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The Cotton tensor in Riemannian spacetimes
Open accessAbstract
Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension n . We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components as n ( n 2 − 4)/3 for the first time. Subsequently, we show its characteristic properties and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einstein's field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw and Templeton. For each class examples are given. Finally, we investigate the relation between the Cotton tensor and the energy–momentum in Einstein's theory and derive a conformally flat perfect fluid solution of Einstein's field equations in three dimensions.
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TL;DR
The paper gives a systematic exterior-form derivation and irreducible decomposition of the Cotton tensor in Riemannian spacetimes, showing that in Riemannian space only its tensor piece survives with \(n(n^2-4)/3\) independent components. It then classifies the Cotton tensor in three dimensions, relates it to Einstein and topologically massive gravity field equations, and derives a conformally flat perfect-fluid solution.
Key findings
- The Cotton 2-form is derived from the Bianchi identities as \(C_\alpha := DL_\alpha\), with \(L_\alpha\) the Schouten 1-form; its trace and axial pieces vanish in Riemannian space, leaving only TENCOT with \(n(n^2-4)/3\) independent components — 5 in \(n=3\) and 16 in \(n=4\).
- In three dimensions the Cotton tensor is conformally invariant, and its vanishing is necessary and sufficient for conformal flatness; in more than three dimensions the vanishing of the Weyl 2-form is the relevant conformal-flatness condition.
- The 3D Cotton tensor is classified by its eigenvalues/Jordan form into classes I, I', D, II, III, N, and O, paralleling the Petrov classification; examples are given for all classes, including a 3D Reissner-Nordström analog, a Gödel solution, and DJT solutions.
- The Cotton tensor is related to matter via the differentiated Einstein equation, \(C_\alpha = (-1)^{n-1+\mathrm{ind}}\ell D\star[\Sigma_\alpha - \frac{1}{n-1}e_\alpha\rfloor(\Sigma_\beta\wedge\vartheta^\beta)]\); all vacuum Einstein solutions have vanishing Cotton 2-form, and in 4D the Bach tensor is recovered as a conserved 3-form.
Methods
- Theoretical derivation using exterior differential forms, Bianchi identities, irreducible decomposition with respect to the Lorentz group, and variational derivation from the 3D Chern-Simons action with Lagrange multipliers enforcing vanishing torsion and nonmetricity.
- Classification of the 3D Cotton tensor via the generalized eigenvalue problem \(C_{\alpha\beta}-\lambda g_{\alpha\beta}=0\), with Jordan normal forms and Segre notation; examples are checked by computer algebra using Reduce-Excalc.
- No empirical sample; the analysis is mathematical, with illustrative metric examples and a conformally flat static spherically symmetric perfect-fluid solution of Einstein’s field equations in 3D.
Limitations
- The paper states that it found no “sensible” solutions to the Einstein or DJT field equations of Class II or III; the examples for those classes are metrics whose Cotton tensor degenerates on surfaces/hypersurfaces, not field-equation solutions.
- Evident from the text: the work is purely theoretical/mathematical, with no observational data or empirical validation.
- Evident from the text: the Lorentzian classification can involve complex eigenvalues, and the paper notes this point “seems to have been overlooked” by some earlier authors.
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“From the construction it is clear that the Weyl 2-form is totally traceless, i. e., e α ⌋Weyl αβ = -e β ⌋Weyl αβ = 0 , e α ⌋e β ⌋Weyl αβ = 0 .(25)”
§ The Cotton tensor in Riemannian spacetimes > II. BIANCHI IDENTITIES AND THE IRREDUCIBLE DECOMPOSITION OF THE CURVATURE
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“The Bianchi identities imply full integrability of this system.”
§ The Cotton tensor in Riemannian spacetimes > C α as a variational derivative
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“for these spaces the Weyl tensor is always zero”
§ The Cotton tensor in Riemannian spacetimes > I. INTRODUCTION
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“Hence, the vanishing of the Weyl 2-form is also the sufficient condition for conformal flatness.”
§ The Cotton tensor in Riemannian spacetimes > n = 2
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In the paper
“However, the Cotton tensor is only conformally invariant in three dimensions.”
- Also stated in: § IV. CONFORMAL CORRESPONDENCE · plain
“Thus, in n = 3, where the Weyl 2-form vanishes, the Cotton 2-form becomes conformally invariant.”
§ The Cotton tensor in Riemannian spacetimes > I. INTRODUCTION
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- Also stated in: § IV. CONFORMAL CORRESPONDENCE · plain
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In the paper
§ The Cotton tensor in Riemannian spacetimes > B. Lorentzian signature
“types D and II with only one independent eigenvalue λ 1 = λ 2 = -2λ 3 are always real.”
§ The Cotton tensor in Riemannian spacetimes > B. Lorentzian signature
“λ 1 = λ 2 = - 1 2 λ 3 = µ 3 3 . (122”
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Metadata
- Authors
- García, A. A., Hehl, F. W., Heinicke, C., & Macías, A.
- Affiliations
- Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (MX)
- University of Cologne (DE)
- Universidad Autónoma Metropolitana (MX)
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2004-01-22
- Volume
- 21
- Issue
- 4
- Pages
- 1099-1118
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2109289785
- Cited by
- 148
- References
- 50
- Citations by year
- 2026: 4 · 2025: 5 · 2024: 8 · 2023: 4 · 2022: 4 · 2021: 10 · 2020: 5 · 2019: 2 · 2018: 4 · 2017: 29 · 2016: 5 · 2015: 3 · 2014: 10 · 2013: 7 · 2012: 5
- FWCI
- 2.67
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Deutsche Forschungsgemeinschaft (528/20-1)
- Consejo Nacional de Ciencia y Tecnología
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Keywords
- Cotton tensor, 2+1 dimensions, conformal flatness, conformal invariance, irreducible decomposition, Bianchi identities, Einstein field equations, topologically massive gravity, energy-momentum tensor, perfect fluid solutions
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