Calculating e.g. Euler density or Riemann tensor's square for a given metric using xAct package

[Debajyoti Sarkar]

Dear all,

I have started to learn about the xAct package to do some general relativistic calculations and I am still in the lower part of the learning curve. As the title suggests, this is a very basic question, but I am having a hard time to get the answer. I have installed the package and all its components properly, so there're no issues there.

I started with 

    << xAct`xTensor`

    

    << xAct`xCoba`

    

    << xAct`xTras`

as I think these will be eventually needed. After that I am simply going ahead in the usual way to define the metric:

    DefManifold[M, 4, IndexRange[a, q]]

    

    DefMetric[-1, metric[-a, -b], CD, PrintAs -> "g"]

    

    DefScalarFunction[σ]

    DefScalarFunction[$N]

    DefScalarFunction[R]

    

    MatrixForm[fm = {

       {-E^(2 σ[z[]]), 0, 0, 0},

       {0, $N[z[]]^2 E^(2 σ[z[]]), 0, 0},

       {0, 0, R[z[]]^2  E^(2 σ[z[]]), 0},

       {0, 0, 0, R[z[]]^2 Sin[θ[]]^2 E^(2 σ[z[]])}

       }]

    

    MetricInBasis[metric, -B, fm]

    

    MetricCompute[metric, B, All, CVSimplify -> Simplify]

This defined the metric tensor and also assigns a basis vector. It also uses the matrix's elements of "fm" as the metric tensor. So I have defined a given metric. The last input calculates several things together such as Riemann Tensor, Christoffel symbol and so on..

From here on, if I want, I can compute Ricci scalar or any function of it. I can also e.g. find out what are the components of Riemann tensor e.g. But what I can't seem to compute is what is the value of $Rs=R_{abcd}R^{abcd}$ for this given metric. I can only write it in a symbolic form, but not evaluated for the given metric and basis.

I tried commands like 

    TensorValues[RiemannCD[-B, -B, -B, -B] RiemannCD[B, B, B, B]] // 

     ToValues[metric]

or 

    TensorValues[Rs] // ToBasis[B] // ToValues

etc.

but none of these are working. I also can't seem to find any such example anywhere!

[Leo Stein]

Dear Debajyoti,

Please see some of the example files, e.g. KerrNewmanExamples.nb for an idea of how to use the 'original' xCoba tools. However I would recommend using the updated CTensor tools, which are not documented as well. There are a large number of examples in the file xCobaDoc2.nb which is part of the distribution. Especially look at some of the examples in Sections 4 and 5. Once you have input a metric `metric` and its Levi-Civita derivative `cd = CovDOfMetric[metric]`, these computations are as simple as e.g. for the Kretschmann

  Riemann[cd][-a,-b,-c,-d] Riemann[cd][a,b,c,d]

and for the Euler density (in 4 dimensions)

  Riemann[cd][-a,-b,-c,-d] Riemann[cd][a,b,c,d] - 4 Ricci[cd][-a,-b] Ricci[cd][a,b] + RicciScalar[cd][]^2

Best

Leo

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[Debajyoti Sarkar]

Dear Leo

Thanks so much for your reply!! I looked at the Kerr-Newman example before (which was on the webpage) but it didn't help me much. However, some of the section 5 examples of xcobadoc2.nb (which I somehow didn't know existed) were really helpful. In fact, the commands there are a bit different from what I knew so far. Anyway, thanks again! Now I can compute any combination of such curvature quantities..

Best,

Deb

--

Dr. Debajyoti Sarkar

Arnold Sommerfeld Center

Ludwig Maximilian University

Room 316

Fakultät für Physik

Theoretische Physik

Theresienstr. 37

80333 München

Tel: +49 (89) 2180-4553