Numerical construction of initial data sets in cosmology

One of the cornerstones of cosmology is that the universe is homogeneous and isotropic. This principle, commonly known as the Copernican principle has lead over the recent decades to the mathematical construction of models that describe the dynamics of our universe. However, due to the existence of experimental evidence, and recent advances in several numerical simulations of sophisticated cosmological models, the community has started to seek more robust and less symmetric universes. Therefore, more general cosmological models than the one of Friedman-Robert-Walker are becoming interesting from both the physical and mathematical points of view.

By model, we refer to a set of equations that define, in certain sense, an initial value problem. In the theory of general relativity, such formulation leads to two sets of partial differential equations that must be simultaneously solved; namely, the evolution and the constraint equations. Because of the non-linear nature of both sets of equations, it is well known that their numerical treatment is, in general, a complex task.

In this project, we are focused on the numerical approach to find initial data sets that allow to describe early stages of universes that are neither homogeneous nor isotropic.

Collaborators: Dr. Cesar Valuenzuela

Students: Alejandro Estrada Llesta (Postgraduate student).

This project is supported by COLCIENCIAS.

Numerical computation of the Bartnitk mass

The notion of quasilocal mass in general relativity remains unclear until today. The main reason for this lies in that the energy carried by the gravitational field is not accounted by the energy-momentum tensor. Thus, defining an appropriate and clear concept of energy in a local region of the spacetime, and consequently, a notion of local mass, is, in general, a convoluted task. This is what is commonly known in the scientific community as the quasi-local mass problem.

Many proposals for a notion of quasi-local mass in general relativity have been put forth in the last decades. Among the most remarkable are Hawking’s mass, Penrose’s mass, Geroch’s mass, and Bartnik’s mass. The latter, in particular, has received special attention from the scientific community during the last years due to the fact that it is based on the well-established concept of the ADM-mass, which is a positive and invariant quantity of the spacetime. In general words, the concept of Bartnik’s mass consists in defining the mass of a local spatial region of the spacetime with 2-dimensional surface boundary, as the infimum of the set formed by all the ADM-masses obtained from all of the possible asymptotically flat three-dimensional manifolds that isometrically contain the region enclosed by the surface boundary. Years later, it was proved that an infimum can only be attained if the minimizing metric corresponds to a static spacetime with some suitable boundary conditions on the inner 2-dimensional surface boundary. Thus, these conditions are colled inner boundary conditions of the asymptotically flat three-dimensional manifold. With the motivation from above, in this project, we are focused on numerically construct asymptotically flat static metrics given some inner boundary conditions and hence, come up with an infrastructure that computes the Bartnitk mass for any closed region of the spacetime.