Abstract
One of the purposes of this chapter is a close comparison of the basic properties of the electromagnetic and gravitational radiations. For concreteness, we will consider both radiations mainly in the non-relativistic limit, i.e. we shall assume that they are generated by objects moving with speeds much smaller than the speed of light. In this case the multipole approximation yields sensible, analytic, results, and so we will have at our disposal sufficiently explicit formulas, allowing for a detailed qualitative as well as quantitative comparison. For obvious reasons we report a few predictions of General Relativity without derivations, providing, however, where possible, heuristic arguments.
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Notes
- 1.
The energy-momentum tensor of a rigid body in axisymmetric rotation is constant in time, and generates, therefore, a time-independent gravitational field.
- 2.
Pulsars are very compact stars with radii of a few kilometers and masses of around one solar mass. This means that at their surface the gravitational field is, actually, rather intense, of the order \(H^{\mu \nu }/c^2\sim 4MG/Rc^2\sim 1\), and so the internal gravitational field is in a strong-field regime. In this case an exact, although implicit, solution of Einstein’s equations (9.17) can be obtained by replacing in (9.30) \(T^{\mu \nu }\) with the exact energy-momentum tensor \(\mathbb {T}^{\mu \nu }\), see Eqs. (9.17) and (9.19). The non-relativistic wave-zone potential is then again of the form (9.53), where the quadrupole moment is now given by (9.39), with the total energy density \(\mathbb {T}^{00}\) in place of \(T^{00}\). Eventually, this amounts thus merely to the identification of \(Mc^2\) with the total energy of the pulsar. See also Ref. [7], written in resolution of the quadrupole-formula controversy.
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Lechner, K. (2018). Gravitational Radiation. In: Classical Electrodynamics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-91809-9_9
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