Daniel de la Fuente

Within the generalized Newton-Cartan theory, Galilean Twisted spacetimes are introduced as dual models of the well-known relativistic twisted spacetimes. As a natural generalization, torqued vector fields in Galilean spacetimes are defined, showing that the local structure of a Galilean spacetime admitting a timelike torqued vector field is given by a Twisted spacetime. In addition, several results assuring the global splitting as Twisted spacetime are obtained. On the other hand, completeness of free falling observers is studied, as well as geodesic completeness.

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.

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.

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.