Cotton Tensor Research Articles

Electric-magnetic duality of dyonic Kerr-Newman-NUT-AdS spacetimes

We study the (anti-)self-duality conditions under which the electric and magnetic parts of the conserved charges of the dyonic Kerr-Newman-NUT-anti-de Sitter solution become equivalent. Within a holographic framework, the stress tensor and the boundary Cotton tensor are computed from the electric/magnetic content of the Weyl tensor. The holographic stress tensor/Cotton tensor duality is recovered along the (anti-)self-dual curve in parameter space. We show that the latter not only implies a duality relation for the mass but also for the angular momentum. The partition function is computed to first order in the saddle-point approximation and a Bogomol’nyi-Prasad-Sommerfield bound is obtained. The ground state of the theory is enlarged to all the (anti-)self-dual configurations when the SO(4) and U(1) Pontryagin densities are introduced. We demonstrate this at the level of the action and variations thereof. Published by the American Physical Society 2024

Characterization of Bach and Cotton Tensors on a Class of Lorentzian Manifolds

In this work, we aim to investigate the characteristics of the Bach and Cotton tensors on Lorentzian manifolds, particularly those admitting a semi-symmetric metric ω-connection. First, we prove that a Lorentzian manifold admitting a semi-symmetric metric ω-connection with a parallel Cotton tensor is quasi-Einstein and Bach flat. Next, we show that any quasi-Einstein Lorentzian manifold admitting a semi-symmetric metric ω-connection is Bach flat.

Radiation in holography

We show how to encode the radiative degrees of freedom in 4-dimensional asymptotically AdS spacetimes, using the boundary Cotton and stress tensors. Background radiation leads to a reduction of the asymptotic symmetry group, in contrast to asymptotically flat spacetimes, where a non-vanishing news tensor does not restrict the asymptotic symmetries. Null gauges, such as Λ-BMS, provide a framework for AdS spacetimes that include radiation in the flat limit. We use this to check that the flat limit of the radiative data matches the expected definition in intrinsically asymptotically flat spacetimes. We further dimensionally reduce our construction to the celestial sphere, and show how the 2-dimensional celestial currents can be extracted from the 3-dimensional boundary data.

Spin-(s, j) projectors and gauge-invariant spin-s actions in maximally symmetric backgrounds

Given a maximally symmetric d-dimensional background with isometry algebra g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{g} $$\\end{document}, a symmetric and traceless rank-s field ϕa(s) satisfying the massive Klein-Gordon equation furnishes a collection of massive g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{g} $$\\end{document}-representations with spins j ∈ {0, 1, · · · , s}. In this paper we construct the spin-(s, j) projectors, which are operators that isolate the part of ϕa(s) that furnishes the representation from this collection carrying spin j. In the case of an (anti-)de Sitter ((A)dSd) background, we find that the poles of the projectors encode information about (partially-)massless representations, in agreement with observations made earlier in d = 3, 4. We then use these projectors to facilitate a systematic derivation of two-derivative actions with a propagating massless spin-s mode. In addition to reproducing the massless spin-s Fronsdal action, this analysis generates new actions possessing higher-depth gauge symmetry. In (A)dSd we also derive the action for a partially-massless spin-s depth-t field with 1 ≤ t ≤ s. The latter utilises the minimum number of auxiliary fields, and corresponds to the action originally proposed by Zinoviev after gauging away all Stückelberg fields. Some higher-derivative actions are also presented, and in d = 3 are used to construct (i) generalised higher-spin Cotton tensors in (A)dS3; and (ii) topologically-massive actions with higher-depth gauge symmetry. Finally, in four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 Minkowski superspace, we provide closed-form expressions for the analogous superprojectors.

Quasi-Bach flow and quasi-Bach solitons on Riemannian manifolds

In this article, we introduce quasi-Bach tensor and correspondingly introduce almost quasi-Bach solitons, thereby generalizing the existing notion of Bach tensor and almost Bach solitons. We explore some properties of gradient quasi Bach solitons with harmonic Weyl curvature tensor. We study the relationships between Weyl tensor, Cotton tensor, tensor D introduced by Cao, and quasi Bach tensor. We also find the evolution of volume, Einstein metric, Ricci curvature and scalar curvature, under the quasi Bach flow. Our results obtained here extends the results of Bach solitons and Bach flow. Finally, we obtain the characterization of gradient quasi-Bach soliton of type I, a particular quasi Bach soliton, on the product manifolds S2×H2 and R2×H2. Our exploration generalizes gradient Bach soliton on R2×H2 obtained by P. T. Ho, while the gradient soliton on S2×H2 is a novel one and is complementary to the results obtained by P. T. Ho.

Algebraic classification of 2+1 geometries: a new approach

We present a convenient method of algebraic classification of 2+1 spacetimes into the types I, II, D, III, N and O, without using any field equations. It is based on the 2+1 analogue of the Newman–Penrose curvature scalars ΨA of distinct boost weights, which are specific projections of the Cotton tensor onto a suitable null triad. The algebraic types are then simply determined by the gradual vanishing of such Cotton scalars, starting with those of the highest boost weight. This classification is directly related to the specific multiplicity of the Cotton-aligned null directions and to the corresponding Bel–Debever criteria. Using a bivector (that is 2-form) decomposition, we demonstrate that our method is fully equivalent to the usual Petrov-type classification of 2+1 spacetimes based on the eigenvalue problem and determining the respective canonical Jordan form of the Cotton–York tensor. We also derive a simple synoptic algorithm of algebraic classification based on the key polynomial curvature invariants. To show the practical usefulness of our approach, we perform the classification of several explicit examples, namely the general class of Robinson–Trautman spacetimes with an aligned electromagnetic field and a cosmological constant, and other metrics of various algebraic types.

Chern-Simons action and the Carrollian Cotton tensors

In three-dimensional pseudo-Riemannian manifolds, the Cotton tensor arises as the variation of the gravitational Chern-Simons action with respect to the metric. It is Weyl-covariant, symmetric, traceless and covariantly conserved. Performing a reduction of the Cotton tensor with respect to Carrollian diffeomorphisms in a suitable frame, one discloses four sets of Cotton Carrollian relatives, which are conformal and obey Carrollian conservation equations. Each set of Carrollian Cotton tensors is alternatively obtained as the variation of a distinct Carroll-Chern-Simons action with respect to the degenerate metric and the clock form of a strong Carroll structure. The four Carroll-Chern-Simons actions emerge in the Carrollian reduction of the original Chern-Simons ascendant. They inherit its anomalous behaviour under diffeomorphisms and Weyl transformations. The extremums of these Carrollian actions are commented and illustrated.

Flat from anti de Sitter

Ricci-flat solutions to Einstein’s equations in four dimensions are obtained as the flat limit of Einstein spacetimes with negative cosmological constant. In the limiting process, the anti-de Sitter energy-momentum tensor is expanded in Laurent series in powers of the cosmological constant, endowing the system with the infinite number of boundary data, characteristic of an asymptotically flat solution space. The governing flat Einstein dynamics is recovered as the limit of the original energy-momentum conservation law and from the additional requirement of the line-element finiteness, providing at each order the necessary set of flux-balance equations for the boundary data. This analysis is conducted using a covariant version of the Newman-Unti gauge designed for taking advantage of the boundary Carrollian structure emerging at vanishing cosmological constant and its Carrollian attributes such as the Cotton tensor.

Conformal flatness of compact three-dimensional Cotton-parallel manifolds

A three-dimensional pseudo-Riemannian manifold is called essentially conformally symmetric (ECS) if its Cotton tensor is parallel but nowhere-vanishing. In this note we prove that three-dimensional ECS manifolds must be noncompact or, equivalently, that every compact three-dimensional Cotton-parallel pseudo-Riemannian manifold must be conformally flat.

Ehlers, Carroll, charges and dual charges

We unravel the boundary manifestation of Ehlers’ hidden Möbius symmetry present in four-dimensional Ricci-flat spacetimes that enjoy a time-like isometry and are Petrov-algebraic. This is achieved in a designated gauge, shaped in the spirit of flat holography, where the Carrollian three-dimensional nature of the null conformal boundary is manifest and covariantly implemented. The action of the Möbius group is local on the space of Carrollian boundary data, among which the Carrollian Cotton tensor plays a predominent role. The Carrollian and Weyl geometric tools introduced for shaping an appropriate gauge, as well as the boundary conformal group, which is BMS4, allow to define electric/magnetic, leading/subleading towers of charges directly from the boundary Carrollian dynamics and explore their behaviour under the action of the Möbius duality group.

Cotton gravity and 84 galaxy rotation curves

Recently, as a generalization of general relativity, a gravity theory has been proposed in which gravitational field equations are described by the Cotton tensor. That theory allows an additional contribution to the gravitational potential of a point mass that rises linearly with radius as $\Phi = -GM/r + \gamma r/2$, where $G$ is the Newton constant. The coefficients $M$ and $\gamma$ are the constants of integration and should be determined individually for each physical system. When applied to galaxies, the coefficient $\gamma$, which has the dimension of acceleration, should be determined for each galaxy. This is the same as having to determine the mass $M$ for each galaxy. If $\gamma$ is small enough, the linear potential term is negligible at short distances, but can become significant at large distances. In fact, it may contribute to the extragalactic systems. In this paper, we derive the effective field equation for Cotton gravity applicable to extragalactic systems. We then use the effective field equation to numerically compute the gravitational potential of a sample of 84 rotating galaxies. The 84 galaxies span a wide range, from stellar disk-dominated spirals to gas-dominated dwarf galaxies. We do not assume the radial density profile of the stellar disk, bulge, or gas; we use only the observed data. We find that the rotation curves of 84 galaxies can be explained by the observed distribution of baryons. This is due to the flexibility of Cotton gravity to allow the integration constant $\gamma$ for each galaxy. In the context of Cotton gravity, "dark matter" is in some sense automatically included as a curvature of spacetime. Consequently, even galaxies that have been assumed to be dominated by dark matter do not need dark matter.

A study on K-paracontact and (k,μ)-paracontact manifold admitting vanishing Cotton tensor and Bach tensor

"The object of the present paper is to study K-paracontact manifold admitting parallel Cotton tensor, vanishing Cotton tensor and to study Bach flatness on K-paracontact manifold. In that we prove for a K-paracontact metric manifold M^{2n+1} has parallel Cotton tensor if and only if M^{2n+1} is an η-Einstein manifold and r=-2n(2n+1). Further we show that if g is an η-Einstein K-paracontact metric and if g is Bach flat then g is an Einstein. Also we study vanishing Cotton tensor on (k,μ)-paracontact manifold for both k>-1 and k<-1. Finally, we prove that if M^{2n+1} is a (k,μ)-paracontact manifold for k ≠ -1 and if M^{2n+1} has vanishing Cotton tensor for μ ≠ k, then M^{2n+1} is an η-Einstein manifold."

Scalarization of Chern-Simons-Kerr black hole solutions and wormholes

Chern-Simons gravity rotating Kerr-type black hole solutions are revisited from the point of view of their scalarization, namely examining the back reaction of pseudoscalar axion fields on the rotating geometry. To lowest order in an angular momentum expansion, and a long range approximation for both the axion field configuration and its back reaction onto the metric, such solutions had been discussed for the first time in the context of string theory. In this work, we extend the analysis to give analytic expressions for slowly rotating black holes outside the horizon, which formally include an all order expansion in inverse powers of the radial distance from the centre of the black hole, which allows to approach arbitrarily close the horizon. We also discuss in some detail the way Chern-Simons gravity violates the energy conditions, which leads to secondary axionic hair of the rotating black hole solutions. This discussion is extended to studies of wormhole solutions, which we construct in this article via the thin-shell procedure and we demonstrate that the presence of the axion field enforces the two black hole solutions to be counter-rotating and for slow rotation, there is no backreaction of the Cotton tensor on the thin-shell of the wormhole.

Killing-Yano Cotton currents

We discuss conserved currents constructed from the Cotton tensor and (conformal) Killing-Yano tensors (KYTs). We consider the corresponding charges generally and then exemplify with the four-dimensional Plebański-Demiański metric where they are proportional to the sum of the squares of the electric and the magnetic charges. As part of the derivation, we also find the two conformal Killing-Yano tensors of the Plebański-Demiański metric in the recently introduced coordinates of Podolsky and Vratny. The construction of asymptotic charges for the Cotton current is elucidated and compared to the three-dimensional construction in Topologically Massive Gravity. For the three-dimensional case, we also give a conformal superspace multiplet that contains the Cotton current in the bosonic sector. In a mathematical section, we derive potentials for the currents, find identities for conformal KYTs and for KYTs in torsionful backgrounds.

Ricci almost solitons with associated projective vector field

Abstract A Ricci almost soliton whose associated vector field is projective is shown to have vanishing Cotton tensor, divergence-free Bach tensor and Ricci tensor as conformal Killing. For the compact case, a sharp inequality is obtained in terms of scalar curvature.We show that every complete gradient Ricci soliton is isometric to the Riemannian product of a Euclidean space and an Einstein space. A complete K-contact Ricci almost soliton whose associated vector field is projective is compact Einstein and Sasakian.

Cotton tensor on Sasakian 3-manifolds admitting eta-Ricci solitons

The object of the present paper is to characterize Cotton tensor on a 3-dimensional Sasakian manifold admitting $\eta$-Ricci solitons. After introduction, we study 3-dimensional Sasakian manifolds and introduce a new notion, namely, Cotton pseudo-symmetric manifolds. Next we deal with the study Cotton tensor on a Sasakian 3-manifold admitting $\eta$-Ricci solitons. Among others we prove that such a manifold is a manifold of constant scalar curvature and Einstein manifold with some appropriate conditions. Also, we classify the nature of the soliton metric. Finally, we give an important remark.

Higher spin analogs of linearized topologically massive gravity and linearized new massive gravity

We suggest a new spin-4 self-dual model (parity singlet) and a new spin-4 parity doublet in $D=2+1$. They are of higher order in derivatives and are described by a totally symmetric rank-4 tensor without extra auxiliary fields. Despite the higher derivatives they are ghost free. We find gauge invariant field combinations which allow us to show that the canonical structure of the spin-4 (spin-3) models follows the same pattern of its spin-2 (spin-1) counterpart after field redefinitions. For $s=1$, 2, 3, 4, the spin-$s$ self-dual models of order $2s\ensuremath{-}1$ and the doublet models of order $2s$ can be written in terms of three gauge invariants. The cases $s=3$ and $s=4$ suggest a restricted conformal higher spin symmetry as a principle for defining linearized topologically massive gravity and linearized ``new massive gravity'' for arbitrary integer spins. A key role in our approach is played by the fact that the Cotton tensor in $D=2+1$ has only two independent components for any integer spin.

Comment on “Emergence of the Cotton tensor for describing gravity”

Recently, it has been claimed by J. Harada [Phys. Rev. D 103, L121502 (2021) that the Cotton tensor can describe the effects of gravity beyond general relativity. In this short comment, we show that the Cotton tensor is already present in general relativity. Moreover, we show that Harada's proposal is equivalent to general relativity concerning the equation of motion both for the sources and for the trace-free part of the curvature. In essence, what remains completely free in Harada's theory are the field equations relating the part of the curvature which is locally determined by the energy-momentum distribution.

Reply to “Comment on ‘Emergence of the Cotton tensor for describing gravity”’

Recently, P. Bargue\~no has commented in Phys. Rev. D 104, 088501 (2021) that the field equations obtained in the original paper [J. Harada, Phys. Rev. D 103, L121502 (2021)] have less information than those of general relativity. This reply provides clarification. The field equations obtained in the original paper have information on gravity that cannot be obtained from general relativity. For instance, the effective potential of gravity has a term that is linearly proportional to the distance. It implies that the magnitude of gravity can be stronger than that of Newton's law of gravitation at large distances.

Emergence of the Cotton tensor for describing gravity

It is shown that the Cotton tensor can describe the effects of gravity beyond general relativity. Any solution of the Einstein equations with or without the cosmological constant satisfies the field equations described by the Cotton tensor. It implies that the cosmological constant is an integration constant. A vacuum of a theory is represented by the vanishing of the Cotton tensor, rather than the vanishing of the Ricci tensor. An exact Schwarzschild-like solution for a static and spherically symmetric source is discovered. Although the field equations involve the third order of derivative, it is found that they reduce to the second-order differential equations due to a variational principle.