Ostrogradsky–Hamilton approach to geodetic brane gravity

Abstract

In this paper, we develop the Ostrogradsky–Hamilton formalism for geodetic brane gravity, described by the Regge–Teitelboim geometric model in higher codimension. We treat this gravity theory as a second-order derivative theory, based on the extrinsic geometric structure of the model. As opposed to previous treatments of geodetic brane gravity, our Lagrangian is linearly dependent on second-order time derivatives of the field variables, the embedding functions. The difference resides in a boundary term in the action, usually discarded. Certainly, this suggests applying an appropriate Ostrogradsky–Hamiltonian approach to this type of theories. The price to pay for this choice is the appearance of second-class constraints. We determine the full set of phase space constraints, as well as the gauge transformations they generate in the reduced phase space. Additionally, we compute the algebra of constraints and explain its physical content. In the same spirit, we deduce the counting of the physical degrees of freedom. We comment briefly on the naive formal canonical quantization emerging from our development.

Lower-dimensional Regge-Teitelboim gravity

“…It possesses non-Einsteinian solutions as it belongs to the same class as mimetic gravity (both theories appearing after a differential field transformation in GR) [3], but has a clear geometric interpretation. Unfortunately, the field equations of the theory (Regge-Teitelboim equations, RT) turn out to be much harder to analyze than Einstein ones [4], and only a few particular solutions has been found (for recent developments, see [5][6][7] and references therein). For that reason in this paper we want to investigate another regime of this theory: we sacrifice the first criteria (applicability to our universe) in favor of the second one (ability to find a solution) and consider a lower-dimensional version of RT gravity.…”