We continue the study of topological nonrelativistic quantum gravity associated with a family of Ricci flow equations on Riemannian manifolds. This topological gravity is of the cohomological type, and it exhibits an ${\cal N}=2$ extended BRST symmetry. In our previous work, we constructed this theory in a two-step procedure in the appropriate nonrelativistic ${\cal N}=2$ superspace, first presenting a topological theory of the spatial metric $g_{ij}$, and then adding the superspace versions of the lapse and shift variables $n$ and $n^i$ while gauging the symmetries of foliation-preserving spacetime diffeomorphisms. In the relation to Perelman's theory of the Ricci flow, the role of Perelman's dilaton is played by our nonprojectable lapse. Here we demonstrate that this construction is equivalent to a standard one-step BRST gauge-fixing of a theory whose fields are $g_{ij}$, $n^i$ and $n$, and whose gauge symmetries consist of (i) the topological deformations of $g_{ij}$, and (ii) the ultralocal nonrelativistic limit of spacetime diffeomorphisms. The supercharge $Q$ of our superspace construction plays the role of the BRST charge. The spacetime diffeomorphism symmetries appear in an interestingly "shifted" form, which may be of broader interest for nonrelativistic quantum gravity outside of the present topological context. In contrast to the foliation-preserving spacetime diffeomorphisms, the gauge symmetries identified in this paper act nonprojectably on time, making it clear that this theory has no local propagating degrees of freedom. We point out an intriguing dual interpretation of the same theory, as a gauge fixing of a dual copy of ultralocal spacetime diffeomorphisms, with the role of ghosts and antighosts interchanged and the second supercharge $\bar Q$ of the ${\cal N}=2$ superalgebra playing the role of the BRST charge in the dual picture.
We provide a proof that nonholonomically constrained Ricci flows of (pseudo) Riemannian metrics positively result into nonsymmetric metrics (as explicit examples, we consider flows of some physically valuable exact solutions in general relativity). There are constructed and analyzed three classes of solutions of Ricci flow evolution equations defining nonholonomic deformations of Taub NUT, Schwarzschild, solitonic and pp--wave symmetric metrics into nonsymmetric ones.
f(Ricci) gravity is a special kind of higher curvature gravity whose bulk Lagrangian density is the trace of a matrix-valued function of the Ricci tensor. It is shown that, under some mild constraints, f(Ricci) gravity admits Einstein manifolds as exact vacuum solutions, and can be ghost free around maximally symmetric Einstein vacua. It is also shown that the entropy for spherically symmetric black holes in f(Ricci) gravity calculated via Wald's method and the boundary Noether charge approach proposed by Majhi and Padmanabhan are in good agreement.
We perform the stability analysis of Schwarzschild-AdS (SAdS) black hole in the Einstein-Ricci cubic gravity. It shows that the Ricci tensor perturbations exhibit unstable modes for small black holes. We call this the mass-induced instability of SAdS black hole because the instability of small black holes arises from the massiveness in the linearized Einstein-Ricci cubic gravity, but not a feature of higher-order derivative theory giving ghost states. Also, we point out that the correlated stability conjecture holds for the SAdS black hole by computing the Wald entropy of SAdS black hole in Ei…
In this work, we have considered the Ricci dark energy model, where the energy density of the universe is proportional to the Ricci scalar curvature, in the dynamic Chern-Simons modified gravity. We show that in this context the evolution of the scale factor is similar to that displayed by the modified Chaplygin gas.
A theory of gravitation is proposed, modeled after the notion of a Ricci flow. In addition to the metric an independent volume enters as a fundamental geometric structure. Einstein gravity is included as a limiting case. Despite being a scalar-tensor theory the coupling to matter is different from Jordan-Brans-Dicke gravity. In particular there is no adjustable coupling constant. For the solar system the effects of Ricci flow gravity cannot be distinguished from Einstein gravity and therefore it passes all classical tests. However for cosmology significant deviations from standard Einstein cos…
It is well-known that there exists a Hawking-Page phase transition between a spherical AdS black hole and a thermal AdS space. The phase transition does not happen between a Ricci flat AdS black hole whose horizon is a Ricci flat space and a thermal AdS space in the Poincare coordinates. However, the Hawking-Page phase transition occurs between a Ricci flat AdS black hole and an AdS soliton if at least one of horizon coordinates for the Ricci flat black hole is compact. We show a similar phase transition betwen the Ricci flat black holes and deformed AdS solitons in the Gauss-Bonnet gravity an…
We show that the space of solutions of a wide family of Ricci-based metric-affine theories of gravity can be put into correspondence with the space of solutions of general relativity (GR). This allows us to use well-established methods and results from GR to explore new gravitational physics beyond it.
We find exact solutions describing Ricci flows of four dimensional pp-waves nonlinearly deformed by two/three dimensional solitons. Such solutions are parametrized by five dimensional metrics with generic off-diagonal terms and connections with nontrivial torsion which can be related, for instance, to antisymmetric tensor sources in string gravity. There are defined nontrivial limits to four dimensional configurations and the Einstein gravity.
A special class of higher curvature theories of gravity, Ricci Cubic Gravity (RCG), in general d dimensional space-time has been investigated in this paper. We have used two different approaches, the linearized equations of motion and auxiliary field formalism to study the massive and massless graviton propagating modes of the AdS background. Using the auxiliary field formalism, we have found the renormalized boundary stress tensor to compute the mass of Schwarzschild AdS and Lifshitz black holes in RCG theory.
The Ricci dark energy is a model inspired by the holographic dark energy models with the dark energy density being proportional to Ricci scalar curvature. Here this model is studied in the bumblebee gravity theory. It is a gravitational theory that exhibit spontaneous Lorentz symmetry breaking. Then the modified Friedmann equation is solved for two cases. In the first case the coupling constant $ξ$ is equal to zero. And in the second case a solution in the vacuum, where the bumblebee field becomes a constant that minimizes the potential, is considered. The coupling constant controls the intera…
Brownian Motion in the Hilbert Space of Quantum States along with the Ricci Flow and Stochastically Emergent Einstein-Hilbert Action: Formulating a Well-Defined Feynman Path-Integral Measure for Quantum Fields in the Presence of Gravity
In a number of physically important cases, the nonholonomically (nonintegrable) constrained Ricci flows can be modelled by exact solutions of Einstein equations with nonhomogeneous (anisotropic) cosmological constants. We develop two geometric methods for constructing such solutions: The first approach applies the formalism of nonholonomic frame deformations when the gravitational evolution and field equations transform into systems of nonlinear partial differential equations which can be integrated in general form. The second approach develops a general scheme when one (two) parameter familie…
We obtain an infinite number of exact static, Ricci-flat spherically symmetric vacuum solutions for a class of f(R) theories of gravity. We analytically derive two exact vacuum black-hole solutions for the same class of f(R) theories. The two black-hole solutions have the event-horizon at the same point; however, their asymptotic features are different. Our results suggest that no-hair theorem may not hold for generic modified gravity theories. We discuss the implications of our work to distinguish modified gravity theories from general relativity in gravitational wave detections.
We extend the correspondence between metric-affine Ricci-Based Gravity theories and General Relativity (GR) to the case in which the matter sector is represented by linear and nonlinear electromagnetic fields. This complements previous studies focused on fluids and scalar fields. We establish the general algorithm that relates the matter fields in the GR and RBG frames and consider some applications. In particular, we find that the so-called Eddington-inspired Born-Infeld gravity theory coupled to Maxwell electromagnetism is in direct correspondence with GR coupled to Born-Infeld electromagnet…
We generalize the algorithm that establishes the correspondence between metric-affine Eddington-inspired Born-Infeld (EiBI) gravity and General Relativity (GR) to any bosonic matter sector. Along the way, a polished version of the proof of existence of a metric compatible frame associated to any metric-affine Ricci-Based Gravity (RBG) is presented. Particular attention is given to the problem of generating solutions of a RBG from the GR conterpart, providing a general recipe for the EiBI case. This extends previous results obtained for specific matter that included anisotropic fluids, scalar f…
We show how minimally-coupled matter fields of arbitrary spin, when coupled to Ricci-Based Gravity theories, develop non-trivial effective interactions that can be treated perturbatively only below a characteristic high-energy scale $Λ_Q$. Our results generalize to arbitrary matter fields those recently obtained for spin 1/2 fields in \cite{Latorre:2017uve}. We then use this interactions to set bounds on the high-energy scale $Λ_Q$ that controls departures of Ricci-Based Gravity theories from General Relativity. Particularly, for Eddington-inspired Born-Infeld gravity we obtain the strong boun…
We study a novel class of higher curvature gravity models in $n$ spacetime dimensions which we call Ricci polynomial gravity. The action is consisted purely of a polynomial in Ricci curvature of order $N$. In the absence of the second order terms in the action, the models are ghost free around the Minkowski vacuum. By appropriately choosing the coupling coefficients in front of each terms in the action, it is shown that the models can have multiple vacua with different effective cosmological constants, and can be made free of ghost and scalar degrees of freedom around at least one of the maxim…
It has been argued that, underlying any given quantum-mechanical model, there exists at least one deterministic system that reproduces, after prequantisation, the given quantum dynamics. For a quantum mechanics with a complex d-dimensional Hilbert space, the Lie group SU(d) represents classical canonical transformations on the projective space CP^{d-1} of quantum states. Let R stand for the Ricci flow of the manifold SU(d-1) down to one point, and let P denote the projection from the Hopf bundle onto its base CP^{d-1}. Then the underlying deterministic model we propose here is the Lie group SU…
We investigate the asymptotic safety conjecture for quantum gravity including curvature invariants beyond Ricci scalars. Our strategy is put to work for families of gravitational actions which depend on functions of the Ricci scalar, the Ricci tensor, and products thereof. Combining functional renormalisation with high order polynomial approximations and full numerical integration we derive the renormalisation group flow for all couplings and analyse their fixed points, scaling exponents, and the fixed point effective action as a function of the background Ricci curvature. The theory is charac…
Recently, Amendola et al. proposed a geometrical theory of gravity containing higher-order derivative terms. The authors introduced anticurvature scalar $(A)$, which is the trace of the inverse of the Ricci tensor ($A^{μν} = R_{μν}^{-1}$). In this work, we consider two classes of Ricci-inverse -- Class I and Class II -- models. Class I models are of the form $f(R, A)$ where $f$ is a function of Ricci and anticurvature scalars. Class II models are of the form ${\cal F}(R, A^{μν}A_{μν})$ where ${\cal F}$ is a function of Ricci scalar and square of anticurvature tensor. For both these classes of …
The Palatini gravitational action is enlarged by an arbitrary function $f(X)$ of the determinants of the Ricci tensor and the metric, $X=|\textbf{det}.R|/|\textbf{det}.g|$. The resulting Ricci-determinant theory exhibits novel deviations from general relativity. We study a particular realization where the extension is characterized by the square-root of the Ricci-determinant, $f(X)=λ_\text{Edd}\sqrt{X}$, which corresponds to the famous Eddington action. We analyze the obtained equations for a perfect fluid source and show that the affine connection can be solved in terms of the energy density …
We study Ricci flows of some classes of physically valuable solutions in Einstein and string gravity. The anholonomic frame method is applied for generic off-diagonal metric ansatz when the field/ evolution equations are transformed into exactly integrable systems of partial differential equations. The integral varieties of such solutions, in four and five dimensional gravity, depend on arbitrary generation and integration functions of one, two and/ or three variables. Certain classes of nonholonomic frame constraints allow us to select vacuum and/or Einstein metrics, to generalize such soluti…
We find constant scalar curvature Type-N and Type-D solutions in all higher curvature gravity theories with actions of the form f(Ricci) that are built on the Ricci tensor, but not on its derivatives. In our construction, these higher derivative theories inherit some of the previously studied solutions of the cosmological topologically massive gravity and the new massive gravity field equations, once the parameters of the theories are adjusted. Besides the generic higher curvature theory, we have considered in some detail the examples of the quadratic curvature theory, the cubic curvature theo…
Taking advantage of a previously developed method, which allows to map solutions of General Relativity into a broad family of theories of gravity based on the Ricci tensor (Ricci-based gravities), we find new exact analytical scalar field solutions by mapping the free-field static, spherically symmetric solution of General Relativity (GR) into quadratic $f(R)$ gravity and the Eddington-inspired Born-Infeld gravity. The obtained solutions have some distinctive feature below the would-be Schwarzschild radius of a configuration with the same mass, though in this case no horizon is present. The co…
We present several new space-periodic solutions of the static vacuum Einstein equations in higher dimensions, both with and without black holes, having Kasner asymptotics. These latter solutions are referred to as gravitational solitons. Further partially compactified solutions are also obtained by taking appropriate quotients, and the topologies are computed explicitly in terms of connected sums of products of spheres. In addition, it is shown that there is a correspondence, via Wick rotation, between the spacelike slices of the solitons and black hole solutions in one dimension less. As a co…
In this paper, we study the cosmological analysis of the modified holographic Ricci dark energy model and reconstruct different scalar field models in the context of Chern-Simon modified gravity. We investigate the deceleration parameter, which shows that the universe is in the accelerating expansion phase. The equation of state parameter in this case also favors the fact that dark energy is the dominant component of universe which is responsible for the accelerated expansion. A number of scalar fields, such as: quintessence, tachyon, K-essence and dilaton models are reconstructed using modifi…
In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off-diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/Ricci flow models with nontrivial torsion to configurations with the Levi-Civita connection is allowed in some specific physical circumstances by constraining the class of integral varieties for the Eins…
The combined effects of the Lorentz-symmetry violating Chern-Simons and Ricci-Cotton actions are investigated for the Einstein-Hilbert gravity in the second order formalism modified by higher derivative terms, and their consequences on the spectrum of excitations are analyzed. We follow the lines of previous works and build up an orthonormal basis of operators that splits the fundamental fields according to their individual degrees of freedom. With this new basis, the attainment of the propagators is remarkably simplified and the identification of the physical and unphysical modes gets a new i…
We consider Topologically Massive Gravity (TMG), which is three dimensional general relativity with a cosmological constant and a gravitational Chern-Simons term. When the cosmological constant is negative the theory has two potential vacuum solutions: Anti-de Sitter space and Warped Anti-de Sitter space. The theory also contains a massive graviton state which renders these solutions unstable for certain values of the parameters and boundary conditions. We study the decay of these solutions due to the condensation of the massive graviton mode using Ricci-Cotton flow, which is the appropriate g…
We investigate the black holes in the new massive conformal gravity which is not invariant under conformal transformations because of the presence of the Einstein-Hilbert term. First, we show that the small Schwarzschild black hole is unstable against the $s$-mode of linearized Ricci tensor by solving the Lichnerowicz-Ricci tensor equation. This instability induces the appearance of the non-BBMB (Bocharova-Bronnikov-Melnikov-Bekenstein) black hole that has both Ricci tensor and conformal scalar hair.
We report exact black hole solutions in asymptotically flat or (A)dS four-dimensional spacetime with a conformally coupled self-interacting scalar field in $f(R)$ gravity. We first consider the asymptotically flat model $f(R) = R -2α\sqrt{R}$ and derive an exact black hole solution. Then, we consider the asymptotically (A)dS model $f(R) =R -2 Λ-2 α\sqrt{R-4 Λ}$ and derive an exact black hole solution. In both cases the modified gravity parameter $α$, which has the dimension of the inverse mass, cannot be set to zero and the self-interacting potential is determined from the Klein-Gordon equatio…
Quantum Ricci curvature has been introduced recently as a new, geometric observable characterizing the curvature properties of metric spaces, without the need for a smooth structure. Besides coordinate invariance, its key features are scalability, computability and robustness. We demonstrate that these properties continue to hold in the context of nonperturbative quantum gravity, by evaluating the quantum Ricci curvature numerically in two-dimensional Euclidean quantum gravity, defined in terms of dynamical triangulations. Despite the well-known, highly nonclassical properties of the underlyin…
We construct the Carrollian equivalent of the relativistic energy--momentum tensor, based on variation of the action with respect to the elementary fields of the Carrollian geometry. We prove that, exactly like in the relativistic case, it satisfies conservation equations that are imposed by general Carrollian covariance. In the flat case we recover the usual non-symmetric energy--momentum tensor obtained using Nœther procedure. We show how Carrollian conservation equations emerge taking the ultra-relativistic limit of the relativistic ones. We introduce Carrollian Killing vectors and build as…
Alternative gravitational theories described by Lagrangians depending on general functions of the Ricci scalar have been proven to give coherent theoretical models to describe the experimental evidence of the acceleration of universe at present time. In this paper we proceed further in this analysis of cosmological applications of alternative gravitational theories depending on (other) curvature invariants. We introduce Ricci squared Lagrangians in minimal interaction with matter (perfect fluid); we find modified Einstein equations and consequently modified Friedmann equations in the Palatini …
Quasi-topological terms in gravity can be viewed as those that give no contribution to the equations of motion for a special subclass of metric ansätze. They therefore play no rôle in constructing these solutions, but can affect the general perturbations. We consider Einstein gravity extended with Ricci tensor polynomial invariants, which admits Einstein metrics with appropriate effective cosmological constants as its vacuum solutions. We construct three types of quasi-topological gravities. The first type is for the most general static metrics with spherical, toroidal or hyperbolic isometries…
We establish an explicit correspondence of Einstein gravity on the squashed spheres that are the $U(1)$ bundles over $\mathbb{CP}^m$ to the Kaluza-Klein AdS gravity on the tori. This allows us to map the Ricci-flat Kerr metrics in odd dimensions with all equal angular momenta to charged Kaluza-Klein AdS black holes that can be lifted to become singly rotating M-branes and D3-branes. Furthermore, we find maps between Ricci-flat gravitational instantons to the AdS domain walls. In particular the supersymmetric bolt instantons correspond to domain walls that can be interpreted as distributed M-br…
Killing vector fields in three dimensions play important role in the construction of the related spacetime geometry. In this work we show that when a three dimensional geometry admits a Killing vector field then the Ricci tensor of the geometry is determined in terms of the Killing vector field and its scalars. In this way we can generate all products and covariant derivatives at any order of the ricci tensor. Using this property we give ways of solving the field equations of Topologically Massive Gravity (TMG) and New Massive Gravity (NMG) introduced recently. In particular when the scalars o…
Recently there has been a proposal for modified gravitational f(R) actions which include a direct coupling between the matter action and the Ricci scalar, R. Of particular interest is the specific case where both the action and the coupling are linear in R. It is shown that such an action leads to a theory of gravity which includes higher order derivatives of the matter fields without introducing more dynamics in the gravity sector and, therefore, cannot be a viable theory for gravitation.
In this work we study a modified version of vacuum $f(R)$ gravity with a kinetic term which consists of the first derivatives of the Ricci scalar. We develop the general formalism of this kinetic Ricci modified $f(R)$ gravity and we emphasize on cosmological applications for a spatially flat cosmological background. By using the formalism of this theory, we investigate how it is possible to realize various cosmological scenarios. Also we demonstrate that this theoretical framework can be treated as a reconstruction method, in the context of which it is possible to realize various exotic cosmol…
A new branch of black hole solutions occurs along with the standard Schwarzschild branch in $n$-dimensional extensions of general relativity including terms quadratic in the Ricci tensor. The standard and new branches cross at a point determined by a static negative-eigenvalue eigenfunction of the Lichnerowicz operator, analogous to the Gross-Perry-Yaffe eigenfunction for the Schwarzschild solution in standard $n=4$ dimensional general relativity. This static eigenfunction has two rôles: both as a perturbation away from Schwarzschild along the new black-hole branch and also as a threshold unst…
The paper proposes a statistical fields theory of quantum reference frame underlying the Perelman's analogies between his formalism of the Ricci flow and the thermodynamics. The theory is based on a d=4-ε quantum non-linear sigma model, interpreted as a quantum reference frame system which a to-be-studied quantum system is relative to. The statistic physics and thermodynamics of the quantum frame fields is studied by the density matrix obtained by the Gaussian approximation. The induced Ricci flow of the frame fields and the Ricci-DeTurck flow of the frame fields associated with the density ma…
We study the Ricci flow for the Lorentzian Einstein-Hilbert action. We show that Einstein gravity emerges as a fixed point of the Einstein-Ricci flow equations and derive a renormalization group flow in Euclidean signature. By considering linearizations near the fixed point, the dynamics of the metric reveal that curvature deformations flow according to a forward heat equation with the stress-energy tensor acting as a source.
We formulate Eddington's affine gravity in a spacetime which is immersed in a larger eight dimensional space endowed with a hypercomplex structure. The dynamical equation of the first immersed Ricci-type tensor leads to gravitational field equations which include matter. We also study the dynamical effects of the second Ricci-type tensor when added to the Lagrangian density. A simple Lagrangian density constructed from combination of the standard Ricci tensor and a new tensor field that appears due to the immersion, leads to gravitational equations in which the vacuum energy gravitates with a …
We elaborate on the inflationary model starting from multidimensional Lagrangian and gravity with second-order curvature terms. The effective scalar field is related to the Ricci scalar of extra dimensions. It is shown that the Kretschmann and the Ricci tensor square terms dominate during inflation. The observable values of the spectral index and the tensor-to-scalar ratio are obtained for specific values of the model parameters.
In this paper we study a class of wormhole solutions called generalized Ellis-Bronnikov wormholes in the context of asymptotically safe gravity (ASG). These solutions are characterized by two parameters: an even number $n$ and the wormhole throat radius $r_t$. The particular case $n=2$ recovers the usual Ellis-Bronnikov spacetime, which has already been addressed in the literature. We analyzed the nature of matter in the wormhole's throat, and in nearby regions, of these generalized solutions with $n>2$, using three curvature scalars in the ASG approach, namely, the Ricci scalar, squared Ricci…
It is well-known that Einstein gravity is kinematic (no non-trivial vacuum solution;i.e. Riemann vanishes whenever Ricci does so) in $3$ dimension because Riemann is entirely given in terms of Ricci. Could this property be universalized for all odd dimensions in a generalized theory? The answer is yes, and this property uniquely singles out pure Lovelock (it has only one $N$th order term in action) gravity for which $N$th order Lovelock Riemann tensor is indeed given in terms of corresponding Ricci for all odd $d=2N+1$ dimensions. This feature of gravity is realized only in higher dimensions a…
In Eddington gravity, the action principle involves only the symmetric parts of the connection and the Ricci tensor, with a metric that emerges proportionally to the latter. Here, we relax this symmetric character, prolong the action with the antisymmetric parts of the Ricci term, and allow for various couplings with scalar fields. We propose two possible invariant actions formed by distinct combinations of the independent Ricci tensors and show that the generated metric must involve an additional antisymmetric part due to the relaxation of the symmetrization property. The comprehensive study …
We investigate brane world models in higher-derivative gravity theories where the gravitational Lagrangian is an arbitrary function of the Ricci scalar. Making use of the conformal equivalence of such gravity models and Einstein-Hilbert gravity with a scalar field, we deduce the main features of higher-derivative gravity brane worlds. We solve for a gravity model that has corrections quadratic in the Ricci scalar and show one can evade both fine-tuning and the need for a bulk cosmological constant. An analysis of tensor and scalar perturbations shows gravity is localized on the brane and we re…
We compute the one-loop divergences in a higher-derivative theory of gravity including Ricci tensor squared and Ricci scalar squared terms, in addition to the Hilbert and cosmological terms, on an (generally off-shell) Einstein background. We work with a two-parameter family of parametrizations of the graviton field, and a two-parameter family of gauges. We find that there are some choices of gauge or parametrization that reduce the dependence on the remaining parameters. The results are invariant under a recently discovered "duality" that involves the replacement of the densitized metric by a…
