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

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.

arXiv comment25 pages

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Cited by in corpus (9)

• The $3+1$ Formalism in the Geometric Trinity of Gravity
• Cosmology of Quadratic Metric-Affine Gravity
• The Full Quadratic Metric-Affine Gravity (Including Parity Odd Terms): Exact solutions for the Affine-Connection
• FLRW Cosmology in Metric-Affine $F(R,Q)$ Gravity
• Myrzakulov $F(T,Q)$ gravity: cosmological implications and constraints
• Quadratic Metric-Affine Gravity: Solving for the Affine-Connection
• Riemann Tensor and Gauss-Bonnet density in Metric-Affine Cosmology
• (3+1)-Formulation for Gravity with Torsion and Non-Metricity II: The Hypermomentum Equation
• Metric-Affine Version of Myrzakulov $F(R,T,Q,\mathcal{T})$ Gravity and Cosmological Applications