Cotton Tensor gives wrong result.

[Alan Stafford]

If I compute the Bach Tensor using the definition via the cotton tensor I get a different result to that using an alternative definition.

M = Manifold(4, 'M')

MChart = M.open_subset('MChart')

Chart.<u,v,x,y>   = MChart.chart(r'u:(-oo,+oo) v:(-oo,+oo) x:(-oo,+oo) y:(-oo,+oo)') 

gT= MChart.riemannian_metric('gT')

var('du','dv','dx','dy')

dsds= -du*dv+dx*dx+dy*dy+e^(x*y)*du*du

dsds=dsds.expand()

g00=dsds.coefficient(du,2)

g11=dsds.coefficient(dv,2)

g22=dsds.coefficient(dx,2)

g33=dsds.coefficient(dy,2)

g01=dsds.coefficient(du*dv,1)

g01=g01/2

g10=g01

gT[0,0] = g00.factor() #du du

gT[1,1] = g11.factor() #dv dv

gT[2,2] = g22.factor() #dx dx

gT[3,3] = g33.factor() #dy dy

gT[0,1] = g01.factor() #du dv 

%display latex

show(gT.display())

Metric=gT

Nabla = Metric.connection()

#https://arxiv.org/pdf/gr-qc/0309008.pdf equation 54

Bach=(Nabla(Metric.cotton()).up(Metric,3)['^u_aub'])+((Metric.schouten().up(Metric))*(Metric.weyl().down(Metric)))['^uv_aubv']

Bach.display()

This gives: 1/2*(x^4 + 2*x^2*y^2 + y^4 + 8*x*y + 4)*e^(x*y) du*du

But

Bach=Nabla(Nabla(Metric.weyl().down(Metric))).up(Metric,4).up(Metric,5)['^bd_abcd']-(1/2)*((Metric.ricci().up(Metric))*(Metric.weyl().down(Metric)))['^bd_abcd'] 

Bach.display()

Gives: -1/4*(x^4 + 2*x^2*y^2 + y^4 + 8*x*y + 4)*e^(x*y) du*du

This agrees with the Maple example Bach Tensor computation at. https://www.maplesoft.com/support/help/Maple/view.aspx?path=DifferentialGeometry/Tensor/BachTensor

[Eric Gourgoulhon]

Hi,

What do you mean exactly by "Cotton tensor gives wrong result" ?

Maybe this is matter of convention... SageMath's definition of the Cotton conformal tensor is here:

https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/metric.html#sage.manifolds.differentiable.metric.PseudoRiemannianMetric.cotton

Best wishes,

Eric.

[Alan Stafford]

I have made a test script for metric derived Tensors. For this to pass I have had to change the Cotton Tensor definition in metric.py. I attach this. I have also added the Bach conformal Tensor as g.bach() .

I think in the past calculations have only been checked for the cotton tensor to be zero for conformally flat metrics so multiplicative factors have not been examined.

As it is I get different results calculating the Bach Tensor when I use the Cotton Tensor to that using another definitions which do not, so if it is a matter of convention they are not consistent.