Hidden symmetries from distortions of the conformal structure

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.

Circularly polarized periodic gravitational wave and the Pais-Uhlenbeck oscillator

“…It is straightforward to calculate the algebra of the charges D and K with the Carroll generators (4.8). Finally, we note that the form of (4.13) is similar to the distorted symmetries given in [45,46] and deserves a further analysis.…”

Schrödinger Symmetry in Gravitational Mini-Superspaces

“…If the gravitational system can be mapped to a free particle, the two copies will commute, as it is the case for FRW cosmology. But in the case of the Schwarzschild mechanics, where the system is mapped onto a particle with a constant potential, deformations of the charges prevent the two copies to commute with each other, paralleling the results presented in [65]. Whether these symmetries can be useful to the quantization of these systems remain to be shown.…”

“…As we have seen, a consequence of adding these corrective terms is that, in general, the two copies of Witt charges one can derive do not commute anymore. See [65] for a related discussion.…”