Canonical formulation of embedding gravity in a form of General Relativity with dark matter

TL;DR

This paper reformulates embedding gravity as a version of General Relativity with extra degrees of freedom that can account for dark matter, analyzing its canonical structure and constraints.

Contribution

It provides a canonical formalism for embedding gravity with dark matter-like contributions, clarifying the constraint structure and linking to previous formulations.

Findings

Hamiltonian reduces to a linear combination of first class constraints.

Six pairs of second class constraints remain to be addressed.

The formulation connects embedding gravity to known canonical theories.

Abstract

We study embedding gravity, a modified theory of gravity, in which our space-time is assumed to be a four-dimensional surface in flat ten-dimensional space. Based on a simple geometric idea, this theory can be reformulated as General Relativity with additional degrees of freedom and contribution to action, which can be interpreted as describing dark matter. We study the canonical formalism for such a formulation of embedding gravity. After solving simple constraints, the Hamiltonian is reduced to a linear combination of four first class constraints with Lagrange multipliers. There still remain six pairs of second class constraints. Possible ways of taking these constraints into account are discussed. We show that one way of solving the constraints leads to the canonical system going into the previously known canonical formulation of the complete embedding theory with an implicitly defined constraint.