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Hi I'm using the package xAct but I'm running into an issue. It seems that before using //ChristoffelToGradMetric the christoffel connection has indices (which can be printed using FindIndices), but after using //ChristoffelToGradMetric the resulting expression is no longer recognized by mathematica as having indices. I have to use //ChristoffelToGradMetric because I need the explicit expressions in terms of the metric (I don't want to do this by hand), then I want to replace the dummy indices in this expression with TraceDummy. However, mathematica does not recognize ChristoffelCD[c,-a,-b]//ChristoffelToGradMetric as an object with indices, so I can't split the dummy indices and this is where I need your help.

Consider a manifold with a metric:

<< xAct`xTensor`
DefManifold[M, D, IndexRange[a, f]]
DefMetric[-1, G[-a, -b], CD]

Now I want to obtain the Christoffel connection in terms of the metric. To my knowledge the only way to do this is using //ChristoffelToGradMetric. However, it seems that this function makes mathematica forget the indices of ChristoffelCD[c,-a,-b]. Before using //ChristoffelToGradMetric we have,

ChristoffelCD[c, -a, -b]
FindIndices[ChristoffelCD[c, -a -b]]

with the output:

$\Gamma^c_{ab}$

IndexList[c, -a - b]

Now when I use //ChristoffelToGradMetric and then FindIndices I get an empty list, so mathematica does not recognize the indices on the connection:

ChristoffelCD[c, -a, -b] // ChristoffelToGradMetric
FindIndices[ChristoffelCD[c,-a,-b]//ChristoffelToGradMetric]

with the following output:

$\frac{1}{2} G^{c f$2982} (\partial_aG_{b f$2982} + \partial_bG_{af$2982} - \partial_{f$2982}G_{ab})$

IndexList[]

I expected FindIndices would return IndexList[c, -a, -b, f$2982, -f$2982], but it doesn't. If I plug the connection in terms of the metric in FindIndices by hand then it does work:

FindIndices[1/2 *G[c, f] *(PD[-a][G[-b, -f]] + PD[-b][G[-a, -f]] - PD[-f][G[-a, -b]])]

output:

IndexList[c, f, -b, -f, a]

What is going on here? It is crucial for my code that mathematica recognizes the indices f$2982 as dummy indices because I want to use TraceDummy to split the f-contractions in two sums over two internal manifolds with indices (a1, b1, ...) and (a2, b2, ...). So I want to do the following:

TraceDummy[ChristoffelCD[c,-a,-b]//ChristoffelToGradMetric, f$2982 -> IndexList[a1, a2]]

which then should replace the terms involving the dummy indices f$2982 by two terms with sums over a1 and a2, e.g. for one of the three terms:

$G^{cf}\partial_fG_{ab} = G^{ca1}\partial_{a1}G_{ab} + G^{ca2}\partial_{a2}G_{ab}$

How do I achieve this?

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1 Answer 1

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The problem here is that FindIndices has attribute HoldFirst, and therefore by default it does not evaluate its first argument:

In[10]:= Attributes[FindIndices]
Out[10]= {HoldFirst, Protected}

To force evaluation in the first argument use Evaluate:

In[11]:= FindIndices[Evaluate[ChristoffelCD[c, -a, -b] // ChristoffelToGradMetric]]
Out[11]= xAct`xTensor`IndexList[c, f$30383, -b, -f$30383, -a]

I recommend to use IndicesOf instead of FindIndices. It is a more powerful function and does evaluate its arguments. Take for example:

In[12]:= expr = ChristoffelCD[c, -a, -b] // ChristoffelToGradMetric

Then you can compute

In[14]:= IndicesOf[][expr]
Out[14]= xAct`xTensor`IndexList[c, f$30389, -b, -f$30389, -a]

In[15]:= IndicesOf[Free][expr]
Out[15]= xAct`xTensor`IndexList[-a, -b, c]

In[16]:= IndicesOf[Dummy][expr]
Out[16]= xAct`xTensor`IndexList[-f$30389, f$30389]

In[18]:= IndicesOf[Up][expr]
Out[18]= xAct`xTensor`IndexList[c, f$30389]

In[19]:= IndicesOf[Down][expr]
Out[19]= xAct`xTensor`IndexList[-b, -f$30389, -a]

or mixtures like

In[20]:= IndicesOf[Free, Down][expr]
Out[20]= xAct`xTensor`IndexList[-a, -b]
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