Warburton, Niels Jamie
(2012)
Frequency domain approach to self-force calculations.
*University of Southampton, School of Mathematics, Doctoral Thesis*, 207pp.

## Abstract

In this thesis, the problem of computing the back-reaction, or self-force, caused by a point particle interacting with its own field is studied. In particular, motivated by the prospect of detecting gravitational waves from extreme mass ratio inspiral systems, we consider the motion of the particle in black hole spacetimes. As a toy model for the most astrophysically relevant scenario of orbits about a rotating black hole we first study the scalar-field self-force (SSF)experienced by a scalar charge moving on a fixed geodesic in Kerr spacetime for a variety of orbits. Our approach is to work in the frequency domain, fully decomposing the scalar field into spheroidal harmonic and frequency modes and numerically solving for the retarded field mode-by-mode. Regularization of the retarded field is performed using the standard mode-sum technique which requires spherical harmonic modes as input, which we obtain by projecting the spheroidal harmonic modes on to a basis of spherical harmonics. We find for circular, equatorial orbits that the black hole spin can have a pronounced effect

on the conservative piece of the SSF, causing it to (with respect to the Schwarzschild scalar-field self-force) change sign for certain spins and orbital radii. For eccentric orbits in the equatorial plane, we make use of the recently introduced method of extended homogeneous solutions to overcome the Gibbs phenomenon associated with a naive approach. As an application of our work we compute the shift to the innermost stable circular orbit due to the conservative piece of the scalar-field self-force for a variety of black hole spins. We also present some preliminary results for the SSF along circular, inclined geodesics. As well as studying the toy model SSF, we also consider the gravitational self-force (GSF) problem in the context of orbits around a Schwarzschild black hole. Our approach is again to work in the frequency domain, and we perform a complete decomposition of the metric perturbation

in tensor spherical harmonics and frequency modes. The ten metric perturbation fields decouple with respect to the multipole indices but remain coupled within each spherical harmonic mode. We solve the resulting coupled sets numerically with a code set up to run on a computer cluster. Regularization is again performed using the mode-sum technique. Our resulting code is extremely efficient for low eccentricity orbits, and using it we compute the GSF for a great many points in the orbital parameter space. With these results we fit an analytic model to our numerical data and then use a relativistic osculating elements scheme to evolve the orbital inspiral. This allows us, for the first time, to assess the contribution to a

complete inspiral from the conservative piece of the gravitational self-force. Finally, as an aside, we investigate the recently discovered phenomenon of isofrequency orbits, whereby it is possible to have pairs of physically distinct bound geodesics about a Kerr black

hole that share the same three orbital frequencies.

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