Mode-sum Regularization Research Articles

Self-force effects around charged (2+1)-dimensional BTZ black hole in the Einstein–Gauss–Bonnet gravity

In this paper, we discuss a mode sum regularization method for the self-force acting on a charged test particle moving around a Banados, Teitelboim and Zanelli (BTZ) black hole. First, we have calculated the self-force on each individual mode. Then we evaluated our result by summing up the contribution on each mode. We studied a general scheme for static black holes and then present the result for radial trajectories around [Formula: see text]-dimensional charged BTZ black hole in Einstein–Gauss–Bonnet (EGB) gravity. We find that the parameter which is coupling ([Formula: see text]) GR to Einstein–Gauss–Bonnet gravity has a significant impact on the self-force.

Electromagnetic self-force on a charged particle on Kerr spacetime: Equatorial circular orbits

We calculate the self-force acting on a charged particle on a circular geodesic orbit in the equatorial plane of a rotating black hole. We show by direct calculation that the dissipative self-force balances with the sum of the flux radiated to infinity and through the black hole horizon. Prograde orbits are found to stimulate black hole superradiance, but we confirm that the condition for floating orbits cannot be met. We calculate the conservative component of the self-force by application of the mode sum regularization method, and we present a selection of numerical results. By numerical fitting, we extract the leading-order coefficients in post-Newtonian expansions. The self-force on the innermost stable circular orbits of the Kerr spacetime is calculated, and comparisons are drawn between the electromagnetic and gravitational self forces.

Gravitational self-force regularization in the Regge-Wheeler and easy gauges

We present numerical results for the gravitational self-force and redshift invariant calculated in the Regge-Wheeler and Easy gauges for circular orbits in a Schwarzschild background, utilizing the regularization framework introduced by Pound, Merlin, and Barack. The numerical calculation is performed in the frequency domain and requires the integration of a single second-order ODE, greatly improving computation times over more traditional Lorenz gauge numerical methods. A sufficiently high-order, analytic expansion of the Detweiler-Whiting singular field is gauge-transformed to both the Regge-Wheeler and Easy gauges and used to construct tensor-harmonic mode-sum regularization parameters. We compare our results to the gravitational self-force calculated in the Lorenz gauge by explicitly gauge-transforming the Lorenz gauge self-force to the Regge-Wheeler and Easy gauges, and find that our results agree to a relative accuracy of $10^{-15}$ for an orbital radius of $r_0=6M$ and $10^{-16}$ for an orbital radius of $r_0=10M$.

Vacuum polarization on topological black holes

We investigate quantum effects on topological black hole space-times within the framework of quantum field theory on curved space-times. Considering a quantum scalar field, we extend a recent mode-sum regularization prescription for the computation of the renormalized vacuum polarization to asymptotically anti-de Sitter black holes with nonspherical event horizon topology. In particular, we calculate the vacuum polarization for a massless, conformally-coupled scalar field on a four-dimensional topological Schwarzschild–anti-de Sitter black hole background, comparing our results with those for a spherically-symmetric black hole.

Accelerated motion and the self-force in Schwarzschild spacetime

We provide expansions of the Detweiler–Whiting singular field for a particle with a scalar field moving 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 particle’s worldline is accelerated with respect to the background spacetime.

Mode-sum prescription for vacuum polarization in black hole spacetimes in even dimensions

We present a mode-sum regularization prescription for computing the vacuum polarization of a scalar field in static spherically-symmetric black hole spacetimes in even dimensions. This is the first general and systematic approach to regularized vacuum polarization in higher even dimensions, building upon a previous scheme we developed for odd dimensions. Things are more complicated here since the even-dimensional propagator possesses logarithmic singularities which must be regularized. However, in spite of this complication, the regularization parameters can be computed in closed form in arbitrary even dimensions and for arbitrary metric function $f(r)$. As an explicit example of our method, we show plots for vacuum polarization of a massless scalar field in the Schwarzschild-Tangherlini spacetime for even $d=4,...,10$. However, the method presented applies straightforwardly to massive fields or to nonvacuum spacetimes.

Renormalized Stress-Energy Tensor of an Evaporating Spinning Black Hole.

We provide the first calculation of the renormalized stress-energy tensor (RSET) of a quantum field in Kerr spacetime (describing a stationary spinning black hole). More specifically, we employ a recently developed mode-sum regularization method to compute the RSET of a minimally coupled massless scalar field in the Unruh vacuum state, the quantum state corresponding to an evaporating black hole. The computation is done here for the case a=0.7M, using two different variants of the method: t splitting and φ splitting, yielding good agreement between the two (in the domain where both are applicable). We briefly discuss possible implications of the results for computing semiclassical corrections to certain quantities, and also for simulating dynamical evaporation of a spinning black hole.

Renormalized stress-energy tensor for stationary black holes

We continue the presentation of the pragmatic mode-sum regularization (PMR) method for computing the renormalized stress-energy tensor (RSET). We show in detail how to employ the $t$-splitting variant of the method, which was first presented for $\left\langle\phi^{2}\right\rangle_{ren}$, to compute the RSET in a stationary, asymptotically-flat background. This variant of the PMR method was recently used to compute the RSET for an evaporating spinning black hole. As an example for regularization, we demonstrate here the computation of the RSET for a minimally-coupled, massless scalar field on Schwarzschild background in all three vacuum states. We discuss future work and possible improvements of the regularization schemes in the PMR method.

Mode-sum prescription for the vacuum polarization in odd dimensions

We present a new mode-sum regularization prescription for computing the vacuum polarization of a scalar field in static spherically-symmetric black hole spacetimes in odd dimensions. This is the first general and systematic approach to regularized vacuum polarization in higher dimensions. Remarkably, the regularization parameters can be computed in closed form in arbitrary dimensions and for arbitrary metric function $f(r)$. In fact, we show that in spite of the increasing severity and number of the divergences to be regularized, the method presented is mostly agnostic to the number of dimensions. Finally, as an explicit example of our method, we show plots for vacuum polarization in the Schwarzschild-Tangherlini spacetime for odd $d=5,...,11$.

Analytical high-order post-Newtonian expansions for spinning extreme mass ratio binaries

We present an analytic computation of Detweiler's redshift invariant for a point mass in a circular orbit around a Kerr black hole, giving results up to 8.5 post-Newtonian order while making no assumptions on the magnitude of the spin of the black hole. Our calculation is based on the functional series method of Mano, Suzuki and Takasugi, and employs a rigorous mode-sum regularization prescription based on the Detweiler-Whiting singular-regular decomposition. The approximations used in our approach are minimal; we use the standard self-force expansion to linear order in the mass ratio, and the standard post-Newtonian expansion in the separation of the binary. A key advantage of this approach is that it produces expressions that include contributions at all orders in the spin of the Kerr black hole. While this work applies the method to the specific case of Detweiler's redshift invariant, it can be readily extended to other gauge invariant quantities and to higher post-Newtonian orders.

Indirect (source-free) integration method. II. Self-force consistent radial fall

We apply our method of indirect integration, described in Part I, at fourth order, to the radial fall affected by the self-force (SF). The Mode-Sum regularization 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 SF 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 (2014) 1450090].

Applying the effective-source approach to frequency-domain self-force calculations: Lorenz-gauge gravitational perturbations

With a view to developing a formalism that will be applicable at second perturbative order, we devise a new practical scheme for computing the gravitational self-force experienced by a point mass moving in a curved background spacetime. Our method works in the frequency domain and employs the effective-source approach, in which a distributional source for the retarded metric perturbation is replaced with an effective source for a certain regularized self-field. A key ingredient of the calculation is the analytic determination of an appropriate puncture field from which the effective source and regularized residual field can be calculated. In addition to its application in our effective-source method, we also show how this puncture field can be used to derive tensor-harmonic mode-sum regularization parameters that improve the efficiency of the traditional mode-sum procedure. To demonstrate the method, we calculate the first-order-in-the-mass-ratio self-force and redshift invariant for a point mass on a circular orbit in Schwarzschild spacetime.

Linear-in-mass-ratio contribution to spin precession and tidal invariants in Schwarzschild spacetime at very high post-Newtonian order

Using black hole perturbation theory and arbitrary-precision computer algebra, we obtain the post-Newtonian (pN) expansions of the linear-in-mass-ratio corrections to the spin-precession angle and tidal invariants for a particle in circular orbit around a Schwarzschild black hole. We extract coefficients up to 20pN order from numerical results that are calculated with an accuracy greater than 1 part in $10^{500}$. These results can be used to calibrate parameters in effective-one-body models of compact binaries, specifically the spin-orbit part of the effective Hamiltonian and the dynamically significant tidal part of the main radial potential of the effective metric. Our calculations are performed in a radiation gauge, which is known to be singular away from the particle. To overcome this irregularity, we define suitable Detweiler-Whiting singular and regular fields in this gauge, and we devise a rigorous mode-sum regularization method to compute the invariants constructed from the regular field.

Pragmatic mode-sum regularization method for semiclassical black-hole spacetimes

Computation of the renormalized stress-energy tensor is the most serious obstacle in studying the dynamical, self-consistent, semiclassical evaporation of a black hole in 4D. The difficulty arises from the delicate regularization procedure for the stress-energy tensor, combined with the fact that in practice the modes of the field need be computed numerically. We have developed a new method for numerical implementation of the point-splitting regularization in 4D, applicable to the renormalized stress-energy tensor as well as to $\left\langle \phi^{2}\right\rangle _{ren}$, namely the renormalized $\left\langle \phi^{2}\right\rangle$. So far we have formulated two variants of this method: t-splitting (aimed for stationary backgrounds) and angular splitting (for spherically-symmetric backgrounds). In this paper we introduce our basic approach, and then focus on the t-splitting variant, which is the simplest of the two (deferring the angular-splitting variant to a forthcoming paper). We then use this variant, as a first stage, to calculate $\left\langle \phi^{2}\right\rangle _{ren}$ in Schwarzschild spacetime, for a massless scalar field in the Boulware state. We compare our results to previous ones, obtained by a different method, and find full agreement. We discuss how this approach can be applied (using the angular-splitting variant) to analyze the dynamical self-consistent evaporation of black holes.

Self-force on a scalar charge in Kerr spacetime: Inclined circular orbits

Accurately modeling astrophysical extreme-mass-ratio inspirals 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.

Lorenz gauge gravitational self-force calculations of eccentric binaries using a frequency domain procedure

We present an algorithm for calculating the metric perturbations and gravitational self-force for extreme-mass-ratio inspirals (EMRIs) with eccentric orbits. The massive black hole is taken to be Schwarzschild and metric perturbations are computed in Lorenz gauge. The perturbation equations are solved as coupled systems of ordinary differential equations in the frequency domain. Accurate local behavior of the metric is attained through use of the method of extended homogeneous solutions and mode-sum regularization is used to find the self-force. We focus on calculating the self-force with sufficient accuracy to ensure its error contributions to the phase in a long term orbital evolution will be $\delta\Phi \lesssim 10^{-2}$ radians. This requires the orbit-averaged force to have fractional errors $\lesssim 10^{-8}$ and the oscillatory part of the self-force to have errors $\lesssim 10^{-3}$ (a level frequently easily exceeded). Our code meets this error requirement in the oscillatory part, extending the reach to EMRIs with eccentricities of $e \lesssim 0.8$, if augmented by use of fluxes for the orbit-averaged force, or to eccentricities of $e \lesssim 0.5$ when used as a stand-alone code. Further, we demonstrate accurate calculations up to orbital separations of $a \simeq 100 M$, beyond that required for EMRI models and useful for comparison with post-Newtonian theory. Our principal developments include (1) use of fully constrained field equations, (2) discovery of analytic solutions for even-parity static modes, (3) finding a pre-conditioning technique for outer homogeneous solutions, (4) adaptive use of quad-precision and (5) jump conditions to handle near-static modes, and (6) a hybrid scheme for high eccentricities.

Combined gravitational and electromagnetic self-force on charged particles in electrovac spacetimes

We consider the self-force on a charged particle moving in a curved spacetime with a background electromagnetic field, extending previous studies to situations in which gravitational and electromagnetic perturbations are comparable. The formal expression $f^{ret}_\alpha$ for the self-force on a particle, written in terms of the retarded perturbed fields, is divergent, and a renormalization is needed to find the particle's acceleration at linear order in its mass $m$ and charge $e$. We assume that, as in previous work in a Lorenz gauge, the renormalization for accelerated motion comprises an angle average and mass renormalization. Using the short distance expansion of the perturbed electromagnetic and gravitational fields, we show that the renormalization is equivalent to that obtained from a mode sum regularization in which one subtracts from the expression for the self-force in terms of the retarded fields a singular part field comprising only the leading and subleading terms in the mode sum. The most striking part of our result, arising from a remarkable cancellation, is that the renormalization involves no mixing of electromagnetic and gravitational fields. In particular, the renormalized mass is obtained by subtracting (1) the purely electromagnetic contribution from a point charge moving along an accelerated trajectory and (2) the purely gravitational contribution from a point mass moving along the same trajectory. In a mode-sum regularization, the same cancellation implies that the required regularization parameters are sums of their purely electromagnetic and gravitational values.

Self-force on an accelerated particle

We calculate the singular field of an accelerated point particle (scalar charge, electric charge or small gravitating mass) moving on an accelerated (non-geodesic) trajectory in a generic background spacetime. Using a mode-sum regularization scheme, we obtain explicit expressions for the self-force regularization parameters. In the electromagnetic and gravitational case, we use a Lorenz gauge. This work extends the work of Barack and Ori [1] who demonstrated that the regularization parameters for a point particle in geodesic motion in a Schwarzschild spacetime can be described solely by the leading and subleading terms in the mode-sum (commonly known as the $A$ and $B$ terms) and that all terms of higher order in $\ell$ vanish upon summation (later they showed the same behavior for geodesic motion in Kerr [2], [3]). We demonstrate that these properties are universal to point particles moving through any smooth spacetime along arbitrary (accelerated) trajectories. Our renormalization scheme is based on, but not identical to, the Quinn-Wald axioms. As we develop our approach, we review and extend work showing that that different definitions of the singular field used in the literature are equivalent to our approach. Because our approach does not assume geodesic motion of the perturbing particle, we are able use our mode-sum formalism to explicitly recover a well-known result: The self-force on static scalar charges near a Schwarzschild black hole vanishes.

Applying the effective-source approach to frequency-domain self-force calculations

The equations of motion of a point particle interacting with its own field are defined in terms of a certain regularized self-field. Two of the leading methods for computing this regularized field are the mode-sum and effective-source approaches. In this work we unite these two distinct regularization schemes by generalizing traditional frequency-domain mode-sum calculations to incorporate effective-source techniques. For a toy scalar-field model we analytically compute an appropriate puncture field from which the regularized residual field can be calculated. To demonstrate the method, we compute the self-force for a scalar particle on a circular orbit in Schwarzschild spacetime. We also demonstrate the relation between the worldtube and window function approaches to localizing the puncture field to the neighborhood of the worldline and show how the method reduces to the well-known mode-sum regularization scheme in a certain limit. This new computational scheme can be applied to cases where traditional mode-sum regularization is inadequate, such as in calculations at second perturbative order.

High-order expansions of the Detweiler-Whiting singular field in Kerr spacetime

In a previous paper, we computed expressions for the Detweiler-Whiting singular field of point scalar, electromagnetic and gravitational charges following a geodesic of the Schwarzschild spacetime. We now extend this to the case of equatorial orbits in Kerr spacetime, using coordinate and covariant approaches to compute expansions of the singular field in scalar, electromagnetic and gravitational cases. As an application, we give the calculation of previously unknown mode-sum regularization parameters. We also propose a new application of high-order approximations to the singular field, showing how they may be used to compute $m$-mode regularization parameters for use in the $m$-mode effective source approach to self-force calculations.