Ricci Decomposition - The Pieces Appearing in The Decomposition

The Pieces Appearing in The Decomposition

The decomposition is

The three pieces are:

  1. the scalar part, the tensor
  2. the semi-traceless part, the tensor
  3. the fully traceless part, the Weyl tensor

Each piece possesses all the algebraic symmetries of the Riemann tensor itself, but has additional properties.

The decomposition can have different signs, depending on the Ricci curvature convention, and only makes sense if the dimension satisfies .

The scalar part

is built using the scalar curvature, where is the Ricci curvature, and a tensor constructed algebraically from the metric tensor ,

The semi-traceless part

E_{abcd} = \frac{1}{n-2} \, \left( g_{ac} \, S_{bd} - g_{ad} \, S_{bc} + g_{bd} \, S_{ac} - g_{bc} \, S_{ad} \right) = \frac{2}{n-2} \, \left( g_{ab} - g_{ba} \right)

is constructed algebraically using the metric tensor and the traceless part of the Ricci tensor

where is the metric tensor.

The Weyl tensor or conformal curvature tensor is completely traceless, in the sense that taking the trace, or contraction, over any pair of indices gives zero. Hermann Weyl showed that this tensor measures the deviation of a semi-Riemannian manifold from conformal flatness; if it vanishes, the manifold is (locally) conformally equivalent to a flat manifold.

No additional differentiation is needed anywhere in this construction.

In the case of a Lorentzian manifold, the Einstein tensor has, by design, a trace which is just the negative of the Ricci scalar, and one may check that the traceless part of the Einstein tensor agrees with the traceless part of the Ricci tensor.

Terminological note: the notation is standard in the modern literature, the notations are commonly used but not standardized, and there is no standard notation for the scalar part.

Read more about this topic:  Ricci Decomposition

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