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Register a new userA new analytical method for self-force regularization II. Testing the efficiency for circular orbits
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Wataru Hikida
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In a previous paper, based on the black hole perturbation approach, we formulated a new analytical method for regularizing the self-force acting on a particle of small mass $mu$ orbiting a Schwarzschild black hole of mass $M$, where $mull M$. In our method, we divide the self-force into the $tilde S$-part and $tilde R$-part. All the singular behaviors are contained in the $tilde S$-part, and hence the $tilde R$-part is guaranteed to be regular. In this paper, focusing on the case of a scalar-charged particle for simplicity, we investigate the precision of both the regularized $tilde S$-part and the $tilde R$-part required for the construction of sufficiently accurate waveforms for almost circular inspiral orbits. For the regularized $tilde S$-part, we calculate it for circular orbits to 18 post-Newtonian (PN) order and investigate the convergence of the post-Newtonian expansion. We also study the convergence of the remaining $tilde{R}$-part in the spherical harmonic expansion. We find that a sufficiently accurate Green function can be obtained by keeping the terms up to $ell=13$.
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Accurately modeling astrophysical extreme-mass-ratio-insprials requires calculating the gravitational self-force for orbits in Kerr spacetime. The necessary calculation techniques are typically very complex and, consequently, toy scalar-field models are often developed in order to establish a particular calculational approach. To that end, I present a calculation of the scalar-field self-force for a particle moving on a (fixed) inclined circular geodesic of a background Kerr black hole. I make the calculation in the frequency-domain and demonstrate how to apply the mode-sum regularization procedure to all four components of the self-force. I present results for a number of strong-field orbits which can be used as benchmarks for emerging self-force calculation techniques in Kerr spacetime.
We apply our method of indirect integration, described in Part I, at fourth order, to the radial fall affected by the self-force. The Mode-Sum regularisation is performed in the Regge-Wheeler gauge using the equivalence with the harmonic gauge for this orbit. We consider also the motion subjected to a self-consistent and iterative correction determined by the self-force through osculating stretches of geodesics. The convergence of the results confirms the validity of the integration method. This work complements and justifies the analysis and the results appeared in Int. J. Geom. Meth. Mod. Phys., 11, 1450090 (2014).
Inspirals of stellar-mass objects into massive black holes will be important sources for the space-based gravitational-wave detector LISA. Modelling these systems requires calculating the metric perturbation due to a point particle orbiting a Kerr black hole. Currently, the linear perturbation is obtained with a metric reconstruction procedure that puts it in a no-string radiation gauge which is singular on a surface surrounding the central black hole. Calculating dynamical quantities in this gauge involves a subtle procedure of gauge completion as well as cancellations of very large numbers. The singularities in the gauge also lead to pathological field equations at second perturbative order. In this paper we re-analyze the point-particle problem in Kerr using the corrector-field reconstruction formalism of Green, Hollands, and Zimmerman (GHZ). We clarify the relationship between the GHZ formalism and previous reconstruction methods, showing that it provides a simple formula for the gauge completion. We then use it to develop a new method of computing the metric in a more regular gauge: a Teukolsky puncture scheme. This scheme should ameliorate the problem of large cancellations, and by constructing the linear metric perturbation in a sufficiently regular gauge, it should provide a first step toward second-order self-force calculations in Kerr. Our methods are developed in generality in Kerr, but we illustrate some key ideas and demonstrate our puncture scheme in the simple setting of a static particle in Minkowski spacetime.
[abridged] The inspiral of a stellar compact object into a massive black hole is one of the main sources of gravitational waves for the future space-based Laser Interferometer Space Antenna. We expect to be able to detect and analyze many cycles of these slowly inspiraling systems. To that end, the use of very precise theoretical waveform templates in the data analysis is required. To build them we need to have a deep understanding of the gravitational backreaction mechanism responsible for the inspiral. The self-force approach describes the inspiral as the action of a local force that can be obtained from the regularization of the perturbations created by the stellar compact object on the massive black hole geometry. In this paper we extend a new time-domain technique for the computation of the self-force from the circular case to the case of eccentric orbits around a non-rotating black hole. The main idea behind our scheme is to use a multidomain framework in which the small compact object, described as a particle, is located at the interface between two subdomains. Then, the equations at each subdomain are homogeneous wave-type equations, without distributional sources. In this particle-without-particle formulation, the solution of the equations is smooth enough to provide good convergence properties for the numerical computations. This formulation is implemented by using a pseudospectral collocation method for the spatial discretization, combined with a Runge Kutta algorithm for the time evolution. We present results from several simulations of eccentric orbits in the case of a scalar charged particle around a Schwarzschild black hole. In particular, we show the convergence of the method and its ability to resolve the field and its derivatives across the particle location. Finally, we provide numerical values of the self-force for different orbital parameters.
We extend the gravitational self-force approach to encompass `self-interaction tidal effects for a compact body of mass $mu$ on a quasi-circular orbit around a black hole of mass $M gg mu$. Specifically, we define and calculate at $O(mu)$ (conservative) shifts in the eigenvalues of the electric- and magnetic-type tidal tensors, and a (dissipative) shift in a scalar product between their eigenbases. This approach yields four gauge-invariant functions, from which one may construct other tidal quantities such as the curvature scalars and the speciality index. First, we analyze the general case of a geodesic in a regular perturbed vacuum spacetime admitting a helical Killing vector and a reflection symmetry. Next, we specialize to focus on circular orbits in the equatorial plane of Kerr spacetime at $O(mu)$. We present accurate numerical results for the Schwarzschild case for orbital radii up to the light-ring, calculated via independent implementations in Lorenz and Regge-Wheeler gauges. We show that our results are consistent with leading-order post-Newtonian expansions, and demonstrate the existence of additional structure in the strong-field regime. We anticipate that our strong-field results will inform (e.g.) effective one-body models for the gravitational two-body problem that are invaluable in the ongoing search for gravitational waves.
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