Gravitational Effective Field Theories and Black Hole Mechanics

TL;DR

This thesis explores gravitational effective field theories, establishing their mathematical foundations, and extends black hole mechanics laws to these theories, including proofs of the zeroth, first, and second laws with necessary modifications.

Contribution

It introduces a hyperbolic formulation for gravitational EFTs and generalizes black hole entropy laws within EFT frameworks, including dynamical cases.

Findings

A strongly hyperbolic formulation of Einstein-Maxwell EFT using a modified harmonic gauge.

Proof of the zeroth law for EFTs of gravity, electromagnetism, and scalar fields.

Extension of the first and second laws of black hole mechanics to EFTs with modified entropy definitions.

Abstract

This PhD thesis is submitted to arXiv as a pedagogical resource on gravitational effective field theories and the laws of black hole mechanics. It begins with an introduction to the mathematics and intuition behind effective field theory (EFT) corrections to General Relativity (GR) pitched at the level of a first year postgraduate course. Chapter 3 contains a similarly accessible discussion of the laws of black hole mechanics, their derivation in GR, and why they are a particularly interesting setting in which to apply EFT. The novel research is contained in Chapters 2 and 4-7. The first of these studies the well-posedness of gravitational EFTs. Specifically, we show that a "modified harmonic" gauge produces a strongly hyperbolic formulation of the leading order Einstein-Maxwell EFT so long as the higher derivative terms are small. The latter Chapters are concerned with proving the validity of the laws of black hole mechanics in EFTs of gravity and broad classes of matter fields. We demonstrate that the zeroth law can be proved for EFTs of gravity, electromagnetism and a charged or uncharged scalar field. We find we must modify the statements of the first and second laws by generalizing the formula of black hole entropy from the Bekenstein-Hawking entropy used in GR. For stationary black holes, the Wald entropy is a sufficient definition to satisfy the first law in EFT. For dynamical black holes we show that with further corrections we can prove a non-perturbative second law. We discuss the gauge dependence of this definition of dynamical black hole entropy and provide explicit constructions for specific EFTs.