Session C13: Analytical Calculations and Approximations in General Relativity

Spikes in the approach to spacetime singularities

An analytic approximation is developed for the small scale structure (spikes) that develops in the approach to spacetime singularities. This approximation indicates that spikes are of co-dimension one, and that spikes are transient. Numerical simulations are performed that test the analytic approximation.   

The breakdown of the orbit-averaging approximation in Eccentric Binaries

As gravitational waves are generated in a binary, energy and angular momentum are carried away, forcing the eccentricity of the system to decrease.~ In order to characterize this evolution in eccentricity two methods are generally used: one involves averaging the change in certain orbital elements over one orbital period, while the other does not employ this averaging and uses multiple-scale analysis instead.~In this talk, I will describe these methods and explain why they yield different answers for the evolution of the eccentricity of a binary in the late stages of inspiral.   

No such thing as a circular orbit

The loss of orbital energy and angular momentum to gravitational waves produced in a binary inspiral forces the orbital eccentricity to evolve. The general belief has been that the eccentricity decreases monotonically in the inspiral and completely circularizes the binary. Contrary to this, in this talk I will show that, once the eccentricity is small enough, orbit averaging breaks down and radiation reaction forces the eccentricity to grow secularly before the binary reaches the last stable orbit and merges. Even if the eccentricity is initially exactly zero, non-linear effects in the late inspiral force the eccentricity to grow secularly, thus rendering the concept of a quasicircular obit void. I will conclude with a short discussion of the implications that such eccentricity growth has on gravitational wave observations and parameter estimation.   

Revisiting the emission from an extreme mass ratio plunge

The final stage of an extreme mass ratio inspiral is the rapid plunge of the small particle into a black hole, leading to ringdown. This ringdown carries information about the spacetime near the light ring, but in principle the final emission of the particle probes the spacetime closer to the horizon. Using a near-horizon expansion, we explore the emission from the last stage of the plunge into a spinning black hole, and its imprint on the ringdown signal.   

Eccentric, Spinning Black Hole Binaries in the Inspiral Regime

In this talk, I present a method for developing and calculating the gravitational waveforms from generically spinning, eccentric black hole binaries. I use the Lagrangian formulation of the post-Newtonian equations of motion in the harmonic gauge for the generation of precessing, eccentric gravitational wave signatures. The equations of motion describing the black hole binary system are also of utmost importance to our understanding of fundamental relativity, for both the context of supermassive black holes, and also stellar mass systems. If LIGO is able to measure a non-negligible eccentricity from the binary, this may point to a unique formation model through relativistic 3-body interactions in dense stellar fields, which will impart occasionally significant eccentricity. This tells us something about the formation history of the binary, and explicitly about the last dynamical effect the binary experienced before merging. While it is not expected that LIGO sources have significant eccentricity in band, there may be some residual eccentricity from dynamical interactions prior to merger.   

Calculating the scalar self-force for generic orbits in the Kerr spacetime

We investigate the generic, bound motion of a scalar-charged point mass in the background spacetime of a more massive Kerr black hole. In the context of black hole perturbation theory, the evolution of these (E/IMRI) systems is quantified by a self-force term. As a developmental model, we consider the scalar self-force problem, where---like the gravitational case---the scalar field perturbation interacts back on the source charge and drives the smaller charged-body’s motion. We calculate the scalar self-force for several inclined, eccentric orbits, with inclinations ranging from $\iota=0$ to $\iota=\pi$ and eccentricities ranging from $e=0$ to $e=0.8$. We use a frequency domain code written in Mathematica, along with a combination of spectral integration and MST function expansion techniques, to make calculations with arbitrary numerical precision. We also investigate various resonant orbital configurations for inclined, eccentric motion and observe previously reported quasinormal mode excitations in the self-force for highly-eccentric orbits around a highly-spinning black hole.   

Eccentric orbit binary black hole inspirals: Informing the post-Newtonian expansion through black hole perturbation theory

We use high precision comparisons between perturbation theory and the post-Newtonian expansion to extract new information on eccentric orbit EMRIs. Fluxes are calculated by combining the MST formalism with spectral source integration (SSI) for a multitude of orbits, whose parameters are then fit to the PN form. This fit is performed on each LMN mode individually, allowing us to exploit the patterns contained therein. The outcome is an ability to fit for combinations of transcendental numbers that far exceeds that of prior work, yielding new analytic coefficients to 9PN order on a Schwarzschild background. Full results for the energy and angular momentum lost to infinity are detailed. In addition, preliminary work expanding perturbative source terms directly is discussed, along with an extension to generic orbits on a Kerr background. We conclude with expectations and outlook.   

Time domain calculations of scalar self-force and radiation from an orbiting point charge in Schwarzschild spacetime

Gravitational wave astronomy is a new window on violent mergers of black holes and neutron~ stars, and promises to eventually provide observations of supernovae, extreme-mass-ratio inspirals~(EMRIs) into supermassive black holes, and fluctuations in the Big Bang.~ We focus on time domain~calculations eventually relevant to understanding EMRIs but studying the scalar self-force model~problem.~ In these calculations, scalar radiation is emitted by a point scalar-charged particle in orbit about a more massive Schwarzschild black hole.~ The time domain calculations use a discontinuous~internal boundary condition representation for the point charge.~ We are implementing hyperboloidal~slicing and compactification to improve the treatment of distant and horizon boundaries.~ Results are~compared to earlier frequency domain calculations of Warburton and Barack, and to another more recently developed frequency domain code.~~