition (“Lorentz”)

Field equations 2hµν = 0 2Aµ = 0in Lorentz gauge

†The 4-vector potential isAµ ≡ (Φ,AAA) andχ is an arbitrary scalar function.

Just as

• gauge transformations in the Maxwell theory lead to

– new potentials but

– the same electric and magnetic fields,

• gauge transformations in gravity lead to

– new potentialshµν but to

– the same fields (the linearized formδRµνβγ(x)of the Riemann curvature tensor).

The analogies between coordinate (“gauge”)transformations in linearized gravity andgauge transformations in classical electro-magnetism are summarized in Table 21.1.


A standard choice permits the linearized gravitationalequations to be replaced by the two equations

2hµν(x) = 0,

∂νhνµ(x)− 1

2∂µhνν(x) = 0,

where the first is the linearized Einstein equation and thesecond is a (Lorentz) gauge constraint, and wherehγ

ν isdefined by

hγν ≡ ηγδ hδν

As in electromagnetism, the “Lorentz” gaugeis really a family of gauges. We shall use thatto simply things even further below.


21.2 Weak Gravitational Waves

Let us now seek solutions to

2hµν(x) = 0,

∂νhνµ(x)− 1

2∂µhνν(x) = 0,

We expect the solution to be a superposition of compo-nents in the form

hµν(x) = αµνeik·x +α∗µνe−ik·x

= αµνeikλ xλ+α∗

µνe−ikλ xλ

whereαµν is a symmetric4 × 4 matrix called thepolar-ization tensor.

21.2. WEAK GRAVITATIONAL WAVES 1195

21.2.1 States of Polarization

Since it is symmetric, the polarization tensor should generally have10 independent components.However, the demand that

hµν(x) = αµνeik·x +α∗µνe−ik·x

= αµνeikλ xλ+α∗

µνe−ikλ xλ

satisfy the wave equation

2hµν(x) = 0,

means thatkµkµ = 0,

and the requirement that it satisfy the gauge condition

∂νhνµ(x)− 1

2∂µhνν(x) = 0,

implies thatkµαµ

ν = 12kναµ

µ ,

wherekµ = ηµνkν and so on (raise and lower indices withηµν inlinearized gravity).


The four equationskµαµ

ν = 12kναµ

µ ,

reduce the number of independent components ofανµ to six. But we

have not yet exhausted the gauge (coordinate) degree of freedom be-cause any coordinate transform

xµ → x′µ = xµ + εµ(x) h′µν = hµν(x)−∂µεν −∂νεµ

that leaves∂νhν

µ(x)− 12∂µhν

ν(x) = 0,

valid does not alter the physical content of linearized gravity.

It is possible to use this freedom to set any four of thehµνto zero.


21.2.2 Polarization Tensor in Transverse–Traceless Gauge

We may use the freedom alluded to at the end of the preceding sectionto transform totransverse traceless gauge (TT)by choosing

h0i = hti = 0 (i = 1,2,3)

Tr h≡ hββ = 0.

In terms of the polarization tensorαµν , preceding conditions corre-spond to

α0i = 0 Trα = αββ = 0.

The gauge conditions

∂νhνµ(x)− 1

2∂µhνν(x) = 0,

with µ = 0 then require thatαtt ≡ α00 = 0 which, coupled with

α0i = 0

implies that four of theαµν vanish:

α0µ = 0.

Furthermore, fori = 1,2,3 the gauge conditions lead to the require-ment thatik jαi j eik·x = 0, which is generally true only if thetransver-sality condition,

k jαi j = 0

is satisfied.


Let us take stock. We started with 10 independent compo-nents of the symmetric polarization tensorαµν . We thenfound that

• The conditionh0i = 0 requiresα01 = α02 = α03 = 0,

• The requirement

∂νhνµ(x)− 1

2∂µhνν(x) = 0,

yieldsα00 = 0.

• Therefore, the four componentsα0µ of the symmet-ric polarization tensor vanish in TT gauge.

• The trace condition

Tr h≡ hββ = 0.

gives one constraint and the transversality condition

k jαi j = 0

gives three additional ones for a total of four.

In TT gauge tenαµν , minus fourαµν thatare identically zero, minus four constraints onαµν , leavetwo independent physical polar-izations for gravity waves.


21.2.3 Helicity Components

As we have seen, there are six independent polarizations satisfyingthe wave equation

2hµν(x) = 0,

and the general Lorentz gauge condition

∂νhνµ(x)− 1

2∂µhνν(x) = 0,

but the TT gauge shows explicitly that only two of these polarizationsare physically meaningful. Further insight comes from asking howthe αµν change under rotations of the coordinate system about thezaxis.

• One generally finds that a gravitational plane wave can be de-composed into helicity components±2, ±1, 0, and0, but

• The components with helicity0 and±1 vanish under a suitablechoice of coordinates. (Helicity is the projection of the angularmomentum on the direction of motion.)

• Thus, only the helicity components±2 are physically relevantfor gravitational waves, explaining why there are two indepen-dent physical states of polarization.

It is in this sense that we associate gravitywith a spin-2 field.


Compare with the analogous situation in electromag-netism, which is described by a 4-vector fieldAµ .

• This suggests that the Maxwell field should have 4independent states of polarizationαµ .

• However,kµαµ = 0 reduces this to 3 and

• the freedom to make gauge transformations thatleave theEEE andBBB fields unchanged demonstrates ex-plicitly that the number of independent polarizationsis actually only 2.

• Furthermore, a decomposition under rotations aboutthez axis yields helicities0 and±1, but only the he-licities ±1 are physically relevant.

• These correspond to the 2 independent states of po-larization for a massless vector (spin-1) field.

That there are only two physical states of polarization forthe photon is tied intimately to its masslessness and asso-ciated local gauge invariance.

• A massivevector field has 3 states of polarization andis not locally gauge invariant.

• Likewise, that the gravitational field exhibits only 2physical states of polarization is a consequence of themasslessness of the graviton.

• A massive spin-2 field would have 5 physical polar-ization states.


21.2.4 General Solution in Transverse–Traceless Gauge

To display polarizations explicitly, we assume the gravitational waveto propagate on thezaxis with energyω (in h= c= 1 units). We have

kµkµ = 0,

and the 4-momentum vector must have the form

kµ = (ω ,0,0,ω)

The transversality conditionk jαi j = 0 is explicitly

k1α11+k2α12+k3α13 = 0

k1α21+k2α22+k3α23 = 0

k1α31+k2α32+k3α33 = 0.

But fromkµ = (ω ,0,0,ω), we havek1 = k2 = 0, so

α13 = α23 = α33 = 0,

and from earlierα0µ = 0. Therefore, for the symmetric matrixαµνthe only nonvanishing components areα11, α12 = α21, andα22, andthese are further constrained by the trace requirement

Tr α = αββ = 0,

soα11 = −α22.


Finally, fromkµ = (ω ,0,0,ω)

we haveik ·x = −i(ωt −ωz)

and the general solution of the linearized Einstein equations forzaxispropagation with fixed frequencyω in transverse–traceless gauge is

hµν(t,z) =

0 0 0 0

0 α11 α12 0

0 α12 −α11 0

0 0 0 0

eiω(z−t),

which exhibits explicitly the transverse and traceless properties, withtwo independent polarization states.


• The part of the wave that is proportional toαxx = α11 is calledtheplus polarization(denoted by+) and

• the part proportional toαxy = α12 = α21 is called thecross po-larization (denoted by×).

• For example a purely cross-polarized plane wave propagating inthez direction may be represented as

hµν(t,z) =

0 0 0 0

0 0 1 0

0 1 0 0

0 0 0 0

eiω(z−t).


• The most general gravitational wave is a superposition of waveshaving the form

hµν(t,z) =

0 0 0 0

0 α11 α12 0

0 α12 −α11 0

0 0 0 0

eiω(z−t),

with different ω , directions of propagation, and amplitudes forthe two polarizations.

• In linearized approximation and TT gauge, it may be expressedas

hµν(t,z) =

0 0 0 0

0 f+(t −z) f×(t −z) 0

0 f×(t −z) − f+(t −z) 0

0 0 0 0

,

assuming propagation of the gravitational wave along thezaxis.

21.3. RESPONSE OF TEST PARTICLES TO GRAVITATIONAL WAVES1205

21.3 Response of Test Particles to Gravitational Waves

We cannot detect a gravitational wave locally

• In a local enough region the effects of gravity may betransformed away(equivalence principle).

• Thus the effect of a gravitational wave on a singlepoint test particle hasno measurable consequences.

Gravitational waves may be detected only by their influ-ence ontwo or more test particle at different locations.


21.3.1 Response of Two Test Masses

• Assume a linearized gravitational wave of+ polarization prop-agating on thezaxis,

hµν(t,z) =

0 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 0

f (t −z),

with a corresponding time-dependent metric

ds2 = −dt2 +(1+ f (t −z))dx2+(1− f (t −z))dy2+dz2.

• Consider two test masses with

– mass A initially at rest at the origin,xiA = (0,0,0),

– mass B initially at rest at a pointxiB = (xB,yB,zB), and

– a gravitational wave propagating on thez axis.

• Since the particles are at rest before the gravitational wave ar-rives, the initial 4-velocities are

uA = uB = (1,0,0,0).

• For the test masses the geodesic equation is given by

d2xi

dτ2 = −Γiµνuµuν = −Γi

µνdxµ

dτdxν

dτ.


The undisturbed spacetime is assumed flat andΓiµν vanishes.

• From

δΓγµν =

12

ηγδ(∂hδ µ

∂xν +∂hδν∂xµ −

∂hµν

∂xδ

)

.

we have to first order

d2(δxi)

dτ2 =−δΓiµνuµuν = −δΓi

00

=−12

η iδ(

∂hδ0

∂x0 +∂hδ0

∂x0 −∂h00

∂xδ

)

,

whereuA = uB = (1,0,0,0).

has been used.

• But in TT gaugehδ0 = 0, so

d2(δxi)

dτ2 = −δΓi00 = 0

To this order, thecoordinate distancebetween A and B isnot changedby the gravitational wave.


However, the proper distancebetween A and B ischanged.

For example, assume A and B to lie on thex axis and tobe separated by a distanceL0. From

ds2 = −dt2+(1+ f (t −z))dx2+(1− f (t −z))dy2+dz2.

the relevant line element is then

ds2 = −dt2 +(1+ f (t −z))dx2,

corresponding to the metric

gµν =

(

−1 0

0 1+hxx(t,0)

)

,

wherehxx = h11 and we have chosenz= 0 at timet.


Time

Ma

ss S

epara

tio

n L

Figure 21.2:Test mass separation perturbed by a gravitational wave.

Then the proper distance between A and B is

L(t) =∫ L0

0(−detg)1/2 dx

=∫ L0

0(1+hxx(t,0))1/2 dx

≃

∫ L0

0(1+ 1

2hxx(t,0)) dx=(

1+ 12hxx(t,0)

)

L0.

Therefore the change in distance between the masses is

δL(t)L0

≃ 12hxx(t,0),

which oscillates with the time dependence of the gravitational wave,as indicated schematically in Fig. 21.2.

The amplitude of the variation is greatly exaggerated inFig. 21.2, however! One expects thatδL/L0 ∼ 10−21 fora typical fluctuation associated with gravitational wavesdetectable on Earth from distant astronomical events.


Gravitational wave incident along z-axis

Test

ParticlesTime

Plus (+)

Polarization

Cross (x)

Polarization

x

y

Before

gravitational

wave

Figure 21.3:Effect of a gravitational wave incident along thezaxis on an array oftest masses in thex–y plane.

21.3.2 The Effect of Polarization

The effect of the gravitational wave polarization may be illustrated bythe effect of a gravitational wave on an planar test mass array.

• Like electromagnetic waves, gravitational waves are transverse;thus only separations in the transverse directions (x andy, if thewave is incident alongz) are changed by the gravitational wave.

• The effect of purely plus-polarized and purely cross-polarizedgravitational waves on an initially circular array of test massesis illustrated in Fig. 21.3.

As in Fig. 21.2, the magnitude of the effect isgreatly exaggeratedrelative to that of gravitational waves detectable on Earthfrom distantsources.


Gravitational wave incident along z-axis

Test

ParticlesTime

Plus (+)

Polarization

Cross (x)

Polarization

x

y

Before

gravitational

wave

Each polarization is seen to give rise to an elliptical oscillation inthe distribution of the test masses, with the cross polarization ellipserotated byπ

4 relative to the corresponding plus polarization ellipse.

• The45o relative rotation results from gravity being described bya rank-2 tensor field (spin-2 field).

• In electromagnetism the fields correspond to a rank-1 tensor Aµ

(spin-1 or vector field) and the rotation angle between the twoindependent states of polarization is instead90o.


Mirrored

Test Mass

Mirrored

Test Mass

Mirrored

Test Mass

Mirrored

Test Mass

Laser

Photodetector

Beam Splitter

Light Storage ArmL

igh

t Sto

rag

e A

rm

Figure 21.4:Schematic laser interferometer gravity wave detector. In the lightstorage arms light is multiply reflected, increasing the effective length of the arms.

21.4 Gravitational Wave Detectors

Gravity wave detectors tend to use Michelson laser inter-ferometers with kilometer or longer arms. A typical im-plementation is illustrated in Fig. 21.4.

• Laser light is split and directed down two arms.

• Suspended, mirrored test masses reflect the light atthe ends of the arms.

• The reflected light is recombined and interferencefringes are analyzed for evidence indicating changesin the distances to the test masses.

21.4. GRAVITATIONAL WAVE DETECTORS 1213

Laser

Detector

Mass

Mass

Test Mass

Distribution

Interferometer

Arms

x

y

z

Figure 21.5:Analogy between interaction of a gravitational wave with a test massdistribution and with an interferometer.

Because one is interfering light over long path lengths, itis possible to detect extremely small changes in distance.

• This is necessary, since to detect gravitational wavesfrom merging neutron stars or core-collapse super-novae, fractional changes in distance of order10−21

or smaller must be measured.

• 10−21 is approximately the ratio of the width of ahuman hair to the distance to Alpha Centauri!

Laser interferometers may be viewed as extremely preciseways to measure the distortions illustrated in Fig. 21.3 fora small number of test masses (Fig. 21.5).


5 x 106 km

5 x

10

6 k

m5 x 10 6 km

Sun

Earth

LISA

Figure 21.6:The proposed LISA space-based gravitational wave antenna.

Ground-based detectors such as LIGO are now beginningto operate. A proposed space-based array, LISA, wouldhave 5 million kilometer interferometer arms and could belaunched within a decade. The schematic arrangement forLISA, and its proposed orbit, are illustrated in Fig. 21.6.

21.4. GRAVITATIONAL WAVE DETECTORS 1215

Figure 21.7:Amplitude and frequency ranges expected for gravitationalwavesfrom various sources. The lower detection ranges for LIGO and LISA are indicatedby dotted lines.

The amplitude and frequency ranges for LIGO and LISA,along with ranges expected for important astrophysicalsources of gravitational waves that may be observable, areillustrated in Fig. 21.7. Space-based interferometers likeLISA can go to much lower frequencies because they canhave very long interferometer arms and because there islittle interference from environmental noise.


The preceding discussion introduced the basic idea ofgravitational waves in terms of a linearized approximationto gravity.

• In the following we consider potential sources ofgravitational waves that might be detectable byEarth-based or space-based gravitational wave detec-tors.

• We begin with the simpler case of weak gravitationalwaves.

We shall conclude with a general introduction to the moredifficult problem of describing strong sources of gravita-tional waves.

21.5. PRODUCTION OF WEAK GRAVITATIONAL WAVES 1217

21.5 Production of Weak Gravitational Waves

To study the production of gravitational waves, we mustin general solve the full non-linear Einstein equations withsource termsTµν .

• This is a formidable problem, generally onlytractable for large-scale computation (numerical rel-ativity).

• However, we can gain considerable insight by study-ing a less complex situation, the linearized Einsteinequation with sources.

• This is an analytically accessible problem that hasmany parallels with the study of sources for electro-magnetic waves.

• It has been shown by numerical simulation that manykey features for the production of weak gravitationalwaves carry over in recognizable form for the pro-duction of gravitational waves in strong-gravity en-vironments.

• Therefore, our approach will be to concentrate on amore quantitative treatment of sources for weak grav-itational waves

We shall then conclude with qualitative order–of–magnitude remarks and some numerically computed ex-amples for strong-gravity wave sources.


21.5.1 Energy Densities

In an electromagnetic field or a Newtonian gravitational field it makessense to talk about local energy densities.

• For example, in a local Newtonian field the energy density atapointxxx is given by

ε(xxx) = −1

8πG(∇∇∇Φ(xxx))2,

whereΦ(xxx) is the Newtonian gravitational potential.

• There isno corresponding local energy densityin general rela-tivity.

• If such a density existed, it wouldcontradict the equivalenceprinciple,which requires gravity tovanishin a sufficiently localregion (local inertial frame).

• However, itdoesmake sense to speak of an approximate energydensity associated with a weak gravitational wave of wavelengthλ , provided thatλ is much shorter than the curvatureR of thebackground spacetimethrough which the wave propagates.

Such approximations become very good atlarge distances from thesourceof a gravitational wave, where curvature associated with thesource becomes negligible andλ/R→ 0.

Therefore, we may formulate a description of energy lossfrom gravitational wave sources at large distances fromthe source where we may associate anapproximate en-ergy densitywith a wave by averaging over several wave-lengths.


21.5.2 Multipolarities

The lowest-order contribution from a source to electro-magnetic radiation corresponds todipole motion of thesource.

• The gravitational field is atensor rather than vectorfield.

• Like the electromagnetic field, the production ofgravitational waves requiresnon-spherical motion ofthe charge(which is electrical charge for the electro-magnetic field and inertial mass for the gravitationalfield).

• However, for the gravitational field no monopole ordipole component contributes to the generation ofgravitational waves.

• The lowest order gravitational wave generationthat is permitted corresponds to time-dependentquadrupole distortions of the source mass.

As a result, many of the formulas for sources of electro-magnetic waves and for weak gravitational waves are sim-ilar, but not identical.


21.5.3 Linearized Einstein Equation with Sources

The metric perturbation for long-wavelength gravitational waves farfrom a non-relativistic source (wavelengths much larger than the char-acteristic source size imply low velocities for mass in the source rela-tive to that of light), is

hi j (t,xxx)r→∞ ≃2rI i j (t − r),

where

• double dots denote the second time derivative and

• hi j (t,xxx) is thetrace-reversed amplitude,

hµν(t,xxx)≡ hµν −12ηµνhγ

γ

= hµν −12ηµνTr h,

which satisfies2hµν = −16πTµν .

Thesecond mass momentI i j (t) is given by

I i j (t) ≡∫

ρ(t,xxx)xix jd3x ,

whereρ(t,xxx) is the mass density of the source.


The stress–energy tensorTµν appearing in the linearized relation

2hµν = −16πTµν .

has non-zero components of the form

T00 =T03

c=

T33

c=

c2

16πG

⟨

(a+)2 +(a×)2⟩

,

where

• 〈. . .〉 denotes an average over several wavelengths,

• a+ anda× denote contributions from the two possible polariza-tions,

• T00 is the energy density,

• T03 is the energy flux (c2 times the momentum density),

• T33 is the momentum flux.


R

Figure 21.8:A contact binary as a source of gravitational waves.

21.5.4 Gravity Wave Amplitudes

The gravity wave amplitudehi j is given by


in linear approximation.

• Let us make some estimates based on this formula using as asimple model for gravitational wave emission a binary star sys-tem in an orbit such that the surfaces of the two stars touch (con-tact binary).

• For simplicity, we shall assume the two stars to be of the sameradius and mass, and to revolve in circular orbits about their cen-ter of mass.

Figure 21.8 illustrates.


R

The second mass moment is

I i j (t) =∫

ρ(t,xxx)xix jd3x = 2MR2.

The system revolves with a periodP and taking the derivative twicewith respect to time gives a factor1/P2

I i j ≃ 2MR2

P2 .

Insertion of this approximation in


leads to

hi j ≃2rI i j =

4MR2

rP2 .


R

The relation

hi j ≃2rI i j =

4MR2

rP2 .

can be expressed in terms of source velocities by noting that

• In one period the center of mass for each star travels a distanceC = 2πR.

• Therefore, the velocity of the center of mass for a star is

v =2πR

P,

• This can be solved for the ratioR/P and used to rewritehi j as

hi j ≃4MR2

rP2 → hi j ≃Mv2

π2r.

This is a specialized result obtained assuming crude ap-proximations, but more careful derivations indicate that ithas broader validity than its derivation would suggest.


R

If we

• Note that the Schwarzschild radius and mass are related byrS =

2M,

• reinsert factors ofc andG, and

• drop the numerical factors (note that the numerical factors droppedare larger than order unity),

the expression

hi j ≃Mv2

π2r.

may be rewritten as

h≃rS

rv2

c2.

The amplitude of the metric perturbation associated withthe gravitational wave is largest for compact sources thathave radii comparable with their Schwarzschild radii.


As noted above, the result

h≃rS

rv2

c2.

is more general than its derivation might suggest and weshall take it as a qualitative guide to the amplitude of grav-itational waves far from a weak source.


Weak gravitational waves are generated primarily by systems that aregravitationally bound or nearly so.

• Thereforethe virial theorem is applicableand to order of mag-nitude we may expect that the kinetic and potential energiesarecomparable, which implies

12Mv2 ∼

GM2

R,

• From this we may write

v2

c2 ≃2GM2

MRc2=

rS

R= ε2/7,

where we have defined a gravitational wave emission efficiencyfactor

ε ≡(rS

R

)7/2

(the justification for terming this the gravitational wave efficiencywill be given below).

Therefore,h may be expressed in the form

h≃rS

rv2

c2 ≃r2S

rR= ε2/7rS

r,

= 9.55×10−17ε2/7(

MM⊙

)(

kpcr

)

,

which is dimensionless.


21.5.5 Amplitudes and Event Rates

We may use the preceding results to estimate amplitudes and cor-responding event rates for candidate gravitational wave events thatmight be detectable by the current generation of gravitational waveinterferometers.

• For estimation purposes, let us assume thatε = (rS/R)7/2 ∼ 0.1and that the mass participating in gravitational wave generationis M ∼ M⊙

• (these would be reasonable guesses for merging neutron stars orasymmetric core collapse supernovae, which are expected tobetwo classes of events with high probability for observationbyLIGO).

• If we first consider events within the galaxy, we may assume thaton averager ∼ 10 kpc.

• Then we may estimate

h= 9.55×10−17ε2/7(

MM⊙

)(

kpcr

)

≃ 5×10−18.

• But such events occur within the galaxy onlyonce every 50 yearsor so,on average.

Therefore, to obtain a reasonable event rate we must lookfor gravitational waves from sources at larger distances.


The nearest rich cluster of galaxies is Virgo, at a distanceof about 15 Mpc.

• Therefore, if we go out to 15 Mpc average distance,we may expect event rates for gravitational wavesfrom merging neutron stars and asymmetric core-collapse supernovae to go up toten or more per yearbecause we are now surveying thousands of galaxies.

• But the average observed metric perturbation (whichis directly related to the strain detected by the detec-tors) becomesh∼ 3×10−21.

Thus, we expect that to detect systematicgravitational wave events the detectors mustbe able to sample strains reliably at the∆L/L0 ≃ 10−21 level or better.


21.5.6 Power in Gravitational Waves

The power radiated in gravitational waves for a system that has veloci-ties well belowc and weak internal gravity is given by thequadrupoleformula,

L =dEdt

=15

⟨

I-...

i j I-...

i j⟩

=15

Gc5

⟨

I-...

i j I-...

i j⟩

,

• 〈〉 denotes a time average over a period,

• the triple dot means a third time derivative, and

• the reduced quadrupole tensorI- is defined by

I- i j ≡ I i j − 13δ i j Tr I ,

whereTr I ≡ Ikk.

This formula is the gravitational analog of the formulafor radiated power in electromagnetism but has a differentfactor (1/5 instead of 1/20) and corresponds to quadrupolerather than dipole radiation.


As shown in an Exercise,

L =dEdt

=15

G

c5

⟨

I-...

i j I-...

i j⟩

,

may be reduced to

L ≃ L0r2S

R2

(vc

)6,

where the scale for radiated gravitational wave power is setby

L0 ≡c5

G= 3.6×1059 erg s−1,

and the total energy∆E emitted in one periodP can then be calculatedas

∆E ≃ LP≃ Mc2(rS

R

)7/2= εMc2.

Therefore, we conclude that

ε =(rS

R

)7/2

parameterizes theefficiency of gravitational wave emis-sionby the massM.


21.6 Gravitational Radiation from Binary Systems

In preceding sections we have made some qualitative esti-mates for gravitational wave emission from binary stars.In this section we derive in a somewhat more rigorousfashion a formalism applicable for such systems.

21.6. GRAVITATIONAL RADIATION FROM BINARY SYSTEMS 1233

x

y

z

R1

R2

M1

M2

Ω t

r

Figure 21.9:Coordinate system for binary stars in circular orbits.

21.6.1 Gravitational Wave Luminosity

A binary system is illustrated in Fig. 21.9. For simplicity we assume

• circular orbits

• M1 = M2 ≡ M, and therefore thatR1 = R2 ≡ R.

Introducing polar coordinates for the center of mass for thefirst mass,

x(t) = RcosΩt y(t) = RsinΩt z(t) = 0,

the components of the second mass moment are given by

I i j (t) ≡∫

ρ(t,xxx)xix jd3x ,

which reduces toI i j = 2Mxi(t)x j(t).

for two discrete masses.


Explicitly, the non-zero components are

I11 = Ixx = 2MR2cos2Ωt = MR2(1+cos2Ωt)

I12 = Ixy = 2MR2cosΩt sinΩt = MR2sin2Ωt

I22 = Iyy = 2MR2sin2Ωt = MR2(1−cos2Ωt)

The trace-reversed amplitude is given by


which requires the second time derivatives of theI i j . These are easilycomputed. For example,

Ixx(t) =ddt

(

MR2(1+cos2Ωt))

= −2ΩMR2sin2Ωt

Ixx(t) =−4Ω2MR2cos2Ωt

hxxr→∞ =

2r

Ixx(t − r) =−8Ω2MR2

rcos2Ω(t − r).

Computing the time derivatives for the other components in like man-ner we obtain

hi jr→∞ =

8Ω2MR2

r

−cos2Ω(t − r) −sin2Ω(t − r) 0

−sin2Ω(t − r) cos2Ω(t − r) 0

0 0 0

.

Note that the appearance of2Ω in the arguments of theabove equation means that the frequencyω of the grav-itational wave radiation will be twice the rotational fre-quencyΩ.


The reduced momentI- i j is given by

I- i j ≡ I i j − 13δ i j Tr I ,

but from

Ixx = 2MR2cos2Ωt = MR2(1+cos2Ωt)

Iyy = 2MR2sin2Ωt = MR2(1−cos2Ωt)

the trace,Tr I = Ixx+ Iyy = MR2,

is independent of time. Therefore,I- i j = I i j and the radiated luminos-ity is

L =15

⟨

I-...

i j I-...

i j⟩

=15

⟨

I...

i j I...

i j⟩

.

From

Ixx(t) =−4Ω2MR2cos2Ωt

and the corresponding expressions for the other components, the tripletime derivatives are

I...

xx(t) = 8Ω3MR2sin2Ωt

I...

xy(t) = I...

yx(t) = −8Ω3MR2cos2Ωt

I...

yy(t) =−8Ω3MR2sin2Ωt


Thus, we have

L =15

⟨

I...

i j I...

i j⟩

=15

⟨

(I...

xx)2 +2(I...

xy)2+(I...

yy)2⟩

=15

[

1P

∫ P

0

(

(I...

xx)2 +2(I...

xy)2+(I...

yy)2)

dt

]

=64Ω6M2R4

5P

∫ P

0

(

sin22Ωt +2cos22Ωt +sin22Ωt)

dt

=1285

Ω6M2R4.

The frequencyΩ and the periodP are related byΩ = 2π/P and fromKepler’s third law,

R=

(

MP2

16π2

)1/3

,

so the gravitational wave luminosity may also be expressed in termsof the mass and period as

L =1285

41/3(

πMP

)10/3

=1285

41/3c5

G

(

πGMc3P

)10/3

= 1.9×1033(

M1M⊙

)10/3(1hP

)10/3

erg s−1.

These formulas have assumed circular orbits and equalmasses for the components of the binary. For generaliza-tions to binary systems for whichM1 6= M2 and non-zeroeccentricity, see Shapiro and Teukolsky §16.4.


21.6.2 Influence of Gravitational Radiation on Binary Orbit

In Newtonian approximation the total energy of a binary is

E = Ekinetic1 +Ekinetic

2 +Egrav= Mv2−M2

2R.

But fromv = 2πR/P and Kepler’s third law,

Mv2 = M

(

4π2R2

P2

)

R=

(

MP2

16π2

)1/3

,

which may be used to rewrite the total energy as

E = −M2

4R= −

M4

(

4πMP

)2/3

.

The total energy is negative (it is a bound system), so if the energy isreduced by gravitational wave emission the periodP mustdecrease.

Thus, gravitational wave emission tends todecrease theperiod (and thus the size) of a binary orbit. Although wewill not prove it, the emission of gravitational wave radia-tion also tends tocircularize elliptical orbits.


Differentiating

E = −M2

4R= −

M4

(

4πMP

)2/3

.

with respect tot assuming the mass to be constant, equating to thenegative of the gravitational wave luminosity ,

L =1285

41/3(

πMP

)10/3

=1285

41/3c5

G

(

πGMc3P

)10/3

= 1.9×1033(

M1M⊙

)10/3(1hP

)10/3

erg s−1.

and solving the resulting eqation fordP/dt gives

dPdt

=−96π

541/3

(

2πMP

)5/3

=−3.4×10−12(

MM⊙

)5/3(1 hP

)5/3

This is a dimensionless measure of how the orbital periodis altered over time by the emission of gravitational waveradiation from the binary system.


21.6.3 Gravitational Wave Emission from the Binary Pulsar

The Binary Pulsar consists of a pulsar and a compact companion(most likely also a neutron star) in orbit around their common cen-ter of mass.

• Each star has a mass near 1.4M⊙ and the orbital period is 7.75hours.

• Although the orbit is elliptical,

dPdt

=−96π

541/3

(

2πMP

)5/3

=−3.4×10−12(

MM⊙

)5/3(1 hP

)5/3

(which we derived for circular orbits) is approximately valid.

• This may be used to estimate that

dPdt

∼−2×10−13.

• Thus, over a period of one year

∆P≃−6.2×10−6 s.

This shift is tiny but easily measured in the Binary Pulsarsystem because the timing afforded by the pulsar clockallows orbits to be determined very precisely.


Puls

es/S

econd

Time (hours)

Radia

l Velo

city

(km

/s)

16.93

16.94

16.95

16.96

0 2 4 6 7.75

0

-100

-200

-300PeriastronPeriastron

Receding

Approaching

Figure 21.10:Pulse rate and inferred radial velocity as a function of timefor theBinary Pulsar.

Figure 21.10 illustrates the pulse rate and the radial veloc-ity as a function of time for the Binary Pulsar.


1975 1980 1985 1990 1995 2000 2005-40

-35

-30

-25

-20

-15

-10

-5

0

Cu

mu

lative

Pe

ria

str

on

Sh

ift (s

)

Year

General Relativity

Figure 21.11:Periastron shift of the Binary Pulsar orbit because of gravitationalwave emission. Data points are the measurements of Taylor and Weisberg and thedashed curve is the prediction from general relativity.

Fig. 21.11 illustrates the cumulative shift of the periastron time (timefor closest approach of the pulsar to its companion) as a function ofelapsed time, compared with a precise calculation assumingemissionof gravitational waves from the system to be responsible forthis shift.

• The quality of the data (note the error bars), and the agreementwith the prediction of general relativity are remarkable.

• The Binary Pulsar provides the strongest evidence that gravita-tional waves exist with properties in quantitative agreement withthe predictions of general relativity.

We have not yet seen gravitational waves directly but theBinary Pulsar leaves little doubt of their existence.


21.7 Gravitational Waves from Strong Sources

Our discussion to this point has been primarily in terms of linearizedgravity

• However, the sources that are conjectured to provide the mostlikely events that LIGO and comparable detectors could see can-not be described in the near-source region by a linear approxi-mation to gravity.

• In events such as

– the core collapse of a massive star, or

– mergers involving some combination of neutron stars andblack holes,

the curvature of spacetime becomes very large in the region wheregravitational waves are produced.

• In this strong-gravity domain, reliable calculations aredifficultand can only be produced through large-scale numerical simula-tions.

• Nevertheless, those simulations show that many of the featuresthat we have inferred about gravity waves in the linear regimesurvive in some form in the strong-gravity regime.

• In particular, available calculations suggest that many basic fea-tures of gravity wave production by strong gravity can be ob-tained by dimensional analysis.

In this section we summarize some likely strong sources of gravita-tional waves and show a few results of numerical simulations.

21.7. GRAVITATIONAL WAVES FROM STRONG SOURCES 1243

21.7.1 Merging Neutron Stars

Gravitational waves from merging neutron stars are expected to bestrong enough that their characteristic signature will be detectable innew Earth-based gravitational wave detectors that are justbeginningto operate.

• The possibility of two neutron stars merging might seem a re-mote one.

• A critical point is that once a neutron star binary is formeditsorbital motion radiates energy as gravitational waves, theorbitsmust shrink, and eventually the two neutron stars must merge.

• Formation of the neutron star binary is not easy, however.

– A binary must form with two stars massive enough to be-come supernovae and produce neutron stars, and the neutronstars thus formed must remain bound to each other throughthe two supernova explosions, or

– the neutron star binary must result from gravitational cap-ture.

• Although these are improbable scenarios, calculations indicatethat they are not impossible.

• Some theoretical estimates indicate that formation of a neutronstar binary can happen often enough to produce about one neu-tron star merger each day in the observable Universe.

The probability of forming a neutron star binary in anyone region of space is very small, but the Universe is avery big place.


100 3000 10000 30000 60000 100000

Temperature (millions of K)

0.875 ms 1.375 ms 2.125 ms

2.875 ms

3.625 ms

4.025 ms

Approximate

Schwarzschild

Radius

Figure 21.12: Simulation of the merger of two neutron stars. The elapsedtime is about 3 ms and the approximate Schwarzschild radius for the com-bined system is indicated. The rapid motion of several solarmasses of materialwith large quadrupole distortion and sufficient density to be compressed near theSchwarzschild radius indicates that this merger should be astrong source of gravi-tational waves (Source: S. Rosswog simulation).

Figure 21.12 shows a numerical simulation of neutron-starmerger in a binary neutron star system.


100 3000 10000 30000 60000 100000


0.875 ms 1.375 ms 2.125 ms

2.875 ms

3.625 ms

4.025 ms

Approximate

Schwarzschild

Radius

In this merger,

• The orbit of the binary has steadily decayed as a re-sult of gravitational wave emission, causing the starsto spiral together at a rapidly increasing rate near theend.

• The sequence of images in Fig. 21.12 illustrates themerger over a period of milliseconds near when thesurfaces of the neutron stars first touch.


100 3000 10000 30000 60000 100000


0.875 ms 1.375 ms 2.125 ms

2.875 ms

3.625 ms

4.025 ms

Approximate

Schwarzschild

Radius

Because of the

• very large quadrupole mass distortion,

• the high velocities generated by the revolution onmillisecond timescales, and

• the highly compact nature of the mass distribution,

mergers of neutron stars are expected to be a very strongsource of gravitational waves.


Figure 21.13:Amplitude and frequency ranges expected for gravitationalwavesfrom various sources. The lower detection ranges for LIGO and LISA are indicatedby dotted lines.

As Fig. 21.13 illustrates,

• the frequencies for gravitational waves emitted from neutron starmergers (and from the stellar-size black holes mergers describedbelow) are expected to lie in the range accessible to Earth-basedinterferometers like LIGO.

• The outcome of such neutron star mergers will likely be a Kerrblack hole, since

– the combined mass of the two neutron stars is generallylarger than the upper limit for a neutron star and

– the merged object will be formed with large angular mo-mentum.


Figure 21.14: Gravitational wave emission from an in-spiraling binary blackholes.

21.7.2 Stellar-Size Black Hole Mergers

Merger of 2 black holes, or a neutron star and a black hole,should also be strong sources of gravitational waves.

• Such mergers are expected to have many qualitativesimilarities to the merger of neutron stars.

• They are expected to leave behind a Kerr black hole.

Figure 21.14 illustrates gravitational wave emission froman inspiraling black hole pair based on numerical simula-tions of general relativity.

• The two strong peaks represent the black holes.

• The spirals in spacetime trailing from them are theoutwardly propagating gravitational waves.


21.7.3 Core Collapse in Massive Stars

In the core collapse of a massive star that leads to a supernova

• Densities are reached comparable to that of a neutron star and

• in the collapse a solar mass or more of material may be set inmotion with velocities of order 10% of light velocity.

• If such a collapse were to proceed with spherical symmetry,nogravitational waves would be produced.

• However, numerical simulations generally find large asymme-tries generated by things like large-scale supersonic convection.

• These asymmetries will likely produce quadrupole distortionsin the mass distribution that vary rapidly in time and thus arepotential sources of strong gravitational waves.

As indicated in the following figure,

gravitational-wave frequencies from core-collapse supernovae are ex-pected to lie in the range accessible to Earth-based interferometers.


21.7.4 Merging Supermassive Black Holes

The current paradigm is that the cores of many, perhaps most,massivegalaxies harbor black holes containing106–109 solar masses.

• For example, fully-resolved orbits of individual stars atthe cen-ter of the Milky Way indicate a3.3×106M⊙ black hole.

• Analysis of velocity fields near the centers of giant ellipticalgalaxies often suggest billions of solar masses packed intonon-luminous regions comparable in size to the Solar System.

• There is also very strong observational evidence that galaxy merg-ers are common in the history of the Universe.

• Therefore, we may expect that the merger of supermassive blackholes can occur as a consequence of galaxy collisions.

As indicated in the following figure,

The expected frequency of the gravitational waves that would be pro-duced is too low for detectors like LIGO but the proposed space-basedLISA array would be optimal to search for such events.


Quantitative investigation of supermassive black hole merger eventscan only be done numerically but dimensional analysis argumentsbased on the quadrupole power formula

L =dEdt

=15

Gc5

⟨

I-...

i j I-...

i j⟩

,

and supported by numerical simulation suggest that the peaklumi-nosity is set by the fundamental scalec5/G defined in

L ≃ L0r2S

R2

(vc

)6L0 ≡

c5

G= 3.6×1059 erg s−1

Current understanding indicates that the peak luminosity for super-massive black hole mergers is approximately

L ∝ L0 ≃ ηc5

G≃ η ×1059 erg s−1,

with the efficiency factorη accounting for details of the merger andbeing of order 1% in typical cases.

If such events occur, their luminosities would likely belarger than for any other event in the Universefor a periodof days.


To set the gravitational-wave luminosity

L ∝ L0 ≃ ηc5

G≃ η ×1059 erg s−1,

expected for merger of supermassive black holes in per-spective, this luminosity (assumingη ∼ 0.01) is about

• a million times larger than the photon luminositiesassociated with supernovae and gamma ray bursts,

• ten billion times larger than quasar luminosities, and

• about 25 orders of magnitude larger than the lumi-nosity of the Sun.

Chapter 21 Gravitational Waves - University of Tennesseeeagle.phys.utk.edu/guidry/astro616/lectures/lecture_ch21.pdf · Chapter 21 Gravitational Waves Just as • Maxwell’s theory

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