(3 + 1)-formulation for gravity with torsion and non-metricity: II. The hypermomentum equation. (26th October 2021) - Ariwahjoedi, Seramika; Suroso, Agus; Zen, Freddy P
Abstract: In this article, we consider a particular 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, ω with respect to the connection ω . Moreover, we show that the hypermomentum tensor H could be entirely constructed from ten hypersurfaces variables that arise from its dilation, shear, and rotational (spin) parts. The (3 + 1) hypermomentum equations consist of one scalar equation, three vector equations, three matrix equations, and one tensor equation of order- 2 1 . Together with the (3 + 1) decomposition of the traceless torsion constraint, which consists of a scalar and a vector equation, we obtain ten hypersurface equations that 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, forcing the affine connection to be Levi-Civita as in the standard general relativity.