Recent Developments in General Relativity pp 373-378 | Cite as

# Analytic Solution of Regge-Wheeler Differential Equation for Black Hole Perturbations in Radial Coordinate and Time Domains

## Abstract

An analytic solution of the Regge-Wheeler (RW) equation has been found via the Frobenius method at the regular singularity of the horizon 2*M*, in the form of a time and radial coordinate dependent series. The RW partial differential equation, derived from the Einstein field equations, represents the first order perturbations of the Schwarzschild metric. The known solutions are numerical in the time domain, and approximate and asymptotic for low or high frequencies in the Fourier domain. The former is of little relevance for comprehension of the geodesic equations for a body in the black hole field, while the latter is mainly useful for the description of emitted gravitational radiation. Instead a time domain solution is essential for the determination of the radiation reaction of a particle falling into a black hole, i.e., the influence of emitted radiation on the motion of the perturbing mass in the black hole field. To this end, a semi-analytic solution of the inhomogeneous RW equation with the sourceterm (Regge-Wheeler-Zerilli equation) shall be attempted where a numerical contribution shall be present. Discussion on the features of the series shall be shown in further work.

## Keywords

Black Hole Gravitational Radiation Geodesic Equation Radiation Reaction Einstein Field Equation## Preview

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## References

- 1.Regge T., Wheeler J.A. (1957): Stability of a Schwarzschild singularity. Phys. Rev.
**108**, 1063–1069MathSciNetADSMATHCrossRefGoogle Scholar - 2.Zerilli F.J. (1970): Gravitational field of a particle falling in a Schwarzschild geometry analyzed in tensor harmonics. Phys. Rev. D
**2**, 2141–2160MathSciNetADSMATHCrossRefGoogle Scholar - 3.Zerilli F.J. (1975): in
*Black Holes, Gravitational Waves and Cosmology: an Introduction to Current Research*. A-7, Gordon & Breach, New York, (errata Fr [2])Google Scholar - 4.Moncrief V. (1974): Gravitational perturbations of spherically symmetric systems I. The exterior problem. Ann. Phys. N.Y
**88**, 323–342MathSciNetCrossRefGoogle Scholar - 5.Ruffini R. (1973): Energetics of Black Holes: Part III Gravitational Radiation in Schwarzschild Geometries, in
*Black Holes*, August 1972, Les Houches, ed. by. C. De Witt, B. De Witt, Gordon & Breach, New York, R 528–137Google Scholar - 6.Spallicci A.D.A.M. (1999): On the perturbed Schwarzschild geometry for determination of particle motion, in
*Proc. Gravitational Waves, 2*^{nd}*Eduardo Amaldi Conference*, 1–4 July 1997, CERN, Geneve, ed. by E. Coccia, G. Veneziano, G. Pizzella, World Scientific, Singapore, pp. 303–308Google Scholar - 7.Spallicci A.D.A.M. (1999): Radiation reaction in free fall from perturbative geodesic equations in spherical coordinates, in
*Proc. 8*^{th}*Marcel Grossmann Meeting*, 22–28 June 1997, Jerusalem, ed. by T. Piran, R. Ruffini, World Scientific, Sinagpore, pp. 1107–1110Google Scholar - 8.Zwillinger D. (1998): Handbook of Differential Equations, Academic Press, BostonMATHGoogle Scholar
- 9.Vvedenski D. (1993): Partial Differential Equations with Mathematica. Addison-Wesley, Redwood CityGoogle Scholar