Isometry germs and related structures

TL;DR

This paper introduces the concept of isometry germs at space-time events, defining their algebraic structure and exploring their relation to local frames and generalized Lorentz transformations.

Contribution

It defines isometry germs and their algebraic structure, extending the Lorentz group to a local, deformed algebra related to space-time symmetries.

Findings

The quotient space of isometry germs forms a bracket algebra.

Each isometry germ defines a local stationary frame.

The structure generalizes the Lorentz group with a deformation related to de Sitter algebra.

Abstract

We define an Isometry germ at any given event $x$ of space-time as a vector field $\xi$ defined in a neighborhood of $x$ such that the Lie derivative of both the metric and the Riemannian connection are zero at this event. Two isometry germs can be said to be equivalent if their values and the values of their first derivatives coincide at $x$. The corresponding quotient space can be endowed with a structure of a bracket algebra which is a deformation of de Sitter's Lie algebra. Each isometry germ defines also a local stationary frame of reference, the consideration of the family of adapted coordinate transformations between any two of them leading to a local novel structure that generalizes the Lorentz group.