Studies of Classical and Quantum Gravity

String theory provides insight and reveals interesting features for gravity at both large and small scales. At large scales, in the realm of classical physics, string theory predicts that the number of spacetime dimensions is larger than four, which means that four dimensional (4d) general relativity could be just an approximation. For the small scale physics, ruled by quantum laws, string theory reveals that the graviton scattering amplitudes and non-abelian gauge boson scattering amplitudes are closely related. This is a nontrivial statement, far from obvious from a field theory perspective, given how different are the Lagrangians of general relativity and Yang-Mills fields.

These two aspects can also be studied outside the framework of string theory. The idea of extra dimensions arose at first as a way to achieve a unification of the fundamental forces, as done in Kaluza-Klein theory. Extra dimensions were also used to solve the hierarchy problem, as done in the Randall-Sundrum model. The "squaring" relation between graviton scattering amplitudes and gauge boson amplitudes, although nowhere to be seen at the level of Lagrangian, can nonetheless be checked by computing the Feynman diagrams order by order. This motivates us to find a way to show them within the framework of particle physics.

This thesis is dedicated to the study of these two aspects of gravity. It consists of two parts: the study of classical gravity, covered in Chapters 1 and 2, focusing on how the extra dimensions affect gravitational waves, and the study of quantum gravity, covered in Chapter 3, focusing on relating tree-level scattering amplitudes of gravitons and spin 0 and spin 1 gauge bosons by using the worldline formalism.

In Chapter 1, we consider the simplest paradigm of extra dimensions: a flat 4d background and compact, periodic extra dimensions. We begin with a toy model, with a compactified fifth dimension, and matter localized in the extra dimension. We work in the context of five dimensional (5d) general relativity, although we do make connections with the corresponding Kaluza-Klein effective 4d theory. We show that the luminosity of the gravitational waves emitted in 5d gravity by a binary with the same characteristics (same masses and separation distance) as a 4d binary is 20.8% less relative to the 4d case, to leading post-Newtonian order. The phase of the gravitational waveform differs by 26% relative to the 4d case, to leading post-Newtonian order. These corrections to the waveform and the luminosity are inconsistent with the gravitational-wave and binary pulsar observations. This effectively rules out the possibility of such a simple compactified higher dimensions scenario. We also comment on how our results change if there are several compactified extra dimensions, and show that the discrepancy with 4d general relativity only increases.

In Chapter 2, we move on to consider the other paradigm of extra dimensions, with non-compact, large extra dimensions and matter localized on a brane, as in the Randall-Sundrum model. The brane backreaction curves the background geometry, which takes the form of a slab of Anti-de Sitter space. To ensure agreement with 4d Newtonian gravity at the shortest distance it has been probed, the background needs to be strongly curved. In the context of gravitational waves, the object of interest is the energy-momentum tensor of the gravitational field. We notice that there are issues with the conventional definition of the energy-momentum tensor of gravitational waves, which is defined using the wavelength averaging scheme, and assumes that the curvature scale of the background is much larger than the wavelength of the gravitational waves. We solve this problem by constructing the energy-momentum tensor out of those metric fluctuations which are gauge-invariant, extracted by performing a Scalar-Vector-Tensor SO(1,3) metric decomposition. We give concrete expressions for T0i, which is the part of the energy-momentum tensor necessary for computing the radiated power of the gravitational waves. We show that our expressions reduce to those conventional ones when the background is flat.

In Chapter 3, we utilize the worldline formalism to study how gravitons interact with other particles with different spins and whether there are relations between different interactions. Worldline formalism is a string-inspired, first quantized approach to computing Quantum Field Theory(QFT) amplitudes in the presence of external fields. Successfully applying worldline formalism to gravity requires a proper regularization scheme and correctly identifying some extra terms in the worldline action, which are conventionally called ”counterterms". These counterterms were determined previously in loop diagram calculation and one might expect them to stay the same in tree diagram calculation. However, through computing the 3-point vertex, we noticed some counterterms are not the same as those calculated for loop diagrams when the background field is non-dynamical. We verify these counterterms with calculations of 4-point tree amplitudes. When the background is free and dynamical, however, we can prove that many coefficients of the counterterms can be arbitrarily chosen. In calculating amplitudes, the most important pieces are vertex operators derived from the worldline action, which contain the information of the interaction. We show that the linear vertex operators of particles with different spins are closely related, and thus the scattering amplitudes they produced will be related. One special class of amplitudes, called maximal helicity violation (MHV) amplitudes, allows those relations among linear vertex operators to be directly shown in the final tree amplitudes. For gauge boson MHV interactions and graviton MHV interactions, the "squaring" relation of the 3-point worldline vertices leads to the just the double copy relation, which states that the numerator of each pole structure in a gravity amplitude is exactly the square of those in a corresponding gauge boson amplitude.