# How can I define a metric in xAct?

I'm using "xAct" package and I want to define a my metric, that is a non standard metric. For example, my metric is diagonal

 g={-e^(2a),e^(2b),e^(2b),e^(2b),e^(2c),e^(2c)sin^2(theta)},


where a,b,c are functions of the time variable t. I cannot understand how I can insert this definition when I use the command DefMetric...Can anyone help me?

• have you looked at mathematica.stackexchange.com/questions/67446/… (see search button top right) Dec 20 '14 at 9:16
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– user9660
Dec 20 '14 at 9:24
• Yes, I've looked, but the problem is that I have a manifold in 6D and at a certain point Mathematica finds errors, even if I execute standard metric! Is there (or Do you know) a more simple package than xAct that calculates Riemann and Ricci tensors and the Gauss Bonnet term? Dec 20 '14 at 9:56

First of all, note that you'll need to use xCoba to do any calculations in terms of coordinate components. From the xTensor documentation page:

xTensor does not perform component calculations. Use the companion package xCoba.

In xCoba, you need to define a "chart" on your manifold with the coordinates you're using, and explicitly call DefScalarFunction for the functions you're going to be using. For example, here's how you'd define a flat FRW metric:

<< xActxCoba
(* Define the manifold *)
DefManifold[M, 4, IndexRange[a, n]]

(* Define coordinates on a chart of M *)
DefChart[ch, M, {0, 1, 2, 3}, {t[], x[], y[], z[]}, ChartColor -> Blue]

(* Define aa (the scale factor) to be a scalar function of one or more coordinates *)
DefScalarFunction[aa]

(* Create the metric tensor in terms of its coordinate components *)
met = CTensor[DiagonalMatrix[{-1, aa[t[]]^2, aa[t[]]^2, aa[t[]]^2}], {-ch, -ch}];

(* Set this metric to be the metric of the manifold on the chart ch *)
SetCMetric[met, ch, SignatureOfMetric -> {3, 1, 0}]

(* Define the covariant derivative operator *)
cd = CovDOfMetric[met];

(* Calculate various curvature tensors and cache *)
MetricCompute[met, ch, All]

(* Display Kretschmann invariant *)
Kretschmann[cd][]

Out[] = (12 (aa'[t[]]^4 + aa[t[]]^2 (aa'')[t[]]^2))/aa[t[]]^4


The Gauss-Bonnet invariant is not automatically computed by MetricCompute, but you could easily define it in terms of the appropriate contractions.

Note that none of this is terribly clear in the main xCobaDoc.nb file that's linked to on the main xCoba page. However, when you download and install the package, there should be a file called xCobaDoc2.nb in the Documentation directory that gives a lot more detail on how to use CTensor to define coordinate tensors, with some helpful examples. That's how I personally figured out how to use the package...

• what is the {-ch,-ch} for in the met=CTensor[...] part?
– jboy
Jul 7 at 11:48
• @jboy: It signifies that you're giving the metric components in terms of the ch coordinates. The - signs signify that you're providing the components of $g_{\mu \nu}$ in this basis; if you wanted to provide $g^{\mu \nu}$, you would use {ch, ch} instead. Jul 7 at 11:58
• thanks for that. How about this line SetCMetric[met, ch, SignatureOfMetric -> {3, 1, 0}] what does {3,1,0} mean? isnt the signature for the robertson-walker metric -1,1,1,1?
– jboy
Jul 7 at 14:07
• @jboy: The order of elements in that list is "numbers of +s", "number of -s", "number of 0s". Jul 7 at 14:48

MetricInBasis[metric,-B,yourmetric] can work. See "Spherical Symmetry" of xCoba examples for more information.