Writing a particular vector equation for a given metric

[Debajyoti Sarkar]

Dear all,

Using Xcoba, Xtensor, I am trying to write down the following equation for a given metric.

\partial_N [\sqrt{g} g^{NN'} g^{MM'} (\partial_N' A_M'-\partial_M' A_N')]= \epsilon^{MBCDE} (\partial_B A_C-\partial_C A_B) (\partial_D A_E-\partial_E A_D) ---- eq. (1)

g is of course the metric and A_M is a vector which depends on (some of) the coordinates. \epsilon is the Levi civita symbol.

The commands I have used so far is:

%%%%%%%%%%%%%%%%%%%%%%%%

<< xAct`xTensor`

<< xAct`xCoba`

DefManifold[M, 5, IndexRange[a, q]]

DefChart[B, M, {0, 1, 2, 3, 4}, {t[], x[], y[], z[], r[]}]

met = CTensor[{

   {-r^2 (1 - 1/r^4), 0, 0, 0, 1},

   {0, r^2, 0, 0, 0},

   {0, 0, r^2, 0, 0},

   {0, 0, 0, r^2, 0},

   {1, 0, 0, 0, 0}

   }, {-B, -B}]

$CVSimplify = Simplify;

SetCMetric[met, B, SignatureOfMetric -> {4, 1, 0}] // AbsoluteTiming

MetricCompute[met, B, All]

CD = CovDOfMetric[met];

%%%%%%%%%%%%%%%%%%%%%

At this point, I am stuck. For example, I don't think in this process, I have defined the Levicivita symbol \epsilon. Also, I know how to define a scalar function of some coordinate variables and take covariant derivatives of those scalar functions in such a way that there are no free indices. However, how do I define vector functions A_M of some variables and write the left and right hand side of equation (1) in a differential equation form?

Sorry if it is a trivial question, but I am still learning these techniques.

Best regards,

[Debajyoti Sarkar]

Additions:

I just want to point out one more thing on which I need help in particular, regarding my question: 

1. Note that equation (1) has a free index and I want to find the differential equation (1) for various values of this free index, e.g. when the free index is t or r etc., 

Thanks again,

-- Deb

[Ben Niehoff]

Hi,

As far as I understand, xCoba wants you to enter each component individually, so you should never really have to worry about "vector valued functions".  I don't think xCoba can separate the components of *equations*, though, so what you may want to do is define a new tensor with a single free index, and set it to LHS - RHS of your equation.  Then xCoba's functions like ComponentArray will work; you can enter the various components, and then set everything ==0 and solve.

By the way, it looks like the equation you're trying to implement is

d \star F = F \wedge F

in which case I recommend you check out xTerior, which knows about differential forms, wedge products, and Hodge duals.  You may find it's less work to set up the problem that way.

Ben