Rafael María Rubio

In this paper we study the geometric structure of Galilean spacetimes that possess a certain type of infinitesimal symmetries. Specifically, we assume the existence of certain vector fields whose flow generates conformal transformations of the Leibnizian structure, particularly highlighting torseforming, torque, and concircular vector fields. Such fields enable a local splitting of spacetime as a Galilean product, which, under certain additional hypotheses, can be extended to a global one. Finally, we delve deeper into the analysis of the multiplicity of irrotational conformally Leibnizian fields of observers with the same gravitational field.

In this note, we present a short and simple proof of both the Bernstein theorem and the Calabi–Bernstein theorem, which allows us to visualize both the common features and the remarkable differences between the two results. The proofs are based on the construction of conformal metrics that ensure the parabolicity of the surfaces. Consequently, they do not rely on complex analysis.

The notion of Unchanged Direction (UD) motion in an arbitrary Galilean spacetime is introduced providing a mathematical framework for the intuitive notion that “proper acceleration does not change its direction”. Given a prescribed scalar acceleration, the corresponding initial value problem is analyzed and used to characterize the UD trajectories as the projected curves on the spacetime of the integral curves of a smooth vector field defined on a certain fiber bundle. This key fact allows us to prove a result on the completeness of inextensible UD motions, that can be physically interpreted saying that observers which obey a UD motion live forever. Concretely, it is shown that a UD observer in a globally synchronizable Galilean spacetime can be extended if there exists a complete spatially conformally Leibnizian field of observers with conformal factor and gravitational field bounded along finite times. Finally, another completeness result is presented in the case of the existence of a compact absolute spacelike leaf.

We introduce semi-Riemannian structures well-adapted to certain fields of observers in a Galilean spacetime. The Levi-Civita connection of such a semi-Riemannian metric will allow us to obtain variational characterizations of the Galilean geodesics as well as global results on the topological and differentiable structure of the spacetime. Moreover, these new semi-Riemannian metrics provide a new way to compare the studied Newton–Cartan models with their relativistic counterparts.

In this work, we consider geometrical models describing spinning particles in the context of generalized Robertson–Walker spacetimes. We study spacelike and timelike trajectories for massive and massless particles governed by two different Lagrangians: one depending on the torsion and the other on the curvature of the trajectory. For both, we are able to relate the curvature and torsion of such trajectories with the curvature of the spatial fiber of the cosmological model. Moreover, several conserved quantities are described as well as their possible relations with physical parameters of the particle.

Obstructions to the existence of trapped submanifolds in spacetimes of arbitrary dimension are given. These obstructions are obtained under natural geometric assumptions, which can be applied to initial data set for Einstein equations, assuring the absence of trapped submanifolds in its development. We highlight that for several of our results the existence of symmetries in the spacetime is not necessary.

The aim of this manuscript is to obtain rigidity and non-existence results for parabolic spacelike submanifolds with causal mean curvature vector field in orthogonally splitted spacetimes, and in particular, in globally hyperbolic spacetimes. We also obtain results regarding the geometry of submanifolds by ensuring, under some mild hypothesis, the non-existence of local minima or maxima of certain distinguished function. Furthermore, in this last case the submanifold does not need to be parabolic or even complete. As an application in General Relativity, we obtain several nice results regarding (non-necessarily closed) trapped surfaces in a huge family of spacetimes. In fact, we show how our technique allows us to recover some relevant previous results for trapped surfaces in both, standard static spacetimes and generalized Robertson–Walker spacetimes.

We analyze the concept of uniformly accelerated observer in Galilean spacetimes in the context of Newton–Cartan theory and find natural geometric assumptions to ensure that an inextensible uniformly accelerated observer in a Galilean spacetime does not disappear in a finite proper time.

In this work, we study spacelike trajectories of a relativistic particle model whose Lagrangian is given by the total torsion of its worldline. Once the field equations are established, we investigate the trajectories in two physically relevant families of four-dimensional spacetimes: generalized Robertson–Walker and standard static spacetimes. For the former we obtain both characterization as well as non-existence results for Frenet curves of different orders, which are the curves capable of modeling trajectories. For standard static spacetimes, we obtain a full characterization of the critical curves and present an example in which we can explicitly compute non-trivial trajectories.