(3+1)-Formulation for Gravity with Torsion and Non-Metricity II: The Hypermomentum Equation

TL;DR

This paper performs a (3+1) decomposition of the hypermomentum equation in Metric-Affine General Relativity, revealing how hypermomentum relates to hypersurface variables and recovering standard GR conditions in special cases.

Contribution

It introduces a detailed (3+1) decomposition of the hypermomentum equation in MAGR, expressing hypermomentum in terms of hypersurface variables and analyzing special cases.

Findings

Decomposition of hypermomentum into hypersurface variables.

Main set of 10 hypersurface equations derived.

Recovery of Levi-Civita connection in special cases.

Abstract

In this article, we consider a special case of Metric-Affine f(R)-gravity for f(R) = R, i.e. the Metric-Affine General Relativity (MAGR). As a companion to the first article in the series, we perform the (3+1) decomposition to the hypermomentum equation, obtained from the minimization of the MAGR action S [g, {\omega}] with respect to the connection {\omega}. Moreover, we show that the hypermomentum tensor H could be constructed completely from 10 hypersurfaces variables that arise from its dilation, shear, and rotational (spin) parts. The (3+1) hypermomentum equations consists of 1 scalar, 3 vector, 3 matrix, and 1 tensor equation of order-(2,1). Together with the (3+1) decomposition of the traceless torsion constraint, consisting of 1 scalar and 1 vector equation, we obtain 10 hypersurface equations, which are the main result in this article. Finally, we consider some special cases of MAGR, namely, the zero hypermomentum, metric, and torsionless cases. For vanishing hypermomentum, we could retrieve the metric compatibility and torsionless condition in the (3+1) framework, hence forcing the affine connection to be Levi-Civita as in the standard General Relativity.