Gravitational wave astronomy has flourished in the last few years, with close to 100

detections of binary coalescences by the LIGO-Virgo-KAGRA collaboration to date.

Furthermore, the detectors are expected to keep increasing in sensitivity and a next

generation of detectors with orders of magnitude more sensitivity are being planned.

We are entering an era of high precision gravitational waves. Models of gravitational

waveforms emitted by these compact binaries lie at the heart of the data analysis pipeline,

and are crucial for interpreting the data. The models must be faithful to general relativity

(GR) for accurate tests of GR and unbiased interpretation of the binary parameters. As

the detectors increase in sensitivity, the waveform models must commensurately increase

their precision. The required precision of the models for third generation detectors

is several orders of magnitude higher than current precision levels. Therefore, we are

presented with a monumental task ahead of the future detectors.

In this dissertation we develop tools and methods from the treatment of gravitational

waves in exact GR to aid this effort of increasing the waveform accuracy. These methods

are based on the theory of asymptotically flat spacetimes, that is spacetimes that

approach Minkowski space along null directions. Surprisingly, the asymptotic symmetry

group of such spacetimes is not the Poincar  group. The symmetries form an infinite

dimension group, known as the Bondi-Metzner-Sachs (BMS) group. This group has

profound consequences on the gravitational radiation, which can broadly be split into

two categories.

First, associated to the symmetries one can find conserved charges of the spacetime.

Gravitational radiation carry a flux associated to these charges. This gives us an infinite

class of balance laws that come from exact GR. Because the balance laws are derived from

exact GR, this provides a method to test the fidelity of waveform models to GR directly

and for any parameters. However, to apply these balance laws we need to calculate

the charges. Thus, we introduce methods to calculate the initial and final charges for

compact binaries, and then apply the balance laws to detections. On the other hand, we

can even use the balance laws to improve the waveforms directly. We use the balance

laws to correct extrapolated numerical waveforms so that they include the memory effect,

dramatically increasing its accuracy.

The balance laws also have implications for the angular momentum of compact binaries.

Angular momentum in full GR cannot be defined in general due to the supertranslation

ambiguity that arises from the BMS group. Nonetheless, for stationary spacetimes the

angular momentum can be defined. We show that for CBCs we expect to be able to get

preferred Poincar  groups in the far past and far future. However, these are in general

different Poincare groups and there cannot be an angular momentum flux balance law

between them. Nonetheless, we show that it surprisingly turns out for compact binaries

that the naive balance law holds. Therefore, we also apply the angular momentum laws

to waveform models, which provides a method to test the accuracy of the waveform

model across parameter space without directly comparing to numerical simulations.

Second, the symmetry group also tells us that there are infinitely many reference

frames at null infinity. In particular, when comparing two different waveforms, it is

important to ensure that they are in the same reference frame. However, these issues have

often been ignored. One example of this is the hybridization, or stitching, of numerical

waveforms in the with inspiraling waveforms from a post-Newtonian approximation.

In general, these are expressed in different reference frames. We develop methods to

hybridize the waveforms my mapping them into identical frames that improves the

effectiveness of hybridization. Furthermore, such issues also arise in comparing ringdown

waveforms from numerical relativity to the expectations from perturbation theory. The

two approaches are in two different frames in general. Consequently, an accurate analysis

of the ringdown regime requires careful treatment of these issues. Therefore, we study

the effects this has on the quasinormal mode analysis.