Abstract
We derive a set of coupled equations for the gravitational and electromagnetic perturbation in the Reissner–Nordström geometry using the Newman–Penrose formalism. We show that the information of the physical gravitational signal is contained in the Weyl scalar function Ψ4, as is well known, but for the electromagnetic signal, the information is encoded in the function χ, which relates the perturbations of the radiative Maxwell scalars φ2 and the Weyl scalar Ψ3. In deriving the perturbation equations, we do not impose any gauge condition and as a limiting case, our analysis contains previously obtained results, for instance, those from Chandrashekhar’s book. In our analysis, we also include the sources for the perturbations and focus on a dust-like charged fluid distribution falling radially into the black hole. Finally, by writing the functions on the basis of spin-weighted spherical harmonics and the Reissner–Nordström spacetime in Kerr–Schild type coordinates, a hyperbolic system of coupled partial differential equations is presented and numerically solved. In this way, we completely solve a system that generates a gravitational signal as well as an electromagnetic/gravitational one, which sets the basis to find correlations between them and thus facilitates gravitational wave detection via electromagnetic signals.
Highlights
Gravitational wave astronomy was born in 2015 with the discovery of the first astronomical source named GW150914 [1,2,3]. This was the first observation of a binary black hole system and was done with the gravitational wave interferometer LIGO [4], which detected the gravitational wave signal produced by the merger of two black holes in a binary system
Chandrasekhar described the scattering of electromagnetic waves on a Reissner–Nordström black hole and the resulting generation of outgoing gravitational waves, and using the gauge freedom of the Maxwell equations in a curved background, he derived the electromagnetic equations by finding a gauge that restores the symmetry to the perturbation equations [21]
Recalling the Peeling theorem [29] that states that the Weyl scalars have the asymptotic decay Ψs ≡ 1/r5−s, it proves convenient to rewrite the equation for the gravitational perturbation in terms of the quantity r Ψ4 (1), which does not decay in the asymptotic region
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Nordström black hole and the resulting generation of outgoing gravitational waves, and using the gauge freedom of the Maxwell equations in a curved background, he derived the electromagnetic equations by finding a gauge that restores the symmetry to the perturbation equations [21]. He showed that the curved spacetime produced by a black hole is sensitive to the electromagnetic field part χ of the spacetime and this awareness (1).
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