Jose Torrente Teruel

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.