TOPICS

# Weyl Tensor

The Weyl tensor is the tensor defined by

 (1)

where is the Riemann tensor, is the scalar curvature, is the metric tensor, and 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,

 (2)

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

 (3)

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

Scalar Curvature, Riemann Tensor

## Explore with Wolfram|Alpha

More things to try:

## 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.

Weyl Tensor

## Cite this as:

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