Terms Of Curvature Invariants Research Articles

Semiclassical analysis of a nonlocal boundary value problem related to magnitude

AbstractWe study a Dirichlet boundary problem related to the fractional Laplacian in a manifold. Its variational formulation arises in the study of magnitude, an invariant of compact metric spaces given by the reciprocal of the ground state energy. Using recent techniques developed for pseudodifferential boundary problems we discuss the structure of the solution operator and resulting properties of the magnitude. In a semiclassical limit we obtain an asymptotic expansion of the magnitude in terms of curvature invariants of the manifold and the boundary, similar to the invariants arising in short-time expansions for heat kernels.

Quantum energy inequalities along stationary worldlines

Quantum energy inequalities (QEIs) are lower bounds on the averaged energy density of a quantum field. They have been proved for various field theories in general curved spacetimes but the explicit lower bound is not easily calculated in closed form. In this paper we study QEIs for the massless minimally coupled scalar field in four-dimensional Minkowski spacetime along stationary worldlines—curves whose velocity evolves under a 1-parameter Lorentz subgroup—and find closed expressions for the QEI bound, in terms of curvature invariants of the worldline. Our general results are illustrated by specific computations for the six prototypical stationary worldlines. When the averaging period is taken to infinity, the QEI bound is consistent with a constant energy density along the worldline. For inertial and uniformly linearly accelerated worldlines, this constant value is attained by the Minkowski and Rindler vacuums respectively. It is an open question as to whether the bounds for other stationary worldlines are attained by other states of interest.

Conflict between some higher-order curvature invariant terms

A viable quantum theory does not allow curvature invariant terms of different higher orders to be accommodated in the gravitational action. We show that there is indeed a conflict between the curvature squared and Gauss-Bonnet squared terms from the point of view of hermiticity. This means one should choose either, in addition to the Einstein-Hilbert term, but never the two together. We explore early cosmic evolution with Gauss-Bonnet squared term.

Isotropic exact solutions in F(R,Y,phi ) gravity via Noether symmetries

The present article investigates the existence of Noether and Noether gauge symmetries of flat Friedman–Robertson–Walker universe model with perfect fluid matter ingredients in a generalized scalar field formulation namely f(R,Y,phi ) gravity, where R is the Ricci scalar and Y denotes the curvature invariant term defined by Y=R_{alpha beta }R^{alpha beta }, while phi represents scalar field. For this purpose, we assume different general cases of generic f(R,Y,phi ) function and explore its possible forms along with field potential V(phi ) by taking constant and variable coupling function of scalar field omega (phi ). In each case, we find non-trivial symmetry generator and its related first integrals of motion (conserved quantities). It is seen that due to complexity of the resulting system of Lagrange dynamical equations, it is difficult to find exact cosmological solutions except for few simple cases. It is found that in each case, the existence of Noether symmetries leads to power law form of scalar field potential and different new types of generic function. For the acquired exact solutions, we discuss the cosmology generated by these solutions graphically and discuss their physical significance which favors the accelerated expanding eras of cosmic evolution.

Renormalized holographic subregion complexity under relevant perturbations

We construct renormalized holographic entanglement entropy (HEE) and subregion complexity (HSC) in the CV conjecture for asymptotically AdS4 and AdS5 geometries under relevant perturbations. Using the holographic renormalization method developed in the gauge/gravity duality, we obtain counter terms which are invariant under coordinate choices. We explicitly define different forms of renormalized HEE and HSC, according to conformal dimensions of relevant operators in the d = 3 and d = 4 dual field theories. We use a general embedding for arbitrary entangling subregions and showed that any choice of the coordinate system gives the same form of the counter terms, since they are written in terms of curvature invariants and scalar fields on the boundaries. We show an explicit example of our general procedure. Intriguingly, we find that a divergent term of the HSC in the asymptotically AdS5 geometry under relevant perturbations with operators of conformal dimensions in the range 0 < ∆ < frac{1}{2} and frac{7}{2} < ∆ < 4 cannot be cancelled out by adding any coordinate invariant counter term. This implies that the HSCs in these ranges of the conformal dimensions are not renormalizable covariantly. We also write the plot of the renormalization procedure in the case of asymptotically AdSd+1 geometries, with d > 4.

The gravitational field of the SdS space-time

The goal of this paper is to study SdS space-time and its gravitational field with consideration of the canonical form and invariants of the curvature tensor. The characteristic of [Formula: see text]-tensor identifies the type of gravitational field. Gaussian curvature quantities enunciated in terms of curvature invariants.

Geometrized quantum Galileons

We investigate the renormalization structure of the scalar Galileon model in flat spacetime by calculating the one-loop divergences in a closed geometric form. The geometric formulation is based on the definition of an effective Galileon metric and allows to apply known heat-kernel techniques. The result for the one-loop divergences is compactly expressed in terms of curvature invariants of the effective Galileon metric and corresponds to a resummation of the divergent one-loop contributions of all n-point functions. The divergent part of the one-loop effective action therefore serves as generating functional for arbitrary n-point counterterms. We discuss our result within the Galileon effective field theory and give a brief outlook on extensions to more general Galileon models in curved spacetime.

History of cosmic evolution with modified Gauss\u2013Bonnet-dilatonic coupled term

Gauss–Bonnet-dilatonic coupling in four dimensions plays an important role to explain late-time cosmic evolution. However, this term is an outcome of the low energy string effective action and thus ought to be important in the early universe too. Unfortunately, a phase-space formulation of such a theory does not exist in the literature due to branching. We therefore consider a modified theory of gravity, which contains a nonminimally coupled scalar–tensor sector in addition to a higher-order scalar curvature invariant term with Gauss–Bonnet-dilatonic coupling. Such an action unifies early inflation with late-time cosmic acceleration. The quantum version of the theory is also well behaved.

Special coordinate systems in pseudo-Finsler geometry and the equivalence principle

Special coordinate systems are constructed in a neighborhood of a point or of a curve. Taylor expansions can then be easily inferred for the metric, the connection, or the Finsler Lagrangian in terms of curvature invariants. These coordinates circumvent the difficulties of the normal and Fermi coordinates in Finsler geometry, which in general are not sufficiently differentiable. They are obtained applying the usual constructions to the pullback of a horizontally torsionless connection. The results so obtained are easily specialized to the Berwald or Chern-Rund connections and have application in the study of the equivalence principle in Finslerian extensions of general relativity.

Generalized second law of thermodynamics on the apparent horizon in modified Gauss–Bonnet gravity

Modified gravity (MG) and generalized second law (GSL) of thermodynamics are interesting topics in the modern cosmology. In this regard, we investigate the GSL of gravitational thermodynamics in the framework of modified Gauss–Bonnet (GB) gravity or [Formula: see text]-gravity. We consider a spatially FRW universe filled with the pressureless matter and radiation enclosed by the dynamical apparent horizon with the Hawking temperature. For two viable [Formula: see text] models, we first numerically solve the set of differential equations governing the dynamics of [Formula: see text]-gravity. Then, we obtain the evolutions of the Hubble parameter, the GB curvature invariant term, the density and equation of state (EoS) parameters as well as the deceleration parameter. In addition, we check the energy conditions for both models and finally examine the validity of the GSL. For the selected [Formula: see text] models, we conclude that both models have a stable de Sitter attractor. The EoS parameters behave quite similar to those of the [Formula: see text]CDM model in the radiation/matter dominated epochs, then they enter the phantom region before reaching the de Sitter attractor with [Formula: see text]. The deceleration parameter starts from the radiation/matter dominated eras, then transits from a cosmic deceleration to acceleration and finally approaches a de Sitter regime at late times, as expected. Furthermore, the GSL is respected for both models during the standard radiation/matter dominated epochs. Thereafter when the universe becomes accelerating, the GSL is violated in some ranges of scale factor. At late times, the evolution of the GSL predicts an adiabatic behavior for the accelerated expansion of the universe.

Can you hear the shape of dual geometries?

We compute the sub-leading terms in the Tian-Yau-Zelditch asymptotic expansion of the partition function for dual giant gravitons on AdS 5 × L 5 and provide a bulk interpretation in terms of curvature invariants. We accomplish this by relating the partition function of dual giant gravitons to the Hilbert series for mesonic operators in the CFT. The coefficients of the subleading terms encode integrated curvature invariants of L 5. In the same spirit of Martelli, Sparks and Yau, we are able to compute these integrated curvature invariants without explicit knowledge of the Sasaki-Einstein metric on L 5. These curvature invariants contribute to the 1/N 2 corrections of the difference of the 4D anomaly coefficients a and c recently found by Liu and Minasian, which we now have a purely field theoretic method of calculating.

Why Noether Symmetry of F(R) Theory Yields Three-Half Power Law?

Noether symmetry of F(R) theory of gravity in vacuum or in matter dominated era yields three-half power law of R. We show that this particular curvature invariant term is very special in the context of isotropic and homogeneous cosmological model as it makes the first fundamental form cyclic. As a result, it allows a unique power law solution, typical for this particular fourth order theory of gravity, both in the vacuum and in the matter dominated era. This power law solution has been found to be quite good to explain the early stage but not so special and useful to explain the late stage of cosmological evolution. The usefulness of Palatini variational technique in this regard has also been discussed.

STUDY OF SYMMETRY IN F(R) THEORY OF GRAVITY

An action in which the Ricci scalar is non-minimally coupled with a scalar field and contains higher order curvature invariant terms carries a conserved current under certain conditions that decouples geometric part from the scalar field. The conserved current relates the pair of arbitrary coupling parameters f(ϕ) and ω(ϕ) with the gravitational field variable, where ω(ϕ) is the Brans–Dicke coupling parameter. The existence of such conserved current may be helpful to sketch the cosmological evolution from its early age till date in a single frame.

Hamiltonian formulation of curvature squared action

It has been observed earlier that, in principle, it is possible to obtain a quantum mechanical interpretation of higher order quantum cosmological models in the spatially homogeneous and isotropic background, if auxiliary variable required for the Hamiltonian formulation of the theory is chosen properly. It was suggested that for such a choice, it is required to get rid of all the total derivative terms from the action containing higher order curvature invariant terms, prior to the introduction of auxiliary variable. Here, the earlier work has been modified and it is shown that the action, $A=\beta\int \sqrt{-g} R^2 d^4 x$ should be supplemented by a boundary term in the form $\sigma=4\beta\int ^{3}R K \sqrt{h} d^{3}x$, where, $^3 R$, $K$ and $h$ are the Ricci scalar for three-space, trace of the extrinsic curvature and determinant of the metric on the three space respectively. The result has been tested in the background of homogeneous and anisotropic models and thus confirmed.

The speciality index as invariant indicator in the BKL mixmaster dynamics**During the revision of this paper, we were saddened by the news of the death of our coauthor Zoltan Perjes. We would like to remember him here as a wonderful human being with undiminished enthusiasm for engaging younger colleagues in discussing a broad range of problems in gravitational physics.

The long-standing difficulty in general relativity of classifying the dynamics of cosmological models, e.g. as chaotic, is directly related to the gauge freedom intrinsic to relativistic spacetime theories: in general the invariance under diffeomorphisms makes any analysis of dynamical evolution dependent on the particular choice of time slicing one uses. We show here that the speciality index, a scalar dimensionless curvature invariant that has been mainly used in numerical relativity as an indicator of the special or non-special Petrov-type character of a spacetime, is a time-independent quantity (a pure number) at each Kasner step of the Belinski–Khalatnikov–Lifshitz (BKL) map approximating the mixmaster cosmology. Thus the BKL dynamics can be characterized in terms of the speciality index, i.e. in terms of curvature invariants directly related to observables. Possible applications for the associated mixmaster dynamics are discussed.

Quantum mechanical probability interpretation in the mini-superspace model of higher order gravity theory

It has been shown that introduction of higher order curvature invariant terms like R 2 or R μν R μν in the Robertson–Walker minisuperspace model of the Einstein–Hilbert action leads to Schrödinger-like equation, whose corresponding effective Hamiltonian is hermitian. Thus, it is possible to write the continuity equation in a straightforward manner which reveals a quantum mechanical probability interpretation of the theory.

Rotating topological black branes in various dimensions and AdS/CFT correspondence

We consider rotating topological black branes with one rotational parameter in various dimensions. Also a general five-dimensional higher genus solution of the Einstein equation with a negative cosmological constant which represents a topological black brane with two rotational parameters is introduced. We find out that the counterterms inspired by conformal field theory introduced by Kraus, Larsen and Sieblink cannot remove the divergences in $r$ of the action in more than five dimensions. We modify the counterterms by adding a curvature invariant term to it. Using the modified counterterms we show that the $r$ divergences of the action, the mass, and the angular momentum densities of these spacetimes are removed. We also find out in the limit of $m=0$ the mass density of these spacetimes in odd dimensions is not zero.