In differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension n is a third-order tensor concomitant of the metric. The vanishing of the Cotton tensor for n = 3 is necessary and sufficient condition for the manifold to be locally conformally flat. By contrast, in dimensions n ≥ 4, the vanishing of the Cotton tensor is necessary but not sufficient for the metric to be conformally flat; instead, the corresponding necessary and sufficient condition in these higher dimensions is the vanishing of the Weyl tensor, while the Cotton tensor just becomes a constant times the divergence of the Weyl tensor. 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

edit

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

 

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

 

sometimes called the Cotton–York tensor.

Properties

edit

Conformal rescaling

edit

Under conformal rescaling of the metric   for some scalar function  . We see that the Christoffel symbols transform as

 

where   is the tensor

 

The Riemann curvature tensor transforms as

 

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

 

Similarly the Ricci scalar transforms as

 

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

 

or using coordinate independent language as

 

where the gradient is contracted with the Weyl tensor W.

Symmetries

edit

The Cotton tensor has the following symmetries:

 

and therefore

 

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

 

where   is the positive divergence in the first component of W.

References

edit
  • 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