Cotton Tensor Research Articles

Conformal Structures of Static Vacuum Data

In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity. For the solution-jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the Cotton tensor of the initial three-metric h satisfies a certain conformally invariant condition (*) at space-like infinity, it is sufficient that h be asymptotically static at space-like infinity. The purpose of this article is to characterize the gap between these conditions. We show that with the class of metrics which satisfy condition (*) on the Cotton tensor and a certain non-degeneracy requirement is associated a one-form $\kappa$ with conformally invariant differential $d\kappa$. We provide two criteria: If $h$ is real analytic, $\kappa$ is closed, and one of it integrals satisfies a certain equation then h is conformal to static data near space-like infinity. If h is smooth, $\kappa$ is asymptotically closed, and one of it integrals satisfies a certain equation asymptotically then h is asymptotically conformal to static data at space-like infinity.

Conformal pure radiation with parallel rays

We define pure radiation metrics with parallel rays to be n-dimensional pseudo-Riemannian metrics that admit a parallel null line bundle K and whose Ricci tensor vanishes on vectors that are orthogonal to K. We give necessary conditions in terms of the Weyl, Cotton and Bach tensors for a pseudo-Riemannian metric to be conformal to a pure radiation metric with parallel rays. Then, we derive conditions in terms of the tractor calculus that are equivalent to the existence of a pure radiation metric with parallel rays in a conformal class. We also give analogous results for n-dimensional pseudo-Riemannian pp-waves.

Axial anomalies of Lifshitz fermions

AbstractWe compute the axial anomaly of a Lifshitz fermion theory with anisotropic scaling z = 3 which is minimally coupled to geometry in 3+1 space‐time dimensions. We find that the result is identical to the relativistic case using path integral methods. An independent verification is provided by showing with spectral methods that the η‐invariant of the Dirac and Lifshitz fermion operators in three dimensions are equal. Thus, by the integrated form of the anomaly, the index of the Dirac operator still accounts for the possible breakdown of chiral symmetry in non‐relativistic theories of gravity. We apply this framework to the recently constructed gravitational instanton backgrounds of Hořava–Lifshitz theory and find that the index is non‐zero provided that the space‐time foliation admits leaves with harmonic spinors. Using Hitchin's construction of harmonic spinors on Berger spheres, we obtain explicit results for the index of the fermion operator on all such gravitational instanton backgrounds with SU(2) × U(1) isometry. In contrast to the instantons of Einstein gravity, chiral symmetry breaking becomes possible in the unimodular phase of Hořava–Lifshitz theory arising at λ = 1/3 provided that the volume of space is bounded from below by the ratio of the Ricci to Cotton tensor couplings raised to the third power. Some other aspects of the anomalies in non‐relativistic quantum field theories are also discussed.

Classification of six derivative Lagrangians of gravity and static spherically symmetric solutions

We classify all the six-derivative Lagrangians of gravity, whose traced field equations are of second or third order, in arbitrary dimensions. In the former case, the Lagrangian in dimensions greater than six reduces to an arbitrary linear combination of the six-dimensional Euler density and the two linearly independent cubic Weyl invariants. In five dimensions, besides the independent cubic Weyl invariant, we obtain an interesting cubic combination, whose field equations for static spherically symmetric spacetimes are of second order. In the latter case, in arbitrary dimensions we obtain two combinations, which in dimension three, are equivalent to the complete contraction of two Cotton tensors. Moreover, we also recover all the conformal anomalies in six dimensions. Finally, we present the general static, spherically symmetric solution for some of these Lagrangians.

Four dimensional string solutions in Hořava-Lifshitz gravity

We investigate string-like solutions in four dimensions based on Hořava-Lifshitz gravity. For a restricted class of solutions where the Cotton tensor vanishes, we find that the string-like solutions in Einstein gravity including the BTZ black strings are solutions in Hořava-Lifshitz gravity as well. The geometry is warped in the same way as in Einstein gravity, but the “conformal” lapse function is not constrained in Hořava-Lifshitz gravity. It turns out that if λ ≠ 1, there exist no other solutions. For the value of model parameter with which Einstein gravity recovers in IR limit (i.e., λ = 1), there exists an additional solution of which the conformal lapse function is determined. Interestingly, this solution admits a uniform BTZ black string along the string direction, which is distinguished from the warped BTZ black string in Einstein gravity. We also point out that this solution is exactly the same with the black string solution obtained in a low-energy string theory action including the dilaton and the Kalb-Ramond fields.

Conserved current for the Cotton tensor, black hole entropy and equivariant Pontryagin forms

The Chern–Simons Lagrangian density in the space of metrics of a three-dimensional manifold M is not invariant under the action of diffeomorphisms on M. However, its Euler–Lagrange operator can be identified with the Cotton tensor, which is invariant under diffeomorphims. As the Lagrangian is not invariant, the Noether theorem cannot be applied to obtain conserved currents. We show that it is possible to obtain an equivariant conserved current for the Cotton tensor by using the first equivariant Pontryagin form on the bundle of metrics. Finally we define a Hamiltonian current which gives the contribution of the Chern–Simons term to the black hole entropy, energy and angular momentum.

Geometric flows in Hořava-Lifshitz gravity

We consider instanton solutions of Euclidean Hořava-Lifshitz gravity in four dimensions satisfying the detailed balance condition. They are described by geometric flows in three dimensions driven by certain combinations of the Cotton and Ricci tensors as well as the cosmological-constant term. The deformation curvature terms can have competing behavior leading to a variety of fixed points. The instantons interpolate between any two fixed points, which are vacua of topologically massive gravity with Λ > 0, and their action is finite. Special emphasis is placed on configurations with SU(2) isometry associated with homogeneous but generally non-isotropic Bianchi IX model geometries. In this case, the combined Ricci-Cotton flow reduces to an autonomous system of ordinary differential equations whose properties are studied in detail for different couplings. The occurrence and stability of isotropic and anisotropic fixed points are investigated analytically and some exact solutions are obtained. The corresponding instantons are classified and they are all globally $ \mathbb{R} \times {S^3} $ and complete spaces. Generalizations to higher-dimensional gravities are also briey discussed.

TIMELIKE GEODESIC MOTION IN HORAVA–LIFSHITZ SPACE–TIME

Recently a nonrelativistic renormalizable theory of gravitation has been proposed by P. Horava. When restricted to satisfy the condition of detailed balance, this theory is intimately related to topologically massive gravity in three dimensions, and the geometry of the Cotton tensor. At long distances, this theory is expected to flow to the relativistic value λ = 1, and could therefore serve as a possible candidate for a UV completion of Einstein's general relativity or an infrared modification thereof. In this paper under allowing the lapse function to depend on the spatial coordinates xias well as t, we obtain the spherically symmetric solutions. And then by analyzing the behavior of the effective potential for the particle, we investigate the timelike geodesic motion of particle in the Horava–Lifshitz space–time. We find that the nonradial particle falls from a finite distance to the center along the timelike geodesics when its energy is in an appropriate range. However, we find that it is complexity for radial particle along the timelike geodesics. There are three different cases due to the energy of radial particle: (i) when the energy of radial particle is higher than a critical value EC, the particle will fall directly from infinity to the singularity; (ii) when the energy of radial particle equals to the critical value EC, the particle orbit at r = rCis unstable, i.e. the particle will escape from r = rCto the infinity or to the singularity, depending on the initial conditions of the particle; (iii) when the energy of radial particle is in a proper range, the particle will rebound to the infinity or plunge to the singularity from a infinite distance, depending on the initial conditions of the particle.

The mixmaster universe in Hořava–Lifshitz gravity

We consider spatially homogeneous (but generally non-isotropic) cosmologies in the recently proposed Hořava–Lifshitz gravity and compare them to those of general relativity using Hamiltonian methods. In all cases, the problem is described by an effective point particle moving in a potential well with exponentially steep walls. Focusing on the closed-space cosmological model (Bianchi type IX), the mixmaster dynamics is now completely dominated by the quadratic Cotton tensor potential term for a very small volume of the universe. Unlike general relativity, where the evolution toward the initial singularity always exhibits chaotic behavior with alternating Kasner epochs, the anisotropic universe in Hořava–Lifshitz gravity (with parameter λ > 1/3) is described by a particle moving in a frozen potential well with fixed (but arbitrary) energy E. Alternating Kasner epochs still provide a good description of the early universe for very large E, but the evolution appears to be non-ergodic. For very small E there are harmonic oscillations around the fully isotropic model. The question of chaos remains open for intermediate energy levels.

On higher derivatives in 3D gravity and higher-spin gauge theories

The general second-order massive field equations for arbitrary positive integer spin in three spacetime dimensions, and their "self-dual" limit to first-order equations, are shown to be equivalent to gauge-invariant higher-derivative field equations. We recover most known equivalences for spins 1 and 2, and find some new ones. In particular, we find a non-unitary massive 3D gravity theory with a 5th order term obtained by contraction of the Ricci and Cotton tensors; this term is part of an N=2 super-invariant that includes the "extended Chern-Simons" term of 3D electrodynamics. We also find a new unitary 6th order gauge theory for "self-dual" spin 3.

Generalized uncertainty principle and Hořava–Lifshitz gravity

We explore a connection between generalized uncertainty principle (GUP) and modified Hořava–Lifshitz (HL) gravity. The GUP density function may be replaced by the cutoff function for the renormalization group of modified Hořava–Lifshitz gravity. We find the GUP-corrected graviton propagators and compare these with tensor propagators in the HL gravity. Two are qualitatively similar, but the p5-term arisen from Cotton tensor is missed in the GUP-corrected graviton propagator.

CHIRAL BLACK HOLE IN THREE-DIMENSIONAL GRAVITATIONAL CHERN–SIMONS

A chiral black hole can be defined by the three-dimensional pure gravitational Chern–Simons action as an independent gravitational theory. The third-order derivative of the Cotton tensor gives a dimensional constant which plays a role of the cosmological constant. The handedness of angular momentum depends on the signature of the Chern–Simons coefficient. Even in the massless black hole which corresponds to the static black hole, it has a nonvanishing angular momentum. We also discuss three different entropies depending on the methods of Wald, Cardy and 't Hooft.

Topological regularization and self-duality in four-dimensional anti–de Sitter gravity

It is shown that the addition of a topological invariant (Gauss-Bonnet term) to the anti-de Sitter (AdS) gravity action in four dimensions recovers the standard regularization given by holographic renormalization procedure. This crucial step makes possible the inclusion of an odd parity invariant (Pontryagin term) whose coupling is fixed by demanding an asymptotic (anti) self-dual condition on the Weyl tensor. This argument allows to find the dual point of the theory where the holographic stress tensor is related to the boundary Cotton tensor as $T_{j}^{i}=\pm (\ell ^{2}/8\pi G)C_{j}^{i}$, which has been observed in recent literature in solitonic solutions and hydrodynamic models. A general procedure to generate the counterterm series for AdS gravity in any even dimension from the corresponding Euler term is also briefly discussed.

Quantum gravity at a Lifshitz point

We present a candidate quantum field theory of gravity with dynamical critical exponent equal to $z=3$ in the UV. (As in condensed-matter systems, $z$ measures the degree of anisotropy between space and time.) This theory, which at short distances describes interacting nonrelativistic gravitons, is power-counting renormalizable in $3+1$ dimensions. When restricted to satisfy the condition of detailed balance, this theory is intimately related to topologically massive gravity in three dimensions, and the geometry of the Cotton tensor. At long distances, this theory flows naturally to the relativistic value $z=1$, and could therefore serve as a possible candidate for a UV completion of Einstein's general relativity or an infrared modification thereof. The effective speed of light, the Newton constant and the cosmological constant all emerge from relevant deformations of the deeply nonrelativistic $z=3$ theory at short distances.

Duality in linearized gravity and holography

We consider spherical gravitational perturbations of AdS4 spacetime satisfying general boundary conditions at spatial infinity. Using the holographic renormalization method, we compute the energy–momentum tensor and show that it can always be cast in the form of a Cotton tensor for a dual boundary metric. In particular, axial and polar perturbations obeying the same boundary conditions for the effective Schrödinger wavefunctions exhibit an energy–momentum/Cotton tensor duality at conformal infinity. We demonstrate explicitly that this is holographic manifestation of the electric/magnetic duality of linearized gravity in the bulk, which simply exchanges axial with polar perturbations of AdS4 spacetime. We note that this particular realization of gravitational duality is also valid for perturbations near flat and dS4 spacetime, depending on the value of a cosmological constant.

Dual gravitons in AdS4/CFT3and the holographic Cotton tensor

We argue that gravity theories in AdS4 are holographically dual to either of two three-dimensional CFT's: the usual Dirichlet CFT1 where the fixed graviton acts as a source for the stress-energy tensor, and a dual CFT2 with a fixed dual graviton which acts as a source for a dual stress-energy tensor. The dual stress-energy tensor is shown to be the Cotton tensor of the Dirichlet CFT. The two CFT's are related by a Legendre transformation generated by a gravitational Chern-Simons coupling. This duality is a gravitational version of electric-magnetic duality valid at any radius r, where the renormalized stress-energy tensor is the electric field and the Cotton tensor is the magnetic field. Generic Robin boundary conditions lead to CFT's coupled to Cotton gravity or topologically massive gravity. Interaction terms with CFT1 lead to a non-zero vev of the stress-energy tensor in CFT2 coupled to gravity even after the source is removed. We point out that the dual graviton also exists beyond the linearized approximation, and spell out some of the details of the non-linear construction.

Energy-momentum/Cotton tensor duality forAdS4black holes

We consider the theory of gravitational quasi-normal modes for general linear perturbations of AdS4 black holes. Special emphasis is placed on the effective Schrodinger problems for axial and polar perturbations that realize supersymmetric partner potential barriers on the half-line. Using the holographic renormalization method, we compute the energy-momentum tensor for perturbations satisfying arbitrary boundary conditions at spatial infinity and discuss some aspects of the problem in the hydrodynamic representation. It is also observed in this general framework that the energy-momentum tensor of black hole perturbations and the energy momentum tensor of the gravitational Chern-Simons action (known as Cotton tensor) exhibit an axial-polar duality with respect to appropriately chosen supersymmetric partner boundary conditions on the effective Schrodinger wave-functions. This correspondence applies to perturbations of very large AdS4 black holes with shear viscosity to entropy density ratio equal to 1/4\pi, thus providing a dual graviton description of their hydrodynamic modes. We also entertain the idea that the purely dissipative modes of black hole hydrodynamics may admit Ricci flow description in the non-linear regime.

Cotton flow

Using the conformally-invariant Cotton tensor, we define a geometric flow, the Cotton flow, which is exclusive to three dimensions. This flow tends to evolve the initial metrics into conformally flat ones, and is somewhat orthogonal to the Yamabe flow, the latter being a flow within a conformal class. We define an entropy functional, and study the flow of nine homogeneous spaces both numerically and analytically. In particular, we show that the arbitrarily deformed homogeneous 3-sphere flows into the round 3-sphere. Two of the nine homogeneous geometries, which are degenerated by the Ricci flow, are left intact by the Cotton flow.

Dimensional Reduction of Conformal Tensors and Einstein-Weyl Spaces

Conformal Weyl and Cotton tensors are dimensionally reduced by a Kaluza- Klein procedure. Explicit formulas are given for reducing from four and three dimensions to three and two dimensions, respectively. When the higher dimensional conformal tensor vanishes because the space is conformallly flat, the lower-dimensional Kaluza-Klein func- tions satisfy equations that coincide with the Einstein-Weyl equations in three dimensions and kink equations in two dimensions.

Chern–Simons theory and three-dimensional surfaces

There are two natural Chern–Simons theories associated with the embedding of a three-dimensional surface in Euclidean space: one is constructed using the induced metric connection and involves only the intrinsic geometry? the other is extrinsic and uses the connection associated with the gauging of normal rotations. As such, the two theories appear to describe very different aspects of the surface geometry. Remarkably, at a classical level, they are equivalent. In particular, it will be shown that their stress tensors differ only by a null contribution. Their Euler–Lagrange equations provide identical constraints on the normal curvature. A new identity for the Cotton tensor is associated with the triviality of the Chern–Simons theory for embedded hypersurfaces implied by this equivalence.