(ed.). Gravitational waves (IOP, 2001)(422s).

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Exercises for chapter 5 57(f)This shows that the divergence-free property of G µν is closely related to thecoordinate invariance of Einstein’s theory.Suppose a plane wave, travelling in the z-direction in linearized theory, hasboth polarization components h + and h × . Show that its energy flux in thez-direction, T (GW)0z ,is〈T (GW)0z 〉= k232π (A2 + + A2 × ),where the angle brackets denote an average over one period of the wave.
Chapter 6Mass- and current-quadrupole radiationIn this lecture we focus on the wave amplitude itself, and how it and thepolarization depend on the motions in the source. Consider an isolated sourcewith a stress-energy tensor T αβ . As in chapter 2, the Einstein equation is( )− ∂2∂t 2 +∇2 h αβ =−16πT αβ (6.1)(h αβ = h αβ − 1 2 ηαβ h and h αβ ,β = 0). Its general solution is the following retardedintegral for the field at a position x i and time t in terms of the source at a positiony i and the retarded time t − R:h αβ (x i , t) = 4∫ 1R T αβ (t − R, y i ) d 3 y, (6.2)where we defineR 2 = (x i − y i )(x i − y i ). (6.3)6.1 Expansion for the far field of a slow-motion sourceLet us suppose that the origin of coordinates is in or near the source, and the fieldpoint x i is far away. Then we define r 2 = x i x i and we have r 2 ≫ y i y i . Wecan, therefore, expand the term R in the dominator in terms of y i . The lowestorder is r, and all higher-order terms are smaller than this by powers of r −1 .Therefore, they contribute terms to the field that fall off faster than r −1 , and theyare negligible in the far zone. Therefore, we can simply replace R by r in thedominator, and take it out of the integral.The R inside the time-argument of the source term is not so simple. If wesuppose that T αβ does not change very fast we can substitute t − R by t − r (theretarded time to the origin of coordinates) and expand( ) 1t − R = t − r + n i y i + O , with n i = x ir r , ni n i = 1. (6.4)58
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GRAVITATIONAL WAVES
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c IOP Publishing Ltd 2001All righ
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viContents4 Astrophysics of gravita
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viiiContents11.3 Gravitational wave
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xContents16.5 Cosmological perturba
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PrefaceGravitational waves today re
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2 Gravitational waves, theory and e
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4 Gravitational waves, theory and e
Page 19 and 20: 6 Gravitational waves, theory and e
Page 21 and 22: 8 Gravitational waves, theory and e
Page 23 and 24: 10 Gravitational waves, theory and
Page 26 and 27: SynopsisGravitational waves and the
Page 28 and 29: Chapter 2Elements of gravitational
Page 30 and 31: Using the TT gauge to understand gr
Page 32 and 33: Interaction of gravitational waves
Page 34 and 35: Analysis of beam detectors 21This d
Page 36 and 37: Exercises for chapter 2 232. Show t
Page 38 and 39: Gravitational-wave detectors 25indi
Page 40 and 41: Gravitational-wave observables 27th
Page 42 and 43: The physics of interferometers 29
Page 44 and 45: The physics of interferometers 31Dr
Page 46 and 47: The physics of interferometers 33Fi
Page 48 and 49: The physics of resonant mass detect
Page 50 and 51: The physics of resonant mass detect
Page 52 and 53: A detector in space 39LISA has been
Page 54 and 55: Gravitational and electromagnetic w
Page 56 and 57: Chapter 4Astrophysics of gravitatio
Page 58 and 59: Sources detectable from ground and
Page 64 and 65: Variational principles and the ener
Page 66 and 67: Variational principles and the ener
Page 68 and 69: Practical applications of the Isaac
Page 72 and 73: Expansion for the far field of a sl
Page 74 and 75: Application of the TT gauge to the
Page 82 and 83: 6.4.1 Mass-quadrupole radiationEner
Page 84 and 85: Chapter 7Source calculationsNow tha
Page 86 and 87: The r-modes 73As we mentioned in ch
Page 88 and 89: The r-modes 75instability. In an id
Page 90 and 91: The r-modes 77Table 7.1. Gravitatio
Page 92 and 93: The r-modes 79evolution turns out t
Page 94 and 95: ReferencesWe have divided the refer
Page 96 and 97: References 83[19] van der Klis M 19
Page 98 and 99: Solutions to exercises 85For this p
Page 100 and 101: Solutions to exercises 87The gauge
Page 103 and 104: Chapter 8Resonant detectors for gra
Page 105 and 106: Sensitivity and bandwidth of resona
Page 107 and 108: 8.2 Sensitivity for various GW sign
Page 109 and 110: Sensitivity for various GW signals
Page 111 and 112: Recent results obtained with the re
Page 113 and 114: Discussion and conclusions 101Figur
Page 115 and 116: Chapter 9The Earth-based large inte
Page 117 and 118: Introduction 105Figure 9.1. The sim
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18.5 cmThe SA suspension and requir
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A few words about the Low Frequency
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Conclusion 113Figure 9.7. The R and
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Chapter 10LISA: A proposed joint ES
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Description of the LISA mission 117
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Expected gravitational-wave results
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References 149[8] Folkner W M 1998
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References 151[68] Miralda-Escuda J
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Introduction 153detect GWs with a s
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Testing theories of gravity 155•
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0Testing theories of gravity 157the
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Taking the scalar product we findGr
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Gravitational wave radiation in the
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The hollow sphere 169and introduced
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Scalar-tensor cross sections 171mon
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Scalar-tensor cross sections 173Tab
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Scalar-tensor cross sections 175Tab
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References 177Frossati G and Coccia
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Chapter 12Generalities on the stoch
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Introduction 183contribution to the
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Definitions 185The value of H 0 is
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Definitions 18712.2.2 The character
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Definitions 189or, dividing by the
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The overlap reduction function 191O
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The overlap reduction function 193T
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The overlap reduction function 195F
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Achievable sensitivities to the SGW
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Observational bounds 207Table 12.10
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equation gives immediately(ρgwρ
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Chapter 13Sources of SGWBHere we re
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Topological defects 213This correla
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Topological defects 215It can be sh
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Topological defects 217(i)A loop ra
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Topological defects 219• A peak n
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Topological defects 22113.1.2 Hybri
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Inflation 223Hubble radius and when
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Inflation 225as a first approximati
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Inflation 227the lower bound to the
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String cosmology 22913.3 String cos
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String cosmology 231HPre-big bangPo
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String cosmology 233Figure 13.6. h
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First-order phase transitions 23580
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Astrophysical sources 237which is t
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References 239of [71]). However, fo
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References 241[39] Liddle A R 2000
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PART 4THEORETICAL DEVELOPMENTSHerma
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246 Infinite-dimensional symmetries
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Chapter 15Gyroscopes and gravitatio
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270 Gyroscopes and gravitational wa
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Chapter 16Elementary introduction t
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282 Elementary introduction to pre-
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Chapter 17Post-Newtonian computatio
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340 Post-Newtonian computation of b
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PART 5NUMERICAL RELATIVITYEdward Se
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362 Numerical relativityspite of mo
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364 Numerical relativityterms, of m
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366 Numerical relativityHere we hav
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368 Numerical relativityinformation
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370 Numerical relativityThe Palma,
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372 Numerical relativityHyperbolic
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374 Numerical relativityEinstein di
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376 Numerical relativityare the abi
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378 Numerical relativityeach cell i
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380 Numerical relativityFigure 18.1
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382 Numerical relativity1D code [45
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384 Numerical relativityfunctions,
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386 Numerical relativityto a very a
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388 Numerical relativity18.5 Comput
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390 Numerical relativitynumerics an
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392 Numerical relativityinitial dis
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396 Numerical relativityof these re
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400 Numerical relativityof Einstein
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402 Numerical relativity18.9.5.2 Co
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404 Numerical relativity[23] Ashby
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406 Numerical relativity[101] Shapi
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408 Numerical relativity(Singapore:
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410 IndexEhlers Lagrangian, 254Eins
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412 Indexscalarspherical harmonics,
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