The Weyl tensor in dimension 
is the tensor 
defined by
![R_(abcd)=C_(abcd)+2/(n-2)(g_(a[c)R_d]b-g_(b[c)R_(d]a))
-2/((n-1)(n-2))Rg_(a[c)g_(d]b),](/images/equations/WeylTensor/NumberedEquation1.svg) |
(1)
|
where 
is the Riemann
tensor of the metric tensor 
, 
is the Ricci curvature
tensor, and 
is the scalar curvature. Square brackets denote
the normalized antisymmetric part, so that
![T_([ab])=1/2(T_(ab)-T_(ba))](/images/equations/WeylTensor/Inline7.svg)
(Wald 1984,
p. 40; Carroll 1997, p. 82).
The Weyl tensor is defined so that every tensor
contraction between indices gives 0. In particular,
 |
(2)
|
(Weinberg 1972, p. 146). The number of independent components of a Weyl tensor at a point in 
dimensions for 
is given by
 |
(3)
|
(Weinberg 1972, p. 146). For 
, 4, ..., this gives 0, 10, 35, 84, 168, ... (OEIS A052472).
See also
Antisymmetric Part,
Metric Tensor,
Ricci Curvature Tensor,
Riemann
Tensor,
Scalar Curvature
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References
Carroll, S. M. "Lecture Notes on General Relativity." 3 Dec 1997. https://arxiv.org/abs/gr-qc/9712019.
https://www.preposterousuniverse.com/wp-content/uploads/grnotes-three.pdf.Eisenhart,
L. P. Riemannian
Geometry. Princeton, NJ: Princeton University Press, 1964.Parker,
L. and Christensen, S. M. "The Ricci, Einstein, and Weyl Tensors."
§2.7.1 in MathTensor:
A System for Doing Tensor Analysis by Computer. Reading, MA: Addison-Wesley,
pp. 30-32, 1994.Sloane, N. J. A. Sequence A052472
in "The On-Line Encyclopedia of Integer Sequences."Wald, R. M.
General
Relativity. Chicago, IL: University of Chicago Press, 1984.Weinberg,
S. Gravitation
and Cosmology: Principles and Applications of the General Theory of Relativity.
New York: Wiley, 1972.Weyl, H. "Reine Infinitesimalgeometrie."
Math. Z. 2, 384-411, 1918.Referenced on Wolfram|Alpha
Weyl
Tensor
Cite this as:
Weisstein, Eric W. "Weyl Tensor." From
MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WeylTensor.html
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