Visualizing, Approximating, and Understanding Black-Hole Binaries

Citation

Nichols, David Andrew (2012) Visualizing, Approximating, and Understanding Black-Hole Binaries. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8D56-9Y02. https://resolver.caltech.edu/CaltechTHESIS:05152012-231012129

Abstract

Numerical-relativity simulations of black-hole binaries and advancements in gravitational-wave detectors now make it possible to learn more about the collisions of compact astrophysical bodies. To be able to infer more about the dynamical behavior of these objects requires a fuller analysis of the connection between the dynamics of pairs of black holes and their emitted gravitational waves. The chapters of this thesis describe three approaches to learn more about the relationship between the dynamics of black-hole binaries and their gravitational waves: modeling momentum flow in binaries with the Landau-Lifshitz formalism, approximating binary dynamics near the time of merger with post-Newtonian and black-hole-perturbation theories, and visualizing spacetime curvature with tidal tendexes and frame-drag vortexes.

In Chapters 2--4, my collaborators and I present a method to quantify the flow of momentum in black-hole binaries using the Landau-Lifshitz formalism. Chapter 2 reviews an intuitive version of the formalism in the first-post-Newtonian approximation that bears a strong resemblance to Maxwell’s theory of electromagnetism. Chapter 3 applies this approximation to relate the simultaneous bobbing motion of rotating black holes in the superkick configuration---equal-mass black holes with their spins anti-aligned and in the orbital plane---to the flow of momentum in the spacetime, prior to the black holes’ merger. Chapter 4 then uses the Landau-Lifshitz formalism to explain the dynamics of a head-on merger of spinning black holes, whose spins are anti-aligned and transverse to the infalling motion. Before they merge, the black holes move with a large, transverse, velocity, which we can explain using the post-Newtonian approximation; as the holes merge and form a single black hole, we can use the Landau-Lifshitz formalism without any approximations to connect the slowing of the final black hole to its absorbing momentum density during the merger.

In Chapters 5--7, we discuss using analytical approximations, such as post-Newtonian and black-hole-perturbation theories, to gain further understanding into how gravitational waves are generated by black-hole binaries. Chapter 5 presents a way of combining post-Newtonian and black-hole-perturbation theories---which we call the hybrid method---for head-on mergers of black holes. It was able to produce gravitational waveforms and gravitational recoils that agreed well with comparable results from numerical-relativity simulations. Chapter 6 discusses a development of the hybrid model to include a radiation-reaction force, which is better suited for studying inspiralling black-hole binaries. The gravitational waveform from the hybrid method for inspiralling mergers agreed qualitatively with that from numerical-relativity simulations; when applied to the superkick configuration, it gave a simplified picture of the formation of the large black-hole kick. Chapter 7 describes an approximate method of calculating the frequencies of the ringdown gravitational waveforms of rotating black holes (quasinormal modes). The method generalizes a geometric interpretation of black-hole quasinormal modes and explains a degeneracy in the spectrum of these modes.

In Chapters 8--11, we describe a new way of visualizing spacetime curvature using tools called tidal tendexes and frame-drag vortexes. This relies upon a time-space split of spacetime, which allows one to break the vacuum Riemann curvature tensor into electric and magnetic parts (symmetric, trace-free tensors that have simple physical interpretations). The regions where the eigenvalues of these tensors are large form the tendexes and vortexes of a spacetime, and the integral curves of their eigenvectors are its tendex and vortex lines, for the electric and magnetic parts, respectively. Chapter 8 provides an overview of these visualization tools and presents initial results from numerical-relativity simulations. Chapter 9 uses topological properties of vortex and tendex lines to classify properties of gravitational waves far from a source. Chapter 10 describes the formalism in more detail, and discusses the vortexes and tendexes of multipolar spacetimes in linearized gravity about flat space. The chapter helps to explain how near-zone vortexes and tendexes become gravitational waves far from a weakly gravitating, time-varying source. Chapter 11 is a detailed investigation of the vortexes and tendexes of stationary and perturbed black holes. It develops insight into how perturbations of (strongly gravitating) black holes extend from near the horizon to become gravitational waves.

Thesis Files