# Mathematical Relativity Seminar

## Past sessions

### 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.

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

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.

Paper by Felix Finster

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

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.

Notes for this talk

### Boundedness of energy for the Wake Klein-Gordon model

We consider the global-in-time existence theory for the Wave-Klein-Gordon model for the Einstein-Klein-Gordon equations introduced by LeFloch and Ma. By using the hyperboloidal foliation method, we prove that a hierarchy of weighted energies of the solutions remain (essentially) bounded for all times.

### Perturbations of the asymptotic region of the Schwarzschild-de Sitter spacetime

Although the study of the Cauchy problem in General Relativity started in the decade of 1950 with the work of Foures-Bruhat, addressing the problem of global non-linear stability of solutions to the Einstein field equations is in general a hard problem. The first non-linear global non-linear stability result in General Relativity was obtained for the de Sitter spacetime by H. Friedrich in the decade of 1980. In this talk the main tool used in the above result is introduced: a conformal (regular) representation of the Einstein field equations — the so-called conformal Einstein field equations (CEFE). Then, the conformal structure of the Schwarzschild-de Sitter spacetime is analysed using the extended conformal Einstein field equations (XCEFE). To this end, initial data for an asymptotic initial value problem for the Schwarzschild-de Sitter spacetime are obtained. This initial data allow to understand the singular behaviour of the conformal structure at the asymptotic points where the horizons of the Schwarzschild-de Sitter spacetime meet the conformal boundary. Using the insights gained from the analysis of the Schwarzschild-de Sitter spacetime in a conformal Gaussian gauge, we consider nonlinear perturbations close to the Schwarzschild-de Sitter spacetime in the asymptotic region. Finally, we'll show that small enough perturbations of asymptotic initial data for the Schwarzschild-de Sitter spacetime give rise to a solution to the Einstein field equations which exists to the future and has an asymptotic structure similar to that of the Schwarzschild-de Sitter spacetime.

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

### Test fields cannot destroy extremal black holes

We prove that (possibly charged) test fields satisfying the null energy condition at the event horizon cannot overspin/overcharge extremal Kerr-Newman or Kerr-Newman-anti de Sitter black holes, that is, the weak cosmic censorship conjecture cannot be violated in the test field approximation. The argument relies on black hole thermodynamics (without assuming cosmic censorship), and does not depend on the precise nature of the fields. We also discuss generalizations of this result to other extremal black holes.

This seminar is joint with CENTRA.

### The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics

The classical singularity theorems of General Relativity show that a Lorentzian manifold with a $C^2$-metric that satisfies physically reasonable conditions cannot be geodesically complete. One of the questions left unanswered by the classical singularity theorems is whether one could extend such a spacetime with a lower regularity Lorentzian metric such that the extension still satisfies these physically reasonable conditions and does no longer contain any incomplete causal geodesics. In other words, the question is if a lower differentiability of the metric is sufficient for the theorems to hold. The natural differentiability class to consider here is $C^{1,1}$. This regularity corresponds, via the field equations, to a finite jump in the matter variables, a situation that is not a priori regarded as singular from the viewpoint of physics and it is the minimal condition that ensures unique solvability of the geodesic equations. Recent progress in low-regularity Lorentzian geometry has allowed one to tackle this question and show that, in fact, the classical singularity theorems of Penrose, Hawking, and Hawking-Penrose remain valid for $C^{1,1}$-metrics. In this talk I will focus on the Hawking-Penrose theorem, being the most recent and in a sense most general of the aforementioned results, and some of the methods from low regularity causality and comparison geometry that were employed in its proof. This is joint work with J. D. E. Grant, M. Kunzinger and R. Steinbauer.

### Wave equations with initial data on compact Cauchy horizons

I will present a new energy estimate for wave equations close to compact non-degenerate Cauchy horizons. The estimate allows one to conclude several existence and uniqueness results for wave equations with initial data on the Cauchy horizon. This generalizes classical results that were proven under the assumption of either analyticity or symmetry of the spacetime or closedness of the generators. In particular, the results are useful in understanding the geometry of vacuum spacetimes with a compact non-degenerate Cauchy horizon (without making any extra assumptions). This problem is closely related to the strong cosmic censorship conjecture.

### On the Cosmic No-Hair Conjecture in $\mathbb{T}^2$-symmetric non-linear scalar field spacetimes

At late times, cosmological spacetimes solving Einstein's field equations, at least when assuming a positive cosmological constant, are conjectured to isotropise and appear like the de Sitter spacetime to late time observers. This is the statement of the Cosmic No-Hair conjecture. In this talk, I consider Einstein's non-linear scalar field equations and spacetimes with $\mathbb{T}^2$-symmetry. I present results on future global existence of such solutions and discuss the conjecture in the setting of a constant potential.

This talk is based on arXiv:1712.01801.

### 07/03/2018, 15:00 — 16:00 — Room P3.10, Mathematics BuildingMasashi Kimura, Instituto Superior Técnico

We introduce a novel type of ladder operators, which map a solution to the massive Klein-Gordon equation into another solution with a different mass. It is shown that such operators are constructed from closed conformal Killing vector fields in arbitrary dimensions if the vector fields are eigenvectors of the Ricci tensor. As an example, we explicitly construct the ladder operators in AdS spacetime. It is shown that the ladder operators exist for masses above the Breitenlohner-Freedman bound. We also discuss their applications, ladder operator for spherical harmonics, the relation between supersymmetric quantum mechanics, and some phenomenon around extremal black holes whose near horizon geometry is AdS2.

### Cosmic no-hair in spherically symmetric black hole spacetimes

We analyze in detail the geometry and dynamics of the cosmological region arising in spherically symmetric black hole solutions of the Einstein-Maxwell-scalar field system with a positive cosmological constant. More precisely, we solve, for such a system, a characteristic initial value problem with data emulating a dynamic cosmological horizon. Our assumptions are fairly weak, in that we only assume that the data approaches that of a subextremal Reissner-Nordström-de Sitter black hole, without imposing any rate of decay. We then show that the radius (of symmetry) blows up along any null ray parallel to the cosmological horizon ("near" $i^+$), in such a way that $r=+\infty$ is, in an appropriate sense, a spacelike hypersurface. We also prove a version of the Cosmic No-Hair Conjecture by showing that in the past of any causal curve reaching infinity both the metric and the Riemann curvature tensor asymptote those of a de Sitter spacetime. Finally, we discuss conditions under which all the previous results can be globalized.

### Free-evolution formulations of GR for numerical relativity

In this talk I will give an overview of the formulations of GR used in numerical relativity. I will summarize what is known about their mathematical properties and explain how local well-posedness of the IVP is achieved. Subsequently I will discuss a dual-foliation formulation of the field equations. The new formalism allows a larger class of coordinates to be employed in applications. These include choices popular in mathematical relativity.

### 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.

### Aspects of cosmological perturbation theory

Cosmological perturbation theory has been around for over 70 years and is the underlying theory for the interpretation of observations that have resulted in several Nobel prizes. Is there really anything new one can say about this field from a mathematical physics perspective? In this talk, which consists of two parts, I will try to convince you that the answer is yes. The first part deals with second order perturbations, which is a field that gives rise to notoriously messy equations. However, I will show that by using underlying physically motivated mathematical structures significant simplifications can be achieved, which give rise to new conserved quantities and simple explicit solutions in the so-called long wavelength limit, and for the currently dominating cosmological paradigm, the $\Lambda$CDM models. The second part is about a new research program where first order cosmological perturbation equations are reformulated as dynamical systems, which allows one to use dynamical systems methods and approximations, complementing previous investigations. Throughout the talk I will focus on ideas rather than technical details.

### The Question of Essential Metric Regularity at General Relativistic Shock Waves

It is an open question whether shock wave solutions of the Einstein Euler equations contain "regularity singularities'', i.e., points where the spacetime metric would be Lipschitz continuous ($C^{0,1}$), but no smoother, in any coordinate system. In 1966, Israel showed that a metric $C^{0,1}$ across a single shock surface can be smoothed to the $C^{1,1}$ regularity sufficient for spacetime to be non-singular and for locally inertial frames to exist. In 2015, B. Temple and I gave the first (and only) extension of Israel's result to shock wave interactions in spherical symmetry by a new constructive proof involving non-local PDE's. In 2016, to address most general shock wave solutions (generated by Glimm's random choice method), we introduced the "Riemann flat condition" on $L^\infty$ connections and proved our condition necessary and sufficient for the essential metric regularity to be smooth (i.e. $C^{1,1}$). In our work in progress, we took the Riemann flat condition to derive an elliptic system which determines the essential metric regularity at shock waves (and beyond). Our preliminary results suggest that our elliptic system is well-posed and we believe this system to provide a systematic way for resolving the problem of regularity singularities completely.

Current organizers: José Natário, João Lopes Costa