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Ricci Decomposition
-------------------
The Ricci decomposition produces an irreducible representation
of the Riemann tensor that consists of 3 different parts.
R_{μναβ} = S_{μναβ} ⊕ E_{μναβ} ⊕ C_{μναβ}
Where:
S_{μναβ} = (R/n(n - 1))(g_{μα}g_{βν} - g_{μβ}g_{αν}) where R = g^{μν}R_{μν} and R_{μν}
is the contraction of the 1st and 3rd indeces of R_{βμαν}, i.e.
R_{μν} = R^{α}_{μαν} = δ^{ρ}_{α}R^{α}_{μρν} = δ^{ρ}_{α}g^{λα}R_{λμρν} = g^{ρλ}R_{λμρν} = R_{μν}.
E_{μναβ} = (1/(n - 2))(g_{μα}Z_{νβ} - g_{μβ}Z_{να} + g_{νβ}Z_{μα} - g_{να}Z_{μβ}) where
Z_{μν} = R_{μν} - (1/n)g_{μν}R is the traceless Ricci tensor.
E_{μναβ} is semi-traceless. Semi-traceless means it is built from
the metric (which is not traceless) and the traceless Ricci
tensor which is.
C_{μναβ} is a the WEYL TENSOR.
Therefore, the first piece, the scalar part, is built out
of the curvature scalar R and the metric. The second piece,
the semi-traceless piece, is built out of the metric and
the traceless Ricci tensor. The third piece is what is left
over and is called the Weyl tensor. This final piece is
completely traceless.
The Riemann Tensor
------------------
Properties:
R_{μναβ} = R_{αβμν} ... symmetric
_{ } = -R_{νμαβ} ... antisymmetric
_{ } = -R_{μνβα} ... antisymmetric
R_{μναβ} + R_{μβbα} + R_{μαβν} = 0 ... 1st BIANCHI IDENTITY.
∇_{ρ}R_{μναβ} + ∇_{α}R_{μνβρ} + ∇_{β}R_{μνρα} = 0 ... 2nd BIANCHI IDENTITY.
The Ricci Tensor
----------------
The Ricci tensor represents that part of the gravitational*
field which is due to the immediate presence of nongravitational
energy and momentum. It describes how the volume of a body
changes due to tidal forces it feels when moving along a
geodesic. Converging geodesics lead to a shrinking of the
volume. Diverging geodesics lead to an expanding volume.
* There is no term for gravitational energy in the Stress-Energy
tensor. The stress-energy tensor is defined locally and only
includes non-gravitational forms of energy (matter) whereas
gravitational energy is a non-local quantity. This argument
comes from the Equivalence Principle. One can always find in
any given locality, a frame of reference in which all local
gravitational fields vanish (i.e. a free falling reference
frame). In other words, in an appropriate coordinate system
a small region of space is free of a gravitational field. GR
states that gravitational energy exists globally but is not
localizable. There is no possibility of defining an energy
density of the gravitational field in GR at one point.
Properties:
- Using the property that R_{βμνα} = R_{ναβμ}
R_{νμ} = g^{βα}R_{ανβμ} ... the metric is symmetric.
Therefore, R_{μν} = R_{νμ} ... symmetric
The Scalar Curvature
--------------------
The Scalar curvature assigns a single real number determined
by the intrinsic geometry of spacetime near that point.
Specifically, it represents the amount by which the volume
of a geodesic ball (a ball made of a series of flat sides)
in a curved spacetime, V_{S}, deviates from that of the standard
ball in Euclidean space, V_{E}. The ratio of the 2 volumes is
given by:
V_{S}/V_{E} = 1 - Rr^{2}/6(n + 2)
When the scalar curvature is positive at a point, the volume
of the ball about the point has smaller volume than a ball
of the same radius in Euclidean space. On the other hand,
when the scalar curvature is negative at a point, the volume
of a small ball is larger than it would be in Euclidean
space.
To reiterate, the difference between the Ricci tensor and
the Scalar curvature is subtle. The former shows how an
arbitrary volume changes along a geodesic. The latter
measures the deviation of the volume of a geodesic ball
from its volume in Euclidean space.
The Weyl Tensor
---------------
The Weyl tensor is associated with changes in the shape of
a body due to the tidal forces it feels when moving along
a geodesic. It does not effect the volume. For example,
a ball moving along a geodesic could be stretched into an
ellipsoid with the same volume as the ball.
Properties:
C_{μναβ} = C_{αβμν} ... symmetric
_{ } = -C_{νμαβ} ... antisymmetric
_{ } = -C_{μνβα} ... antisymmetric
C_{μναβ} + C_{μβbα} + C_{μαβν} = 0 ... 1st BIANCHI IDENTITY.
∇_{ρ}C_{μναβ} + ∇_{α}C_{μνβρ} + ∇_{β}C_{μνρα} = 0 ... 2nd BIANCHI IDENTITY.
The Weyl tensor is completely traceless.
The 3 Tensors
-------------
------------------------------------------
| Riemann | Ricci | Weyl |
+-----+----+----+----+---+---+----+---+---+
Dimension (n) | 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 curvature tensor is a sum of the Weyl curvature
tensor plus terms built out of the Ricci tensor and Scalar
curvature.
- 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 spacetime is said to be Ricci flat.
Ricci flat spacetime will still have curvature unless the
Weyl tensor vanishes. Regions of spacetime in which the
Weyl tensor vanishes contain no gravitational radiation
and are conformally flat. This means that each point has
a neighborhood that can be mapped to flat space by an
angle preserving transformation.
- In 3D it is both necessary and sufficient for the Ricci
tensor to vanish for the manifold to be flat.
- In 2D it is both necessary and sufficient for the Scalar
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.
- Independent components are calculated from:
Riemann = n^{2}(n^{2} - 1)/12
Ricci = n(n + 1)/2 (because of symmetry)
Weyl = n^{2}(n^{2} - 1)/12 - n(n + 1)/2 for n > 2