Index splitting in a metric

[Amin Faraji]

 Hi everybody!

I need to compute the Christoffels,  curvature tensors and ... for a general metric as following

ds^2=dr^2+r^2 d\tau^2+g_{ij}(x)dx^i dx^j

where i,j=1,2,...,d. It is actually a metric in d+2 dimensional space time with coordinates (r,\tau,x^i).

Is it possible to do that using the xAct's packages?

I would really appreciate if someone could show me an example.

Thanks in advance.

AF

[AF]

 Dear Friends,

What I am asking is how we can simultaneously work with implicit and explicit components of the metric in xAct. Is it possible to work with a metric whose just one or two components are specified?

 Regards,

 AF 

[Obinna Umeh]

Hi AF,

Have you had a look at xCoba, there are two examples on spherically symmetric spacetime and KerrNewman spacetime here  http://www.xact.es/xCoba/index.html.

Obinna

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[AF]

 Hi Obinna,

Thanks, but as you see in these examples all of the components of the metric are specified. What I am talking a bout is a typical metric in which only one or two of the components are known, for example g_{11}=dr^2 , g_{22}=r^2 dt^2, and the other ones have been left unspecified as g_{ij}(x).

 Regards,

 AF

[Alfonso García-Parrado Gómez-Lobo]

Hi,

I attach a notebook where an example like the one you mentioned is
worked out. Unfortunately, there aren't many explanations in the code so
it might be difficult to follow for someone not familiar with
xAct/Mathematica.

Alfonso.

[Cyril Pitrou]

Hi,

I also attach an example notebook. It uses products of manifold, products of metrics, and then the function ExpandProductMetric to split the curvature tensors.

The example is different from what you need and it is rather a S2xM product. The S2 manifold/metric is taken from the xAct`Harmonics package. You might be able to get inspiration from it to build your own products of manifolds and metrics. Previously it was designed with M having dimension 2, but also works with an unspecified dimension dim.

For expression which involve not just curvature tensors, you will need to also use TraceProductDummies.

Best,

Cyril Pitrou

[AF]

 Dear Alfonso and Cyril,

Thank you so much, these notes are really helpful.

 Best Regards,

 AF

On Tuesday, April 21, 2015 at 11:46:34 PM UTC+4:30, AF wrote: