Odd Ricci scalar in Sagemanifolds

[Rogerio]

Recently I encountered an odd result while using Sagemanifolds to calculate the Ricci scalar for a specific 2D metric. I was trying to reproduce Eq. (37) of this paper, but the result was quite different. Here is the code

 N = Manifold(2,'N')
 X.<M,J> = N.chart(r' M:(0,oo) J:(-oo,oo)')
 ​
 dM = X.coframe()[0]
 dJ = X.coframe()[1]
 g0 = 2/(1-J^2/M^4)^(3/2)*(-2*((1-J^2/M^4)^(3/2) +1-3*J^2/M^4)*dM*dM - 2*J/M^3*dM*dJ - 2*J/M^3*dJ*dM + dJ*dJ/M^2)
 ​
 g = N.metric('g')
 g[:] = g0[:]
 ric = g.ricci_scalar()

The result is a high order rational function not resembling the paper result. However, taking

 R = N.scalar_field(1/(4*M^2)*(sqrt(1-J^2/M^4)-2)/sqrt(1-J^2/M^4),name='R')
 delta_R = ric-R
 ​
 delta_R == 0

Results True.

How to express the Ricci scalar as shown in the paper?

Thanks in advance!

[Eric Gourgoulhon]

Hi, 

The complicated result that you get is due to a lack of simplification in Sage. More precisely, the default simplification chain automatically applied in tensor calculus on manifolds, namely

https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/utilities.html#sage.manifolds.utilities.simplify_chain_real

yields overcomplicated expressions in your case. 

To get a much simpler result, which looks much closer to Eq. (37) of the paper, you have to replace the default simplification chain by a customized one. In your case, it suffices to use the function factor().

For this, use the manifold's method set_simplify_function() just after the declaration of the chart X

https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/manifold.html#sage.manifolds.manifold.TopologicalManifold.set_simplify_function

i.e. start you notebook with

N = Manifold(2,'N')
X.<M,J> = N.chart(r' M:(0,oo) J:(-oo,oo)')

N.set_simplify_function(factor)

Best wishes,

Eric.

[Rogerio]

Thank you again, Eric!