(3 + 1)-formulation for gravity with torsion and non-metricity: the stress–energy–momentum equation. (5th July 2021)

• Record Type: Journal Article
• Authors: Ariwahjoedi, Seramika; Suroso, Agus; Zen, Freddy P
• Is Part Of: Classical and quantum gravity. Volume 38:Number 15(2021)
• Publication Date: 2021-07-05
• Subjects: 3 + 1 decomposition -- ADM formulation -- metric-affine gravity -- Gauss–Codazzi equation -- torsion -- non-metricity -- Einstein field equation; Quantum gravity -- Periodicals; Gravitation -- Periodicals; Relativity (Physics) -- Periodicals; Space and time -- Periodicals; Periodicals; 521.1
• DOI: 10.1088/1361-6382/abf27a
• Languages: English
• ISSNs: 0264-9381

Abstract

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.