The Hamiltonian constraint in the symmetric teleparallel equivalent of general relativity

TL;DR

This work analyzes the Hamiltonian structure of the symmetric teleparallel equivalent of general relativity (STEGR), focusing on how boundary terms influence the canonical formulation without changing dynamics. The authors perform a 3+1 (ADM) decomposition of the STEGR Lagrangian, derive the Hamiltonian and momentum constraints, and obtain Hamilton's equations, highlighting how a different boundary-term choice modifies the constraint equations and gauge evolution. A concrete spherical-symmetry example demonstrates explicit differences between STEGR and GR in the Hamiltonian constraint, with STEGR yielding a simpler radial dependence and introducing first-order lapse derivatives. The results point to potential numerical-relativity advantages of STEGR and motivate further investigations into hyperbolicity, covariant extensions, and applications to nonlinear gravity in a computational setting.

Abstract

General relativity (GR) admits two alternative formulations with the same dynamics attributing the gravitational phenomena to torsion or nonmetricity of the manifold's connection. They lead, respectively, to the teleparallel equivalent of general relativity (TEGR) and the symmetric teleparallel equivalent of general relativity (STEGR). In this work, we focus on STEGR and present its differences with the conventional, curvature-based GR. We exhibit the 3+1 decomposition of the STEGR Lagrangian in the coincident gauge and present the Hamiltonian, the Hamiltonian and momenta constraints, and Hamilton's equations. For a particular case of spherical symmetry, we explicitly show the differences in the Hamiltonian and the Hamiltonian constraint between GR and STEGR. We finally discuss the implications that these differences, which represent genuine different features between the two formulations of gravity, might encompass to numerical relativity.