The Riemann tensor (Schutz 1985) , 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 .
The Riemann tensor is in some sense the only tensor that can be constructed from the metric tensor and its first and second derivatives,
(1)
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where are Christoffel symbols of the first kind and is a comma derivative (Schmutzer 1968, p. 108; Weinberg 1972). In one dimension, . In four dimensions, there are 256 components. Making use of the symmetry relations,
(2)
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the number of independent components is reduced to 36. Using the condition
(3)
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the number of coordinates reduces to 21. Finally, using
(4)
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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 dimensions is given by
(5)
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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 and is
(6)
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(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 ,
(7)
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Let
(8)
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where the quantity inside the is a Christoffel symbol of the second kind. Then
(9)
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Broken down into its simplest decomposition in dimensions,
(10)
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Here, is the Ricci curvature tensor, is the scalar curvature, and is the Weyl tensor.