Session T16: Self-Force and Post-Geodesic Motion

Dissipation due to the (not-so) conservative self-force for resonant extreme-mass-ratio inspirals

The inspiral of a small compact object into a massive black hole, or an extreme-mass-ratio inspiral (EMRI), is often modeled using self-force theory. In self-force theory, a small body orbiting in a stationary gravitational background experiences a self-force due its own perturbing (gravitational, scalar, and/or electromagnetic) field. This self-force, in turn, sources the small body's motion. One often separates this self-force into two pieces: a (time-antisymmetric) dissipative component, which is responsible for the inspiral, and a (time-symmetric) conservative component, which typically perturbs the small body's orbit without driving its inspiral. However, using a scalar model, we numerically demonstrate that when a scalar point-charge enters a resonant orbit about a Kerr black hole (i.e., its radial and polar orbital frequencies form a rational ratio), the conservative scalar self-force becomes not-so conservative and can contribute to the dissipation of the Carter constant, thus driving the system away from the resonance. We analyze the strength of the conservative self-force contributions for different resonances and discuss how these results inform future EMRI models.   

Kerr self-force via elliptic PDEs

Our long-term goal is to calculate the Lorenz gauge gravitational self-force for an extreme mass-ratio binary system in Kerr spacetime. Prior time domain efforts encountered catastrophic instabilities, which we overcome by entering the frequency domain and solving elliptic PDEs. To develop techniques, we first consider the self-force exerted on a scalar charge in Kerr spacetime via the effective source method. Challenges include handling a large sparse linear system and how imperfect boundary conditions can introduce errors that pollute the entire domain. We have applied various techniques to overcome these challenges, such as analyzing the boundary behavior to impose more sophisticated boundary conditions with improved accuracy. Various preliminary self-force results will be presented and discussed.   

Post-geodesic corrections to the self-force during EMRIs

We quantified how past orbital behavior affects the self-force during an extreme mass-ratio inspiral. We expanded the metric to first order in the mass ratio using black hole perturbation theory, and used a two-timescale expansion of the Regge-Wheeler equation to access non-geodesic corrections to the metric perturbation. The asymptotic coefficients of higher order terms in the two-timescale expansion provide energy and angular momentum fluxes, which we used to calculate post-geodesic self-force corrections. We present preliminary results demonstrating the verified asymptotic coefficients and corrections to the self-force.   

A discontinuous Galerkin method for the distributionally-sourced s=0 Teukolsky equation

Extreme Mass Ratio Inspirals (EMRI) will be an important astrophysical source for the upcoming gravitational wave detector Laser Interferometer Space Antenna (LISA). EMRI systems are characterized by mass ratios greater than 10⁴ and will be in the detector's sensitive band for hundreds of thousands of orbital cycles. Numerical relativity codes cannot be used to simulate such systems, so point-particle black hole perturbation theory (ppBHPT) is typically used instead. For perturbations of Kerr, time-domain numerical solvers for the Teukolsky equation have traditionally been based on low-order finite-difference methods. In this talk I will discuss a spectrally-accurate Discontinuous Galerkin method for the distributionally-forced Teukolsky equation. Importantly, the method maintains spectral accuracy even at the location of the delta function. I will present numerical results from the code, including the computation of Price tails and scalar field evolutions from a perturbing black hole in a circular orbit.   

Particle motion under the conservative piece of the first-order gravitational self-force is Hamiltonian

The two body problem in general relativity is of great theoretical and observational interest, and can be studied in the post-Newtonian, post-Minkowskian and small mass-ratio approximations, as well as with effective one-body and fully numerical techniques. When gravitational wave dissipation is turned off, motion is expected to form a Hamiltonian dynamical system. This has been established to various orders in the post-Newtonian and post-Minkowskian approximations, but not yet in the small mass-ratio regime beyond the leading order of geodesic motion. We show that the motion under the conservative (time even) piece of the first-order self-force is Hamiltonian in any stationary spacetime, and find an explicit expression for the Hamiltonian in terms of a Green's function. In Kerr, this result extends previous results of Fujita et. al. who derived a Hamiltonian description valid only for non-resonant orbits. As applications, we derive a simple necessary condition that the self-force must satisfy for the motion to be integrable and clarify the domain of validity of the first law of binary black hole mechanics in the small mass-ratio regime.

Precisely computing bound orbits of spinning bodies around black holes

Very large mass-ratio binary black hole systems are of interest as a clean limit of the two-body problem in general relativity, as well as for describing important low-frequency gravitational wave (GW) sources. At lowest order, the smaller black hole follows a geodesic of the larger black hole's spacetime. Accurate models of large mass-ratio systems include post-geodesic corrections which account for forces driving the small body away from the geodesic. Spin-curvature forces, which arise due to the coupling of a test body's spin to the spacetime curvature, is an example of such an effect. In a previous talk, we outlined a frequency-domain method for precisely computing bound orbits of spinning test bodies experiencing spin-curvature forces. In this talk, we show how to apply this approach to the fully generic case, in which orbits are inclined and eccentric and with the small body's spin arbitrarily oriented. An osculating geodesic integrator that includes both spin-curvature forces and the backreaction due to GWs can be used to generate an adiabatic spinning-body inspiral. We present preliminary results combining the osculating element description with the tetrad formulation for Kerr parallel transport to build a framework for completely generic worldlines of spinning bodies.   

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 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. As such, the model can be evolved without having to resolve all ~10^6 orbit cycles of an EMRI, yielding an inspiral that can be evaluated in less than a second for any mass-ratio. In the case of a non-rotating black hole, we compare inspirals evolved using GSF data computed in the Lorenz and radiation gauges. We find that the two gauges produce differing inspirals with a deviation of comparable magnitude to the conservative GSF correction. This emphasizes the need for including the dissipative second order GSF to obtain gauge independent, post-adiabatic waveforms.   

Hyperboloidal method for frequency-domain self-force calculations

Gravitational self-force theory is the leading approach for modelling gravitational wave emission from small mass-ratio compact binaries. This method perturbatively expands the metric of the binary in powers of the mass ratio. The source for the perturbations depends on the orbital configuration, calculational approach, and/or the order the perturbative expansion is carried too. These sources fall into three broad classes: distributional, extended and compact, and non-compact. The latter, in particular, is important for emerging second-order in the mass ratio calculations. Traditional frequency domain approaches employ the variation of parameters method and compute the perturbation on constant time slices of the spacetime with numerical boundary conditions supplied at finite radius from series expansions of the asymptotic behaviour. This approach has been very successful but the boundary conditions calculations are tedious and the approach is not well suited to non-compact sources where homogeneous solutions must be computed at all radii.  In this talk, I outline an alternative approach where the spacetime is foliated by horizon-penetrating hyperboloidal slices. Further compactifying the coordinates along these slices allows for simple treatment of the boundary conditions. We implement this approach with a multi-domain spectral solver with analytic mesh refinement and present results for the scalar-field self-force on circular orbits as an example problem. We find the method works efficiently for all three classes of sources encountered in self-force calculations and has some distinct advantages over the traditional approach.   

Gravitational waveforms for compact binaries from second-order self-force theory

We produce gravitational waveforms for nonspinning compact binaries undergoing a quasicircular inspiral. Our approach is based on a two-timescale expansion of the Einstein equations in second-order self-force theory, which allows first-principles waveform production in milliseconds. Although the approach is designed for extreme mass ratios, our waveforms agree remarkably well with those from full numerical relativity, even for comparable-mass systems. Our results will be invaluable in accurately modelling extreme-mass-ratio inspirals for the LISA mission and intermediate-mass-ratio systems currently being observed by the LIGO-Virgo-KAGRA Collaboration.