Research

My main areas of research are the geometric foundations of gravity, modified theories of gravity, constrained Hamiltonian systems, and numerical relativity.

Geometric foundations of gravity

Gravity as understood by the theory of general relativity works under the assumption that matter interacts with the curvature of spacetime. However, this assumes that the underlying connection defining parallel transport is the Levi-Civita one. By allowing more general mathematical structures, we encounter the torsion and non-metricity of spacetime. These geometrical properties of a connection can be understood on how they affect vectors when they are parallel-transported or moving along a curve.

Curvature

It can be understood by its effect on a vector that has been parallel-transported along a closed curve, which is rotated by an amount given by the curvature R of spacetime.

Torsion

If we take two vectors and transport them along each other, the failure in the closure of the parallelogram is given by the torsion T.

Nonmetricity

In a spacetime with nonmetricity, the variation of the length of a vector that is being transported along a curve changes, and is given by the nonmetricity Q.

(Images were taken from Beltrán Jiménez, J.; Heisenberg, L.; Koivisto, T.S. The Geometrical Trinity of Gravity. Universe 2019, 5, 173. )

Modified theories of gravity

Curvature R, torsion T and non-metricity Q allow to define three different, but physically equivalent versions of general relativity. They are called the teleparallel equivalent of general relativity (TEGR) and the symmetric teleparallel equivalent of general relativity (STEGR). Both TEGR and STEGR actions are equivalent to the classic Einstein-Hilbert action for general relativity, since their Lagrangians differ from the Ricci scalar only by four-divergences. This means that both theories are equivalent to general relativity and they describe the same physical content.

Diagram was adapted from Universe 2019, 5, 173.

Modifications of the TEGR and STEGR actions give rise to f(T) and f(Q) gravity theories, that have been studied for several years, with interesting applications in cosmology and high energy physics. The theories are appealing since they have equations of motion of second order in their fundamental variables, but are prone to have theoretical problems related to extra degrees of freedom that are not well defined and depend on the background. There has been controversy in the literature and these issues have not been fully settled yet.

Constrained Hamiltonian systems

Most of physical theories admit a Hamiltonian formulation, which means to reformulate them from the Lagrangian formalism by means of a Legendre transform that maps fields and their velocities into coordinates and their momenta. This reformulation helps to identify constraints and separate spurious from true degrees of freedom, and reveals underlying structures and symmetries. This is performed through Dirac's procedure (also known as Dirac-Bergmann algorithm). A flow of this algorithm can be seen in the following figure, which has been adapted from my PhD thesis.

Numerical relativity

My current research focus lies on developing the mathematical framework of numerical relativity in the metric and symmetric teleparallel equivalents of general relativity. Numerical relativity has become utterly important since the discovery of gravitational waves, due to its necessity to interpret LIGO's measurements in the strong gravity regime. I'm proposing unconventional approaches to the mathematical formalism that can give new insights of how we understand the equations reigning the nonlinear regime and the numerics behind.

My PhD work

In my PhD I worked on the Hamiltonian structure of modified teleparallel theories of gravity. For this, we started developing an independent research of the Hamiltonian formulation of TEGR. We consider a second-order Lagrangian written explicitly in terms of the tetrad and co-tetrad field through a supermetric or constitutive tensor. This mathematical object allows to calculate the eigenvalues and eigenvectors of the matrix relating the canonical velocities and momenta. Then the Hamiltonian was obtained by the Moore-Penrose pseudoinverse method, allowing to calculate the constraint structure of the theory and the time consistency of them. Our work has been published in Phys.Rev.D.94, 104045 (2016). Due to its novelties and the simple structure of the Hamiltonian and the momenta, this procedure will allow the study of the constraint structure of more general gravity theories with teleparallel structure.

The aim of my PhD was to study the issue of the degrees of freedom of the so-called f(T) gravity, a class of modified gravities originated from a Born-Infeld teleparallel gravity model, proposed more than ten years ago (this article is in the top 100 most cited gr-qc articles of 2019). Modified teleparallel and f(T) gravity have become a very active field of research in the gr-qc community in the last years, since some models provide a theoretical framework for inflation and the accelerated expansion of the universe. Through a detailed Hamiltonian analysis, we showed that f(T) gravity has one extra degree of freedom compared with TEGR for Minkowski and FLRW backgrounds, published at Phys.Rev. D97, 104028 (2018). The number of degrees of freedom of the theory presents a bifurcation, and for more complex backgrounds the number has been argued to be five in total. In our particular case we find that one generator of local Lorentz transformation becomes second-class by pairing up with a second-class constraint coming from the introduction of a scalar field. The results of this analysis can be contrasted through the comparison between the teleparallel Jordan and Einstein frames of f(T) gravity, see Phys. Rev. D 98, 124037 (2018). We have also shown that the extra d.o.f. can not be described as a disformal scalar field, since this kind of transformations are unable to remove Lorentz violating terms, see Symmetry 2020, 12, 152. However, it has given signs of being related with the modification of pseudo-invariant systems, see Phys.Rev.D 101, 084017 (2020).

Modified teleparallel theories of gravity inherit a local Lorentz violation, since the TEGR Lagrangian is a Lorentz pseudo-invariant object. This fact has led to some covariant approaches that try to restore such local Lorentz symmetry; we summarize the status of this subject in Universe 5 (2019) 158, and provide discussion and criticism on covariant approaches.

I have also been involved in the study of solutions of general relativity in the context of f(T) gravity, the Kerr and McVittie geometries. In Eur.Phys.J. C75 (2015) 77 we introduced the null tetrad approach that allows to easily find tetrads that are solutions of f(T) equations of motion. We applied this method to the Kerr geometry. Then in Eur.Phys.J. C77 (2017) 825 we applied the same method to the McVittie geometry, and we were able to find new tetrads that reproduce FLRW cosmology but with a vanishing Weitzenböck scalar. Also, an important point in f(T) gravity is that Bianchi identities are satisfied, and using them we were able to prove that spherically symmetric solutions always exist for any functional form, see arXiv:2006.08507 [gr-qc].

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