1

Physics and Astrophysics of Gravitational Waves

Notes to Lectures at Cardiff UniversityFebruary 2005

Bernard Schutz

Cardiff University, Wales & Albert Einstein Institute (AEI), Germany

http://www.aei.mpg.de, Bernard.Schutz@aei.mpg.de

Outline of topics to be covered:

1. Special and general relativity. Review of Minkowski geometry; grav-itation as geometry; curvature of time and space.

2. Elements of gravitational waves. Linearized waves; propagation.

3. Energy and gravitational radiation. How energy is defined; practicalapplications of the energy formula.

4. Mass-quadrupole radiation. Radiation generation in linearized the-

ory; energy radiated in waves; radiation generation in the full Newtonianlimit.

5. Example source calculation. Gravitational waves from a binary sys-tem.

6. Gravitational wave detectors. History and present status of detec-

tor development; interaction of gravitational waves with beam detectors;physics of interferometers; LISA, a detector in space.

7. Data Analysis. Matched filtering; parameter searches.

8. Overview of gravitational wave sources. Binary systems; neutronstars as pulsars and as X-ray sources; supernovae/hypernovae/gamma-

ray bursts; and the big bang.

9. Sources for LISA. Supermassive black holes; extreme mass-ratio inspi-ral; galactic binaries; special challenges of LISA data analysis.

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Chapter 1 – Special and general relativity

Lectures assume familiarity with reltivistic electromagnetism and with Minkowskigeometry. The metric (interval) is

ds2 = ηαβdxαdxβ,

where the symbol η denotes the matrix diag(−1, 1, 1, 1). I will take c = 1

and use the usual summation convention on repeated indices. Greek indicessum over all four coordinates (0..3), Latin over the three spatial coordinates

(1..3).

General relativity describes gravitation as geometry. Inspiration: the prin-ciple of equivalence, roots back to Galileo.

• Any smooth geometry is locally flat, and in GR this means that it islocally Minkowskian. Local means in space and time: the local Minkowski

frame is a freely-falling observer.

• In a general coordinate system the Minkowski equation is replaced by

ds2 = gαβdxαdxβ,

where g is a position-dependent symmetric 4 × 4 matrix. As in specialrelativity, the metric measures proper time and proper distance. Coordi-

nates are arbitrary in GR, but most situations are easier to analyse inappropriately chosen coordinates. A free particle follows a geodesic ofthis metric, defined as a locally straight world-line.

• There are 4 degrees of freedom to choose coordinates, and 10 components

of g: 6 “true” functions left for the geometry. The tensorial descriptionof the geometry is through the Riemann curvature tensor, which contains

second derivatives of g.

• Derived from the Riemann tensor is the Einstein tensor G, which is basis

of the field equationsGαβ = 8πT αβ,

where T is the stress-energy tensor, whose components contain the energydensity, momentum density, and stresses inside the source of the field.(We also take Newton’s constant G = 1. ) In GR momentum and stress

as well as energy density create gravity.

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• In non-relativistic situations, the energy density dominates, which is the

0-0 field equation. The most important curvature for nearly-Newtoniansystems is the curvature of time, which is measureable by the gravita-tional redshift of clocks. All of Newtonian orbital motion can be derived

from time-curvature. Relativistic particles, such as photons, are sensitivealso to the spatial curvature.

• The coordinate invariance of the theory implies that not all field equations

are independent; mathematically the Einstein tensor is divergence-free forany metric:

∇αGαβ = 0.

This is called the Bianchi identity. It implies, from the field equation,

∇αT αβ = 0.

This is the equation of conservation of energy and momentum in thematter sources.

• Derivatives like ∇α are defined so that in a freely-falling frame they arethe derivatives of special relativity. Called covariant derivatives. In a

general coordinate system they involve derivatives of tensor componentsand of basis vectors eα, so expressions are complicated. They involve

Christoffel symbols Γαβδ:

∇δeβ = Γαβδeα.

The Christoffel symbols involve the first derivatives of the metric tensor.They vanish in a local feely falling frame, but only at the single event

where the frame is perfectly freely falling. The second derivatives ofthe metric cannot in general be made to vanish by going to any specialcoordinate system.

• It is important to understand what the conservation law of the stress-

energy tensor means. It is conservation in a freely-falling frame. Thisis the equivalence principle: in a freely falling frame, the local matter

energies are conserved. It does not represent a conservation law withgravitational potential energy, for example. Such global energy conserva-tion laws are valid only if the metric is independent of time. The local

law is an equation of motion. It implies, for example, that isolated smallparticles fall on geodesics of the metric.

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• If a tensor has zero covariant derivative in a given direction, it is said to

be parallel-transported. Thus, a vector V is parallel-transported in thedirection W if

W α∇αV β = 0

for all β. Parallel-transport means that the field is held constant in a

freely-falling frame.

• A special case of parallel transport is the geodesic equation, which is

the statement that the four-velocity U — whose components are Uα =dxα/dτ (where τ is proper time) — is parallel-transported along itself:

dUα

dτ+ Γα

µνUµU ν = 0.

• The second derivatives of the metric contain coordinate-invariant infor-mation that is collected in the Riemann curvature tensor R. An interest-

ing definition of R involves the commutator of covariant derivatives of avector field V :

[∇α,∇β]Vµ = Rµ

ναβV ν.

In Minkowski spacetime, derivatives commute and the curvature is zero.In a curved space(time), covariant derivatives in different directions do

not commute. This is most easily seen in terms of parallel transport:vector fields that are parallel transported tangent to a sphere along dif-

ferent great circles, for example, do not coincide with one another whenthe great circles intersect again.

• The mathematics of tensor calculus can get very complicated. The ex-

pressions for the Riemann tensor in terms of the components of the metrictensor are long and not very informative. We will not go into such things

in these lectures.

• Approximation methods are crucial in general relativity.

– We will deal mostly with linearized theory, where the curvature issmall and spacetime is nearly Minkowskian.

– Another important approximation is the post-Newtonian approxima-

tion. GR contains Newtonian gravity as a (degenerate) limit, andone can develop an asymptotic approximation to GR away from thislimit. The limit links two parameters: the gravitational field is weak

and the velocities of the sources are small. In linearized theory thefield is weak but the sources do not have to be non-relativistic.

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– Perturbation theory is the study of solutions near a known solution.

This is a generalization of linearized theory. It can study stellar stabil-ity, the orbits (with radiation reaction) of particles near black holes,and so on.

– A final approximation method is numerical: simulations of black holes

and other situations are increasingly useful in understanding GR.

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Chapter 2 – Elements of gravitational waves

GR is nonlinear, fully dynamical ⇒ in general no clear distinction betweenwaves and the rest of the metric. The notion of a wave is OK in certain

limits:

• in linearized theory;

• as small perturbations of a smooth background metric;

• in post-Newtonian theory (far zone).

We will concentrate on linearized theory, but much of this work carries overto the other cases in a straightforward way.

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Mathematics of linearized theory

• In linearized theory metric is nearly that of flat spacetime:

ds2 = (ηαβ + hαβ) dxαdxβ, |hαβ | 1.

• Define trace-reversed metric perturbation hαβ = hαβ − 12ηαβη

µνhµν and

adopt Lorentz gauge:hαβ

,β = 0,

where a subscripted comma denotes the partial derivative with respect

to the coordinate associated with the index that follows the comma.Lorentz gauge is just a gauge (coordinate) choice: four equations use up

4 degrees of freedom to specify spacetime coordinates. Initial data forthese equations is still free.

• In Lorentz gauge, the Einstein field equations are just a set of decoupledwave equations ⎛

⎝− ∂2

∂t2+ ∇2

⎞⎠ hαβ = −16πT αβ.

• To understand propagation, it is easiest to look at plane waves:

hαβ = Aαβ exp(2πıkµxµ),

for constant amplitudes Aαβ and wave vector kµ. Then the Einsteinequations imply that the wave vector is null kαkα = 0 (propagation at

the speed of light), and the gauge condition implies that the amplitudeand wave vector are orthogonal, Aαβkβ = 0.

• Further gauge conditions (adjustments of the initial data for the Lorentz

gauge equations) are possible. Just state them here: we will explicitlyconstruct them in Chapter 4. We can demand that

1. A0β = 0 ⇒ Aijkj = 0: Transverse wave; and

2. Ajj = 0: Traceless wave amplitude.

These conditions can only be applied outside a sphere surrounding the

source. Together, they put the metric into the transverse-traceless (TT)gauge. In TT gauge, hαβ = hαβ .

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Using the TT gauge to understand gravitational waves

• Only two independent polarizations The TT gauge leaves only twoindependent wave amplitudes out of the original 10. Take the wave to

move in the z-direction, so that kz = k, kx = ky = 0. Then gauge +transversality ⇒ A0α = Azα = 0, leaving only Axx, Axy = Ayx, and Ayy

nonzero. Tracelessness ⇒ Ayy = −Axx. So there are only 2 independentamplitudes, 2 independent degrees of freedom for polarization.

• A wave for which Axy = 0 produces a metric of the form

ds2 = −dt2 + (1 + h+)dx2 + (1 − h+)dy2 + dz2,

where h+ = Axx exp[ik(z − t)]. This produces opposite effects on properdistance on the two axes, contracting one while expanding the other.

• If Axx = 0 then only the off-diagonal term hxy = h× is non-trivial, and

these can be obtained from the previous case by a 45 rotation.

• A general wave is a linear combination of these two. If one lags the otherin phase, the polarization is circular or elliptical. The existence of onlytwo polarizations is a property of any non-zero spin field that propagates

at the speed of light.

• The effect of a wave in TT gauge on a particle at rest can be computedfrom the Christoffel symbols. Its initial acceleration is

d2

dτ 2xi = −Γi

00 = −1

2(2hi0,0 − h00,i) = 0.

So the particle does not move. The TT gauge represents a coordinatesystem that is comoving with freely-falling particles. Because h0α = 0,

TT-time is proper time on the clock of a freely falling particle at rest.

• Tidal forces show the action of the wave independently of coordinates.For example, the geodesic deviation equation for the separation ξ of twofreely falling particles initially at rest is

d2

dτ 2ξi = Ri

0j0ξj =

1

2hij,00ξ

j.

This contains the same information as we saw in the metric above. The

Riemann tensor is gauge-invariant in linearized theory.

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Chapter 3 – Energy and Gravitational Radiation

• How waves carry energy. Energy has been one of the most confusingaspects of graviational wave theory and hence of general relativity. Itcaused much controversy, and even Einstein himself took different sides

of the controversy at different times in his life. Physicists today havereached a wide consensus. The problem is difficult because of the equiv-

alence principle: in a local frame there are no waves and hence no localdefinition of energy that can be coordinate-invariant. Moreover, a wave

is a time-dependent metric, and in such spacetimes there is no globalenergy conservation law. Energy is only well-defined in certain regimes,

which coincide with those for which waves can be cleanly separated from”background” metrics.

• Asymptotically flat spacetimes. Relativists have introduced theconcept of asymptotic flatness, which idealises an isolated body, one

whose geometry becomes flat far away. If the body is stationary (time-independent) then test fields (falling particles) will have conserved energy.

If there is a small perturbation that can be treated as a wave, then itsenergy will be conserved, in the sense that the total mass-energy of thesystem (as measured by planetary orbits far away) decreases as the waves

leave.

• Wave flux. The energy carried by a wave as it leaves a body or asit moves through a nearly-flat spacetime can be written as an effective

stress-energy tensor for the wave. It is known as the Isaacson tensor, andin linearized theory it has the following expression:

T(GW )αβ =

1

32πhµν,αhµν

,β.

It represents the localisation of energy to regions whose size is of ordera wavelength, but not smaller. It can be defined in the same way if h

represents a perturbation away from a background curved spacetime, inwhich case the derivatives are replaced by covariant derivatives.

• It is important to understand that energy is a useful but not fundamental

concept in gravitational wave studies. The fundamental quantity is themetric perturbation itself. We don’t need energy concepts to calculatethe radiated wave amplitude (below) nor to compute the effect of the

wave on a detector (above). Energy is a useful concept only when it is

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conserved, where it makes source calculations easier and allows one to

understand sources better. But we could get along without it!

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Practical applications of the energy formla

• Relation between typical wave amplitude and the energy radi-ated by a source. If we are far from a source of gravitational waves, we

can treat the waves by linearized theory. Then if we adopt TT gauge andspecialize the stress-energy tensor of the radiation to a flat background,

we get

T(GW )αβ =

1

32πhTT

µν ,αhTTµν,β.

Since there are only two components, a wave travelling with frequency f

(wave number k = 2πf) and with a typical amplitude h in both polar-izations carries an energy flux Fgw equal to (see Exercise 6)

Fgw =π

4f 2h2.

Putting in the factors of c and G and scaling to reasonable values gives

Fgw = 3 mW m−2[

h

1 × 10−22

]2 [ f

1 kHz

]2,

which is a very large energy flux even for this weak a wave. It is twice

the energy flux of a full moon! Integrating over a sphere of radius r,assuming a total duration of the event τ , and solving for h, again with

appropriate normalizations, gives

h = 10−21[

Egw

0.01Mc2

]1/2 [ r

20 Mpc

]−1 [ f

1 kHz

]−1 [ τ

1 ms

]−1/2.

This is the formula for the “burst energy”, normalized to numbers ap-

propriate to a gravitational collapse occuring in the Virgo cluster. Itexplains why physicists and astronomers regard the 10−21 threshold as soimportant. But this formula could be applied to binary systems radiating

away their orbital gravitational binding energy over long periods of timeτ , for example.

• Curvature produced by waves. Although the Isaacson flux tensor

is an approximation, it is a very robust and satisfying approximation.Isaacson showed that the background spacetime will actually exhibit asmall average curvature when the waves are contained on it, and that

this curvature has an Einstein tensor given by.

Gαβ = 8πT(GW )αβ .

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This is the Einstein field equation for a spacetime with the wave-energy

as its source. It only holds to lowest order in the (small) wave amplitude,but we would not expect the Isaacson flux expression to be meaningfulat higher orders of approximation anyway.

• Cosmological background of radiation. This self-consistent picture

allows us to talk about, for example, a cosmological gravitational wavebackground that contributes to the curvature of the Universe. Since the

energy density is the same as the flux (when c = 1), we have

ρgw =π

4f 2h2,

but now we must interpret h in a statistical way. Basically it is done byreplacing h2 by a statistical mean square amplitude per unit frequency(Fourier transform power per unit frequency) called Sh(f), so that the

energy density per unit frequency is proportional to f 2Sh(f). It is thenconventional to talk about the energy density per unit logarithm of the

frequency, which means multiplying by f . The result, after being care-ful about averaging over all directions of the waves and all independent

polarization components, is

dρgw

d ln f= 4π2f 3Sh(f).

Finally, what is of most interest is the energy density as a fraction of theclosure or critical cosmological density, given by the Hubble constant H0

as ρc = 3H20/8π. The resulting ratio is called Ωgw(f):

Ωgw(f) =32π3

3H20f 3Sh(f).

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Chapter 4 – Mass-Quadrupole Radiation

Field of a source in linearized theory.

• Isolated source The Einstein equation is⎛⎝− ∂2

∂t2+ ∇2

⎞⎠ hαβ = −16πT αβ.

Its general solution is the following retarded integral for the field at aposition xi and a time t in terms of the source at a position yi and the

retarded time:

hαβ(xi, t) = 4∫ 1

RT αβ(t − R, yi)d3y,

where we define

R2 = (xi − yi)(xi − yi).

• Expansion for the far field of a slow-motion source. Let us suppose

that the origin of coordinates is in or near the source, and the field pointxi is far away. Then we define r2 = xixi and we have r2 yiyi. We

can therefore expand the term R in the denominator in terms of yi. Thelowest order is r, and all higher-order terms are smaller than this bypowers of r−1. Therefore, they contribute terms to the field that fall off

faster than 1/r, and they are negligible in the far zone. So we can simplyreplace R by r in the denominator, and take it out of the integral.

The R inside the time-argument of the source term is not so simple. Wehandle that in the following way. Let us define t′ = t − r (the retarded

time to the origin of coordinates) and expand

t − R = t − r + niyi + O(1/r), with ni = xi/r, nini = 1.

The terms of order 1/r are negligible for the same reason as above, butthe first term in this expansion must be taken into account. It depends

on the direction to the field point, given by the unit vector ni. We usethis by making a Taylor expansion in time on the time-argument of thesource. The combined effect of these approximations is

hαβ =4

r

∫ [T αβ(t′, yi) + T αβ

,0(t′, yi)njyj +

1

2T αβ

,00(t′, yi)njnkyjyk + . . .

]d3y.

We will need the Taylor expansion out to this order.

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• Moments of the source. The integrals in the above expression contain

moments of the components of the stress-energy. It is useful to give thesenames. Use M for moments of the density T 00, P for moments of themomentum T 0i, and S for moments of the stress T ij. Here is our notation:

M(t′) =∫

T 00(t′, yi)d3y, Mj(t′) =

∫T 00(t′, yi)yjd

3y,

Mjk(t′) =

∫T 00(t′, yi)yjykd

3y;

P (t′) =∫

T 0(t′, yi)d3y, P j(t

′) =∫

T 0(t′, yi)yjd3y;

Sm(t′) =∫

T m(t′, yi)d3y.

These are the moments we will need.

Among these moments there are some identities that follow from theconservation law in linearized theory, T αβ

,β = 0, which we use to replace

time derivatives of components of T by divergences of other componentsand then integrate by parts. The identities we will need are

M = 0, Mk = P k, M jk = P jk + P kj ;

P j = 0, P jk = Sjk.

These can be applied recursively to show, for example, one further very

useful relation:d2M jk

dt2= 2Sjk.

• Radiation zone expansions. Using these relations and notation it is

not hard to show that

h00(t, xi) =4

rM +

4

rP jnj +

4

rSjk(t′) + . . . ;

h0j(t, xi) =4

rP j +

4

rSjk(t′)nk + . . . ;

hjk(t, xi) =4

rSjk(t′) + . . . .

In these expressions, one must remember that the moments are evaluated

at the retarded time t′ = t−r (except for those moments that are constantin time), and they are multiplied by components of the unit vector to the

field point nj = x/r.

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Application of TT gauge to the Mass Quadrupole

Field.

• Gauge transformations. We are already in Lorentz gauge, and thiscan be checked by taking derivatives of the expressions for the field thatwe have derived. But we are manifestly not in TT gauge. Making a gauge

transformation consists of choosing a vector field ξα and modifying themetric by

hαβ → hαβ − ξα,β − ξβ,α.

The corresponding expression for the potential hαβ is

hαβ → hαβ − ξα,β − ξβ,α + ηαβξµµ.

For the different components this implies changes

δh00 = −ξ0,0 + ξj

,j,

δh0j = −ξ0,j + ξj

,0,

δhjk = −ξj,k − ξk,j + δjkξ,

where δjk is the Kronecker delta (unit matrix). In practice, when taking

derivatives, the algebra is vastly simplified by the fact that we are keepingonly the 1/r terms in the potentials. This means that spatial derivatives

do not act on 1/r but only on t′ in the arguments. Since t′ = t − r, itfollows that ∂t′/∂xj = −nj, and ∂h(t′)/∂xj = −h(t′)nj.

• The TT gauge transformation. The following vector field puts themetric into TT gauge to the order we are working:

ξ0 = −1

rP k

k − 1

rP jknjnk,

ξi = −4

rM i − 4

rP ijnj +

1

rP k

kni +

1

rP jknjnkn

i.

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• The wave amplitude in TT gauge. The result of applying this gauge

transformation to the original amplitudes is:

hTT 00 =4M

r;

hTT 0i = 0;

hTT ij =4

r

[⊥ik⊥j Sk +

1

2⊥ij (Skn

kn − Skk)

],

where the notation ⊥jk represents the projection operator perpendicular

to the direction ni to the field point,

⊥jk= δjk − njnk.

It can be verified that this tensor is transverse to the direction ni and isa projection, in the sense that it projects to itself:

⊥jk nk = 0, ⊥jk⊥ k =⊥j .

The time component of the field is not totally eliminated in this gauge

transformation: it must contain the Newtonian field of the source. (Infact we have succeeded in eliminating the momentum part of the field,which is also static. Our gauge transformation has incorporated a Lorentz

transformation that has put us into the rest frame of the source.) But thisis a constant term. Since waves are time-dependent, the time-dependent

part of the field is now purely spatial, transverse (because everything ismultiplied by ⊥), and traceless (as can be verified by explicit calculation).

The expression for the spatial part of the field actually does not dependon the trace of Sjk, as can be seen by constructing the trace-free part of

the tensor, defined as:

S–jk = Sjk − 1

3δjkS

.

In fact, it is more conventional to use the mass moment here instead ofthe stress, so we also define

M—jk = M jk − 1

3δjkM

, S–jk =1

2

d2M—jk

dt2.

In terms of M— the far field is:

hTT ij =2

r

(⊥ik⊥j M—k +

1

2⊥ij M—kn

kn.

)

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This is the mass quadrupole field. In other books the notation is some-

what different than we have adopted here. In particular, our quadrupoletensor M— is what is called I– in Misner, Thorne, and Wheeler (1973) andSchutz (1985).

If we define the TT-part of the quadrupole tensor to be

MTTij =⊥ k

i ⊥ ljMkl − 1

2⊥ij⊥kl Mkl,

then we can rewrite the radiation field as

hTTij =2

r

··M

TTij.

• Interpretation of the radiation. It is useful to look at this expres-sion and ask what actually generates the radiation. The source of the

radiation is the second time-derivative of the second moment of the massdensity T 00. The moments that are relevant are those in the plane per-pendicular to the line of sight. So it is interesting that not only is the

action of the wave transverse, but also the generation of radiation usesonly the transverse distribution of mass. In fact we learn from this two

equally important messages,

– the only motions that produce the radiation are the ones transverseto the line of sight; and

– the induced motions in a detector mirror the motions of the sourceprojected onto the plane of the sky.

If most of the mass is static, then the time-derivatives allow us to con-centrate only on the part that is changing.

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Energy Radiated in Gravitational Waves.

• Mass quadrupole radiation. The radiation field we have computedcan be put into our energy flux formula for the TT gauge, and this can

be integrated over a sphere. It is not a difficult calculation, but it doesrequire some simple angular integrals over the vector ni, which depends

on the angular direction on the sphere. These identities are

∫ninjdΩ =

4π

3δij,

∫ninjnkdΩ = 0,

∫ninjnkndΩ =

4π

15

(δijδk + δikδj + δiδjk

).

Using these, one gets the following simple formula for the total luminosityof the source if only mass-quadrupole radiation is computed:

Lmassgw =

1

5

...M—

jk ...M—jk.

Radiation in the Newtonian limit.

• Relaxation of restrictions of linearized theory. The calculation sofar has been within the assumptions of linearized theory. Real sourcesare likely to have significant self-gravity. This means, in particular, that

there will be a significant component of the source energy in gravitationalpotential energy, and this must be taken into account.

Fortunately, the formulas we have derived are robust. It turns out thatthe leading order radiation field from a Newtonian source has the same

formula as in linearized theory.

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Chapter 5 – Radiation from a binary system

• The quadrupole moment of a binary system. The motion of twostars in a binary is a classic source calculation. We shall calculate hereonly for two equal-mass stars in a circular orbit, governed by Newtonian

dynamics. If the stars have mass m and an orbital radius R, orbiting inthe x−y plane with angular velocity ω, then it is easy to show that their

quadrupole moment components are

Mxx = 2mR2 cos2(ωt), Myy = 2mR2 sin2(ωt), Mxy = 2mR2 cos(ωt) sin(ωt).

By using trigonometric identities, we convert these to functions of a fre-quency 2ω and discard the parts that do not depend on time:

Mxx = mR2 cos(2ωt), Myy = −mR2 cos(2ωt), Mxy = mR2 sin(2ωt).

This shows that the radiation will come out at twice the orbital frequency,essentially because in half an orbital period the mass distribution has

returned to its original configuration.

The trace of the quadrupole tensor is already zero.

• The radiated field in different directions. The general expression

for the radiation field is hTT ij = (2/r)MTT ij.

1. Radiation perpendicular to the orbital plane. This is the z-direction,and the tensor M is already transverse to it. So the radiation compo-

nents can be read off of M . We see that h+ = −(8mω2R2/r) cos(2ωt)and h× = (8mω2R2/r) sin(2ωt). Both polarisations are present butare out of phase, so this represents purely circularly polarised radia-

tion.

2. Radiation along the x-axis. The xx and xy components of M will beprojected out, and when M is made trace-free again its componentsbecome MTTyy = −(mR2/2) cos(2ωt) and MTTzz = (mR2/2) cos(2ωt).

This is pure +-polarised radiation with amplitude 4mω2R2/r. Thisis half the amplitude of each of the polarisation components in the

z-direction, so the radiation is much weaker here. The energy fluxwill be only 1/8 of the flux up the rotation axis. By symmetry this

conclusion holds for any direction in the orbital plane.

At directions between the ones we have calculated there will be a mixture

of polarisations, which leads to a general elliptically polarised wave. By

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measuring the polarisation received, a detector (or network of detectors)can measure the angle of inclination of the orbital plane of the binary

to the line of sight. This is often one of the hardest things to measurewith optical observations of binaries, so gravitational wave observationsare complementary to other observations of binaries.

• The energy radiated by the orbital motion. If we put our quadrupole

moment into the luminosity formula we get

Lgw =16

5m2R4ω6.

The various factors are not independent, however, because the angular

velocity is determined by the masses and radius of the orbit. Whenobserving such a system, we can’t usually measure R directly, but we

can infer ω from the observations and often make a guess at m. So weeliminate R using the Newtonian orbit equation:

R3 =m

4ω2 .

If in addition we use the gravitational wave frequency ωgw = 2ω, we get

Lgw =1

20(mωgw)10/3.

• Back-reaction on the orbit. This energy must come from the orbitalenergy, E = −mω2R2. The result is that we can predict the rate of

change of ωgw:

dE

dt= −L, ⇒ ωgw =

1

10m5/3ω11/3

gw .

• Binaries as standard candles. Remarkably, if we can measure thechirp rate ωgw, we can infer from it the distance to the binary system. Itis very unusual in astronomy to be able to observe a system and infer its

distance; systems for which this is possible are called “standard candles”,since basically one must know their intrinsic luminosity and compare that

with their apparent brightness in order to measure the distance. In ourcase, we simply note that the amplitude of the radiated field depends, as

above, on m, R, and r. Since we can solve for R in terms of the othervariables, we can take the field to depend on m, ωgw, and r. Since ameasurement of the chirp rate allows us to infer m, and since we measure

ωgw, it follows that when we also measure the amplitude of the waves wecan determine r, the distance to the binary.

21

• Generalisations.

– Our assumption of equal masses may seem restrictive, but it actuallyis not. With unequal masses one replaces m by something called the

chirp mass M, formed from the reduced mass µ and the total massM in the following way:

M = µ3/5M2/5.

This combination appears in both the formula for ωgw and for h, so

that it is possible to infer distances from these observations even whenthe masses of the two stars are unequal.

– A more serious assumption is that the orbit is circular. Coalescingbinary neutron stars and black holes have probably evolved into cir-

cular orbits by the time they coalesce, but stellar binaries observedby LISA may not always be circular. The Hulse-Taylor binary pulsaris not in a circular orbit. The eccentricity of an orbit brings the stars

closer together than they get in a circular orbit of the same semi-major axis. Because the gravitational wave energy emission is such a

strong power of the velocity (see the factor ω6 in the formula above),radiation is stronger in eccentric orbits, and they shrink faster.

– We have only worked in Newtonian theory for the orbits. Post-Newtonian orbit corrections will be very important in observations.

This might at first seem puzzling, since ground-based detectors willhave low signal-to-noise ratios for these observations. But the key fact

is that the corrections to the orbital radiation have a cumulative ef-fect on the waveform, steadily changing its phase from what might be

expected from Newtonian orbits. If the phase of an orbit changes byas much as π in the whole evolution (half an orbit) then the templatebeing used to search for the signal becomes useless. Since observa-

tions will follow the orbital evolution of such systems for thousandsof orbits, very precise templates are required. By measuring the post-

Newtonian effects on an orbit, one can measure the individual massesof the stars.

– Even more extreme are orbits of small black holes falling into massiveblack holes, such as are seen in the centers of galaxies. Here one

needs to solve the full relativistic orbit equation with corrections tothe geodesic motion that are first-order in the mass of the infalling

object. This is a problem that has not yet been completely solved.

22

It is key for LISA. (When one says that one corrects the geodesicequation even for a freely falling black hole, that does not mean we

are abandoning the equivalence principle. We are finding correctionstothe geodesic equation of the background black hole; the infallingblack hole follows a geodesic of the time-dependent geometry which

it helps to create.)

– The most difficult binaries are merging black holes. Here no analyticapproaches will help and the problem must be done fully numericallyon the biggest supercomputers. The AEI has the world-leading group

in doing such simulations, but the problem is difficult, and it will stillbe some years before reliable waveform predictions can be expected.

23

Chapter 6 – Gravitational wave detectors

History of Gravitational Wave Detection

• 1960 – 75 J. Weber develops first bar detector at Univ. of Maryland.Announces 2-bar coincidences at unexpected rate.

• 1970 – 80 Other groups develop similar bar detectors, see no unusual

events: Glasgow, Garching, Rome, Bell Labs, Stanford, Rochester, LSU,MIT.

• 1980 – 1994

1. Cryogenic bar detectors developed at Rome, Stanford, LSU, Perth(Aus.) Reach below 10−19. A number of joint observations performed.

2. Prototype interferometers developed at MIT, Garching, Glasgow, Cal-tech. Typical sensitivity 10−18.

3. First long coincidence observation with interferometers: the Glas-

gow/Garching 100-hour experiment.

4. Major collaborations formed to propose large-scale detectors:

– LIGO: Caltech & MIT

– VIRGO: France (CNRS) & Italy (INFN)

– GEO: Germany & UK

– AIGO: Australia

24

• 1990 – 2000

1. Ultracryogenic bars constucted in Rome, Frascati. Expected to reachbelow 10−20. New generation of spherical or icosahedral solid-mass

detectors proposed in USA (LSU), Brasil, Netherlands, Italy. Arraysof smaller bars proposed for the highest frequencies.

2. Large interferometers funded and constructed:

– LIGO: Hanford (WA) and Livingstone (LA), 4 km (plus 1 2-kmin WA)

– VIRGO: Pisa, 3 km

– GEO600: Hannover, 600 m

– TAMA300: Japan, 300 m, longer one expected

3. The space-based low-frequency detector LISA proposed to the Euro-pean Space Agency and adopted as one of four Corenerstone missions

in the Horizons 2000+ forward plan.

• 2000 – present

1. Network of 4 bar detectors (Allegro, Nautilus, Auriga, Perth) reports

upper limits based on simultaneous observing over a 1-year period.

2. Rome group reports unexpected correlated events between two an-

tennas, level of significance not clear, much stronger than expectedGWs.

3. NASA and ESA agree to develop LISA jointly for launch in 2012-13.

4. LIGO and GEO begin to take data and analyse results jointly. Firstupper limits from the large interferometers published.

25

Interaction of gravitational waves with beam detectors

Since the TT gauge is comoving for all freely falling particles, it is not thesame as the locally Minkowskian coordinate system that would be used by

an experimenter to analyze an experiment. It is important to be aware ofhow coordinates are used.

• If a detector is small compared to the wavelength of a gravitational wave,then the geodesic deviation equation can be used to just give a simpleextra force on the equipment. Laser interferometers on the Earth can be

treated this way. Then the gravitational wave simply produces a force tobe measured. The gravitational wave physics is simple. The physics of

the detectors is not!! We will look at that later.

• If a detector is comparable to or larger than a wavelength, then thegeodesic deviation equation is not useful: it is only the first term in a

Taylor expansion that becomes very clumsy. Space-based interferometerslike LISA, accurate ranging to solar-system spacecraft, and pulsar timingare all in this class. They are all beam detectors: they use light (or

radio waves) to register the waves.

• It is easiest then to remain in TT gauge and calculate the effect of thewaves on the speed of light. In the “+” metric earlier, a null geodesic

moving in the x-direction has effective speed(dx

dt

)2

=1

1 + h+.

This is a coordinate speed, no contradiction to special relativity.

26

• Suppose one arm of an interferometer is along the x-direction and the

wave is moving in the z-direction with a +-polarization of any waveformh+(t) along this axis. (It is a plane wave, so its waveform does not dependon x.) Then a photon emitted at time t from the origin reaches the other

end, at a fixed coordinate position x = L, at the coordinate time

tfar = t +∫ L

0[1 + h+(t(x))]1/2 dx,

where the argument t(x) denotes the fact that one must know the timeto reach position x in order to calculate the wave field. This implicit

equation can be solved in linearized theory by using the fact that h+ issmall. Then we can set t(x) = t + x and expand the square root. Theresult is

tfar = t + L +1

2

∫ L

0h+(t + x)dx.

In an interferometer, the light is reflected back, so the return trip takes

treturn = t + 2L +1

2

[∫ L

0h+(t + x)dx +

∫ L

0h+(t + L + x)dx

].

• What one monitors is changes in the time for the return trip as a function

of time at the origin. This is directly what happens in radar rangingor in transponding to spacecraft, and close to what happens in direct

interferometry. So one measures the variation of the return time as afunction of the start time t:

dtreturn

dt= 1 +

1

2[h+(t + 2L) − h+(t)] .

This depends only on the wave amplitude when the beam leaves andwhen it returns. Interestingly it does not involve the wave amplitude at

the other end.

• The wave amplitude at the other end does get involved if the wave travels

at an angle θ to the z-axis in the x−z plane. If we re-do this calculation,allowing the phase of the wave to depend on x in the appropriate way,

and taking into account the fact that hxx is reduced if the wave is notmoving in a direction perpendicular to x, we can find (see the exercise

later)

dtreturn

dt= 1 +

1

2(1 − sin θ)h+(t + 2L) − (1 + sin θ)h+(t)

+2 sin θh+[t + L(1 − sin θ)] .

This three-term relation is the starting point for analyzing the responseof all beam detectors.

27

Analysis of beam detectors

Here we consider the implications of the three-term formula for specific waysin which gravitational waves are being looked for using beam detectors.

• Ranging to spacecraft. Both NASA and ESA perform experiments inwhich they monitor the return time of communication signals with inter-

planetary spacecraft for the characteristic effect of gravitational waves.For missions to Jupiter and Saturn, for example, the return times areof order 2 − 4 × 103 s. Any gravitational wave event shorter than this

will appear 3 times in the time-delay: once when the wave passes theEarth-based transmitter, once when it passes the spacecraft, and once

when it passes the Earth-based receiver. Searches use a form of dataanalysis using pattern matching. Using two transmission frequencies and

very stable atomic clocks, it is possible to achieve sensitivities for h oforder 10−13, and even 10−15 may soon be reached.

• Pulsar timing. Many pulsars, particularly the old millisecond pulsars,are extraordinarily regular clocks, with random timing irregularities too

small for the best atomic clocks to measure. If one assumes that theyemit pulses perfectly regularly, then one can use observations of tim-

ing irregularities of single pulsars to set upper limits on the backgroundgravitational wave field. Here the 3-term formula is replaced by a sim-

pler two-term expression, because we only have a one-way transmission.Moreover, the transit time of a signal to the Earth from the pulsar maybe thousands of years, so we cannot look for correlations between the

two terms in a given signal. Instead, the delay is a combination of theeffects of waves at the pulsar when the signal was emitted and waves at

the Earth when it is received.

If one simultaneously observes two or more pulsars, the Earth-based part

of the delay is correlated, and this offers a means of actually detectinglong-period gravitational waves. Observations require timescales of sev-

eral years in order to achieve the long-period stability of pulse arrivaltimes, so this method is suited to looking for strong gravitational waves

with periods of several years. Observations are currently underway at anumber of observatories.

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• Interferometry. An interferometer essentially measures the difference

in the return times along two different arms. The response of each armwill follow the three-term formula, but with different values of θ, depend-ing in a complicated way on the orientation of the arms relative to the

direction of travel and polarization of the wave. Ground-based interfer-ometers are small enough to use the small-L formulas we derived earlier

(and see Exercise 5(b)). But LISA, the space-based interferometer, islarger than a wavelength of gravitational waves for frequencies above

10 mHz, so a detailed analysis based on the 3-term formula is needed.

29

Physics of Interferometric Detectors

Interferometers use laser light to compare the lengths of two perpendicular

arms. Typically the arms respond differently to a gravitational wave, so theirdifference will indicate the presence of a wave.

A detector with an arm length of 4 km responds to a gravitational wave withan amplitude of 10−21 with

δlgw ∼ hl ∼ 4 × 10−18 m.

To increase the signal (response) of the detector, the light path is “folded”:light passes up and down the arms many times in order to create an effectiveoptical arm-length better matched to the signal period.

The main sources of noise are:

1. Thermal noise. Vibrations of the mirrors and of the suspending pen-dulum can mask gravitational waves. Interferometers minimise effect of

noise by measuring only at frequencies far from the resonant frequency,and making sure all materials have high-Q, so their resonances are sharp

and energy leakage to measurement frequencies is small. Interferometersoperate at room temperature.

2. Shot noise. The photons that are used to do interferometry are quan-

tized, and so they arrive with random phases and hence make randomfluctuations in the interference pattern that can look like a gravitational

wave signal. The more photons one uses, the smoother will be the in-terference signal. As a random process, the error improves with the

square-root of the number N of photons. Using infrared light with awavelength λ ∼ 1 µm, one can expect to measure

δlshot ∼ λ/(2π√

N).

To measure at a frequency f , one has to make at least 2f measurementsper second, so one can accumulate photons for a time 1/2f . With light

power P , one has N = P/(hc/λ)/(2f). In order that δlshot should equalδlgw one needs light power of about 600 kW, far beyond the output of any

continuous laser. (Note that if the optical path is folded, then one needsa length measurement here that is longer by the number of bounces. But

in the end the amount of light stored in an arm, i.e. summed over allfolds, is still very large.

30

Light-recycling techniques overcome this problem, by using light effi-

ciently. Light is allowed to build up in the arms until the laser merelyresupplies the mirror losses, which total about 10−5 of the power in thearms for the very high-quality mirrors that are now available. This re-

duces the power requirement for the laser by the same factor, down toabout 6 W. This is attainable with modern laser technology.

3. Ground vibration. Mechanical vibrations must be screened out. There

are many different ways to do this, but all of them are very sensitive tofrequency, working well down to a lowest effective frequency. The mostambitious isolation system is being developed for the Virgo detector.

4. Quantum effects. Shot noise is a quantum noise, but by increasing

laser power it can be reduced. However, increasing laser power increasesintensity fluctuations and hence random light-pressure fluctuations on

the mirrors. This is a classic Heisenberg trade-off: phase and amplitudeare conjugate variables for the electromagnetic field of the laser light.

As laser power increases, the uncertainty in pressure becomes a limitingfactor, and we reach what is called the “standard quantum limit”.

Development of novel quantum-measurement techniques for interferome-ters is a rapidly developing field. Signal-recycled interferometers reshapethe quantum state of the light and actually can improve the sensitivity.

Clever optical tricks can manipulate the uncertainty ellipse, squeezing itin the measurement direction – so-called “squeezed light”. And new op-

tical arrangements can allow interferometers to measure the momentumof the mirrors rather than their positions (“speed meters”); momentum

can be measured repeatedly with high accuracy even though positionaccuracy is sacrificed.

31

Gravitational Wave Observables

Intrinsic to wave: this is what we want to measure

h+(t), h×(t) : waveform of both polarisations as functions of timeθ, φ : direction on sky (except for stochastic background)

The phase of the wave, i.e. the detailed time-dependence of h+ and h×, con-tain most of the information in the wave.

Detector output: this is what we get

Each detector usually gives just a single amplitude h(t) — the waveform(details of the phase evolution of the wave) is affected by two aspects of the

detector:

The detector response is a projection of h+ and h× onto the detector,

and this can change due to detector motion (amplitude modulation)The received phase of the wave is also affected by the motion of

detector (phase modulation or Doppler effect)

Use of information depends on duration of signal

short “bursts” – detector effectively stationary – network of > 3 de-tectors to triangulate position and find h+, h×.

“Continuous” signals – detector moves during observation of signal –

single detector performs aperture synthesis, finding position of sourceand amplitude by itself (if source’s position and amplitude don’tchange). Second detector useful for reliability.

Stochastic backgrounds – just like noise in a single detector. If the

detector noise is well-understood, then it may detect this excessnoise (just as the original microwave background detection). Co-herent cross-correlation between two detectors eliminates much de-

tector noise and is more sensitive when detectors are closer than awavelength.

32

Capabilities of Gravitational Wave Interferometers

Interferometric detectors work over a broad bandwidth: they do not haveany natural resonances in their observing band. Here is how they work for

various sources:

• Bursts A burst is a wave of short duration and wide spectral bandwidth.Interferometers are ideal for such signals, since one can perform pattern-matching over the whole bandwidth and detect opimally.

• Continuous signals These signals have a long duration and a narrow

bandwidth, but typically a low amplitude. By observing for a long time,interferometers can do well. Optical techniques like signal recycling allow

the interferometer to become narrow-band at any frequency, enhancingits sensitivity to narrow-band sources.

• Stochastic signals Interferomters do cross-correlation in broad bands,can give frequency information.

33

LISA: a gravitational wave detector in space

Ground-based interferometers are limited to frequencies above a few Hz,mainly because local disturbances in the Newtonian gravitational field are

larger than expected gravitational waves at frequencies lower than this. Toobserve below 1 Hz one must go into space.

LISA consists of 3 independent spacecraft arrayed as an equilateral triangle,with laser beams along each of the arms. The spacecraft can be taken to bepoint particles following geodesics, so the analysis is similar to other beam

detectors. However, LISA’s arms are 5 × 106 km, comparable to the wave-length of a gravitational wave of frequency 25 mHz, so for frequencies higher

than this one cannot use the small-arm approximation.

Given its 3 arms, LISA can be regarded as having two independent interfer-ometers, made from the arms that join at any two of the three vertices. The

third interferometer’s response can be found from the other two, at least atlow frequencies where the short-arm approximation holds. The interferome-

ters are independent in terms of their gravitational wave response: they sensetwo different polarisations of the gravitational wave. But since the two de-

tectors share a common arm, their instrumental noise in the two detectors isnot independent. Thus, LISA can determine the polarisation of an incoming

wave without needing another detector, but it cannot do a cross-correlationexperiment between its two detectors in order to improve its sensitivity to astochastic background of gravitational waves.

LISA deals with the same limitations as ground-based interferometers, butin very different ways:

1. Thermal noise. This is not a problem for LISA. The large arm-lengthmeans that the physical displacements being measured are millions of

times larger than for ground-based detectors, and thermal vibrations donot reach this amplitude.

2. Shot noise. LISA collimates the beam from a laser and directs it tothe spacecraft at the other end of the arm using a 30-cm mirror. The

diffraction limit of this for light with a wavelenght of 1 micron is of orderone microradian. Over a distance of 5 × 106 km, therefore, the center

of the beam spreads out to several kilometers. The 30-cm mirror onthe other spacecraft intercepts only a tiny fraction of the transmitted

light, and if LISA relied on this light being reflected, again with a large

34

diffraction pattern, then the intensity of the returned light would be

negligibly small. Instead, LISA will have active optical transponders:the incoming light will be amplified by an on-board laser, which returnsa much stronger beam. With these “active mirrors”, LISA needs only

1 W lasers in order for its shot noise limit to provide excellent sensitivity.This is a very comfortable requirement. Shot noise is the limiting noise

at frequencies above about 1 mHz.

3. Vibration. Space is not totally empty. The radiation pressure fromthe Sun is a constant force that would push the spacecraft out of theirpositions, and on top of that the Sun is not a constant radiator. Fluc-

tuations in its intensity at the level of one part in 103 are normal. Solarstorms produce intense fluxes of particles, which can exert forces on the

spacecraft. LISA copes with these disturbances by using a freely-flying“proof mass” inside the spacecraft at the end of each arm as the reference

point for the interferometry. The spacecraft itself is an active shield, sens-ing the position of the proof mass and firing very weak (micro-Newton)

thrusters to compensate the external forces and prevent the proof massfrom being disturbed. Since each spacecraft carries two proof masses, onefor each arm terminating at it, the control algorithm for the thrusters is

not trivial, but it can be described. This form of disturbance isolation iscalled “drag-free technology”, and will become more and more useful in

experiments in space in the future. The accuracy with which the sensingof the proof mass can be done decreases with decreasing frequency, so

vibration noise dominates shot noise below about 1 mHz.

4. Quantum effects. As with thermal effects, these are not important for

LISA because of the large displacements being measured.

5. Laser frequency noise. We did not discuss this for ground-based de-tectors because there are ways to control this, and in any case a ground

detector has only one laser, so its noise creates correlated disturbancesin both arms than to some extent cancel out when the beams interfere.

But in LISA, there are 6 active lasers, each with independent frequencynoise. LISA copes with this by a very clever algorithm for combiningthe signals from the several beams in linear combinations with various

time-shifts that cancel out the frequency noise. The algorithm for doingthis is called “time-delay interferometry”, or TDI. The result is that the

system can produce three TDI combinations that essentially representthe signals from the three interferometers associated with each of the

35

vertices of the triange.

In addition, LISA has to contend with one further “noise”: confusion noise.We will see later that LISA has enough sensitivity to see a huge number of

sources. For some of them the nearest ones will stand out individually but themore distant ones will blend together into a random confusion background of

gravitational waves. Separating the resolvable sources from the more distantones is a challenge to LISA’s data analysis that the ground-based detectorsdo not have to deal with, and we will discuss this later. Over much of LISA’s

frequency range, the confusion limit is not far below the shot noise limit,which explains why it is not worth putting more powerful lasers on board.

At low frequencies, below 1 mHz, the confusion noise actually dominatesother noise sources, in particular sensor noise in the drag-free system.

Interestingly, at low frequencies the fact that the signals from the three in-

terferometers are not independent leads to a noise veto: there is a linearcombination of the three TDI output signals that cancels out all gravita-

tional wave signals, and therefore can be used to measure the instrumentalnoise. This so-called “Sagnac mode” can be used therefore to monitor the

detector. This does not work at higher frequencies, because when we lost theshort-arm approximation then the three interferometers actually give inde-pendent responses.

LISA’s frequency “window” extends from about 0.1 mHz (below which con-fusion noise and sensor noise make observing difficult) up to about 0.1 Hz.

Its best range is between 1 mHz and 10 mHz. Above this, the cancellationsthat happen because the arms are a good fraction of a wavelength degrades

the sensitivity of LISA.

36

Chapter 7 – Data Analysis

Since ground-based detectors are of relatively low sensitivity, sophisticated

data analysis strategies are necessary even to recognise gravitational waveevents. LISA’s confusion-limited sensitivity presents equally demanding (but

different) challenges. The art and science of data analysis are key componentsof all the searches for gravitational waves.

Matched filtering.

Principle: noise is broadband, spread over spectrum, stationary, and Gaus-sian or nearly Gaussian in its statistics. If you can look for a signal in one

part of spectrum, you fight against less noise. Classic: Fourier transform,where amplitude sensitivity is proportional to T 1/2, where T is the observa-

tion time. Generally, if one has an accurate waveform prediction, the gainin amplitude sensitivity is proportional to the square-root of the number of

cycles in the waveform.

In the simplest case, one just takes a correlation of the data, xj, with apredicted template signal hj. The resulting statistic is

c = Σjxjhj.

If x is uncorrelated with h then the expectation value of this is zero. If x

contains a signal proportional to h then the expectation value is non-zero. Ifthe value of c is large enough, then one should be ready to declare a detection.

Notice that if the signal is a sinusiod of frequency f, h = A sin(2πft), then

the correlation is just the sine-transform of the data at that frequency,

c = AΣjxj sin 2πftj.

Matched filtering is a generalisation of the signal-recognition ability of theFourier transform.

37

In practice one does the correlation in the Fourier domain. By Parseval’s the-

orem on Fourier transforms, c can also be found from the Fourier transformsx and h:

c = Σkxkhk∗.

The reason for preferring this is that one usually does not know when toexpect the signal, so that there is an arbitrary time-of-arrival parameter τ inthe waveform, h(t−τ). The Fourier transform of h(t−τ) is h(f) exp(−2πifτ),

so that the previous equation becomes

cj = Σkxkhk∗e−2πijk/N ,

where j is the integer indexing resolvable times corresponding to τ , i.e. suchthat

jk/N = fτ.

This is just a Fourier transform and can be done with the fast FFT algorithm.That means that in the Fourier domain one can perform the time-sliding of

a template much more efficiently and rapidly than in the time domain.

But this is only an optimum detection statistic if the noise is white. If there

are frequencies where the noise is louder (and this is always the case), thenthese should have less weight. If we define Sk to be the noise power at

frequency index k, then the optimum detection statistic is the matched filter:

C = Σkxkhk

∗

Sk.

The function S(f) is called the spectral noise density, and it is a measure ofdetector noise. It is defined essentially as the power spectrum of the noise n

in the frequency domain. If we denote n(f) as the Fourier transform of thenoise, then the definition is:

< n(f)n∗(f ′) >= S(f)δ(f − f ′).

The angle brackets denote an ensemble average over the noise. The delta

function reflects the fact that the noise at different frequencies is statisticallyuncorrelated. The larger S(f), the poorer the detector sensitivity.

All the detector groups plot S(f)1/2 vs f to show their sensitivity. Theytake the square root so it can be compared directly to the amplitude of anyexpected signal. And they calibrate S to the detector’s strain sensitivity (re-

sponse to a gravitational wave) by making sure that the noise n in the lastequation is measured in strain (not in voltage or some other measure more

38

directly connected to what is sampled from the detector output). This cali-brated strain spectral density is called Sh(f). The number Sk in the matched

frequency equation above is essentially the value of Sh at the frequency cor-responding to the index k.

Matched filtering is the optimal linear detection statistic if the noise is Gaus-

sian. No linear transform of the data can do better. But if the noise is morecomplex then one must be more careful. In the ground-based projects, detec-

tor groups spend large efforts on understanding the statistics of their noise.For LISA the confusion background of weak gravitational wave signals will

be Gaussian at small enough amplitudes (where the central limit theoremholds for the dozens or more sources that are superimposed) but there is anon-Gaussian transition between this level and the signals that are strong

enough to be resolved. This is an issue that needs research.

39

Parameter searches.

The template h is generally only known to be a member of a family withseveral parameters, such as the masses of the objects. The search must be

done individually for each member of this family. This raises problems of

• computer power: can we search over the whole family?

• resolution: are we taking templates closely spaced enough?

• accuracy: how well can we measure the parameters?

• significance: what is the chance of a random correlation?

LIGO and GEO have signed an MOU for data exchange and joint analy-

sis. This is organized through the LSC (LIGO Science Collaboration), whichmanages the creation of joint software (LIGO Applications Library) and su-

pervises the work of four international teams: pulsar searches, inspirallingbinary searches, stochastic background searches, and burst searches. Each

team is chaired by two scientists, a theoretician and an experimentalist. Twoof the theoretician chairs are GEO scientists, one at the AEI (Papa: pulsars)

and one at Cardiff (Romano: stochastic). Access to data is only through theLSC or independent MOU’s with the detector groups. TAMA has joined theanalysis in a limited way, and an MOU with VIRGO is in preparation.

The collaboration also involves pooling computer resources. All searchesare compute-intensive in one way or another. The most demanding search

problem is the search for gravitational wave pulsars over the whole sky. Sincemany nearby pulsars may not be known from radio observations, a search in

gravitational waves is desirable. To get sufficient sensitivity it is felt that thesesearches may last several weeks or months, but during that time the signal

is modulated (Doppler-shifted) by the motion of the Earth. The modulationpattern depends on location, and after a 1-year observation there are 1013

different resolvable locations that have to be searched independently! The

AEI’s Merlin computer, a teraflop-class cluster, works full time on this, buteven this computer power this will only allow us to survey a small fraction

of the sky. Other collaboration clusters will help but not suffice.

To address this problem, the Milwaukee and AEI groups have recently re-

leased Einstein@Home, a screen-saver that may provide significantly morepower than in-house clusters. Already in its test phase it acquired enough

users to become the most powerful computer in the gravitational wave com-munity.

40

Chapter 8 – Overview of gravitational wave sources

There are a large number of possible gravitational wave sources. Since

ground-based detectors are confined to frequencies above 10 Hz, their sourcesmust be highly relativistic and not too massive: a black hole of mass 1000Mhas a characteristic frequency of 10 Hz, and larger holes have lower frequen-cies. Nothing of mass above this size can radiate at frequencies above 10 Hz.

In the low-frequency regime that LISA will study, one can expect very mas-sive black holes or more widely-separated binary stars.

Here is a brief list:

• Gravitational collapse. The longest-sought and least-understood source,we still cannot do more than estimate the amplitude h that would be

emitted if a source lost an assumed amount of energy to waves. Theproblem is that it is still not possible to reliably estimate the amount of

asymmetry in gravitational collapse. Numerical simulations show, how-ever, that significant asymmetry only develops in rotating collapse, and

even then only if the pre-collapse star is differentially rotating. It isdifficult to know what initial conditions for collapse are reasonable.

• Binary stars. Our best proof of the reliability of general relativity forgravitational waves is the behavior of the Hulse-Taylor binary pulsar.

But its radiation will be too weak and at too low a frequency even tobe detectable directly by LISA. LISA will see lots of higher frequency

binaries with compact objects (white dwarfs and neutron stars) in them.Below 1 mHz there are expected to be too many to resolve, and mostwill therefore become part of LISA’s confusion background.

• Chirping binary stars. If a binary gives off enough energy that its

orbit shrinks by an observable amount during an observation, it is saidto “chirp”. LISA will see a few chirping binaries. If a binary is compact

enough to radiate at 10 Hz or above, it will always chirp, and in fact itwill completely coalesce in a short time. These coalescing binaries are

the most promising long-term source of radiation for LIGO and VIRGO.Their emitted radiation can be characterized rather well within the post-Newtonian approximation, but only if the stars are not spinning.

Black-hole binaries will also coalesce, and their numbers may be relatively

high. This is because the formation of a black hole has less likelihood to

41

break up a binary system than the formation of a neutron star, which

involves a much larger loss of mass from the collapsing object. Blackhole binaries may be among the first sources detected by ground-baseddetectors. Binary systems consisting of massive black holes in galactic

centers are a major target of LISA.

• Capture events. Massive black holes will occasionally capture a com-pact object (neutron star or stellar-mass black hole) from their surround-

ing stellar cluster. These fall on almost-geodesic orbits before crossingthe horizon. The dominant frequency for their radiation is set by the or-bital timescale around the massive hole, so this is a low-frequency LISA

source. It has considerable interest because systems may spend hundredsof thousands of orbits in the LISA band before they finally enter the black

hole. The phase evolution of these orbits contains detailed informationabout the geometry of the hole.

• Pulsars and other spinning neutron stars. There are a number of

ways in which a neutron star may give off a continuous stream of gravi-tational waves: shape irregularities in a spinning star, unstable r-modes(also only spinning stars), pulsation modes excited during star formation

or by transient events like starquakes (associated with glitches). Radia-tion from shape irregularities or r-modes will be weak but detectable if

one observes for long periods of time, up to many months. Analyzing thedata for a survey of the sky for such systems (i.e. if one does not know

the star ahead of time from radio or X-ray observations) makes severedemands on computer power, and algorithms to do this efficiently are thesubject of much work in the detector groups today. R-modes of neutron

stars are in principle unstable to the emission of gravitational radiation,but this can be quashed by viscosity inside the star. In practice they may

be effective in limiting the spins of low-mass X-ray binary stars, and ScoX-1 seems to be the best candidate. These modes are interesting theoret-

ically also, because unlike most other sources we discuss, they emit notmass-quadrupole radiation but current-quadrupole, the next order up in

the multipole expansion that we derived earlier.

The pulsation modes of neutron stars are at frequencies of 2 kHz or

higher, where present detectors have poor sensitivity. Nevertheless, theseare interesting systems to observe because the exact frequency of such a

mode would for the first time give us insight into the interior physics ofa neutron star.

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• Random backgrounds. The big bang was the most violent event of all,

and it may have created a significant amount of gravitational radiation.Other events in the early universe may also have created radiation, andthere may be backgrounds from more recent epochs, for example the

r-mode instability (see Chapter 7).

• The unexpected. At some level, we are bound to see things we didnot expect. LISA, with its high signal-to-noise ratios, is well-placed to to

this. Most of the universe is composed of dark matter whose existencewe can infer only from its gravitational effects. It would not be partic-ularly surprizing if this dark matter produced gravitational radiation in

unexcpected ways, such as from binaries of small exotic compact objectsof stellar mass. A particluarly intriguing candidate is the cosmic string.

This is a one-dimensional “defect”, or mass concentration, which wouldarise naturally in some unified field theories where symmetry breaking

occurs as the Universe cools off. The strings (not to be confused withstring-theory strings, which are tiny by comparison) essentially trap a

small region of space where the symmetry did not break. These strings,which can be megaparsecs or more in length, have dynamical featuresthat emit characteristic bursts of gravitational waves. So gravitational

wave observatories may well reveal exotic features of the Universe!

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Chapter 9 – Sources for LISA

LISA will open up the low-frequency band between 0.1 mHz and 0.1 Hz. By

comparison with sources for ground-based detectors, sources in this frequencyrange sources either have to have larger masses (the modes of a massive black

hole of 106M have a frequency around 1 mHz) or larger separations (compactwhite dwarf binaries with orbital periods around an hour are known, and

pre-coalescence neutron-star binaries will pass through this frequency rangethousands of years before they coalesce). LISA’s high sensitivity means thatsome sources will be extremely strong in the LISA data, and more distant

sources will still be strong enough to cause a confusion noise background. Wediscuss some of these problems here.

First, a list of expected sources:

• Massive black holes. One of the great surprises of the late 1990s was

the discovery that most galaxies contain a massive (106M) or even asupermassive (109M) black hole in their centres. The formation history

of these holes is still not clear. They probably formed early in the lifetimeof the galaxy and accreted a lot of gas in the early phases, but they mayalso have grown by merger of smaller black holes. There is growing

evidence that even the small proto-galactic clouds whose merger led tothe galaxies we see today contained black holes. Most galaxies probably

contained thousands of black holes of mass 103M or more, and it is verypossible that these have sunk to the center and merged into the large ones.

Quasars and active galaxies seem to be powered by supermassive holes,and there is evidence from X-ray studies that gas accretion accounted for

most of their growth from 106 to 109M.

LISA has enough sensitivity to see the merger of two black holes in the

mass range 104 − −107M if it occurs during the mission lifetime any-where in the Universe. Even very distant events, say at redshift 10,could have very high signal to noise ratios. LISA should therefore be

able to answer some of the questions about black hole growth, and aboutthe initial mass spectrum of massive black holes. It should also be able

to study in detail the complex merger phase where the two black holescome together. By comparing this radiation with that predicted by su-

percomputer simulations of black hole mergers, LISA will explore andtest general relativity in the strongest possible gravitational fields.

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Supermassive black holes radiate at frequencies below LISA’s band, so

the issue of their growth will not directly be addressed by LISA.

Detecting the radiation from massive black hole binaries will not be a

problem. In some cases it will be strong enough to be seen withoutmatched filtering. The inspiral phase will be so easy to pick up that

LISA will be able to issue a prediction of the merger time and locationon the sky at least a month before it happens. This will enable co-ordinated observing campaigns with X-ray telescopes and ground-based

instruments.

• EMRI: Extreme Mass-Ration Inspiral sources. Massive blackholes in galactic centres do not exist in isolation. They are inside vast

clouds of stars that have, from normal stellar evolution, large numbersof neutron stars and stellar-mass black holes (say 10M). Random en-counters among these stars gradually allow the black holes to sink toward

the centre, so that near the massive hole there should be an overdensityof stellar holes. Occasionally a random close encounter should put one

of these holes on a “plunge” orbit, a highly eccentric orbit aimed nearlydirectly at the central hole. If the impact parameter is small enough,

the first pass near the massive hole will radiate enough energy in gravi-tational waves to change the orbit from unbound to bound. The smaller

hole will then execute a large number of orbits during which its peri-bothon distance (peribothon means closest approach to the hole) remainsroughly constant but its apbothon distance shrinks. During this time it

will be a source of isolated bursts of gravitational waves at peribothon.LISA may pick these up but they will not repeat during the LISA mission

lifetime.

But eventually the smaller hole acquires an orbit where its overall period

is in the LISA band. Even then, it may have 105 orbits remaining beforeeventually crossing the horizon. The details of its orbital behaviour are

contained in the phase evolution of the emitted gravitational waves. Ifit is possible to construct accurate matched filters for this radiation then

LISA will be able to examine in detail the geometry of the black holeexterior. It will test the uniqueness theorem of general relativity, that allblack holes will be members of the Kerr family.

This exquisite test will, however, not be easy to perform. At present, we

do not understand the equation of motion of the small black hole aroundthe large on well enough to construct an accurate filter over so many

45

orbits. The hole follows a geodesic only to lowest order. Corrections to its

motion to first order in the mass-ratio of the two holes are required. Someof these are understood, and in principle we have methods to calculateall of them. But we have no efficient means to do so. This is one of the

outstanding problems of LISA science that must be solved before LISAlaunches.

Even when good filters are available, the search algorithm will be verychallenging. The parameter space for the filter family will be huge and

dense, with small changes in parameters making large changes in thephase evolution by the end of the orbit. Studies indicate that hierarchi-

cal methods, analogous to those being developed for ground-based studiesof the pulsar problem (mentioned earlier), will be able to solve this prob-

lem and detect hundreds of inspiral events per year. But the detailedimplementation has not yet been done.

The events detected will come from nearby galaxies, out to redshifts of afew tenths. More distant events will create some of the confusion noise

LISA must contend with. Exactly how this will affect detection, and justhow to pull as many sources as possible out of the confusion background,are problems that have not yet been adequately studied.

• Compact binary systems. Below about 1 mHz, LISA will be dom-

inated by radiation from binary systems in the Galaxy. These are notnormal systems, because they have to have periods of an hour or so to be

in the LISA band. They consist of white dwarfs and (very occasionally)relativistic objects like neutron stars and black holes. LISA is expectedto be able to resolve thousands of such objects, including perhaps one

black-hole/black-hole binary. But there are so many of them at lowerfrequencies that they will blend together into another confusion noise,

and only the nearby systems will be strong enough to resolve.

LISA will be able to give distances to most of the systems that have fre-

quencies above 1 mHz, because they chirp (see earlier). By coordinatingLISA detections with observations by GAIA, the ESA astrometric mis-

sion planned for launch in 2012, astronomers will be able to identify andcharacterize in detail many of these compact systems. This will greatly

contribute to our understanding of stellar evolution and particularly ofbinary evolution.

Extragalactic binaries may also be seen, including for example a black-hole binary in M31, the Andromeda galaxy. But most extragalactic sys-

46

tems blend into a confusion background at higher frequencies (up to10 mHz) that is thought to lie just below the LISA shot-noise limit. This

is one reason that it does not seem to be worthwhile to raise the power ofLISA’s lasers, since the gain in sensitivity would be limited by confusionnoise.

• Stochastic background. LISA can only detect the stochastic back-ground if its noise is higher than instrumental or confusion noise from

foreground objects. Radiation with a closure density Ωgw ∼ 10−10 wouldbe detectable. On standard inflation models this seems unlikely, but there

are many possibilities for radiation generated either in non-standard cos-mological scenarios (string cosmology) or from phase transitions thathappened after inflation. Intriguingly, LISA’s frequency band contains

radiation that would have been emitted by the Universe when its temper-ature was about 1 TeV, corresponding to the electroweak phase transi-

tion. If an excess noise is seen in LISA, the experimeters must have goodconfidence to ascribe it to gravitational waves and not to instrumental

misbehaviour. The Sagnac mode described earlier will help here.

• Unexpected sources. LISA has such good sensitivity that it may welldetect sources that are not know or expected. Cosmic strings (mentioned

earlier) are a possible candidate.

LISA’s data analysis is more complicated than that for ground-based de-tectors. As described earlier, LISA has actually three (TDI) output data

streams, and these can be combined optimally to increase the sensitivity toparticular directions. They can also be combined in the Sagnac mode to

cancel gravitational waves and characterize instrumental noise.

A more serious complication is the fact that LISA sources overlap and fightconfusion. It will not be possible to filter for an EMRI source, for example,using normal matched filtering, if the data stream also contains the signal

from a massive black hole coalescence. The coalescence signal needs to beremoved in order to make the “noise” against which the EMRI signal is fil-

tered as close to Gaussian as possible. LISA data analysis will consist ofan iterative set of steps in which strong sources are identified and removed,

followed by weaker ones, until most sources are identified. Then this must berepeated, since once the weaker ones are known the stronger ones can be re-moved more accurately. Removal of signals accurately requires, of course, the

measurement of all their parameters, which is the astrophysical informationthat scientists want from LISA.

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Moreover, many of LISA’s sources will be radiating for years. Therefore

after, say, five years of observing the binaries will be better known than inthe first year, and this will improve the identification and measurement of theradiation from coalescences that occurred even in the first year. No detailed

implementation of a successful scheme to do this kind of data analysis hasyet been demonstrated.

The LISA project is beginning now, therefore, to plan its data analysis. Therequirement is to understand how to do the data analysis by 4–5 years beforelaunch, and then use the remaining time to implement it in professional and

robust software.

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Reference List – BF Schutz

To learn more about the subjects covered here, you can consult a number of references.

• General Relativity. There are a number of good textbooks in GR. Both of the following cover, atdifferent levels of difficulty and completeness, linearized theory, gauges, and the definition of energy:

– Misner, C. W., Thorne, K. S., Wheeler, J. L., Gravitation, (Freeman & Co., San Francisco, 1973).

– Schutz, B. F., A First Course in General Relativity, (Cambridge University Press, Cambridge,1985).

• Gravitational wave detectors. Conference volumes on detector progress appear more than onceper year these days. You can find progress reports on detectors on the web sites of the different groups.The two references I give here are more tutorial, aimed at introducing the subject.

– Saulson, P. R., Fundamentals of Interferometric Gravitational Wave Detectors, (World Scientific,Singapore, 1994).

– Blair, D.G., ed., The Detection of Gravitational Waves, (Cambridge University Press, CambridgeEngland, 1991).

• Sources of gravitational waves. Again, there have been a number of conference publications onthis subject. The first two references are recent reviews of sources. The third surveys the problem ofdata analysis, which I only touched on in my lectures here.

– Thorne, K. S., “Gravitational Radiation”, in Hawking, S.W., Israel, W., eds., 300 Years of Grav-itation, (Cambridge University Press, Cambridge, 1987), p. 330–458.

– Thorne, K.S., in Kolb, E.W., Peccei, R., eds., Proceedings of Snowmass 1994 Summer Study onParticle and Nuclear Astrophysics and Cosmology, (World Scientific, Singapore, 1995).

– Schutz, B.F., “The Detection of Gravitational Waves”, in Marck, J.-A., Lasota, J.P., eds., Rela-tivistic Gravitation and Gravitational Radiation, (Cambridge Univ. Press, Cambridge, 1997) p.447-475.

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Exercises

EXERCISE 1 (a) A plane wave with A× = 0 and h+ = A+ sin[ω(t − z)] has the metric

ds2 = −dt2 + (1 + h+)dx2 + (1 − h+)dy2 + dz2.

Show that this wave does not change proper separations of free particles if they are aligned along a linebisecting the angle between the x- and y-axes.

(b) Show that a plane wave with A+ = 0 and h× = A× sin[ω(t − z)] has the metric

ds2 = −dt2 + dx2 + 2h×dxdy + dy2 + dz2.

(c) Show that the wave in (b) does not change proper separations of free particles if they are aligned alongthe coordinate axes.

(d) Show that the wave in (b) produces an elliptical distortion of the circle that is rotated by 45o to that ofthe wave in (a).

EXERCISE 2 Let two balls fall freely in the gravitational field of the (spherical) Earth at starting positionsvery near to one another.

(a) If the balls are separated initially in the horizontal direction, calculate their tidal acceleration relative toone another. Show in particular that they are brought together by the tidal forces.

(b) Do the same on the assumption that the balls are separated initially in the vertical direction. Show thatthey are drawn apart.

(c) Show that an initial sphere of free particles, falling freely toward the Earth, will to first order not changeits volume as it experiences the tidal distortions of parts (a) and (b).

(d) Use (a) and (b) to explain qualitatively ocean tides on the Earth.

(e) Show that the Sun and Moon exert tidal forces to the Earth that are similar in size. You don’t need toknow how far away the Sun and Moon are, provided you use the fact that they are roughly the same angularsize in the sky (hence we have solar eclipses) and that they are similar in mean density (to within factors ofa few).

50

EXERCISE 3 (a) Use the metric for a plane wave with “+” polarisation,

ds2 = −dt2 + (1 + h+)dx2 + (1 − h+)dy2 + dz2,

to show that the square of the coordinate speed (in the TT coordinate system) of a photon moving in thex-direction is (

dx

dt

)2

=1

1 + h+.

This is not identically 1. Does this violate relativity? Why or why not?

(b) Imagine that an experimenter at the center of our circle of particles sends a photon to the particle atcoordinate location x = L on the positive-x axis, and that the photon is reflected when it reaches the particleand returns to the experimenter. Suppose further that this takes such a short time that h+ does not changeduring the experiment. To first order in h+, show that the experimenter’s proper time that elapses betweensending out the photon and receiving it back is (2 + h+)L.

(c) The experimenter says that this proves that the proper distance between herself and the particle is(1 + h+/2)L. Is this a correct interpretation of her experiment? If the experimenter uses an alternativemeasuring process for proper distance, such as laying out a number of standard meter sticks between herlocation and the particle, would that produce the same answer? Why or why not?

(d) Show that if the experimenter simultaneously does the same experiment with a particle on the y-axis aty = L, that photon will return after a proper time of (2 − h+)L.

(e) The difference in these return times is 2h+L and can be used to measure the wave’s amplitude. Doesthis result depend on our use of TT gauge, i.e. would we have obtained the same answer had we used adifferent coordinate system?

EXERCISE 4 A frequently asked question is: if gravitational waves alter the speed of light, as we seem tohave used here, and if they move the ends of the interferometer closer and further apart, might these effectsnot cancel, so that there would be no measurable effect on light? Answer this question. You may want toexamine the calculation in Exercise 3: did we make use of the changing distance between the ends, and why?

EXERCISE 5 Beam detectors. (a) Derive the full three-term return equation for the rate of change of thereturn time for a beam travelling through a plane wave h+ along the x-direction, when the wave is movingat an angle θ to the z-axis in the x − z plane. The formula is reproduced here:

dtreturn

dt= 1 +

12(1 − sin θ)h+(t + 2L)− (1 + sin θ)h+(t)

+2 sin θh+[t + L(1 − sin θ)] .

(b) Show that, in the limit where L is small compared to a wavelength of the gravitational wave, the derivativeof the return time is the derivative of t + δL, where δL = L cos2 θ h(t) is the excess proper distance for smallL. Explain where the factor of cos2 θ comes from.(c) Examine the limit of the three-term formula in (a) when the gravitational wave is travelling along thex-axis too (θ = ±π/2): what happens to light going parallel to a gravitational wave?

EXERCISE 6 Suppose a plane wave, travelling in the z-direction in linearized theory, has both polarizationcomponents h+ and h×. Show that its energy flux in the z-direction, T (GW )0z, is

〈T (GW )0z〉 =k2

32π(A2

+ + A2×)

where the angle brackets denote an average over one period of the wave.

EXERCISE 7 Gauge transformations. The tensor transformation law of the components of the metrictensor from coordinates xα to xµ′

is

gµ′ν′ =∂xα

∂xµ′∂xβ

∂xν′ gαβ .

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Show that a gauge transformation (small coordinate transformation) of the form

xα′= xα + ξα,

where ξ is a small, position-dependent displacement, changes an initial metric tensor gαβ = ηαβ + hαβ into

gα′β′ = ηαβ + hαβ − ξα,β − ξβ,α.

In this expression the numerical values of α′ and α are the same, and the primes just tell us which coordinatesystem we are working in. Argue that the result of this calculation is that in the new coordinate system thedeviation of the metric from special relativity is

h′αβ = hαβ − ξα,β − ξβ,α.

Raising and lowering of indices on ξ is accomplished with the metric η. Work only to first order in ξ andassume that derivatives of ξ are of the same order as the components of h.

EXERCISE 8 Prove the set of identities among moments that are needed for the work on radiation. Theseare

M = 0, Mk = P k, M jk = P jk + P kj ,

P j = 0, P jk = Sjk,d2M jk

dt2= 2Sjk.

All of the identities rely on using, sometimes repeatedly, the conservation law T αβ,β = 0, integration by

parts, and the divergence theorem.