Causal symmetries

TL;DR

This paper introduces causal symmetries as transformations preserving the causal structure of Lorentzian manifolds, characterizes their infinitesimal generators, and discusses potential applications in gravitation theory.

Contribution

It defines causal symmetries, explores their algebraic structure, and provides conditions for their infinitesimal generators, extending the understanding of causal transformations in Lorentzian geometry.

Findings

Causal symmetries form a submonoid containing conformal transformations.

Necessary and sufficient conditions for infinitesimal generators are established.

Potential applications to gravitation theory are discussed.

Abstract

Based on the recent work \cite{PII} we put forward 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 which contains as its maximal subgroup the set of conformal transformations. We find the necessary and sufficient conditions for a vector field $\xiv$ to be the infinitesimal generator of a one-parameter submonoid of pure causal symmetries. We speculate about possible applications to gravitation theory by means of some relevant examples.