DifferentialGeometry/Tensor/BelRobinson

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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"

Description

•	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)

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).

•	The Bel-Robinson tensor is totally symmetric: B_{ijhk} = B_{jihk} = B_{hjik} = B_{kjhi}.

•	The Bel-Robinson tensor is trace-free: g^{ij} B_{ijhk} = 0.

•

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.

•	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.

•	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.

Examples

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).

(2.1)

M >

(2.2)

Calculate the Bel-Robinson tensor for the metric g. The result is clearly a symmetric tensor.

M >

(2.3)

Use the optional keyword argument indexlist to calculate the contravariant form of the Bel-Robinson tensor.

M >

(2.4)

The tensor B is trace-free.

M >

(2.5)

M >

(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.

M >

C := -dx*D_x*dz+(1/6)*(Lambda*exp(z)*dx*D_z*dx)-(1/3)*(Lambda*exp(-2*z)*dx*D_z*du)+(3/2)*(exp(3*z)*dx*D_u*dz)-dy*D_y*dz-(1/3)*(Lambda*exp(-2*z)*dy*D_z*dy)-dz*D_x*dx-dz*D_y*dy+(3/2)*(exp(3*z)*dz*D_u*dx)-dz*D_u*du-(1/3)*(Lambda*exp(-2*z)*du*D_z*dx)-du*D_u*dz

(2.7)

M >

nablaB1 := -(1/12)*(Lambda^3*exp(8*z)*D_z*D_u*D_u*D_u*dx)-(1/12)*(Lambda^3*exp(8*z)*D_u*D_z*D_u*D_u*dx)-(1/12)*(Lambda^3*exp(8*z)*D_u*D_u*D_z*D_u*dx)-(1/12)*(Lambda^3*exp(8*z)*D_u*D_u*D_u*D_z*dx)+(3/2)*(exp(10*z)*Lambda^2*D_u*D_u*D_u*D_u*dz)

(2.8)

M >

(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.

M >

(2.10)

M >

(2.11)

The Weyl tensor, if already calculated, can be used to quickly compute the Bel-Robinson tensor.

M >

W := -(1/2)*(Lambda*exp(-z)*dx*dy*dx*dy)+(1/2)*(Lambda*exp(-z)*dx*dy*dy*dx)-(3/2)*(exp(z)*dx*dz*dx*dz)+(3/2)*(exp(z)*dx*dz*dz*dx)+(1/2)*(Lambda*exp(-z)*dy*dx*dx*dy)-(1/2)*(Lambda*exp(-z)*dy*dx*dy*dx)+(3/2)*(exp(z)*dz*dx*dx*dz)-(3/2)*(exp(z)*dz*dx*dz*dx)