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Weyl Tensor

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The Weyl tensor is the tensor C_(abcd) 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),
(1)

where R_(abcd) is the Riemann tensor, R is the scalar curvature, g_(ab) is the metric tensor, and T_([a_1...a_n]) denotes the antisymmetric tensor part (Wald 1984, p. 40).

The Weyl tensor is defined so that every tensor contraction between indices gives 0. In particular,

C^lambda_(mulambdakappa)=0
(2)

(Weinberg 1972, p. 146). The number of independent components for a Weyl tensor in N-D for N>=3 is given by

C_N=1/(12)N(N+1)(N+2)(N-3)
(3)

(Weinberg 1972, p. 146). For N=3, 4, ..., this gives 0, 10, 35, 84, 168, ... (OEIS A052472).

See also

Scalar Curvature, Riemann Tensor

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References

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.

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Weyl Tensor

Cite this as:

Weisstein, Eric W. "Weyl Tensor." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeylTensor.html

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