Metric-compatible connection in xAct

[x tc]

  Hi,

I am learning to use xAct , and I'm really confused about the "covariant derivative" in xAct. Could someone help me? 

My first question is about the metric compatibility. I try to define a kerr metric with BL coordinate . Then I test the metric compatibility by symbolic computation, which yields zero. But When I try to calculate it in  BL coordinate, the result isn't zero. Here is the code.

In[1]:= << xAct`xCoba`

In[2]:= DefManifold[M, 4, {a, b, c, d, e, f}]

$PrePrint = ScreenDollarIndices;

DefChart[BL, M, {0, 1, 2, 3}, {t[], r[], \[Theta][], \[Phi][]}, 

 FormatBasis -> {"Partials", "Differentials"}]

$Assumptions = 

  t[] \[Element] Reals && r[] > 0 && 0 < \[Theta][] < Pi && 

   0 < \[Phi][] < 2 Pi;

DefConstantSymbol[Mass, PrintAs -> "M"]

DefConstantSymbol[Rotation, PrintAs -> "a"]

metricBL = {{-(1 - 2 Mass r[]/(r[]^2 + Rotation^2 Cos[\[Theta][]]^2)),

    0, 0, -2 Mass r[] Rotation Sin[\[Theta][]]^2/(r[]^2 + 

       Rotation^2 Cos[\[Theta][]]^2)}, {0, (r[]^2 + 

      Rotation^2 Cos[\[Theta][]]^2)/(r[]^2 - 2 Mass r[] + Rotation^2),

    0, 0}, {0, 0, r[]^2 + Rotation^2 Cos[\[Theta][]]^2, 

   0}, {-2 Mass r[] Rotation Sin[\[Theta][]]^2/(r[]^2 + 

       Rotation^2 Cos[\[Theta][]]^2), 0, 

   0, ((r[]^2 + Rotation^2)^2 - (r[]^2 - 2 Mass r[] + 

         Rotation^2) Rotation^2 Sin[\[Theta][]]^2) \

Sin[\[Theta][]]^2/(r[]^2 + Rotation^2 Cos[\[Theta][]]^2)}}

metricBL = CTensor[metricBL, {-BL, -BL}, 0];

SetCMetric[metricBL, BL, SignatureOfMetric -> {3, 1, 0}];

MetricCompute[metricBL, BL, All, Verbose -> True];

cdBL = CovDOfMetric[metricBL];

In[17]:= cdBL[a][metricBL[b, c]]

Out[17]= 0

In[18]:= cdBL[{1, -BL}][metricBL[{1, BL}, {1, BL}]]

Out[18]= (-2 Mass + 2 r[])/(Rotation^2 Cos[

\[Theta][]]^2 + r[]^2) - (

 2 r[] (Rotation^2 - 2 Mass r[] + r[]^2))/(Rotation^2 Cos[

\[Theta][]]^2 + r[]^2)^2

In[19]:= metricBL[{1, BL}, {1, BL}]

Out[19]= (Rotation^2 - 2 Mass r[] + r[]^2)/(Rotation^2 Cos[

\[Theta][]]^2 + r[]^2)

I thought cd represent the covariant derivative, but now I feel if I specify a coordinate to the command, then it becomes a partial derivative . If I want to calculate the real covariant derivate of the metric in a coordinate, which command should I use ? 

My second question is about partial derivative. From my understanding, if i define a chart BL or a basis NP, then the corresponding parallel(partial) derivative(which is a covariant derivative with zero affine connection field) will be PDBL or PDNP, right? 

The last question is about the Torsion tensor. I think in xAct  torsion is defined as the difference of affine connection, which is different from the definition i find in wiki. The definition in xact contains the commutator term, which means the nonholonomic coordinate will have torsion in xact , this is why some turorial like "Spherical-vs-Cartesian-Coordinates" in documentation say there is torsion in orthonomal basis ?  

I apologize for my possible  grammatical  error in text, and appreciate your patience for reading my length questions.  

[Jose]

Hi,

There are several different objects here:

    metricBL[-a, -b] is the abstract or geometric metric tensor.

    cdBL[-a][metricBL[-b, -c]] is another abstract or geometric tensor, the covariant derivative of the previous one, which happens to be zero.

    metricBL[{1, -BL}, {1, -BL}] is a component of the metric, i.e. a **scalar** field.

    metricBL[{1, BL}, {1, BL}] is a component of the **inverse** metric, another **scalar** field.

    cd[-a][ metricBL[{1, BL}, {1, BL}] ] is a covariant derivative of a scalar field. All covariant derivatives (including ordinary partial derivatives) coincide on scalar fields.

What you need is the CTensor that expresses the covariant derivative of the metric. That is computed with

    TensorDerivative[metricBL, cdBL]

and in this case it returns the tensor Zero. It's components are all 0. Try Zero[{1, -BL}, {1, -BL}] for example.

Regarding your second question, yes, the covariant derivative associated to a chart or basis is its PD object, so your PDBL or PDNP objects.

I'm not sure I understand your third question. It is true that in xTensor the Torsion tensor is just the non-symmetric part of the affine connection, so a difference of two connection terms if you want. xAct's definition may indeed be different from the definition used in other references. xAct follows Wald's General Relativity definitions in general.

Cheers,

Jose.