Session T16: Approximate Methods in Gravitational Astrophysics

A Spheroidal Harmonic Picture for GWs from Astrophysical Sources I: Bi-Orthogonality

The study of isolated perturbed Kerr black holes plays an important role in gravitational wave (GW) signal modeling for coalescing binary black holes. In this, the natural multipolar structure of Kerr black holes has informed both signal models (Numerical Relativity and approximants) and related tests of General Relativity. In this talk we present new results on the multipolar structure of GWs from Kerr black holes; in particular, we demonstrate for the first time that the natural spherical harmonics for Kerr, the spheroidal harmonics, display bi-orthogonality and so obey a spectral theorem. We thereby present a method for the general calculation of spheroidal harmonic GW multipole moments. This marks a departure from the spherical multipole moments that are currently ubiquitous, despite being most appropriate for only non-spinning spacetimes. Noting that all astrophysical systems with angular momentum are asymptotically similar to the Kerr metric, we discuss possible applications for the spheroidal harmonic decomposition of gravitational waves from general astrophysical sources.   

On Demand

A vast majority of extreme-mass-ratio black hole binaries (EMRIs) will encounter at least one strong $r\theta$-resonance as they evolve through LISA's passband. These resonances occur when the frequencies of the librating radial and polar motion of the EMRI's smaller body form a low-integer ratio, and they drive significant `kicks' in the amount of energy and angular momentum that EMRIs radiate through gravitational waves. These kicks, if not properly accounted for, can amplify errors in modeled EMRI waveforms by factors of $\sim 100$. Despite the importance of modeling these resonant dynamics, researchers have not yet calculated the gravitational self-force experienced by EMRIs during $r\theta$-resonances. As a first step in quantifying these effects, we calculate the scalar self-force (the scalar analog to the gravitational self-force) experienced by a scalar-charged particle following an $r\theta$-resonant geodesic around a Kerr black hole. We present how local and global radiation-reaction effects vary with respect to initial conditions. We also demonstrate, numerically, that conservative self-force effects do not contribute to the leading-order evolution of the system, as hypothesized by previous researchers.   

An extended-body approach to self-force in curved spacetime

Self-force describes the effect of an object's own field on its motion. If we model a physical object as a point particle, then the self-force diverges when evaluated at the particle's location. One way to avoid this difficulty is to model the physical object as an extended body with detailed internal structure. We consider an extended body, modeled as an elastic material, moving in an arbitrary background gravitational field. We express the spacetime coordinates of each point in the body in terms of the Fermi normal coordinates tied to a fiducial observer whose basis vectors are Fermi--Walker transported. We construct the action and equations of motion for the elastic body, expanding in powers of the spatial Fermi normal coordinates $\bar{x}^a$. A center of mass condition ties the motion of the fiducial observer to the motion of the elastic body. We show that (a) the elastic body follows a geodesic at zeroth order in $\bar{x}^a$ and (b) we recover the Mathisson-Papapetrou-Dixon (MPD) equations at first order in $\bar{x}^a$. In each case, the details of the body's elastic interactions are absorbed into the body's multipole moments. In future work, we will remove the restriction to background gravitational fields and derive the gravitational self-force on the extended body.   

On Demand

We present on progress to compute the second order metric perturbation of a Kerr black hole. Out motivation for pursuing this project includes (1) to understand the regime of applicability of the Teukolsky equation in describing the ringdown of Kerr black hole formed after the merger of two similar mass compact objects, and (2) to understand the proposed onset of "gravitational wave turbulence" around very rapidly spinning Kerr black holes. Our procedure for numerically computing the second order perturbation incorporates several observations made by researchers in the mathematical relativity community in their work towards a proof of the nonlinear stability of Kerr black holes.   

On Demand

We address a hole in the space of existing toy models for the astrophysical black hole (BH) interior by simulating the gravitational collapse of \textit{rotating} matter in (2+1)D classical general relativity. We compare and contrast our time-dependent numerical solution to the stationary analytic solution due to Ba\~nados, Teitelboim, and Zanelli (BTZ), as well as to the celebrated Kerr solution. We focus on three features in the dynamical case: the singularity structure, the regularity of the Cauchy horizon, and the geodesic-focusing effect first described by Marolf \& Ori. We observe the latter effect for the first time in a BH formed from gravitational collapse. We also find that curvature singularities form at the origin and Cauchy horizon for low spin, but disappear entirely for sufficiently high spins, signaling a violation of the $C^0$ and $C^2$ formulations of the strong cosmic censorship conjecture.   

Not Participating

Asymptotic waveforms of the gravitational wave strain and the Weyl scalars have infinite-dimensional gauge freedoms expressed by the BMS group. In order to compare numerical relativity waveforms across different simulations, or even different grid resolutions of a simulation, it is critical to systematically understand and fix these gauge freedoms. This also will become an important factor for surrogate waveforms and phenomenological models that require numerical waveforms. We present preliminary results using a method for fixing these BMS gauge freedoms.   

On Demand

Gravitational waveforms for compact binary coalescences (CBCs) have been invaluable for detections by the LIGO-Virgo collaboration. They are produced by powerful combinations of analytical approximations and numerical simulations. So far systematic errors arising from these approximations have been less than statistical errors. However, we are now entering an era of abundant detections, and the third generation observatories, as well as LISA, are on the horizon. Therefore, it is highly desirable to have more accurate waveforms. The goal of this talk is to show that full non-linear general relativity (GR) imposes an infinite number of sharp constraints on the CBC waveforms. These can be used as additional measures both to evaluate the accuracy of candidate waveforms against exact GR, and to discriminate between various avenues currently used to generate waveforms.   

Relativistic Corrections for Asteroseismology to First Post-Newtonian Order

In light of the precision available from the K2 and TESS missions in determining stellar pulsation periods, general relativistic effects are in principle measurable contributors in white dwarf light curves. We provide a demonstration of the appearance of general relativistic effects by considering initially simplified polytropic models for stars of mass and radius comparable to white dwarfs. Relativistic effects are calculated to sufficient accuracy using a first post-Newtonian (1PN) correction to the standard Dziembowski form of the pulsation equations. Background models and linear nonradial pulsations are calculated in both the 1PN and Newtonian regimes and the periods of g-modes and p-modes are compared. Future work will include more astrophysically accurate white dwarf models with g-modes confined near the stellar surface.