Binary Black Hole Mergers: Conservation Laws and Gravitational Wave Memory Effects

Studying binary-black-hole (BBH) systems has long been of great interest in general relativity, especially after the first observed event in 2015 discovered by LIGO, which has revolutionized our understanding of the universe and opened up new insights that were not accessible to electromagnetic astronomy. These detections offer an insightful way to test general relativity in the strong-field and high-luminosity regime. In this dissertation, we cover two ways of characterizing these systems that turn out to be connected; the symmetries of these asymptotic systems and a relativistic strong-field phenomenon associated with them.
We begin by studying BBHs as isolated systems in an empty space that approaches flatness at infinity. Changes in the spacetime can then be quantified using the changes in charges conjugate to the asymptotic symmetries of the spacetime. The symmetries of asymptotically flat spacetimes in general relativity are described by the Bondi-Metzner-Sachs group (or its proposed extensions). Associated with these symmetries are conserved charges, which include the energy-momentum, supermomentum, and relativistic angular momentum (or super-angular momentum). In flat spacetime, there is an agreed upon well defined notion of angular momentum. However, in the non-linear theory of gravity in the presence of gravitational waves several formalisms have been used to compute the spacetime angular momentum. These angular momenta do not always agree, but the different definitions have been summarized in a two-parameter family of angular momenta. We found that a reasonable physical requirement for the angular momentum to vanish in flat spacetime restricts the two parameters to be equal, solving a part of the discrepancy that appears in the angular momentum definition. We examined the effect of this free parameter on the values of the angular momentum and super-angular momentum of nonprecessing binary-black-hole mergers. We found the definitions of angular momentum differ only when these systems are radiating gravitational waves (GWs). The definitions of super-angular momentum differ even after the GWs pass, because of a lasting effect called the GW memory effect. Using numerical-relativity surrogate waveforms, we estimate these differences to be small, but of the order of the accuracy of the angular momentum computed from these simulations.
A significant part of this thesis focuses on the relativistic non-linear GW memory effect and its detection prospects. This effect causes a permanent relative displacement between two freely falling test masses that persists after the passage of a GW signal, leaving a ``memory'' of the event. The memory effect has been computed first in the 1970's, but only with upcoming improvements to the LIGO, Virgo, and KAGRA detectors will the prospects of detecting the effect in a population of BBH mergers be promising. Searches for the memory effect in GW detector data require accurate waveform models, which must be evaluated many times (and, thus, need to be evaluated rapidly). Current analytical waveform models and many numerical-relativity waveforms and surrogates of BBH mergers do not include the memory effect. Instead, GW memory is computed from waveforms without memory by using conservation laws in asymptotically flat spacetimes, which is relatively slow. We present the first time- and frequency-domain waveform models of the GW memory effect for nonspinning BBH mergers for comparable-mass systems that can be evaluated more rapidly. A part of the model involves computing a fit for the final memory offset that incorporates data from both comparable and extreme mass-ratio limits, and which could be applied in both contexts to understand the remnant properties of BBH mergers more fully. In addition to speeding up GW searches, having these analytic models give analytical insights into the time- and frequency-domain properties of the GW memory signal.