# Taught Courses

## Covariant Physics and Black Hole Thermodynamics

Time: Fridays, 1pm to 2pm. Starts 17th September 2021.

Location: Lab 4 F15.

Lecturers: Dr. Josh Kirklin, Dr. Isha Kotecha and Dr. Fabio Mele

Description: General covariance is one of the fundamental principles underlying gravity. It says that the laws of physics do not depend on our choice of coordinates. Geometrically speaking, this means that we can apply any diffeomorphism to a physical system without changing its properties. In other words, diffeomorphisms are a kind of gauge symmetry. In this course, we will explore an elegant modern perspective on general covariance, using an approach known as the covariant phase space formalism. This formalism tells you how to treat covariant theories using classical Hamiltonian mechanics. Quantum gravity must be a quantisation of this classical theory. This means that we can learn a lot about some aspects of the quantum theory using the covariant phase space approach. For example, the laws of black hole thermodynamics are a reflection of the fact that black holes are quantum objects. We will show how key fundamental thermodynamical properties of the black hole, such as its entropy, can be understood using the covariant phase space. We will also discuss more general properties of black hole thermodynamics. Finally, we will explore the connection between gauge symmetry and quantum entanglement, and how this relates to the thermodynamics of spacetime itself.

### $$\mathrm{d}E = \frac{\kappa}{8\pi G}\mathrm{d}A + \Omega\,\mathrm{d}J+\Phi\,\mathrm{d}Q$$

Content:

1. Hamiltonian mechanics

2. Geometry of phase space, the symplectic form

3. Gauge symmetry and constraints in mechanics

4. Covariant field theories

5. Geometry of field space

6. Gauge symmetry and constraints in field theory

7. Global symmetries and large gauge symmetries

8. Conserved charges in general relativity

9. Black hole spacetimes and symmetries

10. Energy, angular momentum and electric charge

11. Black hole entropy as a Noether charge

12. The laws of black hole thermodynamics

13. Spacetime thermodynamics and the Einstein equations

14. Entanglement equilibrium and semiclassical Einstein equations

15. Entanglement and gauge symmetry

Notes:

See attachments below.

Recorded lectures: