# Talks

If you can’t explain it to a six year old, you don’t understand it yourself.

A list of scientific talks I have given over the years on my various projects.

Contrary to the above quote, these are not for six year olds; they only have the slides and nothing of what I said during the talks.

## Supertranslation charge & flux on null surfaces

## Canonical energy & linear stability of Schwarzschild

Consider linearised perturbations of the Schwarzschild black hole in 4 dimensions. Using the linearised Newman-Penrose curvature component, which satisfies the Teukolsky equation, as a Hertz potential we generate a ‘new’ metric perturbation satisfying the linearised Einstein equation. We show that the canonical energy, given by Hollands and Wald, of the ‘new’ metric perturbation is the conserved Regge-Wheeler-like energy used by Dafermos, Holzegel and Rodnianski to prove linear stability and decay of perturbations of Schwarzschild. We comment on a generalisation of this strategy to prove the linear stability of the Kerr black hole.

## The first law of black hole mechanics for fields with internal gauge freedom

We derive the first law of black hole mechanics for physical theories based on a local, covariant and gauge-invariant Lagrangian where the dynamical fields transform non-trivially under the action of some internal gauge transformations. The theories of interest include General Relativity formulated in terms of tetrads, Einstein-Yang-Mills theory and Einstein-Dirac theory. Since the dynamical fields of these theories have some internal gauge freedom, we argue that there is no natural group action of diffeomorphisms of spacetime on such dynamical fields. In general, such fields cannot even be represented as smooth, globally well-defined tensor fields on spacetime. Consequently the derivation of the first law by Iyer and Wald cannot be used directly. Nevertheless, we show how such theories can be formulated on a principal bundle and that there is a natural action of automorphisms of the bundle on the fields. These bundle automorphisms encode both spacetime diffeomorphisms and internal gauge transformations. Using this reformulation we define the Noether charge associated to an infinitesimal automorphism and the corresponding notion of stationarity and axisymmetry of the dynamical fields. We first show that we can define certain potentials and charges at the horizon of a black hole so that the potentials are constant on the bifurcate Killing horizon, giving a generalised zeroth law for bifurcate Killing horizons. We further identify the gravitational potential and perturbed charge as the temperature and perturbed entropy of the black hole which gives an explicit formula for the perturbed entropy analogous to the Wald entropy formula. We then obtain a general first law of black hole mechanics for such theories. The first law relates the perturbed Hamiltonians at spatial infinity and the horizon, and the horizon contributions take the form of a “potential times perturbed charge” term. We also comment on the ambiguities in defining a prescription for the total entropy for black holes.

## First Law for fields with Internal Gauge

We extend the analysis of Iyer and Wald to derive the First Law of blackhole mechanics in the presence of fields charged under an ‘internal gauge group’. We treat diffeomorphisms and gauge transformations in a unified way by formulating the theory on a principal bundle. The first law then relates the energy and angular momentum at infinity to a potential times charge term at the horizon. The gravitational potential and charge give a notion of temperature and entropy respectively.

## First Law for fields with Internal Gauge

## Growth rate of Black Hole instabilities

## Growth rate of Black Hole instabilities

Hollands and Wald showed that dynamic stability of stationary axisymmetric black holes is equivalent to positivity of canonical energy on a space of linearised axisymmetric perturbations satisfying certain boundary and gauge conditions. Using a reflection isometry of the background, we split the energy into kinetic and potential parts. We show that the kinetic energy is positive. In the case that potential energy is negative, we show existence of exponentially growing perturbations and further obtain a variational formula for the growth rate.

## Growth rate of Black Hole instabilities

## Positive energy & Black Hole stability

Hollands and Wald showed that dynamic stability of stationary axisymmetric black holes is equivalent to positivity of canonical energy on a space of linearised axisymmetric perturbations satisfying certain boundary and gauge conditions. We show that the “kinetic energy” — the energy of the perturbations that are odd under reflection in \( t \) and \(\phi \) — is positive. We discuss implications of having a positive kinetic energy for proving exponential growth in the case where the “potential energy” can be made negative.

## Gauge, energy & Black Hole stability

## Gauge conditions & Black Hole stability

Hollands and Wald showed that dynamic stability of a black hole is equivalent to the positivity of canonical energy on a space of linearised perturbations satisfying certain boundary conditions and gauge conditions. The boundary/gauge conditions are naturally formulated on the space of initial data for the perturbations in terms of orthogonality to gauge transformations. These perturbations can be uniquely specified in terms of transverse-traceless tensors. Using these transverse-traceless data, positivity of kinetic energy for perturbations can be proven.

## Gauge conditions & Black Hole stability

Hollands and Wald showed that dynamic stability of a black hole is equivalent to the positivity of canonical energy on a space of linearised perturbations satisfying certain boundary conditions and gauge conditions. The boundary/gauge conditions are naturally formulated on the space of initial data for the perturbations in terms of orthogonality to gauge transformations. These perturbations can be uniquely specified in terms of transverse-traceless tensors. Using these transverse-traceless data, positivity of kinetic energy for perturbations can be proven.