The Einstein-Hilbert type action on metric-affine almost-product manifolds

TL;DR

This paper explores a generalized Einstein-Hilbert action on metric-affine manifolds with distributions, deriving variational formulas, Euler-Lagrange equations, and characterizing critical points, extending classical theories to more complex geometric settings.

Contribution

It introduces variational formulas and Euler-Lagrange equations for a mixed Einstein-Hilbert action on metric-affine manifolds with distributions, including new equations analogous to Einstein-Cartan theory.

Findings

Derived Euler-Lagrange equations for the action.

Characterized critical points on vacuum space-time.

Explicit form of Ricci-type tensor for semi-symmetric connections.

Abstract

We continue our study of the mixed Einstein-Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metric-affine space and use them to derive Euler-Lagrange equations (which in the case of space-time are analogous to those in Einstein-Cartan theory) and to characterize critical points of this action on vacuum space-time. Together with arbitrary variations of metric and connection, we consider also variations that partially preserve the metric, e.g., along the distribution, and also variations among distinguished classes of connections (e.g., statistical and metric compatible, and this is expressed in terms of restrictions on contorsion tensor). One of Euler-Lagrange equations of the mixed Einstein-Hilbert action is an analog of the Cartan spin connection equation, and the other can be presented in the form similar to the Einstein equation, with Ricci curvature replaced by the new Ricci type tensor. This tensor generally has a complicated form, but is given in the paper explicitly for variations among semi-symmetric~connections.