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Ricci Decomposition
-------------------
The Ricci Decomposition is a way of breaking up the the Riemann
tensor it 3 different parts as follws:
R_{abcd} = S_{abcd} + E_{abcd} + C_{abcd}
E_{abcd} is related to the RICCI TENSOR, R_{ab}, as follows:
E_{abcd} = (1/(n - 2))(g_{ac}S_{bd} - g_{ad}S_{bc} - g_{bd}S_{ac} - g_{bc}S_{ad})
Where,
S_{ab} = R_{ab} - (1/n)g_{ab}R ... the traceless Ricci tensor.
We can define R_{ab} as the contraction of the first and third
indeces of R_{dacb} (the 1st and 4th or 2nd and 4th are also
possible) as follows:
R_{ab} = g^{cd}R_{dacb}
_{ } = R^{c}_{acb}
Note: g^{da}R_{dabc} = g^{bc}R_{dabc} = 0.
E_{abcd} is semi-traceless because it is built from the metric and
the traceless Ricci tensor.
S_{abcd} is related to the CURVATURE SCALAR, R, as follows:
S_{abcd} = (R/n(n - 1))(g_{ac}g_{db} - g_{ad}g_{cb})
Where R is the metric contraction g^{ab}R_{ab}.
C_{abcd} is a the WEYL TENSOR.
Notes:
Properties of R_{abcd}:
- R_{abcd} = R_{cdab} ... symmetric
_{ } = -R_{bacd} ... antisymmetric
_{ } = -R_{abdc} ... antisymmetric
- R_{abcd} + R_{adbc} + R_{acdb} = 0 ... 1st BIANCHI IDENTITY.
- ∇_{e}R_{abcd} + ∇_{c}R_{abde} + ∇_{d}R_{abec} = 0 ... 2nd BIANCHI IDENTITY.
Properties of R_{ab}:
- Using the property that R_{dabc} = R_{bcda}
R_{ba} = g^{dc}R_{cbda} ... the metric is symmetric.
Therefore, R_{ab} = R_{ba} ... symmetric
- R_{ab} is associated with changes in the volume of a body due to
tidal forces it feels when moving along a geodesic.
Properties of C_{abcd}:
- C_{abcd} ... same symmetry properties as R_{abcd}
- C_{abcd} is completely traceless. It is associated with changes
in the shape of a body due to the tidal forces it feels when
moving along a geodesic.
The 3 Tensors
-------------
-----------------------------------------------------
| Riemann | Ricci | R | Weyl |
+-----+-----+-----+-----+-----+-----+-----+-----+-----+
Dimension | 4 | 3 | 2 | 4 | 3 | 2 | 4 | 3 | 2 |
+-----+-----+-----+-----+-----+-----+-----+-----+-----+
Total Comps. | 256 | 81 | 16 | 16 | 9 | 1 | | | |
+-----+-----+-----+-----+-----+-----+-----+-----+-----+
Indep. Comps. | 20 | 6 | 1 | 10 | 6 | 1 | 10 | 0 | 0 |
-----------------------------------------------------
Notes:
- The Riemann tensor is the sum of the Weyl and Ricci tensors.
- In 4D it is both necessary and sufficient for the Riemann tensor
to vanish for the manifold to be flat. If the Ricci tensor vanishes
the space is said to be RICCI FLAT.
- In 3D it is both necessary and sufficient for the Ricci tensor
to vanish for the manifold to be flat.
- The Ricci tensor can describe everything about 3D curvature and all
the 81 elements of the Riemann tensor can be calculated from the 6
independent components of the Ricci tensor.
- In 2D it is both necessary and sufficient for the Scalar tensor
to vanish for the manifold to be flat.
- Independent components are calculated from:
Riemann = n^{2}(n^{2} - 1)/12
Weyl = n^{2}(n^{2} - 1)/12 - n(n + 1)/2 for n > 2