2020/07/24 by Vladimir Rovenski, Rovenski, Vladimir, Tomasz Zawadzki +1
Physics and Astronomy · Mathematics · #Cosmology and Gravitation Theories #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2007.12406
We continue our study of the mixed Einstein-Hilbert action as a functional of\na pseudo-Riemannian metric and a linear connection. Its geometrical part is the\ntotal mixed scalar curvature on a smooth manifold endowed with a distribution\nor a foliation. We develop variational formulas for quantities of extrinsic\ngeometry of a distribution on a metric-affine space and use them to derive\nEuler-Lagrange equations (which in the case of space-time are analogous to\nthose in Einstein-Cartan theory) and to characterize critical points of this\naction on vacuum space-time. Together with arbitrary variations of metric and\nconnection, we consider also variations that partially preserve the metric,≠.g., along the distribution, and also variations among distinguished classes\nof connections (e.g., statistical and metric compatible, and this is expressed\nin terms of restrictions on contorsion tensor). One of Euler-Lagrange equations\nof the mixed Einstein-Hilbert action is an analog of the Cartan spin connection\nequation, and the other can be presented in the form similar to the Einstein\nequation, with Ricci curvature replaced by the new Ricci type tensor. This\ntensor generally has a complicated form, but is given in the paper explicitly\nfor variations among semi-symmetric~connections.\n