# Mathematical Relativity Seminar

## Past sessions

### Elastic shocks in relativistic rigid rods and balls

We study the free boundary problem for the "hard phase" material introduced by Christodoulou, both for rods in $(1+1)$-dimensional Minkowski spacetime and for spherically symmetric balls in $(3+1)$-dimensional Minkowski spacetime. Unlike Christodoulou, we do not consider a "soft phase", and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks, defined as hypersurfaces where the material's density, pressure and velocity are discontinuous, must be null hypersurfaces. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if the pressure or its time derivative do not vanish at the free boundary initially. These shocks interact with the free boundary, causing it to lose regularity.

### The Friedrich-Butscher method for the construction of initial data in General Relativity

The construction of initial data for the Cauchy problem in General Relativity is an interesting problem from both the mathematical and physical points of view. As such, there have been numerous methods studied in the literature the "Conformal Method" of Lichnerowicz-Choquet-Bruhat-York and the "gluing" method of Corvino-Schoen being perhaps the best-explored. In this talk I will describe an alternative, perturbative, approach proposed by A. Butscher and H. Friedrich, and show how it can be used to construct non-linear perturbations of initial data for spatially-closed analogues of the $k = -1$ FLRW spacetime. Time permitting, I will discuss possible renements/extensions of the method, along with its generalisation to the full Conformal Constraint Equations of H. Friedrich.

This seminar is joint with CENTRA, and will take place on the Physics Department (seminar room, 2nd floor).

### Introduction to the Theory of Shock Waves (VI)

I plan to cover the following topics: Euler equations; Burger's equation; $p$-system; symmetric hyperbolic PDEs; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

### Bibliography

• Joel Smoller, Shock waves and Reaction Diffusion Equations.
• Constantine Dafermos, Hyperbolic Conservation Laws in Continuum Physics.
• Alexandre Chorin and Jerrold Marsden, A Mathematical Introduction to Fluid Mechanics.
• Lecture notes of Blake Temple.

Notes for this talk

Planned duration: 6 x 1.5 hours.

### Multi-body spherically symmetric steady states of Newtonian self-gravitating elastic matter

We study the problem of static, spherically symmetric, self-gravitating elastic matter distributions in Newtonian gravity. To this purpose we first introduce a new definition of homogeneous, spherically symmetric (hyper)elastic body in Euler coordinates, i.e., in terms of matter fields defined on the current physical state of the body. We show that our definition is equivalent to the classical one existing in the literature and which is given in Lagrangian coordinates, i.e., in terms of the deformation of the body from a given reference state. After a number of well-known examples of constitutive functions of elastic bodies are re-defined in our new formulation, a detailed study of the Seth model is presented. For this type of material the existence of single and multi-body solutions is established.

### Solutions of the wave equation bounded at the Big Bang

By solving a singular initial value problem, we prove the existence of solutions of the wave equation $\Box_g\phi=0$ which are bounded at the Big Bang in the Friedmann-Lemaitre-Robertson-Walker cosmological models. More precisely, we show that given any function $A \in H^3(\Sigma)$ (where $\Sigma=\mathbb{R}^n$, $\mathbb{S}^n$ or $\mathbb{H}^n$ models the spatial hypersurfaces) there exists a unique solution $\phi$ of the wave equation converging to $A$ in $H^1(\Sigma)$ at the Big Bang, and whose time derivative is suitably controlled in $L^2(\Sigma)$.

### Construction of anti de Sitter-like spacetimes using the metric conformal field equations

In this talk I with describe how to make use of the metric version of the conformal Einstein field equations to construct anti-de Sitter-like spacetimes by means of a suitably posed initial-boundary value problem. The evolution system associated to this initial-boundary value problem consists of a set of conformal wave equations for a number of conformal fields and the conformal metric. This formulation makes use of generalised wave coordinates and allows the free specification of the Ricci scalar of the conformal metric via a conformal gauge source function. I will consider Dirichlet boundary conditions for the evolution equations at the conformal boundary and show that these boundary conditions can, in turn, be constructed from the 3-dimensional Lorentzian metric of the conformal boundary and a linear combination of the incoming and outgoing radiation as measured by certain components of the Weyl tensor. To show that a solution to the conformal evolution equations implies a solution to the Einstein field equations we also provide a discussion of the propagation of the constraints for this initial-boundary value problem. The existence of local solutions to the initial-boundary value problem in a neighbourhood of the corner where the initial hypersurface and the conformal boundary intersect is subject to compatibility conditions between the initial and boundary data. The construction described is amenable to numerical implementation and should allow the systematic exploration of boundary conditions. I will also discuss extensions of this analysis to the case of the Einstein equations coupled with various tracefree matter models. This is work in collaboration with Diego Carranza.

This seminar is joint with CENTRA, and will take place on the Physics Department (seminar room, 2nd floor).

### Quantum effects near the inner horizon of a black hole

The analytically extended Kerr and Reissner-Nordström metrics, describing respectively spinning or spherical charged black holes (BHs), reveal a traversable passage through an inner horizon (IH) to another external universe. Consider a traveler intending to access this other universe. What will she encounter along the way? Is her mission doomed to fail? Does this other external universe actually exist?

Answering these questions requires one to understand the manner in which quantum fields influence the internal geometry of BHs. In particular, this would include the computation of the renormalized stress-energy tensor (RSET) on BH backgrounds - primarily near the IH. Although a theoretical framework for such a computation does exist, this has been a serious challenge for decades (partially due to its inevitable numerical implementation). However, the recently developed pragmatic mode-sum regularization method has made the RSET computation more accessible.

In this talk, we will first consider the computation of the simpler quantity $\langle\phi^2\rangle_{ren}$, for a minimally-coupled massless scalar field inside a (4d) Reissner-Nordström BH. We shall then proceed with the long sought-after RSET, focusing on the computation of the semi-classical fluxes near the IH. Our novel results for the latter will be presented, with a closer look at the extremal limit. Finally - we will discuss possible implications to the fate of our traveler.

This seminar is joint with CENTRA.

### Flat FLRW and Kasner Big Bang singularities analyzed on the level of scalar waves

We consider the wave equation, $\square_g\psi=0$, in fixed flat Friedmann-Lemaitre-Robertson-Walker and Kasner spacetimes with topology $\mathbb{R}_+\times\mathbb{T}^3$. We obtain generic blow up results for solutions to the wave equation towards the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to solutions that blow up in an open set of the Big Bang hypersurface $\{t=0\}$. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate $L^2$-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the $L^2(\mathbb{T}^3)$ norms of the solutions blow up towards the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

### Introduction to the Theory of Shock Waves (V)

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

### Bibliography

• Joel Smoller, Shock waves and Reaction Diffusion Equations.
• Constantine Dafermos, Hyperbolic Conservation Laws in Continuum Physics.
• Alexandre Chorin and Jerrold Marsden, A Mathematical Introduction to Fluid Mechanics.
• Lecture notes of Blake Temple.

Notes for this talk

Planned duration: 6 x 1.5 hours.

### Introduction to the Theory of Shock Waves (IV)

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

### Bibliography

• Joel Smoller, Shock waves and Reaction Diffusion Equations.
• Constantine Dafermos, Hyperbolic Conservation Laws in Continuum Physics.
• Alexandre Chorin and Jerrold Marsden, A Mathematical Introduction to Fluid Mechanics.
• Lecture notes of Blake Temple.

Notes for this talk

Planned duration: 6 x 1.5 hours.

### Introduction to the Theory of Shock Waves (III)

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

### Bibliography

• Joel Smoller, Shock waves and Reaction Diffusion Equations.
• Constantine Dafermos, Hyperbolic Conservation Laws in Continuum Physics.
• Alexandre Chorin and Jerrold Marsden, A Mathematical Introduction to Fluid Mechanics.
• Lecture notes of Blake Temple.

Notes for this talk

Planned duration: 6 x 1.5 hours.

### Introduction to the Theory of Shock Waves (II)

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

### Bibliography

• Joel Smoller, Shock waves and Reaction Diffusion Equations.
• Constantine Dafermos, Hyperbolic Conservation Laws in Continuum Physics.
• Alexandre Chorin and Jerrold Marsden, A Mathematical Introduction to Fluid Mechanics.
• Lecture notes of Blake Temple.

Notes for this talk

Planned duration: 6 x 1.5 hours.

### The nonlinear stability of Schwarzschild

I'll discuss a joint work with Sergiu Klainerman on the stability of Schwarzschild as a solution to the Einstein vacuum equations with initial data subject to a certain symmetry class.

### Introduction to the Theory of Shock Waves (I)

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

### Bibliography

• Joel Smoller, Shock waves and Reaction Diffusion Equations.
• Constantine Dafermos, Hyperbolic Conservation Laws in Continuum Physics.
• Alexandre Chorin and Jerrold Marsden, A Mathematical Introduction to Fluid Mechanics.
• Lecture notes of Blake Temple.

Notes for this talk

Planned duration: 6 x 1.5 hours.

Bibliography:

Joel Smoller, “Shock waves and Reaction Diffusion Equations”

Constantine Dafermos, “Hyperbolic Conservation Laws in Continuum Physics”

Alexandre Chorin and Jerrold Marsden, “A Mathematical Introduction to Fluid Mechanics”

Lecture notes of Blake Temple (on his webpage)

### Optimal metric regularity in General Relativity follows from the RT-equations by elliptic regularity theory

I am going to present Blake Temple's and my recent breakthrough regarding optimal metric regularity: We recently derived a set of nonlinear elliptic equations, with differential forms as unknowns, (the "Regularity Transformation equations" or "RT-equations"), and proved existence of solutions. The RT-equations determine whether optimal metric regularity can be achieved in General Relativity. Our existence result applies to connections in and Riemann curvature in $W^{m,p}$, $m\geq1$, $p>n$, and thus yields that such connections can always be smoothed to optimal regularity (one derivative above their curvature) by coordinate transformation. Extending this existence theory to the case of GR shock waves, when the connection is in $L^{\infty}$, is subject of our ongoing research. Our current existence result demonstrates that the method of determining optimal metric regularity by the RT-equations works.

### Stability and instability in spherical symmetry of Reissner-Nordström black holes for the Einstein-Maxwell-Klein-Gordon model

Penrose’s Strong Cosmic Censorship Conjecture is one of the central problems of Mathematical General Relativity. Its proof for the Einstein-Maxwell-Uncharged-Scalar-Field (EMSF) model in spherical symmetry relies on the formation of a Cauchy horizon that is $C^0$ regular but $C^2$ singular for generic Cauchy data. EMSF model however only admits two-ended black holes, unlike its charged analogue that allows for one-ended black holes, relevant to the study of charged gravitational collapse in spherical symmetry. In this talk, I will present my work about spherically symmetric charged and massive scalar fields on black holes. This includes a study of the black hole interior, that relates the behaviour of fields on the event horizon to the formation of a $C^0$ regular and $C^2$ singular Cauchy horizon. I will also mention my more recent work on the black hole exterior stability, for weakly charged massless scalar fields.

### On some stability and instability problems for hard stars in spherical symmetry

I will review Christodoulou's two phase model of relativistic fluids in the context of gravitational collapse, and describe the properties of static solutions to the hard phase with vacuum boundary (stars). We expect small stars to be (orbitally) stable, but limiting configurations with large central density to be unstable, and I will explain some of the underlying heuristics, and related results for these scenarios.

### Correspondence and Rigidity Results on Asymptotically Anti-de Sitter Spacetimes (II)

In theoretical physics, it is often conjectured that a correspondence exists between the gravitational dynamics of asymptotically Anti-de Sitter (aAdS) spacetimes and a conformal field theory of their boundaries. In the context of classical relativity, one can attempt to rigorously formulate such a correspondence statement as a unique continuation problem for PDEs: Is an aAdS solution of the Einstein equations uniquely determined by its data on its conformal boundary?

In these talks, we report on recent progress in this direction, and we highlight the connections between correspondence conjectures in physics, unique continuation theory for wave equations, and the geometry of aAdS spacetimes. We discuss recent unique continuation theorems for waves on aAdS spacetimes that form the key step toward correspondence results, as well as novel geometric obstructions to these results. As an application, we provide an answer to the following symmetry extension question: when can a symmetry on the conformal boundary be extended into the interior?

This is joint work with Gustav Holzegel (Imperial College London).

### Correspondence and Rigidity Results on Asymptotically Anti-de Sitter Spacetimes (I)

In theoretical physics, it is often conjectured that a correspondence exists between the gravitational dynamics of asymptotically Anti-de Sitter (aAdS) spacetimes and a conformal field theory of their boundaries. In the context of classical relativity, one can attempt to rigorously formulate such a correspondence statement as a unique continuation problem for PDEs: Is an aAdS solution of the Einstein equations uniquely determined by its data on its conformal boundary?

In these talks, we report on recent progress in this direction, and we highlight the connections between correspondence conjectures in physics, unique continuation theory for wave equations, and the geometry of aAdS spacetimes. We discuss recent unique continuation theorems for waves on aAdS spacetimes that form the key step toward correspondence results, as well as novel geometric obstructions to these results. As an application, we provide an answer to the following symmetry extension question: when can a symmetry on the conformal boundary be extended into the interior?

This is joint work with Gustav Holzegel (Imperial College London).

### The quantised Dirac field and the fermionic signature operator (III)

In this mini course I am going to introduce the Dirac equation (describing fermions) in Minkowski spacetime and explain how to extend the equation to curved spacetimes (Lecture 1). In Lecture 2, I will introduce the canonical quantisation of the Dirac field in Minkowski spacetime and describe the problem of time-dependent external fields in the canonical quantisation formalism. In Lecture 3, I will present a proposal of myself and Felix Finster addressing the problem of time-dependent external fields and spacetimes, based on the fermionic signature operator.

Older session pages: Previous 2 Oldest

Current organizers: José Natário, João Lopes Costa