# Algebra of Implicitly Defined Constraints for Gravity as the General Form of Embedding Theory

## TL;DR

This paper analyzes the canonical structure of the embedding theory of gravity, treating 4D space-time as a surface in higher-dimensional flat space, and demonstrates that its constraints form a closed algebra of first-class constraints.

## Contribution

It provides a detailed canonical analysis of the general embedding theory of gravity without artificially imposed constraints, showing the algebra of constraints is closed and first-class.

## Key findings

- The algebra of four constraints is closed.
- All constraints are first-class.
- Explicit form of the constraint algebra is obtained.

## Abstract

We consider the embedding theory, the approach to gravity proposed by Regge and Teitelboim, in which 4D space-time is treated as a surface in high-dimensional flat ambient space. In its general form, which does not contain artificially imposed constraints, this theory can be viewed as an extension of GR. In the present paper we study the canonical description of the embedding theory in this general form. In this case, one of the natural constraints cannot be written explicitly, in contrast to the case where additional Einsteinian constraints are imposed. Nevertheless, it is possible to calculate all Poisson brackets with this constraint. We prove that the algebra of four emerging constraints is closed, i.e., all of them are first-class constraints. The explicit form of this algebra is also obtained.