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08/11/2019, 16:00 — 17:00 — Room P3.10, Mathematics Building

Gabriel Lopes Cardoso, *Instituto Superior Técnico*

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Weyl metrics and Wiener-Hopf factorization

We consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics in four space-time dimensions. We present, for the first time, a rigorous proof of the remarkable fact that the canonical Wiener-Hopf factorization of a matrix obtained from a general (possibly unbounded) monodromy matrix, with respect to an appropriately chosen contour, yields a solution to the non-linear gravitational field equations. This holds regardless of whether the dimensionally reduced metric in two dimensions has Minkowski or Euclidean signature. We show moreover that, by taking advantage of a certain degree of freedom in the choice of the contour, the same monodromy matrix generally yields various distinct solutions to the field equations. Our proof, which fills various gaps in the existing literature, is based on the solution of a second Riemann-Hilbert problem and highlights the deep role of the spectral curve, the normalization condition in the factorization and the choice of the contour. This approach allows for the explicit construction of solutions, including new ones, to the non-linear gravitational field equations, using simple complex analytic results. As an illustration, we show that by factorizing a simple rational diagonal matrix, we explicitly obtain a class of Weyl metrics which includes, in particular, the solution describing the interior region of the Schwarzschild black hole, a cosmological Kasner solution and the Rindler metric.

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21/10/2019, 11:00 — 12:00 — Room P3.10, Mathematics Building

Amol Sasane, *London School of Economics*

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Decay of solutions to the Klein-Gordon equation on some expanding cosmological spacetimes

The decay of solutions to the Klein-Gordon equation is studied in two expanding cosmological spacetimes, namely the de Sitter universe in flat Friedmann-Lemaître-Robertson-Walker (FLRW) form, and the cosmological region of the Reissner-Nordström-de Sitter (RNdS) model. Using energy methods, for initial data with finite higher order energies, decay rates for the solution are obtained. Also, a previously established decay rate of the time derivative of the solution to the wave equation, in an expanding de Sitter universe in flat FLRW form, is improved, proving Rendall's conjecture. A similar improvement is also given for the wave equation in the cosmological region of the RNdS spacetime.

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20/09/2019, 14:00 — 15:00 — Room P4.35, Mathematics Building

Yafet Sanchez Sanchez, *Leibniz University Hannover*

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Quantum Observables in low regularity spacetimes

In this talk I will describe how to construct the algebra of observables of a quantized scalar field when the spacetime metric is not smooth. I will show the main difference with the smooth case, the technical difficulties that arise and how we addressed them.

This is joint work with G. Hörmann, C. Spreitzer and J. Vickers.

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27/06/2019, 14:00 — 15:00 — Seminar room (2.8.3), Physics Building

Rodrigo Fontana, *Universidade Federal da Fronteira do Sul - UFFS Chapecó*

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Quasinormal modes of black holes: field propagation and stability

The propagation of probe fields around black hole geometries is an interesting tool for the investigation of two important aspects: the stability of these geometry and the quasinormal modes spectra. In general, for simple enough metrics (e. g. spherically symmetric), the field equation reduces to a wave-like form with a specific potential. This turns the problem of integration into a wave scattering problem with a potential barrier similar to that of a Schrödinger equation in quantum mechanics. With the proper boundary conditions the spectrum is that of damped waves, the quasinormal modes. These represents the characteristic response of black holes in general to linear perturbations and give information about the stability of the geometry. In this talk I will refer some examples of quasinormal modes and destabilization of geometries in black holes in which a non-minimally coupled scalar field act, in the context of Horndeski theory. By using the same tools of field propagation, I still present some of our previous results in relation to the violation of strong cosmic censorship in charged geometries.

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

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22/05/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Phillipo Lappicy, *Universidade de São Paulo*

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Space of initial data for self-similar Schwarzschild solutions

The Einstein constraint equations describe the space of initial data for the evolution equations, dictating how space should curve within spacetime. Under certain assumptions, the constraints reduce to a scalar quasilinear parabolic equation on the sphere with various singularities and nonlinearity being the prescribed scalar curvature of space. We focus on self-similar Schwarzschild solutions. Those describe, for example, the initial data for the interior of black holes. We construct the space of initial data for such solutions and show that the metric at the event horizon is constrained to the global attractors of such parabolic equations. Lastly, some properties of those attractors and its solutions are explored.

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15/05/2019, 15:30 — 16:30 — Room P4.35, Mathematics Building

Carlos Herdeiro, *Instituto Superior Técnico*

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Light ring stability in ultra-compact objects

We prove the following theorem: axisymmetric, stationary solutions of the Einstein field equations formed from classical gravitational collapse of matter obeying the null energy condition, that are everywhere smooth and ultracompact (i.e., they have a light ring) must have at least two light rings, and one of them is stable. It has been argued that stable light rings generally lead to nonlinear spacetime instabilities. Our result implies that smooth, physically and dynamically reasonable ultracompact objects are not viable as observational alternatives to black holes whenever these instabilities occur on astrophysically short time scales. The proof of the theorem has two parts: (i) We show that light rings always come in pairs, one being a saddle point and the other a local extremum of an effective potential. This result follows from a topological argument based on the Brouwer degree of a continuous map, with no assumptions on the spacetime dynamics, and hence it is applicable to any metric gravity theory where photons follow null geodesics. (ii) Assuming Einstein's equations, we show that the extremum is a local minimum of the potential (i.e., a stable light ring) if the energy-momentum tensor satisfies the null energy condition.

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20/03/2019, 15:00 — 16:00 — Room P5.18, Mathematics Building

José Natário, *Instituto Superior Técnico*

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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.

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14/03/2019, 14:30 — 15:30 — Seminar room (2.8.3), Physics Building

Jarrod Williams, *Queen Mary, University of London*

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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).

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08/03/2019, 10:00 — 12:00 — Room P3.10, Mathematics Building

Moritz Reintjes, *Instituto Superior Técnico*

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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.

#### See also

Notes for this talk

Planned duration: 6 x 1.5 hours.

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27/02/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Artur Alho, *Instituto Superior Técnico*

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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.

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22/02/2019, 11:30 — 12:30 — Room P3.10, Mathematics Building

Pedro Girão, *Instituto Superior Técnico*

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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 $L^2(\Sigma)$.

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13/02/2019, 11:00 — 12:00 — Seminar room (2.8.3), Physics Building

Juan Antonio Valiente Kroon, *Queen Mary, University of London*

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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).

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07/02/2019, 14:30 — 15:30 — Room P3.10, Mathematics Building

Noa Zilberman, *Technion*

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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.

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30/01/2019, 11:00 — 12:00 — Room P3.10, Mathematics Building

Anne Franzen, *Instituto Superior Técnico*

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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.

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10/12/2018, 14:00 — 15:30 — Room P3.10, Mathematics Building

Moritz Reintjes, *Instituto Superior Técnico*

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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.

#### See also

Notes for this talk

Planned duration: 6 x 1.5 hours.

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07/12/2018, 14:00 — 15:30 — Room P4.35, Mathematics Building

Moritz Reintjes, *Instituto Superior Técnico*

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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.

#### See also

Notes for this talk

Planned duration: 6 x 1.5 hours.

###
30/11/2018, 14:00 — 15:30 — Room P3.10, Mathematics Building

Moritz Reintjes, *Instituto Superior Técnico*

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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.

#### See also

Notes for this talk

Planned duration: 6 x 1.5 hours.

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16/11/2018, 14:00 — 15:30 — Room P3.10, Mathematics Building

Moritz Reintjes, *Instituto Superior Técnico*

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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.

#### See also

Notes for this talk

Planned duration: 6 x 1.5 hours.

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12/11/2018, 10:30 — 12:00 — Room P3.10, Mathematics Building

Jeremie Szeftel, *Laboratoire Jacques-Louis Lions de l'Université Pierre et Marie Curie*

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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.

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09/11/2018, 15:00 — 16:30 — Room P3.10, Mathematics Building

Moritz Reintjes, *Instituto Superior Técnico*

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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.

#### See also

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)