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On Twisted Spacetimes: a new class of Galilean cosmological models

Daniel de la Fuente, Rafael M. Rubio, Jose Torrente

TL;DR

The paper develops a nonrelativistic geometric framework for twisted cosmological models by introducing Galilean Twisted (GT) spacetimes as Newton–Cartan analogues of relativistic twisted spacetimes. GT spacetimes are defined on $M=I\times F$ with $\Omega=\mathrm{d}t$ and a warped metric $g=f^2\pi_F^* h$, together with a symmetric Galilean connection $\nabla$ satisfying $\nabla_{\partial_t}\partial_t=0$ and $\mathrm{Rot}(\partial_t)=0$, enabling a detailed local and global analysis. The work establishes a local GT structure from timelike torqued vector fields, and proves global splitting results: a simply connected Galilean spacetime globally decomposes as GT if and only if there exists a torqued vector field whose associated observer field is a uniform global generator; geodesic completeness results are obtained under bounds on the twist factor $f$, with strong physical interpretation of geodesics and relative velocities in this nonrelativistic setting. Together, these results provide a robust nonrelativistic cosmological model framework with explicit criteria for completeness and global decomposition, useful for Newton–Cartan cosmology and related nonrelativistic holographic contexts.

Abstract

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 general geodesic completeness.

On Twisted Spacetimes: a new class of Galilean cosmological models

TL;DR

The paper develops a nonrelativistic geometric framework for twisted cosmological models by introducing Galilean Twisted (GT) spacetimes as Newton–Cartan analogues of relativistic twisted spacetimes. GT spacetimes are defined on with and a warped metric , together with a symmetric Galilean connection satisfying and , enabling a detailed local and global analysis. The work establishes a local GT structure from timelike torqued vector fields, and proves global splitting results: a simply connected Galilean spacetime globally decomposes as GT if and only if there exists a torqued vector field whose associated observer field is a uniform global generator; geodesic completeness results are obtained under bounds on the twist factor , with strong physical interpretation of geodesics and relative velocities in this nonrelativistic setting. Together, these results provide a robust nonrelativistic cosmological model framework with explicit criteria for completeness and global decomposition, useful for Newton–Cartan cosmology and related nonrelativistic holographic contexts.

Abstract

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 general geodesic completeness.
Paper Structure (9 sections, 14 theorems, 86 equations, 1 figure)

This paper contains 9 sections, 14 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Premilinaries
  3. Spatially conformally Leibnizian spacetimes
  4. Spacelike differential map and spacelike gradient
  5. Galilean Twisted Spacetimes
  6. Geodesic Completeness of GT spacetimes
  7. Physical interpretation of geodesics
  8. Galilean Torqued Vector Fields
  9. Global splitting

Key Result

Proposition 2.2

Let $(M,\Omega,g,\nabla)$ be a Galilean spacetime with symmetric connection $\nabla$. Then, a vector field $K$ satisfies equation (wdefi) if and only if the function $\Omega(K)$ is spatially invariant, i.e., $V(\Omega(K))=0$, $\forall V\in\Gamma(\mathrm{An}(\Omega))$.

Figures (1)

  • Figure :

Theorems & Definitions (36)

  • Definition 2.1
  • Proposition 2.2
  • Definition 3.1
  • Example 3.2
  • Remark 4.1
  • Lemma 4.2
  • proof
  • Remark 4.3
  • Lemma 4.4
  • Theorem 4.5
  • ...and 26 more