Building on the recently introduced notion of quantum Ricci curvature and motivated by considerations in nonperturbative quantum gravity, we advocate a new, global observable for curved metric spaces, the curvature profile. It is obtained by integrating the scale-dependent, quasi-local quantum Ricci curvature, and therefore also depends on a coarse-graining scale. To understand how the distribution of local, Gaussian curvature is reflected in the curvature profile, we compute it on a class of regular polygons with isolated conical singularities. We focus on the case of the tetrahedron, for which we have a good computational control of its geodesics, and compare its curvature profile to that of a smooth sphere. The two are distinct, but qualitatively similar, which confirms that the curvature profile has averaging properties which are interesting from a quantum point of view.
We discuss a recently proposed limiting curvature theory of gravity and its application to the problem of singularities inside black holes. In this theory the growth of the curvature is suppressed by specially chosen inequality constraints included in the gravity action. We consider a case of a spherically symmetric four dimensional black hole and demonstrated that imposed curvature constraints modify a solution in the black hole interior. Instead of forming the curvature singularity the modified metric describes a space which is exponentially expanding in one direction and oscillating the oth…
We compute the entropies for general curvature squared gravities in arbitrary dimensions using the conserved charge and Virasoro algebra from surface term. We introduce an auxiliary tensor field in order to obtain the boundary action which establish a variational principle. Our results reproduce those in some specific models, that is, the Gauss-Bonnet, new massive gravity (NMG), and critical gravity.
We compute curvature-dependent graviton correlation functions and couplings as well as the full curvature potential $f(R)$ in asymptotically safe quantum gravity coupled to scalars. The setup is based on a systematic vertex expansion about metric backgrounds with constant curvatures initiated in arXiv:1711.09259 for positive curvatures. We extend these results to negative curvature and investigate the influence of minimally coupled scalars. The quantum equation of motion has two solutions for all accessible numbers of scalar fields. We observe that the solution at negative curvature is a minim…
The zeroth law of black hole mechanics is an assertion of constancy of the surface gravity on a stationary Killing horizon. The Hawking temperature of the black hole horizon is proportional to the surface gravity. Therefore, the constancy of the surface gravity is reminiscent of the zeroth law of ordinary thermodynamics. In this work, we provide a proof of the zeroth law in Lanczos-Lovelock theories of gravity, where the Einstein Hilbert action is supplemented by higher curvature terms.
We define and compute the energy of higher curvature gravity theories in arbitrary dimensions. Generically, these theories admit constant curvature vacua (even in the absence of an explicit cosmological constant), and asymptotically constant curvature solutions with non-trivial energy properties. For concreteness, we study quadratic curvature models in detail. Among them, the one whose action is the square of the traceless Ricci tensor always has zero energy, unlike conformal (Weyl) gravity. We also study the string-inspired Einstein-Gauss-Bonnet model and show that both its flat and Anti-de-S…
We construct not only an induced gravity model with the restricted diffeomorphisms, that is, transverse diffeomorphisms which preserves the curvature density, but also that with the full diffeomorphisms. By solving the equations of motion, it turns out that these models produce Einstein's equations with a certain Newton's constant in addition to the constraint for the curvature density. In the limit of the infinite Newton constant, the models give rise to induced gravity. Moreover, we discuss cosmological solutions on the basis of the gravitational models at hand.
Partial results are obtained for Schwarzschild- like solutions in a gravity theory with action density (-g)^(1/2)[Rik^2+bR^2]. A seven parameter family of implicit solutions is found. A number of explicit solutions are also exhibited.
We extract exact charged black-hole solutions with flat transverse sections in the framework of D-dimensional Maxwell-f(T) gravity, and we analyze the singularities and horizons based on both torsion and curvature invariants. Interestingly enough, we find that in some particular solution subclasses there appear more singularities in the curvature scalars than in the torsion ones. This difference disappears in the uncharged case, or in the case where f(T) gravity becomes the usual linear-in-T teleparallel gravity, that is General Relativity. Curvature and torsion invariants behave very differen…
We derive the field equations for topologically massive gravity coupled with the most general quadratic curvature terms using the language of exterior differential forms and a first order constrained variational principle. We find variational field equations both in the presence and absence of torsion. We then show that spaces of constant negative curvature (i.e. the anti de-Sitter space $AdS_3$) and constant torsion provide exact solutions.
Recently, in the Einstein gravity, Majhi and Padmanabhan proposed a straightforward and transparent way in obtaining the Bekenstein-Hawking entropy by using the approach based on the Virasoro algebra and central charge. In this work, we generalize their approach to the modified gravity with higher curvature corrections and show that their approach can successfully lead to the corresponding Wald entropy in the higher curvature gravity. Our result shows that the approach is physically general.
We continue our study of gravity described by the action density (-g)^(1/2)(R_ik^2+bR^2); and look for cosmological solutions of gravity coupled to dust, for the closed isotropic model. There is a solution that at t approaches zero has for the radius, a(t)=t/3^(1/2); in the absence of dust this solution holds for all time.
A low-energy perturbation theory is developed from the nonperturbative framework of covariant Loop Quantum Gravity (LQG) by employing the background field method. The resulting perturbation theory is a 2-parameter expansion in the semiclassical and low-energy regime. The two expansion parameters are the large spin and small curvature. The leading order effective action coincides with the Einstein-Hilbert action. The subleading corrections organized by the two expansion parameters give the modifications of Einstein gravity in quantum and high-energy regime from LQG. The perturbation theory deve…
In this paper we describe a model of a four-dimensional spherically symmetric black hole in a limiting curvature theory of gravity. In this theory the Einstein-Hilbert action is modified by adding to the action terms providing inequality constraints on chosen curvature invariants. We demonstrated that in such a model, instead of formation of the curvature singularity, the spacetime remains regular in the black hole interior. For black holes with gravitational radius much larger than the radius $\ell$ of the critical curvature the obtained solutions describe a space exponentially expanding in o…
By generalizing different recent works to the context of higher curvature gravity, we provide a unifying framework for three related results: (i) If an asymptotically AdS spacetime computes the entanglement entropies of ball-shaped regions in a CFT using a generalized Ryu-Takayanagi formula up to second order in state deformations around the vacuum, then the spacetime satisfies the correct gravitational equations of motion up to second order around AdS; (ii) The holographic dual of entanglement entropy in higher curvature theories of gravity is given by Wald entropy plus a particular correctio…
We consider the effects of higher curvature terms on a holographic dual description of boundary conformal field theory. Specifically, we consider three-dimensional gravity with a specific combination of Ricci tensor square and curvature scalar square, so called, new massive gravity. We show that a boundary entropy and an entanglement entropy are given by similar expression with those of the Einstein gravity case when we introduce an {\it effective} Newton's constant and an {\it effective} cosmological constant. We also show that the holographic g-theorem still holds in this extension, and we g…
We provide new formulas for the energy of black hole solutions in the anti-de Sitter (AdS) sector of a generic gravity theory with curvature-squared terms. This is achieved by the addition of counterterms of the extrinsic type (\emph{Kounterterms}) which produces a finite variation of the total action. The procedure is compatible with the boundary conditions in asymptotically AdS spaces within the framework of Quadratic Curvature Gravity (QCG) theory. As a consequence, without resorting to background-substraction, or linearized methods, the conserved charges correctly reproduce the energy of s…
In this note, we evaluate the Weyl-invariant quadratic curvature tensors for the particular Weyl's gauge field constructed in the $3+1$-dimensional noncompact Weyl-Einstein-Yang-Mills model. We subsequently extend the model to its higher curvature version. Here, we also compute Weyl-invariant extension of topological Gauss-Bonnet term for this specific choice of vector field.
We consider f(R) gravity theories in the presence of a scalar field minimally coupled to gravity with a self-interacting potential. When the scalar field backreacts to the metric we find at large distances scalarized Schwarzschild-AdS and Schwarzschild-AdS-like black hole solutions. At small distances due to strong curvature effects and the scalar dynamis we find a rich structure of scalarized black hole solutions. When the scalar field is conformally coupled to gravity we also find scalarized black hole solutions at small distances.
We disprove the widespread belief that higher order curvature theories of gravity in the metric-affine formalism are generally ghost-free. This is clarified by considering a sub-class of theories constructed only with the Ricci tensor and showing that the non-projectively invariant sector propagates ghost-like degrees of freedom. We also explain how these pathologies can be avoided either by imposing a projective symmetry or additional constraints in the gravity sector. Our results put forward that higher order curvature gravity theories generally remain pathological in the metric-affine (and …
The curvature of brane solutions in Yang-Mills matrix models is expressed in terms of conserved currents associated with global symmetries of the model. This implies a relation between the Ricci tensor and the energy-momentum tensor due to the basic matrix model action, without invoking an Einstein-Hilbert term. The coupling is governed by the extrinsic curvature of the brane embedding, which arises naturally for compactified brane solutions. The effective gravity on the brane is thereby related to the compactification moduli, and protected from quantum corrections due to the relation with glo…
The Second Law of black hole thermodynamics is shown to hold for arbitrarily complicated theories of higher curvature gravity, so long as we allow only linearized perturbations to stationary black holes. Some ambiguities in Wald's Noether charge method are resolved. The increasing quantity turns out to be the same as the holographic entanglement entropy calculated by Dong. It is suggested that only the linearization of the higher-curvature Second Law is important, when consistently truncating a UV-complete quantum gravity theory.
Standard methods for calculating the black hole entropy beyond general relativity are ambiguous when the horizon is non stationary. We fix these ambiguities in all quadratic curvature gravity theories, by demanding that the entropy be increasing at every time, for linear perturbations to a stationary black hole. Our result matches with the entropy formula found previously in holographic entanglement entropy calculations. We explicitly calculate the entropy increase for Vaidya-like solutions in Ricci-tensor gravity to show that (unlike the Wald entropy) the holographic entropy obeys a Second La…
We review the possible violation of the Equivalence Principle at finite temperature T in the framework of curvature based Extended Theories of Gravity. Specifically, we first show how it is possible to derive Equivalence Principle violation from Quantum Field Theory at non-vanishing T. Subsequently, we exhibit how this result can be precisely recovered by following an alternative path that envisages the employment of generalized Einstein equations with a temperature-dependent energy-momentum tensor. Finally, we adopt the latter formalism in the context of some Extended Gravity models to quanti…
Solving field equations in the context of higher curvature gravity theories is a formidable task. However in many situations, e.g., in the context of $f(R)$ theories the higher curvature gravity action can be written as Einstein-Hilbert action plus a scalar field action. We show that not only the action but the field equations derived from the action are also equivalent provided the spacetime is regular. We also demonstrate that such equivalence continues to hold even when gravitational field equations are projected on a lower dimensional hypersurface. We have further depicted explicit example…
A general geometrical scheme is presented for the construction of novel classical gravity theories whose solutions obey two-sided bounds on the sectional curvatures along certain subvarieties of the Grassmannian of two-planes. The motivation to study sectional curvature bounds comes from their equivalence to bounds on the acceleration between nearby geodesics. A universal minimal length scale is a necessary ingredient of the construction, and an application of the kinematical framework to static, spherically symmetric spacetimes shows drastic differences to the Schwarzschild solution of genera…
For the BTZ black hole in the Einstein gravity, a statistical entropy has been calculated to be equal to the Bekenstein-Hawking entropy. In this paper, the statistical entropy of the BTZ black hole in the higher curvature gravity is calculated and shown to be equal to the one derived by using the Noether charge method. This suggests that the equivalence of the geometrical and statistical entropies of the black hole is retained in the general diffeomorphism invariant theories of gravity. A relation between the cosmic censorship conjecture and the unitarity of the conformal field theory on the b…
We consider possible resolutions of singularities in a contracting anisotropic universe for a class of higher derivative gravity theories. We give evidence that for our models the big crunch singularity may be replaced by a nearly flat Minkowski-like phase before the Universe reenters an anisotropic expansion in a time-symmetric manner.
We consider the combined Yang Mills-Dilaton-Gravity system in the presence of a Gauss-Bonnet term as it appears in the $4D$ Effective Superstring Action. We give analytical arguments and demonstrate numerically the existence of black hole solutions with non-trivial dilaton and Yang Mills hair for the particular case of SU(2) gauge fields. The thermodynamical properties of the solutions are also discussed.
We study physical properties of constant curvature black holes (CCBHs) in Einstein anti-de Sitter (AdS) gravity. These objects, which are locally AdS throughout the space, are constructed from identifications of global AdS spacetime, in a similar fashion as Banados-Teitelboim-Zanelli (BTZ) black hole in three dimensions. We find that, in dimensions equal or greater than four, CCBHs have zero mass and angular momentum. Only in odd dimensions we are able to associate a nonvanishing conserved quantity to these solutions, which corresponds to the vacuum (Casimir) energy of the spacetime.
We find unitary and local theories of higher curvature gravity in the vielbein formalism, known as the Poincaré gauge theory by utilizing the equivalence to the ghost-free massive bigravity. We especially focus on three and four dimensions but extensions into a higher dimensional spacetime are straightforward. In three dimensions, a quadratic gravity $\mathcal{L}=R+T^2+R^2$, where $R$ is the curvature and $T$ is the torsion with indices omitted, is shown to be equivalent to zwei-dreibein gravity and free from the ghost at fully non-linear orders. In a special limit, new massive gravity is reco…
We study the cosmological perturbations of the recently proposed extension of non-linear massive gravity with a scalar field. The added scalar field ensures a new symmetry on the field space of the theory. The theory has the property of having a flat dS solution, in contrast to the standard dRGT massive gravity. The tensor part is the same as that of the standard dRGT and shows gravitational waves with a time dependent mass which modifies the dispersion relation. We obtain the curvature perturbation of the model on superhorizon scales for a specific choice of $ω=0$ and find that the theory doe…
In modified theories of gravity, higher curvature terms may be added to the Einstein-Hilbert action. Conventionally, the effects of the higher curvature terms on the black hole thermodynamics are rather difficult to obtain. In this paper, we show that, at least at the first order level, the corrections to the thermodynamics of the Schwarzschild black hole can be easily obtained, without solving the modified black hole metric. We examine some specific examples, which produces the known results in the literature, as well as those previously unknown. Furthermore, we find some nonperturbative exac…
It is shown that static solutions with a finite curvature at the horizon may exist in dilaton gravity at temperatures $T\neq T_{H}$ (including T=0) where $T_{H} $is the Hawking one. Hawking radiation is absent and the state of a system represents thermal excitation over the Boulware vacuum. The horizon remains unattainable for a observer because of thermal divergences in the stress-energy of quantum fields there. However, the curvature at the horizon is finite, when measured from outside, since these divergences are compensated by those in gradients of a dilaton field. Spacetimes under conside…
Constructs from conformal geometry are important in low dimensional gravity models, while in higher dimensions the higher curvature interactions of Lovelock gravity are similarly prominent. Considering conformal invariance in the context of Lovelock gravity leads to natural, higher-curvature generalizations of the Weyl, Schouten, Cotton and Bach tensors, with properties that straightforwardly extend those of their familiar counterparts. As a first application, we introduce a new set of conformally invariant gravity theories in D=4k dimensions, based on the squares of the higher curvature Weyl …
It is known that spatial curvature can stabilize extra dimensions in Lovelock gravity. In the present paper we study stability of the stabilization solutions in 3-d order Lovelock gravity. We show that in the case of negative spatial curvature of extra dimension space the stabilization solution is always stable. On the contrary, for positive spatial curvature the stability depends on the coupling constant values.
We study the particle spectrum and the unitarity of the generic n-dimensional Weyl-invariant quadratic curvature gravity theories around their (anti-)de Sitter [(A)dS] and flat vacua. Weyl symmetry is spontaneously broken in (A)dS and radiatively broken at the loop level in flat space. Save the three dimensional theory (which is the Weyl-invariant extension of the new massive gravity), the graviton remains massless and the unitarity requires that the only viable Weyl-invariant quadratic theory is the Weyl-invariant extension of the Einstein-Gauss-Bonnet theory. The Weyl gauge field on the othe…
Probing higher curvature gravity via ringdown with overtones
Kaichi Tokorozawa Secondary School（东京都立小平高等学校）；Department of Physics, Rikkyo University（立教大学物理系）；Department of Information, Artificial Intelligence and Data Science, Daiichi Institute of Technology（第一工业大学信息、人工智能与数据科学系）；Department of Physics, Tokyo Metropolitan University（东京都立大学物理系）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Department of Physics, College of Humanities and Sciences, Nihon University（日本大学人文学部物理系）
研究高曲率引力中黑洞的度规扰动，发现高曲率修正使有效势近邻区域变形，泛音模式的 quasinormal mode 频率偏离 GR 值更显著，分析高曲率引力模型中的回波波形。
We extend the Abbott-Deser-Tekin procedure of defining conserved quantities of asymptotically constant-curvature spacetimes, and give an analogous expression for the conserved charges of geometries that are solutions of quadratic curvature gravity models in generic D-dimensions and that have arbitrary asymptotes possessing at least one Killing isometry. We show that the resulting charge expression correctly reduces to its counterpart when the background is taken to be a space of constant curvature and, moreover, is background gauge invariant. As applications, we compute and comment on the ener…
A new numerical integration method for examining a black hole structure was realized. Black hole solutions with dilatonic hair of 4D low energy effective SuperString Theory action with Gauss-Bonnet quadratic curvature contribution were studied, using this method, inside and outside the event horizon. Thermodynamical properties of this solution were also studied.
It is a known fact that the Kerr-Schild type solutions in general relativity satisfy both exact and linearized Einstein field equations. We show that this property remains valid also for a special class of the Kerr-Schild metrics in arbitrary dimensions in generic quadratic curvature theory. In addition to the AdS-wave (or Siklos) metric which represents plane waves in an AdS background, we present here a new exact solution, in this class, to the quadratic gravity in D-dimensions which represents a spherical wave in an AdS background. The solution is a special case of the Kundt metrics belongi…
We compute the Euclidean action for constant curvature black holes (CCBHs), as an attempt to associate thermodynamic quantities to these solutions of Einstein anti-de Sitter (AdS) gravity. CCBHs are gravitational configurations obtained by identifications along isometries of a $D$-dimensional globally AdS space, such that the Riemann tensor remains constant. Here, these solutions are interpreted as extended objects, which contain a $(D-2)$-dimensional de-Sitter brane as a subspace. Nevertheless, the computation of the free energy for these solutions shows that they do not obey standard thermod…
We demonstrate linear stability of the dilatonic Black Holes appearing in a string-inspired higher-derivative gravity theory with a Gauss- Bonnet curvature-squared term. The proof is accomplished by mapping the system to a one-dimensional Schrodinger problem which admits no bound states. This result is important in that it constitutes a linearly stable example of a black hole that bypasses the `no-hair conjecture'. However, the dilaton hair is `secondary'in the sense that it is not accompanied by any new quantum number for the black hole solution.
We show that causality constrains the sign of quartic Riemann corrections to the Einstein-Hilbert action. Our constraint constitutes a restriction on candidate theories of quantum gravity.
We investigate the phase diagram of quantum gravity with a vertex expansion about constantly-curved backgrounds. The graviton two- and three-point function are evaluated with a spectral sum on a sphere. We obtain, for the first time, curvature-dependent UV fixed point functions of the dynamical fluctuation couplings $g^*(R)$, $μ^*(R)$, and $λ_3^*(R)$, and the background $f(R)$-potential. Based on these fixed point functions we compute solutions to the quantum and the background equation of motion with and without Standard Model matter. We have checked that the solutions are robust against chan…
We show how gravitational actions, in particular the Einstein-Hilbert action, can be obtained from additional terms in Yang-Mills matrix models. This is consistent with recent results on induced gravitational actions in these matrix models, realizing space-time as 4-dimensional brane solutions. It opens up the possibility for a controlled non-perturbative description of gravity through simple matrix models, with interesting perspectives for the problem of vacuum energy. The relation with UV/IR mixing and non-commutative gauge theory is discussed.
Weyl-invariant extensions of three-dimensional New Massive Gravity, generic n-dimensional Quadratic Curvature Gravity theories and three-dimensional Born-Infeld gravity theory are analyzed in details. As required by Weyl-invariance, the actions of these gauge theories do not contain any dimensionful parameter; therefore the local symmetry is spontaneously broken in (Anti) de Sitter vacua in analogy with the Standard Model Higgs mechanism. About the flat vacuum, symmetry breaking mechanism is more complicated: The conformal symmetry is radiatively broken (at two loop level in 3-dimensions and a…
The second order perturbation calculations for gravity wave and Einstein equation for space time and matter are presented for the FRW metric cosmological model. While exact equations are found, suitable approximations are made to obtain definite results. In the gravity wave case the small wavelength case allows nearly locally flat background for obtaining a fit to the WMAP data. In the density and curvature case the FRW background is retained for the length scale of WMAP. Clustering and inhomogeneity are understood. The gravity wave ripples from Big Bang couple nonlinearly and redistribute the…
In the context of CA conjecture for holographic complexity, we study the action growth rate at late time approximation for general quadratic curvature theory of gravity. We show how the Lloyd's bound saturates for charged and neutral black hole solutions. We observe that a second singular point may modify the action growth rate to a value other than the Lloyd's bound. Moreover, we find the universal terms that appear in the divergent part of complexity from computing the bulk and joint terms on a regulated WDW patch.
Curvature squared terms, when added to the Einstein-Hilbert action and treated non-perturbatively, generically result in the propagation of an extra massive scalar state and an extra massive spin-2 ghost state. Using the Stueckelberg trick, we study the high-energy limit in which the mass of the spin-2 state is taken to zero, with strong- coupling scales held fixed. The Stueckelberg approach makes transparent the interplay between the ghost graviton and the healthy graviton which allows the theory to evade the usual lambda-3 strong coupling scale of massive gravity and become renormalizable, a…
We suggest a thick braneworld model in the squared curvature gravity theory. Despite the appearance of higher order derivatives, the localization of gravity and various bulk matter fields is shown to be possible. The existence of the normalizable gravitational zero mode indicates that our four-dimensional gravity is reproduced. In order to localize the chiral fermions on the brane, two types of coupling between the fermions and the brane forming scalar is introduced. The first coupling leads us to a Schrödinger equation with a volcano potential, and the other a Pöschl-Teller potential. In both…
Timelike Entanglement Entropy in Higher Curvature Gravity
Jilin University（吉林大学）；Ningbo University（宁波大学）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（阿尔伯特·爱因斯坦研究所））；Inner Mongolia University（内蒙古大学）
In this paper we discuss singularity theorems in quantum gravity using effective field theory methods. To second order in curvature, this effective field theory contains two new degrees of freedom which have important implications for the derivation of these energy theorems: a massive spin-2 field and a massive spin-0 field. Using an explicit mapping of this theory from the Jordan frame to the Einstein frame, we show that both the cosmological and the black hole singularity theorems may not hold due to the presence of a massive spin-2 field in the particle spectrum of quantum gravity. Furtherm…
