Self-force Corrections Research Articles

Black hole scattering near the transition to plunge: Self-force and resummation of post-Minkowskian theory

Geodesic scattering of a test particle off a Schwarzschild black hole can be parametrized by the speed-at-infinity v and the impact parameter b, with a “separatrix,” b=bc(v), marking the threshold between scattering and plunge. Near the separatrix, the scattering angle diverges as ∼log(b−bc). The self-force correction to the scattering angle (at fixed v, b) diverges even faster, like ∼A1(v)bc/(b−bc). Here we numerically calculate the divergence coefficient A1(v) in a scalar-charge toy model. We then use our knowledge of A1(v) to inform a resummation of the post-Minkowskian expansion for the scattering angle, and demonstrate that the resummed series agrees remarkably well with numerical self-force results even in the strong-field regime. We propose that a similar resummation technique, applied to a mass particle subject to a gravitational self-force, can significantly enhance the utility and regime of validity of post-Minkowskian calculations for black-hole scattering. Published by the American Physical Society 2024

Gravitational self force from scattering amplitudes in curved space

We employ scattering amplitudes in curved space to model the dynamics of a light probe particle with mass m orbiting in the background spacetime induced by a heavy gravitational source with mass M. Observables are organized as an expansion in m/M to all orders in G — the gravitational self-force expansion. An essential component of our analysis is the backreaction of the heavy source which we capture by including the associated light degrees of freedom. As illustration we consider a Schwarzschild background and verify geodesic motion as well as the first-order self-force correction to two-body scattering through \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}$$\\end{document}(G3). Amplitudes in curved space offer several advantages, and further developments along these lines may advance the computation of gravitational-wave signals for extreme-mass-ratio inspirals.

Derivative corrections to extremal black holes with moduli

We derive formulas for the leading mass, entropy, and long-range self-force corrections to extremal black holes due to higher-derivative operators. These formulas hold for black holes with arbitrary couplings to gauge fields and moduli, provided that the leading-order solutions are static, spherically-symmetric, extremal, and have nonzero horizon area. To use these formulas, both the leading-order black hole solution and the higher-derivative effective action must be known, but there is no need to solve the derivative-corrected equations of motion. We demonstrate that the mass, entropy and self-force corrections involve linearly-independent combinations of the higher-derivative couplings at any given point in the moduli space, and comment on their relations to various swampland conjectures.

Self-force correction to the deflection angle in black-hole scattering: A scalar charge toy model

Using self-force methods, we consider the hyperbolic-type scattering of a pointlike particle carrying a scalar charge $Q$ off a Schwarzschild black hole. For given initial velocity and impact parameter, back-reaction from the scalar field modifies the scattering angle by an amount $\propto\! Q^2$, which we calculate numerically for a large sample of orbits (neglecting the gravitational self-force). Our results probe both strong-field and field-weak scenarios, and in the latter case we find a good agreement with post-Minkowskian expressions. The scalar-field self-force has a component tangent to the four-velocity that exchanges particle's mass with scalar-field energy, and we also compute this mass exchange as a function along the orbit. The expressions we derive for the scattering angle (in terms of certain integrals of the self-force along the orbit) can be used to obtain the gravitational self-force correction to the angle in the physical problem of a binary black hole with a large mass ratio. We discuss the remaining steps necessary to achieve this goal.

Eccentric self-forced inspirals into a rotating black hole

We develop the first model for extreme mass-ratio inspirals (EMRIs) into a rotating massive black hole driven by the gravitational self-force (GSF). Our model is based on an action angle formulation of the method of osculating geodesics for eccentric, equatorial (i.e., spin-aligned) motion in Kerr spacetime. The forcing terms are provided by an efficient spectral interpolation of the first-order GSF in the outgoing radiation gauge. We apply a near-identity (averaging) transformation to eliminate all dependence of the orbital phases from the equations of motion, while maintaining all secular effects of the first-order GSF at post-adiabatic order. This implies that the model can be evolved without having to resolve all orbit cycles of an EMRI, yielding an inspiral model that can be evaluated in less than a second for any mass-ratio. In the case of a non-rotating central black hole, we compare inspirals evolved using self-force data computed in the Lorenz and radiation gauges. We find that the two gauges generally produce differing inspirals with a deviation of comparable magnitude to the conservative self-force correction. This emphasizes the need for including the (currently unknown) dissipative second order self-force to obtain gauge independent, post-adiabatic waveforms.

Analytical determination of the periastron advance in spinning binaries from self-force computations

We present the first analytical computation of the (conservative) gravitational self-force correction to the periastron advance around a spinning black hole. Our result is accurate to the second order in the rotational parameter and through the 9.5 post-Newtonian level. It has been obtained as the circular limit of the correction to the gyroscope precession invariant along slightly eccentric equatorial orbits in the Kerr spacetime. The latter result is also new and we anticipate here the first few terms only of the corresponding post-Newtonian expansion.

Spin self-force

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.

New gravitational self-force analytical results for eccentric equatorial orbits around a Kerr black hole: Gyroscope precession

We analytically compute the gravitational self-force correction to the gyroscope precession along slightly eccentric equatorial orbits in the Kerr spacetime, generalizing previous results for the Schwarzschild spacetime. Our results are accurate through the 9.5 post-Newtonian order and to second order in both eccentricity and rotation parameter. We also provide a post-Newtonian check of our results based on the currently known Hamiltonian for spinning binaries.

Gravitational self-force corrections to gyroscope precession along circular orbits in the Kerr spacetime

We generalize to Kerr spacetime previous gravitational self-force results on gyroscope precession along circular orbits in the Schwarzschild spacetime. In particular we present high order post- Newtonian expansions for the gauge invariant precession function along circular geodesics valid for arbitrary Kerr spin parameter and show agreement between these results and those derived from the full post-Newtonian conservative dynamics. Finally we present strong field numerical data for a range of the Kerr spin parameter, showing agreement with the GSF-PN results, and the expected lightring divergent behaviour. These results provide useful testing benchmarks for self- force calculations in Kerr spacetime, and provide an avenue for translating self-force data into the spin-spin coupling in effective-one-body models.

Gravitational self-force corrections to tidal invariants for spinning particles on circular orbits in a Schwarzschild spacetime

We compute gravitational self-force (conservative) corrections to tidal invariants for spinning particles moving along circular orbits in a Schwarzschild spacetime. In particular, we consider the square and the cube of the gravitoelectric quadrupolar tidal tensor and the square of the gravitomagnetic quadrupolar tidal tensor. Our results are accurate to first order in spin and through the 9.5 post-Newtonian order. We also compute the associated electric-type and magnetic-type eigenvalues.

Gravitational self-force corrections to tidal invariants for particles on circular orbits in a Kerr spacetime

We generalize to the Kerr spacetime existing self-force results on tidal invariants for particles moving along circular orbits around a Schwarzschild black hole. We obtain linear-in-mass-ratio corrections to the quadratic and cubic electric-type invariants and the quadratic magnetic-type invariant in series of the rotation parameter up to the fourth order. We then construct the eigenvalues of both electric and magnetic tidal tensors and analytically compute them through high post-Newtonian orders.

Gravitational self-force corrections to tidal invariants for particles on eccentric orbits in a Schwarzschild spacetime

We study tidal effects induced by a particle moving along a slightly eccentric equatorial orbits in a Schwarzschild spacetime within the gravitational self-force framework. We compute the first order (conservative) corrections in the mass-ratio to the eigenvalues of the electric-type and magnetic-type tidal tensors up to the second order in eccentricity and through the 9.5 post-Newtonian order. Previous results on circular orbits are thus generalized and recovered in a proper limit.

Spin-orbit precession along eccentric orbits: Improving the knowledge of self-force corrections and of their effective-one-body counterparts

The (first-order) gravitational self-force correction to the spin-orbit precession of a spinning compact body along a slightly eccentric orbit around a Schwarzschild black hole is computed through the ninth post-Newtonian order, improving recent results by Kavanagh et al. [Phys.\ Rev.\ D {\bf 96}, 064012 (2017).] This information is then converted into its corresponding Effective-One-Body counterpart, thereby determining several new post-Newtonian terms in the gyrogravitomagnetic ratio $g_{S*}$.

Self-force correction to geodetic spin precession in Kerr spacetime

We present an expression for the gravitational self-force correction to the geodetic spin precession of a spinning compact object with small, but non-negligible mass in a bound, equatorial orbit around a Kerr black hole. We consider only conservative back-reaction effects due to the mass of the compact object ($m_1$) thus neglecting the effects of its spin $s_1$ on its motion, i.e, we impose $s_1 \ll G m_1^2/c$ and $m_1 \ll m_2$, where $m_2$ is the mass parameter of the background Kerr spacetime. We encapsulate the correction to the spin precession in $\psi$, the ratio of the accumulated spin-precession angle to the total azimuthal angle over one radial orbit in the equatorial plane. Our formulation considers the gauge-invariant $\ord(m_1)$ part of the correction to $\psi$, denoted by $\Delta\psi$, and is a generalization of the results of [Class. Quan. Grav., 34, 084001 (2017)] to Kerr spacetime. Additionally, we compute the zero-eccentricity limit of $\Delta\psi$ and show that this quantity differs from the circular-orbit $\Delta\psi^\text{circ}$ by a gauge-invariant quantity containing the gravitational self-force correction to general relativistic periapsis advance in Kerr spacetime. Our result for $\Delta\psi$ is expressed in a manner that readily accommodates numerical/analytical self-force computations, e.g., in radiation gauge, and paves the way for the computation of a new eccentric-orbit Kerr gauge invariant beyond the generalized redshift.

Hamiltonian formulation of the conservative self-force dynamics in the Kerr geometry

We formulate a Hamiltonian description of the orbital motion of a point particle in Kerr spacetime for generic (eccentric, inclined) orbits, which accounts for the effects of the conservative part of the gravitational self-force. This formulation relies on a description of the particle’s motion as geodesic in a certain smooth effective spacetime, in terms of (generalized) action-angle variables. Clarifying the role played by the gauge freedom in the Hamiltonian dynamics, we extract the gauge-invariant information contained in the conservative self-force. We also propose a possible gauge choice for which the orbital dynamics can be described by an effective Hamiltonian, written solely in terms of the action variables. As an application of our Hamiltonian formulation in this gauge, we derive the conservative self-force correction to the orbital frequencies of Kerr innermost stable spherical (inclined or circular) orbits. This gauge choice also allows us to establish a ‘first law of mechanics’ for black-hole-particle binary systems, at leading order beyond the test-mass approximation.

Fundamental frequencies and resonances from eccentric and precessing binary black hole inspirals

Binary black holes which are both eccentric and undergo precession remain unexplored in numerical simulations. We present simulations of such systems which cover about 50 orbits at comparatively high mass ratios 5 and 7. The configurations correspond to the generic motion of a nonspinning body in a Kerr spacetime, and are chosen to study the transition from finite mass-ratio inspirals to point particle motion in Kerr. We develop techniques to extract analogs of the three fundamental frequencies of Kerr geodesics, compare our frequencies to those of Kerr, and show that the differences are consistent with self-force corrections entering at first order in mass ratio. This analysis also locates orbital resonances where the ratios of our frequencies take rational values. At the considered mass ratios, the binaries pass through resonances in one to two resonant cycles, and we find no discernible effects on the orbital evolution. We also compute the decay of eccentricity during the inspiral and find good agreement with the leading order post-Newtonian prediction.

Complete waveform model for compact binaries on eccentric orbits

We present a time domain waveform model that describes the inspiral-merger-ringdown (IMR) of compact binary systems whose components are non-spinning, and which evolve on orbits with low to moderate eccentricity. The inspiral evolution is described using third order post-Newtonian equations both for the equations of motion of the binary, and its far-zone radiation field. This latter component also includes instantaneous, tails and tails-of-tails contributions, and a contribution due to non-linear memory. This framework reduces to the post-Newtonian approximant TaylorT4 at third post-Newtonian order in the zero eccentricity limit. To improve phase accuracy, we incorporate higher-order post-Newtonian corrections for the energy flux of quasi-circular binaries and gravitational self-force corrections to the binding energy of compact binaries. This enhanced inspiral evolution prescription is combined with an analytical prescription for the merger-ringdown evolution using a catalog of numerical relativity simulations. This IMR waveform model reproduces effective-one-body waveforms for systems with mass-ratios between 1 to 15 in the zero eccentricity limit. Using a set of eccentric numerical relativity simulations, not used during calibration, we show that our eccentric model accurately reproduces the features of eccentric compact binary coalescence throughout the merger. Using this model we show that the gravitational wave transients GW150914 and GW151226 can be effectively recovered with template banks of quasi-circular, spin-aligned waveforms if the eccentricity $e_0$ of these systems when they enter the aLIGO band at a gravitational wave frequency of 14 Hz satisfies $e_0^{\rm GW150914}\leq0.15$ and $e_0^{\rm GW151226}\leq0.1$.

Self-Force Corrections to the Periapsis Advance around a Spinning Black Hole.

The linear in mass ratio correction to the periapsis advance of equatorial nearly circular orbits around a spinning black hole is calculated for the first time and to a very high precision, providing a key benchmark for different approaches modeling spinning binaries. The high precision of the calculation is leveraged to discriminate between two recent incompatible derivations of the 4 post-Newtonian equations of motion. Finally, the limit of the periapsis advance near the innermost stable orbit (ISCO) allows the determination of the ISCO shift, validating previous calculations using the first law of binary mechanics. Calculation of the ISCO shift is further extended into the near-extremal regime (with spins up to 1-a=10^{-20}), revealing new unexpected phenomenology. In particular, we find that the shift of the ISCO does not have a well-defined extremal limit but instead continues to oscillate.

Spin-dependent two-body interactions from gravitational self-force computations

We analytically compute, through the eight-and-a-half post-Newtonian order and the fourth-order in spin, the gravitational self-force correction to Detweiler's gauge invariant redshift function for a small mass in circular orbit around a Kerr black hole. Using the first law of mechanics for black hole binaries with spin [L.~Blanchet, A.~Buonanno and A.~Le Tiec, Phys.\ Rev.\ D {\bf 87}, 024030 (2013)] we transcribe our results into a knowledge of various spin-dependent couplings, as encoded within the spinning effective-one-body model of T.~Damour and A.~Nagar [Phys.\ Rev.\ D {\bf 90}, 044018 (2014)]. We also compare our analytical results to the (corrected) numerical self-force results of A.~G.~Shah, J.~L.~Friedman and T.~S.~Keidl [Phys.\ Rev.\ D {\bf 86}, 084059 (2012)], from which we show how to directly extract physically relevant spin-dependent couplings.

Self-force as a cosmic censor in the Kerr overspinning problem

It is known that a near-extremal Kerr black hole can be spun up beyond its extremal limit by capturing a test particle. Here we show that overspinning is always averted once backreaction from the particle’s own gravity is properly taken into account. We focus on nonspinning, uncharged, massive particles thrown in along the equatorial plane and work in the first-order self-force approximation (i.e., we include all relevant corrections to the particle’s acceleration through linear order in the ratio, assumed small, between the particle’s energy and the black hole’s mass). Our calculation is a numerical implementation of a recent analysis by two of us [Phys. Rev. D 91, 104024 (2015)], in which a necessary and sufficient “censorship” condition was formulated for the capture scenario, involving certain self-force quantities calculated on the one-parameter family of unstable circular geodesics in the extremal limit. The self-force information accounts both for radiative losses and for the finite-mass correction to the critical value of the impact parameter. Here we obtain the required self-force data and present strong evidence to suggest that captured particles never drive the black hole beyond its extremal limit. We show, however, that, within our first-order self-force approximation, it is possible to reach the extremal limit with a suitable choice of initial orbital parameters. To rule out such a possibility would require (currently unavailable) information about higher-order self-force corrections.