On the cosmological solutions in Weyl geometry

We investigated the possibility of construction the homogeneous and isotropic cosmological solutions in Weyl geometry. We derived the self-consistency condition which ensures the conformal invariance of the complete set of equations of motion. There is the special gauge in choosing the conformal factor when the Weyl vector equals zero. In this gauge we found new vacuum cosmological solutions absent in General Relativity. Also, we found new solution in Weyl geometry for the radiation dominated universe with the cosmological term, corresponding to the constant curvature scalar in our special gauge. Possible relation of our results to the understanding both dark matter and dark energy is discussed.

Perturbed thermodynamics and thermodynamic geometry of a static black hole in f (R) gravity

In this paper, we consider a static black hole in [Formula: see text] gravity. We recapitulate the expression for corrected thermodynamic entropy of this black hole due to small fluctuations around equilibrium. Also, we study the geometrothermodynamics (GTD) of this black hole and investigate the adaptability of the curvature scalar of geothermodynamic methods with phase transition points of this black hole. Moreover, we study the effect of correction parameter on thermodynamic behavior of this black hole. We observe that the singular point of the curvature scalar of Ruppeiner metric coincides completely with zero point of the heat capacity and the deviation occurs with increasing correction parameter.

On the geometry of the tangent bundle with gradient Sasaki metric

PurposeLet (M, g) be a n-dimensional smooth Riemannian manifold. In the present paper, the authors introduce a new class of natural metrics denoted by gf and called gradient Sasaki metric on the tangent bundle TM. The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature, scalar and sectional curvatures.Design/methodology/approachIn this paper the authors introduce a new class of natural metrics called gradient Sasaki metric on tangent bundle.FindingsThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.Originality/valueThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.

The point free Colombeau geometry in singular general relativity

The problem statement. We argue that the canonical interpretation of the Schwarzschild spacetime in contemporary general relativity is wrong and that revision is needed. And we argue that the Schwarzschild solution is impossible to treat classically, since the Levi-Cività connection is not available for the whole Schwarzschild spacetime (Sch,gijSch (t r, , ,θϕ)) ; where Sch=×(({r ≥ 2m} {∪ ≤ ≤0 r 2m})×S2) ; but it can only be treated by using an embedding of the classical Schwarzschild metric tensor gijSch; ,i j =1,2,3,4 into Colombeau algebra δ(4,Σ),Σ= ={r 2m} {∪ =r 0} supergeneralized functions. The classical Schwarzschild spacetime could be extended up to the distributional semi-Riemannian manifold endowed on the tangent bundle with the Colombeau distributional metric tensor. The aim. The development of new physical interpretation for the distributional curvature scalar (Rε( )r )ε and square scalar (Rεμν( )r Rμνε, ( )r )ε, (Rερσμν( )r Rρδμνε, ( )r )εis aimed. Results. The Schwarzschild solution using Colombeau distributional geometry without leaving Schwarzschild coordinates (t r, , ,θϕ) is studied. We obtain that the distributional Ricci tensor and the curvature scalar are δ-type, (R rε( ))ε=−m rδ( − 2m) ,>0 . The practical value. As distributional square scalars are essentially nonclassical Colombeau type distributions: (Rεμν( )r Rμνε, ( )r )ε, (Rερσμν( )r Rρσμνε, ( )r )ε∈(3 )\ ′(3 ), this provides a new physical interpretation for the distributional curvature scalar (R rε( ))ε and square scalars (Rεμν( )r Rμνε, ( )r )ε, (Rερσμν( )r Rρδμνε, ( )r )ε.

Thermodynamic Curvature of the Binary van der Waals Fluid

The thermodynamic Ricci curvature scalar R has been applied in a number of contexts, mostly for systems characterized by 2D thermodynamic geometries. Calculations of R in thermodynamic geometries of dimension three or greater have been very few, especially in the fluid regime. In this paper, we calculate R for two examples involving binary fluid mixtures: a binary mixture of a van der Waals (vdW) fluid with only repulsive interactions, and a binary vdW mixture with attractive interactions added. In both of these examples, we evaluate R for full 3D thermodynamic geometries. Our finding is that basic physical patterns found for R in the pure fluid are reproduced to a large extent for the binary fluid.

Derivative couplings in gravitational production in the early universe

Gravitational particle production in the early universe is due to the coupling of matter fields to curvature. This coupling may include derivative terms that modify the kinetic term. The most general first order action contains derivative couplings to the curvature scalar and to the traceless Ricci tensor, which can be dominant in the case of (pseudo-)Nambu-Goldstone bosons or disformal scalars, such as branons. In the presence of these derivative couplings, the density of produced particles for the adiabatic regime in the de Sitter phase (which mimics inflation) is constant in time and decays with the inverse effective mass (which in turn depends on the coupling to the curvature scalar). In the reheating phase following inflation, the presence of derivative couplings to the background curvature modifies in a nontrivial way the gravitational production even in the perturbative regime. We also show that the two couplings — to the curvature scalar and to the traceless Ricci tensor — are drastically different, specially for large masses. In this regime, the production becomes highly sensitive to the former coupling while it becomes independent of the latter.

General properties of f(R) gravity vacuum solutions

General properties of vacuum solutions of [Formula: see text] gravity are obtained by the condition that the divergence of the Weyl tensor is zero and [Formula: see text]. Specifically, a theorem states that the gradient of the curvature scalar, [Formula: see text], is an eigenvector of the Ricci tensor and, if it is timelike, the spacetime is a Generalized Friedman–Robertson–Walker metric; in dimension four, it is Friedman–Robertson–Walker.

Higher Dimensional Static and Spherically Symmetric Solutions in Extended Gauss–Bonnet Gravity

We study a theory of gravity of the form f ( G ) where G is the Gauss–Bonnet topological invariant without considering the standard Einstein–Hilbert term as common in the literature, in arbitrary ( d + 1 ) dimensions. The approach is motivated by the fact that, in particular conditions, the Ricci curvature scalar can be easily recovered and then a pure f ( G ) gravity can be considered a further generalization of General Relativity like f ( R ) gravity. Searching for Noether symmetries, we specify the functional forms invariant under point transformations in a static and spherically symmetric spacetime and, with the help of these symmetries, we find exact solutions showing that Gauss–Bonnet gravity is significant without assuming the Ricci scalar in the action.