Hidden symmetries from distortions of the conformal structure

TL;DR

This paper explores how mass and symmetry breaking in spacetime geometries lead to distortions of conformal symmetries, revealing new conservation laws and hidden symmetries in various metrics including pp-waves and de Sitter space.

Contribution

It introduces the concept of symmetry distortions as generators of disformal transformations, linking them to higher order hidden symmetries and extending the understanding of conserved quantities for massive geodesics.

Findings

Distortions of conformal Killing vectors generate conserved quantities for massive geodesics.

Disformal transformations are connected to higher order hidden symmetries.

Symmetry breaking via Lorentz-violating parameters leads to modified generators and conservation laws.

Abstract

It is well established that the mass parameter breaks the conformal symmetries in the case of geodesic motion. The proper conformal Killing vectors cease to generate conserved charges when non-null geodesics are considered. We examine how the introduction of the mass is actually related to the appearance of appropriate distortions in the conformal sector, which lead to new conservation laws. As a prominent example we use a general pp-wave metric, which exploits this property to the maximum. We study the necessary geometric conditions, so that such types of distortions are applicable. We show that the relative vectors are generators of disformal transformations and prove their connection to higher order (hidden) symmetries. Except from the pp-wave geometry, we also provide an additional example in the form of the de Sitter metric. Again, the proper conformal Killing vectors can be appropriately distorted to generate conserved quantities for massive geodesics. Subsequently, we proceed by introducing an additional symmetry breaking effect. The latter is realized by considering a Bogoslovsky type of line-element, which involves a Lorentz violating parameter. We utilize once more the pp-wave case as a guide to study how the broken symmetries - this time also related to Killing vectors - are substituted by distortions of the original generators. We further analyze and discuss the necessary geometric conditions that lead to the emergence of these distortions.