The Cotton tensor in Riemannian spacetimes

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.