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Tensor[BelRobinson] - calculate the Bel-Robinson tensor
Calling Sequences
BelRobinson(g, W, indexlist)
Parameters
g - a metric tensor on a 4-dimensional manifold
W - (optional) the Weyl tensor of the metric g
indexlist - (optional) the keyword argument indexlist = ind, where ind is a list of 4 index types "con" or "cov"
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Description
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The Bel-Robinson tensor B_{ijhk} is a covariant rank 4 tensor defined in terms of the Weyl tensor W_{ijhk} on a 4-dimensional manifold by (see, for example, Penrose and Rindler Vol. 1)
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B_{ijhk} = 1/4 (W_{ilhm}W_j^l_k^m - 1/2 (g_{ij} W_{lmhn} + g_{il} W_{mjhn} + g_{im} W_{jlhn}) W^{lm}_k^n).
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The Bel-Robinson tensor is totally symmetric: B_{ijhk} = B_{jihk} = B_{hjik} = B_{kjhi}.
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The Bel-Robinson tensor is trace-free: g^{ij} B_{ijhk} = 0.
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If g is an Einstein metric, that is, R_{ij} = Lambda g_{ij} (where R_{ij} is the Ricci tensor for the metric g and Lambda is a constant), then the covariant divergence of Bel-Robinson vanishes: g^{il} nabla_l(B_{ijhk}) = 0. Here nabla_l denotes the covariant derivative with respect to the Christoffel connection for g.
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The keyword argument indexlist = ind allows the user to specify the index structure for the Bel-Robinson tensor. For example, with indexlist = ["con", "con", "con", "con"], the contravariant form B^{ijhk} is returned.
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This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form BelRobinson(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-BelRobinson.
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Examples
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Example 1.
First create a 4-dimensional manifold M and define a metric g on M. The metric shown below is a homogenous Einstein metric (see (12.34) in Stephani, Kramer et al).
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![DGsetup([x, y, z, u], M)](/support/helpjp/helpview.aspx?si=5667/file05851/math128.png)
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| (2.1) |
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Calculate the Bel-Robinson tensor for the metric g. The result is clearly a symmetric tensor.
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| (2.3) |
Use the optional keyword argument indexlist to calculate the contravariant form of the Bel-Robinson tensor.
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![B1 := BelRobinson(g, indexlist = ["con", "con", "con", "con"])](/support/helpjp/helpview.aspx?si=5667/file05851/math161.png)
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| (2.4) |
The tensor B is trace-free.
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| (2.5) |
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![ContractIndices(h, B, [[1, 1], [2, 2]])](/support/helpjp/helpview.aspx?si=5667/file05851/math181.png)
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| (2.6) |
The covariant divergence of the tensor B1 vanishes. To check this, first calculate the Christoffel connection C for the metric g and then calculate the covariant derivative of B1.
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| (2.8) |
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![Divergence := ContractIndices(nablaB1, [[1, 5]])](/support/helpjp/helpview.aspx?si=5667/file05851/math214.png)
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| (2.9) |
The divergence of the Bel-Robinson tensor is not automatically zero; the divergence vanishes when the metric g is an Einstein metric. To check this, compute the Ricci tensor of g.
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| (2.10) |
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| (2.11) |
The Weyl tensor, if already calculated, can be used to quickly compute the Bel-Robinson tensor.
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| (2.12) |
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| (2.13) |
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