Causal symmetries

TL;DR

This paper introduces causal symmetries as transformations on Lorentzian manifolds that preserve the causal future direction, exploring their properties, structure, and infinitesimal generators.

Contribution

It defines causal symmetries, analyzes their mathematical structure, and provides conditions for their infinitesimal generators, advancing the understanding of causal transformations in Lorentzian geometry.

Findings

Causal symmetries form a submonoid of transformations.

Necessary conditions are established for vector fields generating causal symmetries.

Examples illustrate the properties and applications of causal symmetries.

Abstract

We define a new type of transformation for Lorentzian manifolds characterized by mapping every causal future-directed vector onto a causal future-directed vector. The set of all such transformations, which we call causal symmetries, has the structure of a submonoid. Some of their properties are investigated and we give necessary conditions for a vector field $\xiv$ to be the infinitesimal generator of a one-parameter group of causal symmetries. Some examples are discussed.