Gravitational Waves
An essential presentation of the theoretical and phenomenological aspects concerning emission, propagation and detection of gravitational waves in the weak-field approximation. Includes, in particular: the explicit computation of the energy-momentum tenso
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Gravitational Waves
The linearized Einstein equations (8.10) describe the gravitational dynamics in the approximation in which the deviations from the Minkowski geometry, represented by hμν , are small enough to be treated as first-order perturbations. Such an approximation can be successfully applied to the static gravitational field of many astrophysical sources, as we have seen in the previous chapter. The linearized approximation holds, however, even if the perturbations hμν are time-dependent. In that case they describe geometric fluctuations which in vacuum propagate at the speed of light, and interact with a coupling strength determined by the Newton constant: they are the so-called gravitational waves, another new and important prediction of Einstein’s theory of general relativity. In this chapter we will introduce the main properties of these waves, focusing on the aspects that are at the grounds of the present techniques of detection. Since the coupling strength is extremely weak, a direct experimental detection of the gravitational waves is still lacking. However, thanks to the instruments allowed by present technology—some of which are already operating, other in the design stage—we can reasonably expect that such a detection will not be long awaited for (see for instance the books [8, 36] of the bibliography). In any case, we should not forget that the gravitational waves have been already detected—although indirectly—through the observation of the orbital period of binary astrophysical systems. The emission of gravitational radiation from those systems, in fact, produces a decrease of the period which has been observed, experimentally measured, and found to be in agreement with the predictions of general relativity (see Sect. 9.2.4).
9.1 Propagation of Metric Fluctuations in Vacuum In the absence of sources, Tμν = 0, the linearized equation (8.10) reduces to a wave equation for the propagation in the Minkowski vacuum of the symmetric tensor field hμν , hμν = 0,
hμν = hνμ ,
M. Gasperini, Theory of Gravitational Interactions, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-88-470-2691-9_9, © Springer-Verlag Italia 2013
(9.1) 151
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9 Gravitational Waves
satisfying the harmonic gauge condition (8.9): 1 ∂ ν hμν = ∂μ h. 2
(9.2)
The above set of equations is formally very similar to the set of equations for the vacuum propagation of electromagnetic waves, Aμ = 0, where Aμ is the vector potential in the Lorenz gauge ∂ μ Aμ = 0. Since the d’Alembert operator is the same, in both cases the solutions describe signals propagating at the speed of light. There are important dynamical differences, however, due to the fact that hμν is a tensor while Aμ is a vector. In fact, as already stressed in Chap. 2, the forces generated by identical static sources are attractive if they are transmitted by a tensor, repulsive if they are transmitted by a vector. The basic reason for this difference traces back to the fact that a tensor field, when quantized, corresponds to massless spin-2 particles (the gr
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