(3 + 1)-formulation for gravity with torsion and non-metricity: II. The hypermomentum 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 Gravity (MAG). 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 MAG, 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.

Quadratic metric-affine gravity: solving for the affine-connection

“…We show that in the absence of matter the Theory always reduces to GR. Finally we generalize our result and find the form of the connection for a wider class of quadratic Theories.1 Some recent developments in MAG and applications include [8,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] . 0123456789().…”

“…1 Some recent developments in MAG and applications include [8,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] . 0123456789().…”

Cosmology of Metric‐Affine R+βR2$R + \beta R^2$ Gravity with Pure Shear Hypermomentum

“…As a result, torsion and non-metricity are typically involved in MAG. Moreover, couplings of matter to the general affine connection are expressed by means of the so-called hypermomentum tensor, [22][23][24] which describes dilation, spin, and shear, encompassing the microstructure of matter.…”

The full quadratic metric-affine gravity (including parity odd terms): exact solutions for the affine-connection

“…There are many reasons that make MAG an appealing gravity theory 1 but probably the most astonishing one is the link it provides in relation to the microstructure of matter [6,7]. In recent years, there is an ever-increasing interest in the MAG formulation [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] with a special emphasis given in cosmological applications [22,[25][26][27][28][29][30][31][32][33][34][35][36][37][38]. 2 Let us also mention that MAG has some promising features in regards to the Quantization of Gravity (see reference [40]).…”