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We consider the motion of charged and spinning bodies on the symmetry axis of a non-extremal Kerr-Newman black hole. If one treats the body as a test point particle of mass, $m$, charge $q$, and spin $S$, then by dropping the body into the black hole from sufficiently near the horizon, the first order area increase, $delta A$, of the black hole can be made arbitrarily small, i.e., the process can be done in a ``reversible manner. At second order, there may be effects on the energy delivered to the black hole---quadratic in $q$ and $S$---resulting from (i) the finite size of the body and (ii) self-force corrections to the energy. Sorce and Wald have calculated these effects for a charged, non-spinning body on the symmetry axis of an uncharged Kerr black hole. We consider the generalization of this process for a charged and spinning body on the symmetry axis of a Kerr-Newman black hole, where the self-force effects have not been calculated. A spinning body (with negligible extent along the spin axis) must have a mass quadrupole moment $gtrsim S^2/m$, so at quadratic order in the spin, we must take into account quadrupole effects on the motion. After taking into account all such finite size effects, we find that the condition $delta^2 A geq 0$ yields a nontrivial lower bound on the self-force energy, $E_{SF}$, at the horizon. In particular, for an uncharged, spinning body on the axis of a Kerr black hole of mass $M$, the net contribution of spin self-force to the energy of the body at the horizon is $E_{SF} geq S^2/8M^3$, corresponding to an overall repulsive spin self-force. A lower bound for the self-force energy, $E_{SF}$, for a body with both charge and spin at the horizon of a Kerr-Newman black hole is given. This lower bound will be the correct formula for $E_{SF}$ if the dropping process can be done reversibly to second order.
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Much of the success of gravitational-wave astronomy rests on perturbation theory. Historically, perturbative analysis of gravitational-wave sources has largely focused on post-Newtonian theory. However, strong-field perturbation theory is essential in many cases such as the quasinormal ringdown following the merger of a binary system, tidally perturbed compact objects, and extreme-mass-ratio inspirals. In this review, motivated primarily by small-mass-ratio binaries but not limited to them, we provide an overview of essential methods in (i) black hole perturbation theory, (ii) orbital mechanics in Kerr spacetime, and (iii) gravitational self-force theory. Our treatment of black hole perturbation theory covers most common methods, including the Teukolsky and Regge-Wheeler-Zerilli equations, methods of metric reconstruction, and Lorenz-gauge formulations, presenting them in a new consistent and self-contained form. Our treatment of orbital mechanics covers quasi-Keplerian and action-angle descriptions of bound geodesics and accelerated orbits, osculating geodesics, near-identity averaging transformations, multiscale expansions, and orbital resonances. Our summary of self-force theorys foundations is brief, covering the main ideas and results of matched asymptotic expansions, local expansion methods, puncture schemes, and point particle descriptions. We conclude by combining the above methods in a multiscale expansion of the perturbative Einstein equations, leading to adiabatic and post-adiabatic evolution schemes. Our presentation is intended primarily as a reference for practitioners but includes a variety of new results. In particular, we present the first complete post-adiabatic waveform-generation framework for generic (nonresonant) orbits in Kerr.
Through detection by low gravitational wave space interferometers, the capture of stars by supermassive black holes will constitute a giant step forward in the understanding of gravitation in strong field. The impact of the perturbations on the motion of the star is computed via the tail, the back-scattered part of the perturbations, or via a radiative Green function. In the former approach, the self-force acts upon the background geodesic, while in the latter, the geodesic is conceived in the total (background plus perturbations) field. Regularisations (mode-sum and Riemann-Hurwitz $zeta$ function) intervene to cancel divergencies coming from the infinitesimal size of the particle. The non-adiabatic trajectories require the most sophisticated techniques for studying the evolution of the motion, like the self-consistent approach.
We provide expansions of the Detweiler-Whiting singular field for motion along arbitrary, planar accelerated trajectories in Schwarzschild spacetime. We transcribe these results into mode-sum regularization parameters, computing previously unknown terms that increase the convergence rate of the mode-sum. We test our results by computing the self-force along a variety of accelerated trajectories. For non-uniformly accelerated circular orbits we present results from a new 1+1D discontinuous Galerkin time-domain code which employs an effective-source. We also present results for uniformly accelerated circular orbits and accelerated bound eccentric orbits computed within a frequency-domain treatment. Our regularization results will be useful for computing self-consistent self-force inspirals where the particles worldline is accelerated with respect to the background spacetime.
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.
The calculation of the self force in the modeling of the gravitational-wave emission from extreme-mass-ratio binaries is a challenging task. Here we address the question of the possible emergence of a persistent spurious solution in time-domain schemes, referred to as a {em Jost junk solution} in the literature, that may contaminate self force calculations. Previous studies suggested that Jost solutions are due to the use of zero initial data, which is inconsistent with the singular sources associated with the small object, described as a point mass. However, in this work we show that the specific origin is an inconsistency in the translation of the singular sources into jump conditions. More importantly, we identify the correct implementation of the sources at late times as the sufficient condition guaranteeing the absence of Jost junk solutions.
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