Contents26 Gravitational Waves and Experimental Tests of General Relativity 126.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 Experimental Tests of General Relativity . . . . . . . . . . . . . . . . . . . . 226.2.1 Equivalence Principle, Gravitational redshift, and Global PositioningSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2.2 Perihelion advance of Mercury . . . . . . . . . . . . . . . . . . . . . . 426.2.3 Gravitational deflection of light, Fermat’s principle and GravitationalLenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.2.4 Shapiro time delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.2.5 Frame dragging and Gravity Probe B . . . . . . . . . . . . . . . . . . 826.2.6 Binary Pulsar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.3 Gravitational Waves and their Propagation . . . . . . . . . . . . . . . . . . 1126.3.1 The gravitational wave equation . . . . . . . . . . . . . . . . . . . . . 1126.3.2 The waves’ two polarizations: + and × . . . . . . . . . . . . . . . . . 1 426.3.3 Gravitons and their spin . . . . . . . . . . . . . . . . . . . . . . . . . 1826.3.4 Energy and Momentum in Gravitational Waves . . . . . . . . . . . . 1926.3.5 Wave propagation in a source’s local asymptotic rest frame . . . . . . 2126.3.6 Wave propagation via geometric optics . . . . . . . . . . . . . . . . . 2326.3.7 Metric perturbation; TT gauge . . . . . . . . . . . . . . . . . . . . . 2526.4 The Generation of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . 2826.4.1 Multip ole-moment expansion . . . . . . . . . . . . . . . . . . . . . . 2826.4.2 Quadrupole-moment formalism . . . . . . . . . . . . . . . . . . . . . 2926.4.3 Gravitational waves from a binary star system . . . . . . . . . . . . . 3326.5 The Detection of G ravitational Waves . . . . . . . . . . . . . . . . . . . . . . 3826.5.1 Interferometer analyzed in TT gauge . . . . . . . . . . . . . . . . . . 4026.5.2 Interferometer analyzed in proper reference frame of beam splitter . . 4326.5.3 Realistic Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . 450Chapter 26Gravitational Waves andExperimental Tests of GeneralRelativityVersion 0626.1 .K.pdf, 16 May 2007.Please send comments, suggestions, and errata via email to kip@tapi r.caltech . edu or on paperto Kip Thorne, 130-33 Ca l tech, Pasadena CA 91 125Box 26.1Reader’s Guide• This chapter relies significantly on– The special relativity portions of Chap. 1.– Chapter 23, on the transition from special relativity to general relativity.– Chapter 24, on the fundamental concepts of general relativity, especially Sec.24.9 on weak, relativistic gravitational fields.– Chapter 25, on relativistic stars and black holes.– Sec. 6 .3 on geometric o ptics.• In addition, Sec. 26.2.3 o n Fermat’s principle and gravitational lenses is closelylinked to Sec. 6.4 on gravitational lenses and Sec. 7 .6 on diffraction a t a caustic.• Portions of this chapter are a foundation for Chap. 27, Cosmology.26.1 Introducti onIn 1915, when Einstein formulated general relativity, human technology was incapable ofproviding definitive experimental tests of his theory. Only a half century later did technologybegin t o catch up. In the remaining 35 years of the century, experiments improved f r om12accuracies of a few tens of per cent to a part in 1000 or even 10,000; and general relativitypassed the tests with flying colors. In Sec. 26.2 we shall describe some of these tests, derivegeneral relativity’s predictions for them, and discuss t he experimental results.In the early twenty-first century, observations of gravitational waves will radically changethe character of research on general relativity. They will produce, for the first time, tests ofgeneral relativity in strong-gravity situations. They will permit us to study relativistic effectsin neutron-star and black-hole binaries with exquisite accuracies. They will enable us to mapthe spacetime geometries of black holes with high precision, and study observat io nally thelarge-amplitude, highly nonlinear vibrations of curved spacetime that occur when two blackholes collide and merge. And (as we shall see in Chap. 27), they may enable us to probethe singularity in which the universe was born and the universe’s evolution in its first tinyfraction of a second.In this chapter we shall develop the theory of gravitational waves in much detail andshall describe the efforts to detect the waves and the sources that may be seen. More specif-ically, in Sec. 26.3 we shall develop the mathematical description of gravitational waves,both classically and quantum mechanically (in the language o f gravitons), and shall studytheir propagation through flat spacetime and also, via the t ools of geometric o ptics, throughcurved spacetime. Then in Sec. 26.4 we shall develop the simplest approximate method forcomputing the generation of gravitational waves, the “quadrupole-moment formalism”; andwe shall describe and present a few details o f other, more sophisticated and accurate meth-ods based on multipolar expansions, post-Newtonian techniques, and numerical simulationson supercomputers (“numerical relativity”). In Sec. 26.5, we shall turn to gravitational-wave detection, focusing especially on detectors such as LIGO and LISA that rely on laserinterferometry.26.2 Experimental Tests of General RelativityIn this section we shall describe briefly some of the most importa nt experimental tests ofgeneral relativity. For greater detail, see Will (1 993, 2001, 2005)26.2.1 Equivalence Principle, Gravitational redshift, and GlobalPositioning SystemA key aspect of the equivalence principle is the prediction that all objects, whose size isextremely small compared to the radius of curvature of spacetime and on which no non-gravitational forces act, should …
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