Extrinsic geometry and Hamiltonian analysis of symmetric teleparallel gravity

TL;DR

This paper explores the geometry and Hamiltonian structure of symmetric teleparallel gravity, revealing that it has the same degrees of freedom as General Relativity through a detailed analysis of non-metricity and boundary conditions.

Contribution

It derives generalized Gauss-Codazzi relations in non-metric geometry and constructs the Hamiltonian for symmetric teleparallel gravity, establishing its equivalence to General Relativity.

Findings

Derived generalized Gauss-Codazzi relations for non-metric foliations.

Identified the extrinsic symmetric tensor as the key geometric object.

Proved symmetric teleparallel gravity has the same degrees of freedom as GR.

Abstract

We analyze the properties of foliations in presence of non-metricity, deriving the generalized Gauss-Codazzi relations in full generality. These results are employed to study the teleparallel framework of non-metric geometry, obtaining constraints on the extrinsic and intrinsic tensors. In particular, an extrinsic symmetric two-tensor plays the role of the extrinsic curvature in Riemannian geometry, whereas no other geometric object can induce new dynamical degrees of freedom. Furthermore, we analyze the variational principle in presence of non-metricity, obtaining the boundary terms for the well-posed and well-defined Cauchy problem. Finally, we exploit the previous results to construct the Hamiltonian of the symmetric teleparallel equivalent of General Relativity, providing a proof that this theory shares the same number of degrees of freedom with its Riemannian counterpart.