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

 

Calling Sequences

     BachTensor(g)

     BachTensor(g, Γ)

     BachTensor(g, Γ, 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 &s dv+dx &t dx+dy &t dy+ⅇxydu &t du

g:=ⅇxydudu12dudv12dvdu+dxdx+dydy

(2.2)
M > 

BBachTensorg

B:=ⅇxyx4+2x2y2+y4+8xy+44dudu

(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 > 

ΓChristoffelg

Γ:=yⅇxyD_vdudxxⅇxyD_vdudyyⅇxyD_vdxduxⅇxyD_vdyduyⅇxy2D_xduduxⅇxy2D_ydudu

(2.5)
M > 

BachTensorg,Γ

ⅇxyx4+2x2y2+y4+8xy+44dudu

(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:=y2ⅇxyD_vdxdudx+ⅇxyxy+1D_vdxdudyy2ⅇxyD_vdxdxduⅇxyxy+1D_vdxdydu+ⅇxyxy+1D_vdydudx+x2ⅇxyD_vdydudyⅇxyxy+1D_vdydxdux2ⅇxyD_vdydydu+y2ⅇxy2D_xdududx+ⅇxyxy+12D_xdududyy2ⅇxy2D_xdudxduⅇxyxy+12D_xdudydu+ⅇxyxy+12D_ydududx+x2ⅇxy2D_ydududyⅇxyxy+12D_ydudxdux2ⅇxy2D_ydudydu

(2.7)
M > 

CCottonTensorg

C:=ⅇxyx2y+y3+2x4dududx+ⅇxyx3+xy2+2y4dududyⅇxyx2y+y3+2x4dudxduⅇxyx3+xy2+2y4dudydu

(2.8)
M > 

BachTensorg,Γ,R,C

ⅇxyx4+2x2y2+y4+8xy+44dudu

(2.9)

Example 3.

In four dimensions, the Bach tensor is an obstruction to a metric being conformal to an Einstein metric. Here we check that the Bach tensor vanishes on a metric conformal to a Ricci-flat metric in four dimensions.

 

M > 

DGsetupu,v,x,y,M

frame name: M

(2.10)
M > 

g0evalDGdu &s dv+dx &t dx+dy &t dy+xydu &t du

g0:=xydudu12dudv12dvdu+dxdx+dydy

(2.11)
M > 

RicciTensorg0

0dudu

(2.12)
M > 

g1evalDGⅇfu,v,x,yg0

g1:=ⅇfu,v,x,yxyduduⅇfu,v,x,y2dudvⅇfu,v,x,y2dvdu+ⅇfu,v,x,ydxdx+ⅇfu,v,x,ydydy

(2.13)
M > 

BachTensorg1

0dudu

(2.14)

See Also

DifferentialGeometry

CurvatureTensor

RicciTensor

RicciScalar

SchoutenTensor

WeylTensor

ProjectiveCurvature

 


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