Unable to simplify tensor expressions with xAct

Question

Using xAct package in Mathematica, I am trying to obtain expansions of Ricci scalar upto second order in metric perturbations. Although I am able to obtain the perturbed expressions, Mathematica/xAct does not seem to simplify some tensor expressions. For example, there are terms like:

    p2[a, -a] p2[b, -b] - (p2[a, -a])^2

where p2[a,b] is a second rank tensor. I would like such expressions to be automatically simplified; in the above case, expected result is 0. However, when I use the command //Simplification, I get the same expression. For instance, this is what I get for the above expression:

    In[42]:= p2[a, -a] * p2[b, -b] - (p2[a, -a])^2 // Simplification

    Out[42]= p2[a, -a]   p2[b, -b] - xAct`xTensor`Scalar[p2[a, -a]]^2

Why is xAct/Mathematica not recognizing the two terms as equal?

What is xActxTensorScalar[] object?

A prescription to simplify such expressions in general would be appreciated.

Answer 1

An expression like p[a, -a]^2 is strictly incorrect in xTensor, because in WL this is p[a, -a]*p[a, -a], which has repeated indices. The way to avoid this problem is to hide internal scalar indices inside the Scalar head. That is, the expression Scalar[p[a, -a]]^2 is the correct way to express that object in xTensor.

Your Simplification call finds an improper expression and tries to fix it by returning an unambiguous expression (this doesn't always work, because ambiguous input may be incorrectly interpreted sometimes). Use then the command NoScalar, which will rename indices to avoid collisions, and finally use Simplification or ToCanonical again. You'll get the expected zero output.

Alternative, the command PutScalar (internally used by Simplification) adds the Scalar heads, and then you can call Simplification:

In[7]:= p[a, -a] p[b, -b] - p[a, -a]^2 // PutScalar // Simplification
Out[7]= 0