A scale invariant, Weyl geometric, Lagrangian approach to cosmology is explored, with a a scalar field φ of (scale) weight −1 as a crucial ingredient besides classical matter (Tann 1998, Drechsler 1999). For a particularly simple class of Weyl geometric models (called Einstein-Weyl universes) the Klein-Gordon equation for φ is explicitly solvable. In this case the energy-stress tensor of the sc...

In this paper, we prove that a compact quasi-Einstein manifold $(M^n,\,g,\,u)$ of dimension $n\geq 4$ with boundary $\partial M,$ nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, the standard hemisphere $\Bbb {S}^n_+,$ or $g=dt^{2}+\psi ^{2}(t)g_{L}$ $u=u(t),$ where $g_{L}$ Einstein Ricci curvature. A similar classification result obtained by assum...

Current generalizations of the classical Einstein–Hilbert Lagrangian formulation of General Relativity are reviewed. Some alternative variational principles, based on different choices of the gravitational field variable (metric tensor, affine connection, or both) are known to reproduce – more or less directly – Einstein’s gravitational equations, and should therefore be regarded as equivalent ...

We consider a modified gravity framework for inflation by adding to the Einstein-Hilbert action direct $f(\phi)T$ term, where $\phi$ is identified as inflaton and $T$ trace of energy-momentum tensor. The goes Einstein naturally when decays out. investigate dynamics in this (not be confused with torsion-scalar coupled theories) on general basis then apply it three well-motivated inflationary mod...

Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity is based on the Einstein-Langevin equation, which has in addition sources due to the noise kernel. The noise kernel is the vacuum expectation value of the (operatorvalued) stress-energy bi-tensor...

We study gravitational aspects of Brane-World scenarios. We show that the bulk Einstein equations together with the junction condition imply that the induced metric on the brane satisfies the full non-linear Einstein equations with a specific effective stress energy tensor. This result holds for any value of the bulk cosmological constant. The analysis is done by either placing the brane close ...

In this paper after introducing a model of binary data matrix for physical measurements of an evolving system (of particles), we develop a Hilbert space as an ambient space to derive induced metric tensor on embedded parametric manifold identified by associated joint probabilities of particles observables (parameters). Parameter manifold assumed as space-like hypersurface evolving along time ax...

The aim of this paper is to introduce and justify a possible generalization the classic Bach field equations on four-dimensional smooth manifold [Formula: see text] in presence text], given by map with source target another Riemannian manifold. Those are characterized vanishing two times covariant, symmetric, traceless conformally invariant tensor field, called text]-Bach tensor, that absence r...

We prove that given a solution of the Einstein equations gab for the matter field Tab, an autoparallel null vector field l a and a solution (lalc, Tac) of the linearized Einstein equation on the given background, the Kerr-Schild metric gac + λlalc (λ arbitrary constant) is an exact solution of the Einstein equation for the energy-momentum tensor Tac + λTac + λl(aTc)bl . The mixed form of the Ei...

It is shown that the Cotton tensor can describe effects of gravity beyond general relativity. Any solution Einstein equations with or without cosmological constant satisfies field described by tensor. implies an integration constant. A vacuum a theory represented vanishing tensor, rather than Ricci An exact Schwarzschild-like for static and spherically symmetric source discovered. Although invo...