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Riemann Tensor


The Riemann tensor (Schutz 1985) R^alpha_(betagammadelta), also known the Riemann-Christoffel curvature tensor (Weinberg 1972, p. 133; Arfken 1985, p. 123) or Riemann curvature tensor (Misner et al. 1973, p. 218), is a four-index tensor that is useful in general relativity. Other important general relativistic tensors such that the Ricci curvature tensor and scalar curvature can be defined in terms of R^alpha_(betagammadelta).

The Riemann tensor is in some sense the only tensor that can be constructed from the metric tensor and its first and second derivatives,

 R^alpha_(betagammadelta)=Gamma_(betadelta,gamma)^alpha-Gamma_(betagamma,delta)^alpha+Gamma_(betadelta)^muGamma_(mugamma)^alpha-Gamma_(betagamma)^muGamma_(mudelta)^alpha,
(1)

where Gamma_(alphabeta)^gamma are Christoffel symbols of the first kind and A_(,k) is a comma derivative (Schmutzer 1968, p. 108; Weinberg 1972). In one dimension, R_(1111)=0. In four dimensions, there are 256 components. Making use of the symmetry relations,

 R_(iklm)=-R_(ikml)=-R_(kilm),
(2)

the number of independent components is reduced to 36. Using the condition

 R_(iklm)=R_(lmik),
(3)

the number of coordinates reduces to 21. Finally, using

 R_(iklm)+R_(ilmk)+R_(imkl)=0,
(4)

20 independent components are left (Misner et al. 1973, pp. 220-221; Arfken 1985, pp. 123-124).

In general, the number of independent components in n dimensions is given by

 C_n=1/(12)n^2(n^2-1),
(5)

the "four-dimensional pyramidal numbers," the first few values of which are 0, 1, 6, 20, 50, 105, 196, 336, 540, 825, ... (OEIS A002415). The number of scalars which can be constructed from R_(lambdamunukappa) and g_(munu) is

 S_n={1   for n=2; 1/(12)n(n-1)(n-2)(n+3)   for n=1,n>2
(6)

(Weinberg 1972). The first few values are then 0, 1, 3, 14, 40, 90, 175, 308, 504, 780, ... (OEIS A050297).

In terms of the Jacobi tensor J^mu_(nualphabeta),

 R^mu_(alphanubeta)=2/3(J_(nualphabeta)^mu-J_(betaalphanu)^mu).
(7)

Let

 D^~_s=partial/(partialx^s)-sum_(l){s  u; l},
(8)

where the quantity inside the {s  u; l} is a Christoffel symbol of the second kind. Then

 R_(pqrs)=D^~_q{p  r; s}-D^~_r{r  q; s}.
(9)

Broken down into its simplest decomposition in N dimensions,

 R_(lambdamunukappa)=1/(N-2)(g_(lambdanu)R_(mukappa)-g_(lambdakappa)R_(munu)-g_(munu)R_(lambdakappa)+g_(mukappa)R_(lambdanu))-R/((N-1)(N-2))(g_(lambdanu)g_(mukappa)-g_(lambdakappa)g_(munu))+C_(lambdamunukappa).
(10)

Here, R_(munu) is the Ricci curvature tensor, R is the scalar curvature, and C_(lambdamunukappa) is the Weyl tensor.


See also

Bianchi Identities, Christoffel Symbol of the First Kind, Christoffel Symbol of the Second Kind, Commutation Coefficient, Gaussian Curvature, Jacobi Tensor, Petrov Notation, Ricci Curvature Tensor, Riemannian Geometry, Riemannian Metric, Scalar Curvature, Weyl Tensor

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. "Geodesic Deviation and the Riemann Curvature Tensor." §8.7 in Gravitation. San Francisco: W. H. Freeman, pp. 218-224, 1973.Parker, L. and Christensen, S. M. "The Riemann Curvature Tensor." §2.7 in MathTensor: A System for Doing Tensor Analysis by Computer. Reading, MA: Addison-Wesley, pp. 28-32, 1994.Schutz, B. F. "Riemann Tensor" and "Geometric Interpretation of the Riemann Tensor." §6.8 in A First Course in General Relativity. Cambridge, England: Cambridge University Press, pp. 210-214, 1985.Schmutzer, E. Relativistische Physik (Klassische Theorie). Leipzig, Germany: Akademische Verlagsgesellschaft, 1968.Sloane, N. J. A. Sequences A002415/M4135 and A050297 in "The On-Line Encyclopedia of Integer Sequences."Weinberg, S. "Definition of the Curvature Tensor" and "Uniqueness of the Curvature Tensor." §6.1 and 6.2 in Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, pp. 131-135, 1972.

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Riemann Tensor

Cite this as:

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

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