Gravitational Effective Field Theories and Black Hole Mechanics (2411.14023v1)

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.

• The paper develops a Modified Harmonic Gauge to secure strong hyperbolicity and a well-posed initial value formulation for higher-derivative gravitational EFTs.
• The paper extends classical black hole mechanics by proving that the zeroth, first, and second laws hold up to higher-order corrections in the EFT framework.
• The paper’s rigorous proofs bridge traditional General Relativity with modern EFT extensions, offering insights for potential gravitational wave observations.

An Overview of "Gravitational Effective Field Theories and Black Hole Mechanics"

In "Gravitational Effective Field Theories and Black Hole Mechanics," Iain Davies undertakes a rigorous investigation into the mathematical foundations and implications of black hole mechanics within the framework of gravitational effective field theories (EFTs). The focus is on addressing two primary concerns: the well-posedness of the equations of motion and the validation of the laws of black hole mechanics beyond General Relativity (GR).

Gravitational Effective Field Theories

The work begins by situating gravitational effective field theories as extensions of General Relativity. These EFTs incorporate higher-derivative corrections parameterized by a UV length scale $l$. The Lagrangian for such theories is an expansion: $$\mathcal{L} = \mathcal{L}_{2} + l \mathcal{L}_3 + l^2 \mathcal{L}_4 + \ldots$$. This expansion allows for the description of gravitational phenomena beyond the low-energy assumptions of GR, accommodating potentially observable corrections at higher energies.

The Well-Posedness of EFTs

Davies tackles the challenge of formulating a well-posed initial value problem (IVP) for the equations of gravitational theories when extended to include higher-derivative terms, particularly with matter fields like electromagnetic fields. The challenge stems from the presence of higher derivatives in the Lagrangian, which can lead to equations of motion that are ill-posed or that necessitate additional initial data beyond what is classically required.

The thesis develops a formulation called Modified Harmonic Gauge (MHG) to address this. By introducing auxiliary metrics, this gauge transforms the equations into a form that maintains strong hyperbolicity—a condition necessary for well-posedness. The methodology is demonstrated for the Einstein-Maxwell effective field theory, showing that it is possible to construct an initial value formulation that provides unique solutions in the regime of weak coupling, i.e., when the effects of higher derivatives remain perturbatively small.

The Laws of Black Hole Mechanics in EFTs

A significant portion of the dissertation is devoted to adapting the classical laws of black hole mechanics—zeroth, first, and second laws—to the setting of gravitational EFTs:

1. Zeroth Law: In GR, the surface gravity $\kappa$ of a stationary black hole is constant. Davies extends the proof to vacuum gravitational EFTs, demonstrating that $\kappa$ remains constant up to order $O(l^N)$ under the assumption of analyticity in $l$, a condition that aligns with the solution lying in the regime of validity of EFT.
2. First Law: Traditionally involving variations in mass, angular momentum, and area, the first law's extension in beyond-GR theories is anchored on the Wald entropy, which abstracts the notion of black hole entropy suitable for theories with higher-derivative corrections. It is emphasized that the appropriate notion of entropy can differ from mere area-proportional forms.
3. Second Law: The classical second law states the non-decreasing nature of the event horizon area. In extending this law, Davies employs the Iyer-Wald and HKR entropies adjusted for the gravitational EFT context. This involves manipulating the equations of motion to isolate various contributions to the change in entropy and show that under a suitable definition, the entropy remains non-decreasing up to $O(l^N)$.

Conclusions and Implications

Davies' work highlights the robustness of black hole mechanics' laws when adapted to gravitational EFTs. Through meticulous mathematical proofs, the research underscores the potential for classical GR results to hold within more complex theoretical frameworks, provided careful consideration of higher-order corrections. The methods and insights developed have implications for both theoretical investigations and potential empirical studies as gravitational wave observatories advance, opening possibilities for observing nuances predicted by EFTs. The dissertation not only strengthens the theoretical backbone of gravitational EFTs but also bridges a crucial gap between classical black hole mechanics and modern theoretical extensions of general relativity.