Cotton tensor

In differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension n is a third-order tensor concomitant of the metric, like the Weyl tensor. The vanishing of the Cotton tensor for n = 3 is necessary and sufficient condition for the manifold to be conformally flat, as with the Weyl tensor for n ≥ 4. For n < 3 the Cotton tensor is identically zero. The concept is named after Émile Cotton.

The proof of the classical result that for n = 3 the vanishing of the Cotton tensor is equivalent to the metric being conformally flat is given by Eisenhart using a standard integrability argument. This tensor density is uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics, as shown by (Aldersley 1979).

Recently, the study of three-dimensional spaces is becoming of great interest, because the Cotton tensor restricts the relation between the Ricci tensor and the energy–momentum tensor of matter in the Einstein equations and plays an important role in the Hamiltonian formalism of general relativity.

Definition

In coordinates, and denoting the Ricci tensor by R_ij and the scalar curvature by R, the components of the Cotton tensor are

C_{{ijk}}=\nabla _{{k}}R_{{ij}}-\nabla _{{j}}R_{{ik}}+{\frac {1}{2(n-1)}}\left(\nabla _{{j}}Rg_{{ik}}-\nabla _{{k}}Rg_{{ij}}\right).

The Cotton tensor can be regarded as a vector valued 2-form, and for n = 3 one can use the Hodge star operator to convert this into a second order trace free tensor density

C_{i}^{j}=\nabla _{{k}}\left(R_{{li}}-{\frac {1}{4}}Rg_{{li}}\right)\epsilon ^{{klj}},

sometimes called the Cotton–York tensor.

Properties

Conformal rescaling

Under conformal rescaling of the metric ${\tilde {g}}=e^{{2\omega }}g$ for some scalar function $\omega$ . We see that the Christoffel symbols transform as

\widetilde {\Gamma }_{{\beta \gamma }}^{{\alpha }}=\Gamma _{{\beta \gamma }}^{{\alpha }}+S_{{\beta \gamma }}^{{\alpha }}

where $S_{{\beta \gamma }}^{{\alpha }}$ is the tensor

S_{{\beta \gamma }}^{{\alpha }}=\delta _{{\gamma }}^{{\alpha }}\partial _{{\beta }}\omega +\delta _{{\beta }}^{{\alpha }}\partial _{{\gamma }}\omega -g_{{\beta \gamma }}\partial ^{{\alpha }}\omega

The Riemann curvature tensor transforms as

{\widetilde {R}^{{\lambda }}}{}_{{\mu \alpha \beta }}={R^{{\lambda }}}_{{\mu \alpha \beta }}+\nabla _{{\alpha }}S_{{\beta \mu }}^{{\lambda }}-\nabla _{{\beta }}S_{{\alpha \mu }}^{{\lambda }}+S_{{\alpha \rho }}^{{\lambda }}S_{{\beta \mu }}^{{\rho }}-S_{{\beta \rho }}^{{\lambda }}S_{{\alpha \mu }}^{{\rho }}

In $n$ -dimensional manifolds, we obtain the Ricci tensor by contracting the transformed Riemann tensor to see it transform as

\widetilde {R}_{{\beta \mu }}=R_{{\beta \mu }}-g_{{\beta \mu }}\nabla ^{{\alpha }}\partial _{{\alpha }}\omega -(n-2)\nabla _{{\mu }}\partial _{{\beta }}\omega +(n-2)(\partial _{{\mu }}\omega \partial _{{\beta }}\omega -g_{{\beta \mu }}\partial ^{{\lambda }}\omega \partial _{{\lambda }}\omega )

Similarly the Ricci scalar transforms as

\widetilde {R}=e^{{-2\omega }}R-2e^{{-2\omega }}(n-1)\nabla ^{{\alpha }}\partial _{{\alpha }}\omega -(n-2)(n-1)e^{{-2\omega }}\partial ^{{\lambda }}\omega \partial _{{\lambda }}\omega

Combining all these facts together permits us to conclude the Cotton-York tensor transforms as

\widetilde {C}_{{\alpha \beta \gamma }}=C_{{\alpha \beta \gamma }}+(n-2)\partial _{{\lambda }}\omega {W_{{\beta \gamma \alpha }}}^{{\lambda }}

or using coordinate independent language as

{\tilde {C}}=C\;+\;\operatorname {grad}\,\omega \;\lrcorner \;W,

where the gradient is plugged into the symmetric part of the Weyl tensor W.

Symmetries

The Cotton tensor has the following symmetries:

C_{{ijk}}=-C_{{ikj}}\,

and therefore

C_{{[ijk]}}=0.\,

In addition the Bianchi formula for the Weyl tensor can be rewritten as

\delta W=(3-n)C,\,

where $\delta$ is the positive divergence in the first component of W.

References

Aldersley, S. J. (1979). "Comments on certain divergence-free tensor densities in a 3-space". Journal of Mathematical Physics. 20 (9): 1905–1907. Bibcode:1979JMP....20.1905A. doi:10.1063/1.524289.
Choquet-Bruhat, Yvonne (2009). General Relativity and the Einstein Equations. Oxford, England: Oxford University Press. ISBN 978-0-19-923072-3.
Cotton, É. (1899). "Sur les variétés à trois dimensions". Annales de la Faculté des Sciences de Toulouse. II. 1 (4): 385–438. Archived from the original on 2007-10-10.
Eisenhart, Luther P. (1977) [1925]. Riemannian Geometry. Princeton, NJ: Princeton University Press. ISBN 0-691-08026-7.
A. Garcia, F.W. Hehl, C. Heinicke, A. Macias (2004) "The Cotton tensor in Riemannian spacetimes", Classical and Quantum Gravity 21: 1099–1118, Eprint arXiv:gr-qc/0309008

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