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Tensor[BachTensor] - calculate the Bach tensor of a metric

 

Calling Sequences

     BachTensor(g)

     BachTensor(g, Γ)

     BachTensor(g, G, R, C)

Parameters

     g       - a metric tensor on the tangent bundle of a manifold

     Γ       - (optional) the Christoffel connection of g

     R       - (optional) the curvature tensor of g

     C       - (optional) the Cotton tensor of g

 

Description

Examples

Description

• 

Let gab be a metric (of any signature) on the tangent bundle of a manifold M of dimensionn>2. The metric determines: the covariant derivative a, the Schouten tensor Pab, the Weyl tensor Wabcd and the Cotton tensor Cabc. The Bach tensor is defined as

Bab= cCacb+PdcWdacb.

he Bach tensor is trace-free: gabBab=0. See A. Grover and P. Nurowski, J. Geom. Phys. 56, 450-484 (2006) for additional properties, applications and references.

• 

The first calling sequence computes Bab directly from the given metric using the formula above. The second calling sequence computes Bab from the given metric and Christoffel connection. The third calling sequence computes Bab directly from the given metric Christoffel connection, curvature and Cotton tensors.

• 

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form BachTensor(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-BachTensor.

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

Calculate the Bach tensor of a metric and check that it is trace-free.

 

DGsetupu,v,x,y,M

frame name: M

(2.1)
M > 

gevalDGdu&sdv+dx&tdx+dy&tdy+expxydu&tdu

g:=_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇxy,1,2,12,2,1,12,3,3,1,4,4,1,_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇxy,1,2,12,2,1,12,3,3,1,4,4,1,_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇxy,1,2,12,2,1,12,3,3,1,4,4,1,_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇxy,1,2,12,2,1,12,3,3,1,4,4,1

(2.2)
M > 

BBachTensorg

B:=_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇxyx4+2x2y2+y4+8xy+44,_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇxyx4+2x2y2+y4+8xy+44,_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇxyx4+2x2y2+y4+8xy+44,_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇxyx4+2x2y2+y4+8xy+44

(2.3)
M > 

TensorInnerProductg,g,B

0

(2.4)

 

Example 2.

Calculate the Bach tensor of a metric and Christoffel connection. We use the metric from the previous example.

 

 

M > 

GammaChristoffelg

Γ:=_DGconnection,M,con_bas,cov_bas,cov_bas,,2,1,3,yⅇxy,2,1,4,xⅇxy,2,3,1,yⅇxy,2,4,1,xⅇxy,3,1,1,yⅇxy2,4,1,1,xⅇxy2,_DGconnection,M,con_bas,cov_bas,cov_bas,,2,1,3,yⅇxy,2,1,4,xⅇxy,2,3,1,yⅇxy,2,4,1,xⅇxy,3,1,1,yⅇxy2,4,1,1,xⅇxy2,_DGconnection,M,con_bas,cov_bas,cov_bas,,2,1,3,yⅇxy,2,1,4,xⅇxy,2,3,1,yⅇxy,2,4,1,xⅇxy,3,1,1,yⅇxy2,4,1,1,xⅇxy2,_DGconnection,M,con_bas,cov_bas,cov_bas,,2,1,3,yⅇxy,2,1,4,xⅇxy,2,3,1,yⅇxy,2,4,1,xⅇxy,3,1,1,yⅇxy2,4,1,1,xⅇxy2

(2.5)
M > 

BachTensorg,Gamma

_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇxyx4+2x2y2+y4+8xy+44,_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇxyx4+2x2y2+y4+8xy+44,_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇxyx4+2x2y2+y4+8xy+44,_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇxyx4+2x2y2+y4+8xy+44

(2.6)

Example 3.

Calculate the Bach tensor of a metric Christoffel connection, curvature tensor and Cotton tensor. We use the metric and connection from the previous examples.

 

M > 

RCurvatureTensorg

R:=_DGtensor,M,con_bas,cov_bas,cov_bas,cov_bas,,2,3,1,3,y2ⅇxy,2,3,1,4,ⅇxyxy+1,2,3,3,1,y2ⅇxy,2,3,4,1,ⅇxyxy+1,2,4,1,3,ⅇxyxy+1,2,4,1,4,x2ⅇxy,2,4,3,1,ⅇxyxy+1,2,4,4,1,x2ⅇxy,3,1,1,3,y2ⅇxy2,3,1,1,4,ⅇxyxy+12,3,1,3,1,y2ⅇxy2,3,1,4,1,ⅇxyxy+12,4,1,1,3,ⅇxyxy+12,4,1,1,4,x2ⅇxy2,4,1,3,1,ⅇxyxy+12,4,1,4,1,x2ⅇxy2,_DGtensor,M,con_bas,cov_bas,cov_bas,cov_bas,,2,3,1,3,y2ⅇxy,2,3,1,4,ⅇxyxy+1,2,3,3,1,y2ⅇxy,2,3,4,1,ⅇxyxy+1,2,4,1,3,ⅇxyxy+1,2,4,1,4,x2ⅇxy,2,4,3,1,ⅇxyxy+1,2,4,4,1,x2ⅇxy,3,1,1,3,y2ⅇxy2,3,1,1,4,ⅇxyxy+12,3,1,3,1,y2ⅇxy2,3,1,4,1,ⅇxyxy+12,4,1,1,3,ⅇxyxy+12,4,1,1,4,x2ⅇxy2,4,1,3,1,ⅇxyxy+12,4,1,4,1,x2ⅇx