In differential geometry, the Weyl curvature tensor, named after
Hermann Weyl, is a measure of the curvature of spacetime or, more
generally, a pseudo-Riemannian manifold. Like the Riemann curvature
tensor, the
Contents 1 Definition 2 Properties 2.1 Conformal rescaling 2.2 Symmetries 2.3 Bianchi identity 3 See also 4 References Definition[edit]
The
C = R − 1 n − 2 ( R i c − s n g ) ∧ ◯ g − s 2 n ( n − 1 ) g ∧ ◯ g displaystyle C=R- frac 1 n-2 left(mathrm Ric - frac s n gright)wedge !!!!!!bigcirc g- frac s 2n(n-1) gwedge !!!!!!bigcirc g where n is the dimension of the manifold, g is the metric, R is the Riemann tensor, Ric is the Ricci tensor, s is the scalar curvature, and h ∧ ◯ k displaystyle hwedge !!!!!!bigcirc k denotes the
( h ∧ ◯ k ) ( v 1 , v 2 , v 3 , v 4 ) = displaystyle (hwedge !!!!!!bigcirc k)(v_ 1 ,v_ 2 ,v_ 3 ,v_ 4 )= h ( v 1 , v 3 ) k ( v 2 , v 4 ) + h ( v 2 , v 4 ) k ( v 1 , v 3 ) displaystyle h(v_ 1 ,v_ 3 )k(v_ 2 ,v_ 4 )+h(v_ 2 ,v_ 4 )k(v_ 1 ,v_ 3 ), − h ( v 1 , v 4 ) k ( v 2 , v 3 ) − h ( v 2 , v 3 ) k ( v 1 , v 4 ) displaystyle -h(v_ 1 ,v_ 4 )k(v_ 2 ,v_ 3 )-h(v_ 2 ,v_ 3 )k(v_ 1 ,v_ 4 ), In full tensor notation, this can be written as C i k ℓ m = R i k ℓ m + 1 n − 2 ( R i m g k ℓ − R i ℓ g k m + R k ℓ g i m − R k m g i ℓ ) + 1 ( n − 1 ) ( n − 2 ) R ( g i ℓ g k m − g i m g k ℓ ) . displaystyle C_ ikell m =R_ ikell m + frac 1 n-2 left(R_ im g_ kell -R_ iell g_ km +R_ kell g_ im -R_ km g_ iell right)+ frac 1 (n-1)(n-2) Rleft(g_ iell g_ km -g_ im g_ kell right). The ordinary (1,3) valent
R
2 =
C
2 +
1 n − 2 ( R i c − s n g ) ∧ ◯ g
2 +
s 2 n ( n − 1 ) g ∧ ◯ g
2 . displaystyle R^ 2 =C^ 2 +left frac 1 n-2 left(mathrm Ric - frac s n gright)wedge !!!!!!bigcirc gright^ 2 +left frac s 2n(n-1) gwedge !!!!!!bigcirc gright^ 2 . This decomposition, known as the Ricci decomposition, expresses the
P = 1 n − 2 ( R i c − s 2 ( n − 1 ) g ) . displaystyle P= frac 1 n-2 left(mathrm Ric - frac s 2(n-1) gright). Then C = R − P ∧ ◯ g . displaystyle C=R-Pwedge !!!!!!bigcirc g. In indices,[2] C a b c d = R a b c d − 2 n − 2 ( g a [ c R d ] b − g b [ c R d ] a ) + 2 ( n − 1 ) ( n − 2 ) R g a [ c g d ] b displaystyle C_ abcd =R_ abcd - frac 2 n-2 (g_ a[c R_ d]b -g_ b[c R_ d]a )+ frac 2 (n-1)(n-2) R~g_ a[c g_ d]b where R a b c d displaystyle R_ abcd is the Riemann tensor, R a b displaystyle R_ ab is the Ricci tensor, R displaystyle R is the Ricci scalar (the scalar curvature) and brackets around indices refers to the antisymmetric part. Equivalently, C a b c d = R a b c d − 4 S [ a [ c δ b ] d ] displaystyle C_ ab ^ cd = R_ ab ^ cd -4S_ [a ^ [c delta _ b] ^ d] where S denotes the Schouten tensor.
Properties[edit]
Conformal rescaling[edit]
The
D d f − d f ⊗ d f + (
d f
2 + Δ f n − 2 ) g = Ric . displaystyle Ddf-dfotimes df+left(df^ 2 + frac Delta f n-2 right)g=operatorname Ric . In dimension ≥ 4, the vanishing of the
C ( u , v ) = − C ( v , u ) displaystyle C(u,v)=-C(v,u)_ ^ ⟨ C ( u , v ) w , z ⟩ = − ⟨ C ( u , v ) z , w ⟩ displaystyle langle C(u,v)w,zrangle =-langle C(u,v)z,wrangle _ ^ C ( u , v ) w + C ( v , w ) u + C ( w , u ) v = 0 . displaystyle C(u,v)w+C(v,w)u+C(w,u)v=0_ ^ . In addition, of course, the
tr C ( u , ⋅ ) v = 0 displaystyle operatorname tr C(u,cdot )v=0 for all u, v. In indices these four conditions are C a b c d = − C b a c d = − C a b d c displaystyle C_ abcd ^ =-C_ bacd =-C_ abdc C a b c d + C a c d b + C a d b c = 0 displaystyle C_ abcd +C_ acdb +C_ adbc ^ =0 C a b a c = 0. displaystyle C^ a _ bac =0. Bianchi identity[edit] Taking traces of the usual second Bianchi identity of the Riemann tensor eventually shows that ∇ a C a b c d = 2 ( n − 3 ) ∇ [ c S d ] b displaystyle nabla _ a C^ a _ bcd =2(n-3)nabla _ [c S_ d]b where S is the Schouten tensor. The valence (0,3) tensor on the right-hand side is the Cotton tensor, apart from the initial factor. See also[edit]
References[edit] ^ a b Danehkar, A. (2009). "On the Significance of the Weyl Curvature in a Relativistic Cosmological Model". Mod. Phys. Lett. A. 24 (38): 3113–3127. arXiv:0707.2987 . Bibcode:2009MPLA...24.3113D. doi:10.1142/S0217732309032046. ^ Grøn & Hervik 2007, p. 490 Hawking, Stephen W.; Ellis, George F. R. (1973), The Large Scale Structure of Space-Time, Cambridge University Press, ISBN 0-521-09906-4 Petersen, Peter (2006), Riemannian geometry, Graduate Texts in Mathematics, 171 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 0387292462, MR 2243772 . Sharpe, R.W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 0-387-94732-9 . Singer, I.M.; Thorpe, J.A. (1969), "The curvature of 4-dimensional Einstein spaces", Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, pp. 355–365 Hazewinkel, Michiel, ed. (2001) [1994], "Weyl tensor", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 Grøn, Øyvind; Hervik, Sigbjørn (2007), Einstein's General Theory of Relativity, New York: Springer, ISBN 978-0-387-69199-2 v t e Tensors Glossary of tensor theory Scope Mathematics coordinate system multilinear algebra Euclidean geometry tensor algebra dyadic algebra differential geometry exterior calculus tensor calculus Physics Engineering continuum mechanics electromagnetism transport phenomena general relativity computer vision Notation index notation multi-index notation Einstein notation Ricci calculus Penrose graphical notation Voigt notation abstract index notation tetrad (index notation) Van der Waerden notation
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