We formulate the most general gravitational models with constant negative curvature ("hyperbolic gravity") on an arbitrary orientable two-dimensional surface of genus $g$ with $b$ circle boundaries in terms of a $\text{PSL}(2,\mathbb R)_\partial$ gauge theory of flat connections. This includes the usual JT gravity with Dirichlet boundary conditions for the dilaton field as a special case. A key ingredient is to realize that the correct gauge group is not the full $\text{PSL}(2,\mathbb R)$, but a subgroup $\text{PSL}(2,\mathbb R)_{\partial}$ of gauge transformations that go to $\text{U}(1)$ local rotations on the boundary. We find four possible classes of boundary conditions, with associated boundary terms, that can be applied to each boundary component independently. Class I has five inequivalent variants, corresponding to geodesic boundaries of fixed length, cusps, conical defects of fixed angle or large cylinder-shaped asymptotic regions with boundaries of fixed lengths and extrinsic curvatures one or greater than one. Class II precisely reproduces the usual JT gravity. In particular, the crucial extrinsic curvature boundary term of the usual second order formulation is automatically generated by the gauge theory boundary term. Class III is a more exotic possibility for which the integrated extrinsic curvature is fixed on the boundary. Class IV is the Legendre transform of class II; the constraint of fixed length is replaced by a boundary cosmological constant term.
In the context of a canonical quantization of general relativity, one can deform the loop gravity phase space on a graph by replacing the T*SU(2) phase space attached to each edge by SL(2,C) seen as a phase space. This deformation is supposed to encode the presence of a non-zero cosmological constant. Here we show how to parametrize this phase space in terms of spinor variables, thus obtaining deformed spinor networks for loop gravity, with a deformed action of the gauge group SU(2) at the vertices. These are to be formally interpreted as the generalization of loop gravity twisted geometries t…
We prove that semiclassical gravity in conformally static, globally hyperbolic spacetimes with a massless, conformally coupled Klein-Gordon field is well posed, when viewed as a coupled theory for the dynamical conformal factor of the metric and the Klein-Gordon theory. Namely, it admits unique and stable solutions whenever constrained fourth-order initial data for the conformal factor and suitably defined Hadamard initial data for the Klein-Gordon state are provided on a spacelike Cauchy surface. As no spacetime symmetries are imposed on the conformal factor, the present result implies that, …
We consider complex invariants associated with compact real three-dimensional hyperbolic spaces. The contribution of the Chern-Simons invariants of irreducible U(n)-flat connections on hyperbolic fibered manifolds to the low order expansion of the quantum gravitational path integral is analyzed.
It is shown here that symmetric hyperbolicity, which guarantees well-posedness, leads to a set of two inequalities for matrices whose elements are determined by a given theory. As a part of the calculation, carried out in a mostly-covariant formalism, the general form for the symmetrizer, valid for a general Lagrangian theory, was obtained. When applied to nonlinear electromagnetism linearly coupled to curvature, the inequalities lead to strong constraints on the relevant quantities, which were illustrated with applications to particular cases. The examples show that non-linearity leads to con…
Two first order strongly hyperbolic formulations of scalar-tensor theories of gravity (STT) allowing nonminimal couplings (Jordan frame) are presented along the lines of the 3+1 decomposition of spacetime. One is based on the Bona-Masso formulation while the other one employs a conformal decomposition similar to that of Baumgarte-Shapiro-Shibata-Nakamura. A modified Bona-Masso slicing condition adapted to the scalar-tensor theory is proposed for the analysis. This study confirms that the scalar-tensor theory posses a well posed Cauchy problem even when formulated in the Jordan frame.
The well-posedness of the gravitational equations of $f(R)$ gravity are studied in this paper. Three formulations of the $f(R)$ gravity with dynamical shifts (which are all based on the Arnowitt-Deser-Misner (ADM) formalism of the equations) are investigated. These three formulations are all proved to be strongly hyperbolic by pseudodifferential reduction. The first one is the Baumagarte-Shapiro-Shibata-Nakamura (BSSN) formulation with the so-called "hyperbolic $K$-driver" condition and the "hyperbolic Gamma driver" condition. The second one is the ADM formulation with modified harmonic gauge …
We consider a description of lattice gravity in six dimensions, where the two extra dimensions have been compactified on a warped hyperbolic disk of constant curvature. We analyze a fine-grained latticization of the hyperbolic disk in the context of an effective theory for massive gravitons. We find that in six-dimensional warped hyperbolic space, lattice gravity appears near the boundary of the disk more weakly coupled than in discretized five-dimensional flat or warped space. Specifically, near the IR branes, the local strong coupling scale can become as large as the local Planck scale.
We study a geometry of the partition function which is defined in terms of a solution of the five-term relation. It is shown that the 3-dimensional hyperbolic structure or Euclidean AdS_3 naturally arises in the classical limit of this invariant. We discuss that the oriented ideal tetrahedron can be assigned to the partition function of string.
4-manifolds have special topological properties which can be used to get a different view on quantum mechanics. One important property (connected with exotic smoothness) is the natural appearance of 3-manifold wild embeddings (Alexanders horned sphere) which can be interpreted as quantum states. This relation can be confirmed by using the Turaev-Drinfeld quantization procedure. Every part of the wild embedding admits a hyperbolic geometry uncovering a deep connection between quantum mechanics and hyperbolic geometry. Then the corresponding symmetry is used to get a dimensional reduction from 4…
The recent formulation of locally covariant quantum field theory may open the way towards a background independent perturbative formulation of Quantum Gravity.
We discuss a simple symplectic formulation for tetrad gravity that leads to the real Ashtekar variables in a direct and transparent way. It also sheds light on the role of the Immirzi parameter and the time gauge.
Solutions of Horava gravity that are asymptotically Lifshitz are explored. General near boundary expansions allow the calculation of the mass of these spacetimes via a Hamiltonian method. Both analytic and numeric solutions are studied which exhibit a causal boundary called the universal horizon, and are therefore black holes of the theory. The thermodynamics of an asymptotically Anti-de Sitter Horava black hole are verified.
We proceed to study a (1+1)-dimensional dilaton gravity system with a hyperbolic dilaton potential. Introducing a couple of new variables leads to two copies of Liouville equations with two constraint conditions. In particular, in conformal gauge, the constraints can be expressed with Schwarzian derivatives. We revisit the vacuum solutions in light of the new variables and reveal its dipole-like structure. Then we present a time-dependent solution which describes formation of a black hole with a pulse. Finally, the black hole thermodynamics is considered by taking account of conformal matters …
This work is mainly devoted to constructing a multisymplectic description of Lovelock's gravity, which is an extension of General Relativity. We establish a Griffiths variational problem for the Lovelock Lagrangian, obtaining the geometric form of the corresponding field equations. We give the unified Lagrangian--Hamiltonian formulation of this model and we study the correspondence between the unified formulations for the Einstein--Hilbert and the Einstein--Palatini models of gravity.
Loop Quantum Gravity (LQG) is an attempt to describe the quantum gravity regime. Introducing a non-zero cosmological constant $Λ$ in this context has been a withstanding problem. Other approaches, such as Chern-Simons gravity, suggest that quantum groups can be used to introduce $Λ$ in the game. Not much is known when defining LQG with a quantum group. Tensor operators can be used to construct observables in any type of discrete quantum gauge theory with a classical/quantum gauge group. We illustrate this by constructing explicitly geometric observables for LQG defined with a quantum group and…
The classical and quantum formulations for open systems related to dissipative dynamics are constructed on a complex hyperbolic ring, following universal symmetry principles, and considering the double thermal fields approach for modeling the system of interest, and the environment. The hyperbolic rotations are revealed as an underlying internal symmetry for the dissipative dynamics, and a chemical potential is identified as conjugate variable to the charge operator, and thus a grand partition function is constructed. As opposed to the standard scheme, there are not patologies associated with …
In this paper we study the hyperbolicity of the equations of motion for the most general Horndeski theory of gravity in a generic "weak field" background. We first show that a special case of this theory, namely Einstein-dilaton-Gauss-Bonnet gravity, fails to be strongly hyperbolic in any generalised harmonic gauge. We then complete the proof that the most general Horndeski theory which, for weak fields, is strongly hyperbolic in a generalised harmonic gauge is simply a "k-essence" theory coupled to Einstein gravity and that adding any more general Horndeski term will result in a weakly, but n…
A first-order formulation of gravity is developed in which the fundamental fields consist of an SL(2,C) connection and two spinor-valued 1-forms. It is shown that the first term of an expansion of the Einstein-Hilbert action leads to an action for these fields which consists of dynamic L2 inner products of their covariant derivatives, resembling reasonable generalisations of the terms found in the actions of typical gauge theories on Minkowski spacetime. If additional terms corresponding to other forces and matter, formulated in the same manner, are then included, this approach may shed new li…
We study the characteristic structure of the Einstein-Hilbert (EH) action when modifications of the form of $R^2,~ R_{μν}^2$, $R_{μνρσ}^2$ and $C_{μνρσ}^2$ are included. We show that when these quadratic terms are significant, the initial value problem is generically ill-posed. We do so by demanding the hyperbolicity of the effective metric for propagation of perturbations. Here, we find a general expression for the effective metric in field space and calculate it explicitly about the cosmological Friedman-Robertson-Walker (FRW) spacetime, and the spherically symmetric Schwarzschild solution. …
A novel realization of the gravity-mediated SUSY breaking is presented taking into account a continuous global R symmetry. Consistently with it, we employ a linear superpotential for the hidden sector superfield and a Kaehler potential parameterizing the SU(1,1)/U(1) Kaehler manifold with constant curvature -1/2. The classical vacuum energy vanishes without unnatural fine tuning and non-vanishing soft SUSY-breaking parameters, of the order of the gravitino mass, arise. A solution to the mu problem of MSSM may be also achieved by conveniently applying the Giudice-Masiero mechanism. The potentia…
First order lagrangians for the Weyl invariant formulation of Unimodular Gravity are proposed. Several alternatives are examined; in some of them first and second order are equivalent in a certain gauge only.
We apply the ADM approach to obtain a Hamiltonian description of the Einstein-Hilbert action. In doing so we add four new ingredients: (i) We eliminate the diffeomorphism constraints. (ii) We replace the densities $\sqrt g$ by a function $\f(x,g_{ij})$ with the help of a fixed metric $χ$ such that the Lagrangian and hence the Hamiltonian are functions. (iii) We consider the Lagrangian to be defined in a fiber bundle with base space $\so$ and fibers F(x) which can be treated as Lorentzian manifolds equipped with the Wheeler-DeWitt metric. It turns out that the fibers are globally hyperbolic. (i…
We provide a metric-like formulation of the spin-3 gravity in three dimensions. It is shown that the Chern-Simons formulation of the spin-3 gravity can be reformulated as a Einstein-Cartan-Sciama-Kibble theory coupled with the higher-spin matter fields. A duality-like transformation is also identified from this metric-like formulation.
We derive the flow equation for the gravitational effective average action in an $f(R)$ truncation on hyperbolic spacetimes using the exponential parametrization of the metric. In contrast to previous works on compact spaces, we are able to evaluate traces exactly using the optimised cutoff. This reveals in particular that all modes can be integrated out for a finite value of the cutoff due to a gap in the spectrum of the Laplacian, leading to the effective action. Studying polynomial solutions, we find poorer convergence than has been found on compact spacetimes even though at small curvature…
The conformal structure of Brans-Dicke gravity action is carefully studied. It is discussed that Brans-Dicke gravity action has definitely no conformal invariance. It is shown, however, that this lack of conformal invariance enables us to demonstrate that Brans-Dicke theory appears to have a better short-distance behavior than Einstein gravity as far as Euclidean path integral formulation for quantum gravity is concerned.
In this paper we discuss gravity in the light-front formulation (light-cone gauge) and show how possible counterterms arise. We find that Poincare invariance is not enough to find the three-point counterterms uniquely. Higher-spin fields can intrude and mimic three-point higher derivative gravity terms. To select the correct term we have to use the remaining reparametrization invariance that exists after the gauge choice. We finally sketch how the corresponding programme for N=8 Supergravity should work.
Differential geometric formulation of quantum gauge theory of gravity is studied in this paper. The quantum gauge theory of gravity which is proposed in the references hep-th/0109145 and hep-th/0112062 is formulated completely in the framework of traditional quantum field theory. In order to study the relationship between quantum gauge theory of gravity and traditional quantum gravity which is formulated in curved space, it is important to find the differential geometric formulation of quantum gauge theory of gravity. We first give out the correspondence between quantum gauge theory of gravity…
This paper is devoted to the Hamiltonian analysis of bimetric gravity in vierbein formulation. We identify all constraints and determine their nature. We also show an existence of additional constraint so that the scalar mode can be eliminated.
We make it precise what it means to have a connection with torsion as solution of the Einstein equations. While locally the theory remains the same, the new formulation allows for topologies that would have been excluded in the standard formulation of gravity. In this formulation it is possible to couple arbitrary torsion to gauge fields without breaking the gauge invariance.
We address some recent concerns about the absence of the Boulware-Deser ghost in the Stuckelberg formulation of nonlinear massive gravity. First we provide general arguments for why any ghost analysis in the Stuckelberg formulation has to agree with existing consistency proofs that have been carried out without using Stuckelberg fields. We then demonstrate the absence of the ghost at the completely nonlinear level in the Stuckelberg formulation of the minimal massive gravity action. The constraint that removes the ghost field and the associated secondary constraint that eliminates its conjugat…
Recently, corrections to the standard Einstein-Hilbert action are proposed to explain the current cosmic acceleration in stead of introducing dark energy. We discuss the Palatini formulation of the modified gravity with a $\ln R$ term suggested by Nojiri and Odintsov. We show that in the Palatini formulation, the $\ln R$ gravity can drive a current exponential accelerated expansion and it reduces to the standard Friedmann evolution for high redshift region. We also discuss the equivalent scalar-tensor formulation of the theory. We indicate that the $\ln R$ gravity may still have a conflict wit…
We present a canonical formulation of gravity theories whose Lagrangian is an arbitrary function of the Riemann tensor. Our approach allows a unified treatment of various subcases and an easy identification of the degrees of freedom of the theory.
In this note we construct a dual formulation of gravity where the main dynamical object is affine connection. We start with the well known first order Palatini formulation but in (Anti) de Sitter space instead of flat Minkowski space as a background. The final result obtained by solving equations for the metric is the Lagrangian written by Eddington in his book in 1924. Also there is an interesting connection with attempts to construct gravitational analog of Born-Infeld electrodynamics.
Recently, corrections to the standard Einstein-Hilbert action are proposed to explain the current cosmic acceleration in stead of introducing dark energy. In the Palatini formulation of those modified gravity models, there is an important observation due to Arkani-Hamed: matter loops will give rise to a correction to the modified gravity action proportional to the Ricci scalar of the metric. In the presence of such term, we show that the current forms of modified gravity models in Palatini formulation, specifically, the 1/R gravity and $\ln R$ gravity, will have phantoms. Then we study the pos…
Starting from a knowledge of certain identities in the Lagrangian description, the diffeomorphism transformations of metric and connection are obtained for both the second order (metric) and the first order (Palatini) formulations of gravity. The transformation laws of the connection and the metric are derived independently in the Palatini formulation in contrast to the metric formulation where the gauge variation of the connection is deduced from the gauge variation of the metric.
In this paper we consider a possibility to construct dual formulation of gravity where the main dynamical field is the Lorentz connection ω_μ^{ab} and not that of tetrad e_μ^a or metric g_μν. Our approach is based on the usual dualization procedure which uses first order parent Lagrangians but in (Anti) de Sitter space and not in the flat Minkowski one. It turns out that in d=3 dimensions such dual formulation is related with the so called exotic parity-violating interactions for massless spin-2 particles.
We present several theories of four-dimensional gravity in the Plebanski formulation, in which the tetrads and the connections are the independent dynamical variables. We consider the relation between different versions of gravitational theories: Einstenian, dual, 'mirror' gravities and gravity with torsion. According to Plebanski's assumption, our world, in which we live, is described by the self-dual left-handed gravity. We propose that if the Mirror World exists in Nature, then the 'mirror gravity' is the right-handed anti-self-dual gravity with broken mirror parity. Considering a special v…
The quantization of the gravitational Chern-Simons coefficient is investigated in the framework of $ISO(2,1)$ gauge gravity. Some paradoxes involved are cured. The resolution is largely based on the inequivalence of $ISO(2,1)$ gauge gravity and the metric formulation. Both the Lorentzian scheme and the Euclidean scheme lead to the coefficient quantization, which means that the induced spin is not quite exotic in this context.
In this contribution, we re-assess the subject of topological gravity by following the Shift Supersymmetry formalism. The gauge-fixing of the theory goes under the Batallin-Vilkovisky (BV) prescription based on a diagram that contains both ghost and anti-ghost superfields, associated to the super-vielbein and the super-Lorentz connection. We extend the formulation of the topological gravity action to an arbitrary number of dimensions of the shift superspace by adopting a formulation based on the gauge-fixing for BF-type models.
We present a construction employing a type IIA supergravity and 3-form flux background together with an NS5-brane that localises massless gravity near the 5-brane worldvolume. The nonsingular underlying type IIA solution is a lift to 10D of the vacuum solution of the 6D Salam-Sezgin model and has a hyperbolic ${\cal H}^{(2,2)}\times S^1$ structure in the lifting dimensions. A fully back-reacted solution including the NS5-brane is constructed by recognising the 10D Salam-Sezgin vacuum solution as a "brane resolved through transgression." The background hyperbolic structure plays a key rôle in g…
We show that the recent claim that the 2+1 dimensional Ashtekar formulation for General Relativity has a finite number of physical degrees of freedom is not correct.
In the attempts toward a quantum gravity theory, general relativity faces a serious difficulty since it is non-renormalizable theory. Hořava-Lifshitz gravity offers a framework to circumvent this difficulty, by sacrificing the local Lorentz invariance at ultra-high energy scales in exchange of power-counting renormalizability. The Lorentz symmetry is expected to be recovered at low and medium energy scales. If gravitation is to be described by a Hořava-Lifshitz gravity theory there are a number of issues that ought to be reexamined in its context, including the question as to whether this grav…
Various Hamiltonian formulations of f(R) gravity can be found in the literature. Some authors follow the Ostrogradsky treatment of higher derivative theories and introduce as extra variables first order time derivatives of the metric (typically the extrinsic curvature). Some others take advantage of the conformal equivalence of f(R) theory with Einstein's gravity coupled to a scalar field and introduce as an extra variable the scalar curvature R itself, which includes second time derivatives of the metric. We show that, contrarily to some claims, these formulations are related by canonical tra…
We propose a simple method for deriving the constraints of the de Rham-Gabadadze-Tolley model in the metric and the Lagrangian formulation, as possible as keeping the Lorentz covariance. In our formulation, it is not necessary to use the Hamilton analysis, the vielbein formulation, nor the Stuckelberg trick for showing the Boulwer-Deser ghost-freeness. It realizes the Lorentz covariant expressions of the constraints in a certain parameter region.
We show that two-dimensional topological BF theories coupled to particles carrying non-Abelian charge admit a new coupling involving the Lagrange multiplier field. When applied to the gauge theoretic formulation of dilatonic gravity it gives rise to a source term for the gravitational field. We show that the system admits black hole solutions.
We consider extensions of the Einstein-Hilbert Lagrangian to a general functional of metric and Riemann curvature tensor. A given such Lagrangian describes two different theories depending on considering connection and metric (Palatini formulation), or only the metric (metric formulation) as independent dynamical degrees of freedom. Equivalence of the Palatini and metric formulations at the level of equations of motion, which as we will argue is a manifestation of the Equivalence Principle, is the physical criterion that restricts form of the Lagrangians of modified gravity theories. We prove …
We show that the asymptotic structures of topologically massive gravity in the spacelike stretched AdS sector and an $SL(2,R)\times U(1)$ Chern-Simons gauge theory can be identified by adopting a natural correspondence between their fields and coupling constants.
The Hamiltonian formulation of Mimetic Gravity is formulated. Although there are two more equations than those of general relativity, these are proved to be the constraint equation and the conservation of energy-momentum tensor. The Poisson brackets are then computed and closure is proved. At the end, Wheeler-DeWitt equation was solved for a homogeneous and isotropic universe. This was done first for a vanishing potential where agreement with the dust case was shown, and then for a constant potential.
