Higher-order Gravity Theories Research Articles

Finite action principle revisited

We extend the cosmological predictions from assuming the total action of the universe is finite. When initial and final singularities in curvature invariants are avoided, it leads to singularities in the gravitational action of the universe. The following properties are required of a universe with finite action: Compact spatial sections (ie a closed universe) giving a finite total lifetime for the universe. Compactification of flat and open universes is excluded. The universe can contain perfect fluids with -1<p/\r{ho}<2 on approach to singularities. The universe cannot display a bounce' or indefinite cyclic behaviour to the past or the future. We find new consequences of imposing finite action: the universe cannot be dominated by massless scalar fields, the kinetic energy of scalar fields, or a p=\r{ho} perfect fluid on approach to the initial or final singularity The ekpyrotic scenario with an effective fluid obeying p/\r{ho}>2 in a closed, flat or open universe is excluded. Any bouncing model with indefinite past or future evolution is ruled out. Einstein static and steady-state universes are ruled out along with past or future eternal inflating universes. Anisotropies cannot dominate the dynamics at singularities. This excludes density inhomogeneity spectra versus mass, of the form {\delta}\r{ho}/\r{ho}proportional to M^{-q}, with q>2/3.Higher-order lagrangian theories of gravity are constrained. The Gauss-Bonnet combination causes an action singularity even though it does not contribute terms to the field equations. Scalar-tensor theories dominated by the scalar field on approach to singularities have action singularities. Dark energy cannot be a cosmological constant. The dark energy must evolve in a closed universe that collapses to a future singularity that cannot be dominated by the kinetic energy of the scalar field.

Degenerate Hamiltonian operator in higher-order canonical gravity — The problem and a remedy

Different routes towards canonical formulation of a classical theory result in different canonically equivalent Hamiltonians, while their quantum counterparts are related through appropriate unitary transformation. However, for higher-order theory of gravity although two Hamiltonians emerging from the same action differing by total derivative terms are related through canonical transformation, the difference transpires while attempting canonical quantization, which is predominant in non-minimally coupled higher-order theory of gravity. We follow Dirac’s constraint analysis to formulate phase-space structures, in the presence (case-I) and absence (case-II) of total derivative terms. While the coupling parameter plays no significant role as such for case-I, quantization depends on its form explicitly in case-II, and as a result unitary transformation relating the two is not unique. We also find certain mathematical inconsistencies in case-I, for modified Gauss–Bonnet-Dilatonic coupled action, in particular. Thus, we conclude that total derivative terms indeed play a major role in the quantum domain and should be taken care of a-priori, for consistency.

Conservation laws and exact solutions in Brans–Dicke cosmology with a scalar field

Scalar-tensor theories have drawn the attention of cosmologist for the past few years because they can provide mechanism to explain the observable phenomena. Moreover, the results of scalar-tensor theories can be applied in higher-order theories of gravity. In this work, we consider a Brans-Dicke scalar field cosmological model that includes a second scalar field minimally coupled to gravity. For the second scalar field we consider the two cases that there exist and there is not any interaction with the Brans-Dicke scalar field. For the two models of our consideration we prove the existence of scaling and de Sitter solutions. Furthermore, we perform an Obsiannikov's classification for the field equations, where we find all the point and contact transformations which leave the gravitational Action Integral invariant. Conservation laws are determined and we show how these can be applied to infer about the integrability of the cosmological models and find new analytic solutions.

Obtaining analytical approximations to black hole solutions in higher-derivative gravity using the homotopy analysis method

The Homotopy Analysis Method (HAM) is considered as a very useful method for obtaining analytical approximate solutions to various nonlinear differential equations arising in many different areas of science and engineering. Despite this, it is seldom used to obtain solutions in General Relativity and particularly higher order theories of gravity, where due to the complexity and nonlinearity of the field equations, most of the known solutions are numerical. We consider the case of a non-Schwarzschild static and spherically symmetric black hole solution in higher derivative gravity that has been studied recently. We obtain an analytical approximation using HAM and compare it with the numerical solution.

Almost universal spacetimes in higher-order gravity theories

We study almost universal spacetimes - spacetimes for which the field equations of any generalized gravity with the Lagrangian constructed from the metric, the Riemann tensor and its covariant derivatives of arbitrary order reduce to one single differential equation and one algebraic condition for the Ricci scalar. We prove that all d-dimensional Kundt spacetimes of Weyl type III and traceless Ricci type N are almost universal. Explicit examples of Weyl type II almost universal Kundt metrics are also given. The considerable simplification of the field equations of higher-order gravity theories for almost universal spacetimes is then employed to study new Weyl type II, III, and N vacuum solutions to quadratic gravity in arbitrary dimension and six-dimensional conformal gravity. Necessary conditions for almost universal metrics are also studied.

Erratum: Equivalent and inequivalent canonical structures of higher order theories of gravity [Phys. Rev. D 96 , 084025 (2017)

In the manuscript, we concluded that Horowitz's as well as Dirac's and Ostrogradski's formalisms fail to produce a viable quantum description of the modified Einstein-Gauss-Bonnet-dilatonic (MEGBD) action. However, later we note that such conclusion stems from an error in the solution to the classical field equations. Here, we rectify the error, along with all the expressions associated with it. In the process, we find that both Horowitz's formalism (as well as Dirac's and Ostrogradski's formalisms) and modified Horowitz's formalism produce a viable, although inequivalent, quantum description of the MEGBD action. Nevertheless, the main motivation of the manuscript, viz. ``canonical formulation of higher-order theories of gravity following different prescriptions in some cases lead to equivalent and in some other cases inequivalent phase-space structures'', remains unaltered.

Equivalent and inequivalent canonical structures of higher order theories of gravity

Canonical formulation of higher order theory of gravity can only be accomplished associating additional degrees of freedom, which are extrinsic curvature tensor. Consequently, to match Cauchy data with the boundary data, terms in addition to the three-space metric, must also be fixed at the boundary. While, in all the three, viz. Ostrogradski's, Dirac's and Horowitz' formalisms, extrinsic curvature tensor is kept fixed at the boundary, a modified Horowitz' formalism fixes Ricci scalar, instead. It has been taken as granted that the Hamiltonian structure corresponding to all the formalisms with different end-point data are either the same or are canonically equivalent. In the present study, we show that indeed it is true, but only for a class of higher order theory. However, for more general higher order theories, e.g. dilatonic coupled Gauss-Bonnet gravity in the presence of curvature squared term, the Hamiltonian obtained following modified Horowitz' formalism is found to be different from the others, and is not related under canonical transformation. Further, it has also been demonstrated that although all the formalisms produce viable quantum description, the dynamics is different and not canonically related to modified Horowitz' formalism. Therefore it is not possible to choose the correct formalism which leads to degeneracy in Hamiltonian.

Universal spacetimes in four dimensions

Universal spacetimes are exact solutions to all higher-order theories of gravity. We study these spacetimes in four dimensions and provide necessary and sufficient conditions for universality for all Petrov types except of type II. We show that all universal spacetimes in four dimensions are algebraically special and Kundt. Petrov type D universal spacetimes are necessarily direct products of two 2-spaces of constant and equal curvature. Furthermore, type II universal spacetimes necessarily possess a null recurrent direction and they admit the above type D direct product metrics as a limit. Such spacetimes represent gravitational waves propagating on these backgrounds. Type III universal spacetimes are also investigated. We determine necessary and sufficient conditions for universality and present an explicit example of a type III universal Kundt non-recurrent metric.

Evolution of compact stars and dark dynamical variables

This work aims to explore the dark dynamical effects of the f(R, T) modified gravity theory on the dynamics of a compact celestial star. We have taken the interior geometry of a spherical star which is filled with an imperfect fluid distribution. The modified field equations are explored by taking a particular form of the f(R, T) model, i.e. f(R,T)=f_1(R)+f_2(R)f_3(T). These equations are utilized to formulate the well-known structure scalars under the dark dynamical effects of this higher-order gravity theory. Also, with the help of these scalar variables, the evolution equations for expansion and shear are formulated. The whole analysis is made under the condition of a constant R and T. We found a crucial significance of dark source terms and dynamical variables on the evolution and density inhomogeneity of compact objects.

Scattering of Ricci scalar perturbations from Schwarzschild black holes in modified gravity

It has already been shown that the gravitational waves emitted from a Schwarzschild black hole in f(R) gravity have no signatures of the modification of gravity from General Relativity, as the Regge–Wheeler equation remains invariant. In this paper we consider the perturbations of Ricci scalar in a vacuum Schwarzschild spacetime, which is unique to higher order theories of gravity and is absent in General Relativity. We show that the equation that governs these perturbations can be reduced to a Volterra integral equation. We explicitly calculate the reflection coefficients for the Ricci scalar perturbations, when they are scattered by the black hole potential barrier. Our analysis shows that a larger fraction of these Ricci scalar waves are reflected compared to the gravitational waves. This may provide a novel observational signature for fourth order gravity.

Higher-order gravity in higher dimensions: geometrical origins of four-dimensional cosmology?

Determining the cosmological field equations is still very much debated and led to a wide discussion around different theoretical proposals. A suitable conceptual scheme could be represented by gravity models that naturally generalize Einstein theory like higher-order gravity theories and higher-dimensional ones. Both of these two different approaches allow one to define, at the effective level, Einstein field equations equipped with source-like energy-momentum tensors of geometrical origin. In this paper, the possibility is discussed to develop a five-dimensional fourth-order gravity model whose lower-dimensional reduction could provide an interpretation of cosmological four-dimensional matter–energy components. We describe the basic concepts of the model, the complete field equations formalism and the 5-D to 4-D reduction procedure. Five-dimensional f(R) field equations turn out to be equivalent, on the four-dimensional hypersurfaces orthogonal to the extra coordinate, to an Einstein-like cosmological model with three matter–energy tensors related with higher derivative and higher-dimensional counter-terms. By considering the gravity model with f(R)=f_0R^n the possibility is investigated to obtain five-dimensional power law solutions. The effective four-dimensional picture and the behaviour of the geometrically induced sources are finally outlined in correspondence to simple cases of such higher-dimensional solutions.

Canonical formulation of Pais–Uhlenbeck action and resolving the issue of branched Hamiltonian

Canonical formulation of higher order theory of gravity requires to fix (in addition to the metric), the scalar curvature, which is acceleration in disguise, at the boundary. On the contrary, for the same purpose, Ostrogradski's or Dirac's technique of constrained analysis, and Horowit'z formalism, tacitly assume velocity (in addition to the co-ordinate) to be fixed at the end points. In the process when applied to gravity, Gibbons–Hawking–York term disappears. To remove such contradiction and to set different higher order theories on the same footing, we propose to fix acceleration at the endpoints/boundary. However, such proposition is not compatible to Ostrogradski's or Dirac's technique. Here, we have modified Horowitz's technique of using an auxiliary variable, to establish a one-to-one correspondence between different higher order theories. Although, the resulting Hamiltonian is related to the others under canonical transformation, we have proved that this is not true in general. We have also demonstrated how higher order terms can regulate the issue of branched Hamiltonian.

Some classes of gravitational shock waves from higher order theories of gravity

We study the gravitational shock wave generated by a massless high energy particle in the context of higher order gravities of the form $F(R,R_{\mu \nu}R^{\mu \nu},R_{\mu \nu \alpha \beta}R^{\mu \nu \alpha \beta})$. In the case of $F(R)$ gravity, we investigate the gravitational shock wave solutions corresponding to various cosmologically viable gravities, and as we demonstrate the solutions are rescaled versions of the Einstein-Hilbert gravity solution. Interestingly enough, other higher order gravities result to the general relativistic solution, except for some specific gravities of the form $F(R_{\mu \nu}R^{\mu \nu})$ and $F(R,R_{\mu \nu}R^{\mu \nu})$, which we study in detail. In addition, when realistic Gauss-Bonnet gravities of the form $R+F(\mathcal{G})$ are considered, the gravitational shock wave solutions are identical to the general relativistic solution. Finally, the singularity structure of the gravitational shock waves solutions is studied, and it is shown that the effect of higher order gravities makes the singularities milder in comparison to the general relativistic solutions, and in some particular cases the singularities seem to be absent.

Einsteinian cubic gravity

We drastically simplify the problem of linearizing a general higher-order theory of gravity. We reduce it to the evaluation of its Lagrangian on a particular Riemann tensor depending on two parameters, and the computation of two derivatives with respect to one of those parameters. We use our method to construct a D-dimensional cubic theory of gravity which satisfies the following properties: 1) it shares the spectrum of Einstein gravity, i.e., it only propagates a transverse and massless graviton on a maximally symmetric background; 2) the relative coefficients of the different curvature invariants involved are the same in all dimensions; 3) it is neither trivial nor topological in four dimensions. Up to cubic order in curvature, the only previously known theories satisfying the first two requirements are the Lovelock ones: Einstein gravity, Gauss-Bonnet and cubic-Lovelock. Of course, the last two theories fail to satisfy requirement 3 as they are, respectively, topological and trivial in four dimensions. We show that, up to cubic order, there exists only one additional theory satisfying requirements 1 and 2. Interestingly, this theory is, along with Einstein gravity, the only theory which also satisfies 3.

The regular state in higher order gravity

We consider the higher-order gravity theory derived from the quadratic Lagrangian [Formula: see text] in vacuum as a first-order (ADM-type) system with constraints, and build time developments of solutions of an initial value formulation of the theory. We show that all such solutions, if analytic, contain the right number of free functions to qualify as general solutions of the theory. We further show that any regular analytic solution which satisfies the constraints and the evolution equations can be given in the form of an asymptotic formal power series expansion.

Analytical solutions in R + qRn cosmology from singularity analysis

The integrability of higher-order theories of gravity is of importance in the determining the properties of these models and so their viability as models of reality. An important tool in the establishment of integrability is the singularity analysis. We apply this analysis to the case of fourth-order theory of gravity f(R)=R+qRn to establish those values of the free parameters q and n for which integrability in this sense exists. As a preliminary we examine the well-known case of n=4/3.

Kerr-Newman-dS/AdS solution and anti-evaporation in higher-order torsion scalar gravity theories

We derive a null tetrad from axially-symmetric vierbein field. The \(f(T)\)-Maxwell field equations with cosmological constant, where T is the scalar torsion, are applied to the null tetrad. An exact non-vacuum solution having three constants of integration is derived which is a solution to the f (T) -Maxwell field equations provided that \(f(T)=T_{0}\) and \( f_{T}=\frac{d f(T)}{d T}=1\), where \( T_{0}\) is a constant. The scalar torsion related to this solution is constant, i.e., \( T=T_{0}\), and differs from the classical general relativity when \( f(T)\neq T_{0}\). We study the singularities of this solution using curvature and torsion invariants. We consider a slow rotation and show that the derived solution behaves asymptotically as de Sitter spacetime and display the existence of Nariai spacetime as a background solution. We assume a perturbation of Nariai spacetime till the first order and investigate the behavior of the black hole horizon. Finally, we explain that the anti-evaporation occurs on the classical level in the f (T) gravitational theories.

Testing modified gravity and no-hair relations for the Kerr-Newman metric through quasiperiodic oscillations of galactic microquasars

We construct multipole moments for stationary, asymptotically flat, spacetime solutions to higher-order curvature theories of gravity. The moments are defined using $3+1$ techniques involving timelike Killing vector constructions as in the classic papers by Geroch and Hansen. Using the fact that the Kerr-Newman metric is a vacuum solution to a particular class of $f(R)$ theories of gravity, we compute all its moments, and find that they admit recurrence relations similar to those for the Kerr solution in general relativity. It has been proposed previously that modeling the measured frequencies of quasiperiodic oscillations from galactic microquasars enables experimental tests of the no-hair theorem. We explore the possibility that, even if the no-hair relation is found to break down in the context of general relativity, there may be an $f(R)$ counterpart that is preserved. We apply the results to the microquasars GRS $1915+105$ and GRO J1655-40 using the diskoseismology and kinematic resonance models, and constrain the spins and ``charges'' of their black holes.

Validating variational principle for higher order theory of gravity

Metric variation of higher order theory of gravity requires fixing of the Ricci scalar in addition to the metric tensor at the boundary. Fixing Ricci scalar at the boundary implies that the classical solutions are fixed once and forever to the de Sitter or anti-de Sitter (dS/AdS) solutions. Here, we justify such requirement from the standpoint of Noether symmetry.

Cosmology: The search for twenty-four (or more) functions

We enumerate the $4(1+F)+2S$ independent arbitrary functions of space required to describe a general relativistic cosmology containing an arbitrary number of noninteracting fluid ($F$) and scalar fields ($S$). Results are also given for arbitrary space dimension and for higher-order gravity theories, where the number increases to $16+4F+2S$. Both counts are subject to assumptions about whether the dark energy is a cosmological constant. A more detailed analysis is provided when global homogeneity is assumed and the functions become constants. This situation is also studied in the case where the flat and open universes have compact spatial topologies. This changes the relative generalities significantly and places new constraints on the types of expansion anisotropy that are permitted. The most general compact homogeneous universes containing Friedmann models are spatially flat and described by $8+4F+2S$ constants. Comparisons are made with the simple six-parameter lambda-CDM model and physical interpretations provided for the parameters needed to describe the most general cosmological models.