# Mathematical Relativity Seminar

## Planned sessions

### 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 $L^2(\Sigma)$.

### 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.

### 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).

### 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.

Current organizers: José Natário, João Lopes Costa