A non-smooth theory for sub-Lorentzian geometry

Descripción del proyecto

Sentar las bases de la geometría sublorentziana

La geometría sublorentziana es el análogo no holonómico de la geometría lorentziana y la contraparte lorentziana de la geometría subriemanniana. A pesar de los avances en geometría lorentziana no suave y de la consolidación de la geometría subriemanniana, este campo sigue estando poco explorado, aunque resulta relevante para teorías de dimensiones superiores, la gravedad cuántica y la gravedad análoga. En el proyecto SUBLOR, financiado por las acciones Marie Skłodowska-Curie, se integrarán estructuras sublorentzianas en espacios de longitud lorentzianos y espacios-tiempos métricos, y se analizarán sus propiedades causales y las geodésicas mediante el principio del máximo de Pontryagin. Asimismo, se caracterizará la geometría infinitesimal mediante tangentes métricas temporales y un teorema lorentziano de bola-caja, y se desarrollará una teoría de la curvatura basada en el transporte óptimo y la curvatura hamiltoniana. SUBLOR contribuirá a sistematizar este ámbito de investigación y a esclarecer su papel en los espacios-tiempo con restricciones y en la física fundamental.

Objetivo

"The project SUBLOR aims to establish the foundations of sub-Lorentzian geometry, the non-holonomic analogue of Lorentzian geometry, within the new synthetic theory of Lorentzian length spaces and metric spacetimes. While recent developments have extended Lorentzian geometry to the non-smooth setting, spacetimes with non-holonomic constraints have been overlooked, despite their central role in modern geometry. Sub-Riemannian geometry has flourished into a thriving field of geometric analysis, whereas its Lorentzian counterpart has seen little development. SUBLOR will address this gap by:

(1) Integrating sub-Lorentzian structures into the framework of Lorentzian length spaces and metric spacetimes. Sub-Lorentzian geometry will be defined in the general control-theoretic setting, its causal structure and topologies analysed, and Pontryagin's maximum principle applied to the study of geodesics, showing that sub-Lorentzian manifolds are Lorentzian length spaces.

(2) Characterising the infinitesimal geometry of sub-Lorentzian manifolds. The timelike metric tangents will be shown to be quotients of sub-Lorentzian Carnot groups, and a Lorentzian version of the Ball-Box theorem (a ""Diamond-Box"" theorem) will be established.

(3) Developing a theory of curvature for sub-Lorentzian geometry. The optimal-transport-based timelike curvature-dimension conditions and their variants will be investigated, and the more intrinsic Hamiltonian curvature will be introduced through Jacobian fields and expansions of the time-separation function.

By building these three foundational layers, which also underpin the success of sub-Riemannian geometry, SUBLOR will systematise sub-Lorentzian geometry, highlight a novel class of spacetimes, and open the way for future applications in higher-dimensional models, quantum gravity, and analogue gravity."