(3+1)-Formulation for Gravity with Torsion and Non-Metricity: The Stress-Energy-Momentum Equation

TL;DR

This paper develops a (3+1) formulation of gravity incorporating torsion and non-metricity, deriving generalized equations and applying them to Metric-Affine gravity, revealing dynamical properties of energy, momentum, and stress-energy.

Contribution

It introduces a (3+1) framework for gravity with torsion and non-metricity, deriving generalized Gauss-Codazzi-Mainardi equations and formulating the Einstein Field Equations in Metric-Affine gravity.

Findings

Derived generalized Gauss-Codazzi-Mainardi equations for affine connections with torsion and non-metricity.

Showed that additional variables on hypersurfaces become dynamical in Metric-Affine gravity.

Recovered standard GR constraints and dynamics in the Levi-Civita limit.

Abstract

We derive the generalized Gauss-Codazzi-Mainardi (GCM) equation for a general affine connection with torsion and non-metricity. Moreover, we show that the metric compatibility and torsionless condition of a connection on a manifold are inherited to the connection of its hypersurface. As a physical application to these results, we derive the (3+1)-Einstein Field Equation (EFE) for a special case of Metric-Affine f(R)-gravity when f(R)=R, the Metric-Affine General Relativity (MAGR). Motivated by the concept of geometrodynamics, we introduce additional variables on the hypersurface as a consequence of non-vanishing torsion and non-metricity. With these additional variables, we show that for MAGR, the energy, momentum, and the stress-energy part of the EFE are dynamical, i.e., all of them contain the derivative of a quantity with respect to the time coordinate. For the Levi-Civita connection, one could recover the Hamiltonian and the momentum (diffeomorphism) constraint, and obtain the standard dynamics of GR.