Lie - point and variational symmetries in minisuperspace Einstein's gravity

TL;DR

This paper applies symmetry theory to minisuperspace Einstein gravity, revealing that including the lapse as a degree of freedom and allowing symmetries to produce constraints maximizes symmetry generators, linking variational symmetries to conformal Killing fields.

Contribution

It extends the standard symmetry analysis by incorporating lapse variables and constraints, showing that variational symmetries correspond to conformal Killing fields in minisuperspace models.

Findings

Maximum symmetry generators include lapse as a degree of freedom.

Variational symmetries coincide with conformal Killing fields.

Symmetry generators relate to Killing and homothetic fields in scaled metrics.

Abstract

We consider the application of the theory of symmetries of coupled ordinary differential equations to the case of reparametrisation invariant Lagrangians quadratic in the velocities; such Lagrangians encompass all minisuperspace models. We find that, in order to acquire the maximum number of symmetry generators, one must (a) consider the lapse $N(t)$ among the degrees of freedom and (b) allow the action of the generator on the Lagrangian and/or the equations of motion to produce a multiple of the constraint, rather than strictly zero. The result of this necessary modification of the standard theory (concerning regular systems) is that the Lie - point symmetries of the equations of motion are exactly the variational symmetries (containing the time reparametrisation symmetry) plus the well known scaling symmetry. These variational symmetries are seen to be the simultaneous conformal Killing fields of both the metric and the potential, thus coinciding with the conditional symmetries defined in phase space. In a parametrisation of the lapse for which the potential becomes constant, the generators of the aforementioned symmetries become the Killing fields of the scaled supermetric and the homothetic field respectively.