18.111.68.236: /* Einstein tensor */ The Einstein tensor given disagreed with the eigenvalue equation below, and gave a stress energy tensor with negative pressures.

2012-05-21T00:02:03Z

Einstein tensor: The Einstein tensor given disagreed with the eigenvalue equation below, and gave a stress energy tensor with negative pressures.

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In [[semi-Riemannian geometry]], the '''Ricci decomposition''' is a way of breaking up the [[Riemann tensor|Riemann curvature tensor]] of a [[pseudo-Riemannian manifold]] into pieces with useful individual algebraic properties. This decomposition is of fundamental importance in Riemannian- and pseudo-Riemannian geometry.

==The pieces appearing in the decomposition==
The decomposition is
:<math>R_{abcd}= \, S_{abcd}+E_{abcd}+C_{abcd}.</math>
The three pieces are:
# the ''scalar part'', the tensor <math>S_{abcd}</math>
# the ''semi-[[traceless]] part'', the tensor <math>E_{abcd}</math>
# the ''fully traceless part'', the [[Weyl tensor]] <math>C_{abcd}</math>
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 <math>n>2</math>.

The scalar part
:<math> S_{abcd} = \frac{R}{n \, (n-1)} \, H_{abcd}</math>
is built using the [[scalar curvature]] <math>R = {R^m}_m</math>, where <math>R_{ab}={R^c}_{acb}</math> is the [[Ricci curvature]], and a tensor constructed algebraically from the [[metric tensor]] <math>g_{ab}</math>,
:<math>H_{abcd} = g_{ac} \, g_{db} - g_{ad} \, g_{cb} = 2g_{a[c} \, g_{d]b}.</math>

The semi-traceless part
:<math>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_{a[c} \, S_{d]b} - g_{b[c} \, S_{d]a} \right) </math>
is constructed algebraically using the metric tensor and the ''traceless part'' of the Ricci tensor
:<math> S_{ab} = R_{ab} - \frac{1}{n} \, g_{ab} \, R</math>
where <math>g_{ab}</math> is the [[metric tensor]].

The [[Weyl tensor]] <math>C_{abcd}</math> or ''conformal curvature tensor'' is completely traceless, in the sense that taking the trace, or [[tensor contraction|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) [[conformal equivalence|conformally equivalent]] to a flat manifold.

No additional differentiation is needed anywhere in this construction.

In the case of a [[Lorentzian manifold]], <math>n=4</math>, the [[Einstein tensor]] <math>G_{ab} = R_{ab} - 1/2 \, g_{ab} R</math> 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.

:<math> S_{ab} = R_{ab} - \frac{1}{4} \, g_{ab} \, R = G_{ab} - \frac{1}{4} \, g_{ab} \, G</math>

''Terminological note:'' the notation <math>R_{abcd}, \, C_{abcd}</math> is standard in the modern literature, the notations <math>S_{ab}, \, E_{abcd}</math> are commonly used but not standardized, and there is no standard notation for the scalar part.

==Mathematical definition==
Mathematically, the Ricci decomposition is the decomposition of the space of all [[tensor]]s having the symmetries of the Riemann tensor into its [[irreducible representation]]s for the action of the [[orthogonal group]] {{harv|Besse|1987|loc=Chapter 1, §G}}. Let ''V'' be an ''n''-dimensional [[vector space]], equipped with a [[metric tensor]] (of possibly mixed signature). Here ''V'' is modeled on the [[cotangent space]] at a point, so that a curvature tensor ''R'' (with all indices lowered) is an element of the [[tensor product]] ''V''&otimes;''V''&otimes;''V''&otimes;''V''. The curvature tensor is skew symmetric in its first and last two entries:
:<math>R(x,y,z,w)=-R(y,x,z,w)=-R(x,y,w,z)\,</math>
and obeys the interchange symmetry
:<math>R(x,y,z,w) = R(z,w,x,y),\,</math>
for all ''x'',''y'',''z'',''w'' ∈ ''V''&lowast;. As a result ''R'' is an element of the subspace ''S''2Λ2''V'', the second [[symmetric power]] of the second [[exterior power]] of ''V''. A curvature tensor must also satisfy the Bianchi identity, meaning that it is in the [[kernel (algebra)|kernel]] of the linear map
:<math>b(R)(x,y,z,w) = R(x,y,z,w) + R(y,z,x,w) + R(z,x,y,w).\,</math>

The space {{nowrap|1='''R'''''V'' = ker ''b''}} in ''S''2Λ2''V'' is the space of algebraic curvature tensors. The Ricci decomposition is the decomposition of this space into irreducible factors. The Ricci contraction mapping
:<math>c : S^2\Lambda^2 V \to S^2V</math>
is given by
:<math>c(R)(x,y) = \operatorname{tr}R(x,\cdot,y,\cdot).</math>
This associates a symmetric 2-form to an algebraic curvature tensor. Conversely, given a pair of symmetric 2-forms ''h'' and ''k'', the [[Kulkarni–Nomizu product]] of ''h'' and ''k''
:<math>(h {~\wedge\!\!\!\!\!\!\bigcirc~} k)(x,y,z,w) = h(x,z)k(y,w)+h(y,w)k(x,z) -h(x,w)k(y,z)-h(y,z)k(x,w)</math>
produces an algebraic curvature tensor.

If ''n'' > 4, then there is an orthogonal decomposition into (unique) irreducible subspaces
:{{nowrap|1='''R'''''V'' = '''S'''''V'' &oplus; '''E'''''V'' &oplus; '''C'''''V''}}
where
:<math>\mathbf{S}V = \mathbb{R} g {~\wedge\!\!\!\!\!\!\bigcirc~} g</math>, where <math>\mathbb{R}</math> is the space of [[real number|real]] scalars
:<math>\mathbf{E}V = g {~\wedge\!\!\!\!\!\!\bigcirc~} S^2_0V</math>, where ''S''{{su|p=2|b=0}}''V'' is the space of trace-free symmetric 2-forms
:<math>\mathbf{C}V = \ker c \cap \ker b.</math>

The parts ''S'', ''E'', and ''C'' of the Ricci decomposition of a given Riemann tensor ''R'' are the orthogonal projections of ''R'' onto these invariant factors. In particular,
:<math>R = S + E + C</math>
is an orthogonal decomposition in the sense that
:<math>|R|^2 = |S|^2 + |E|^2 + |C|^2.</math>
This decomposition expresses the space of tensors with Riemann symmetries as a direct sum of the scalar submodule, the Ricci submodule, and Weyl submodule, respectively. Each of these modules is an [[irreducible representation]] for the [[orthogonal group]] {{harv|Singer|Thorpe|1968}}, and thus the Ricci decomposition is a special case of the splitting of a module for a [[semisimple Lie group]] into its irreducible factors. In dimension 4, the Weyl module decomposes further into a pair of irreducible factors for the [[special orthogonal group]]: the [[self-dual]] and [[antiself-dual]] parts ''W''+ and ''W''−.

==Physical interpretation==
The Ricci decomposition can be interpreted physically in Einstein's theory of [[general relativity]], where it is sometimes called the ''Géhéniau-Debever decomposition''. In this theory, the [[Einstein field equation]]
:<math> G_{ab} = 8 \pi \, T_{ab}</math>
where <math>T_{ab}</math> is the [[stress–energy tensor]] describing the amount and motion of all matter and all nongravitational field energy and momentum, states that the Ricci tensor—or equivalently, the Einstein tensor—represents that part of the gravitational field which is due to the ''immediate presence'' of nongravitational energy and momentum. The Weyl tensor represents the part of the gravitational field which can propagate as a [[gravitational wave]] through a region containing no matter or nongravitational fields. Regions of spacetime in which the Weyl tensor vanishes contain no [[gravitational radiation]] and are also conformally flat.

==See also==

*[[Bel decomposition]] of the [[Riemann tensor]]
*[[Conformal geometry]]
*[[Petrov classification]]
*[[Plebanski tensor]]
*[[Ricci calculus]]
*[[Schouten tensor]]
*[[Trace-free Ricci tensor]]

==References==
*{{Citation | last1=Besse | first1=Arthur L. | title=Einstein manifolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10 | isbn=978-3-540-15279-8 | year=1987 | pages=xii+510}}.

*{{cite book | author=Hawking, S. W.; and Ellis, G. F. R.| title = The Large Scale Structure of Space-Time | publisher=Cambridge: Cambridge University Press | year=1973| isbn=0-521-09906-4}} See ''section 2.6'' for the decomposition. This book uses opposite signature but the same ''Landau-Lifshitz spacelike'' sign convention used in the Wikipedia.

*{{cite book | author=Weinberg, Steven | title=Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity | publisher=New York: John Wiley & Sons | year=1972 | isbn=0-471-92567-5}} See ''section 6.7'' for a discussion of the decomposition (but note different sign conventions).

*{{cite book | author=Wald, Robert M. | title=General Relativity | publisher=The University of Chicago Press | year=1984 | isbn=0-226-87033-2}} See ''section 3.2'' for a discussion of the decomposition.

* {{citation | first = R.W. | last = Sharpe | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer-Verlag, New York | year = 1997 | id = ISBN 0-387-94732-9}}. Section 6.1 discusses the decomposition. Versions of the decomposition also enter into the discussion of conformal and projective geometries, in chapters 7 and 8.

* {{citation | first1=I.M.|last1=Singer|authorlink1=Isadore Singer|first2=J.A.|last2=Thorpe|title=Global Analysis (Papers in Honor of K. Kodaira)|contribution=The curvature of 4-dimensional Einstein spaces|publisher=Univ. Tokyo Press|year=1969|pages=355–365}}.

{{DEFAULTSORT:Ricci Decomposition}}
[[Category:Differential geometry]]
[[Category:Riemannian geometry]]
[[Category:Tensors in general relativity]]

Scalar field solution - Revision history

18.111.68.236: /* Einstein tensor */ The Einstein tensor given disagreed with the eigenvalue equation below, and gave a stress energy tensor with negative pressures.