Can't raise index on a partial derivative of a vector

[Sashwat Tanay]

Hi everybody,

My understanding of upper-index partial derivative is as per this definition.

This is exactly what I want to do using xAct -- raise/lower indices on partial derivatives.

I can lower the indices with ContractMetric; ToCanonical is useless (does nothing). But I can't raise the indices: ContractMetric does nothing, and ToCanonical produces the right result, plus some extra Christoffel terms which should not be there.  

See the pic below. Any help is appreciated. The notebook is attached.

[Thomas Bäckdahl]

Hi!

Canonicalizing expressions with partial derivatives is often slightly problematic because the derivative does not commute with the metric.

ContractMetric does by default not allow upper derivatives. This can be changed with
SetOptions[ContractMetric, AllowUpperDerivatives -> True]

If you have metrics in your expression I would in general recommend
expr// ContractMetric // ToCanonical
instead of just one of them.

In your last example the result is not wrong, but not properly canonicalized due to a symmetry of the Christoffel symbol. You can canonicalize twice to get rid of it.
You could also try
SetOptions[ToCanonical, Method -> {ChangeCovD, "ExpandChristoffel" -> True}]

Regards
Thomas

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[Sashwat Tanay]

Hi Thomas,

This solves the problem. Thanks a lot. Now I have stumbled upon some xAct documentation issues.

Here is what the documentation for AllowUpperDerivatives says

AllowUpperDerivatives: "AllowUpperDerivatives is a Boolean option for ContractMetric. If True, an inverse metric can be contracted with a covariant derivative givien a derivative with a contravariant index. If False, that cannot be done."

In the example above, you showed me how to contract with a partial (not covariant) derivative with a covariant (not contravariant) index. Also "givien" is a typo.

So, is the documentation incomplete/wrong?

[Jose]

Hi,

In xAct the partial derivative PD is just a special case of covariant derivative. It is the "ordinary derivative" of Wald, a covariant derivative with zero curvature and zero torsion. Therefore we have:

In[2]:= CovDQ[PD]
Out[2]= True

The zero torsion and zero curvature conditions imply locally the existence of a coordinate system on which the derivative can be constructed as partial differentiation with respect to the coordinates. PD in xTensor is a "fiducial derivative" defined with respect to some generic, unspecified, coordinate system.

The AllowUpperDerivatives option of ContractMetric applies to all covariant derivatives. For example take

In[3]:= DefManifold[M, 3, {a, b, c}]
In[4]:= DefMetric[-1, g[-a, -b], cd]
In[5]:= DefTensor[v[a], M]

Then this does not work by default:

In[6]:= ContractMetric[g[c, a] cd[-a][v[b]]]
Out[6]= g[c, a]  cd[-a][v[b]]

If you really want to contract that derivative then you need this:

In[7]:= ContractMetric[g[c, a] cd[-a][v[b]], AllowUpperDerivatives -> True]
Out[7]= cd[c][v[b]]

So in this sense the documentation is correct. Thanks for pointing out the typo: the "givien" should be "giving".

Cheers,

Jose.