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I am using xAct package and I would like to get geodesic equations in components for Schwarzshild metric (for instance).

DefManifold[M4, 4, {\[Alpha], \[Beta], \[Rho], \[Sigma], \[Mu], \[Nu], \[Lambda]}]
DefScalarFunction /@ {f, A};
DefConstantSymbol[M]
DefChart[B, M4, {0, 1, 2, 3}, {t[], r[], \[Theta][], \[Phi][]}, ChartColor -> Red]
f[expr_] := (1 - 2 M/expr)
g = CTensor[DiagonalMatrix[{-f[r[]], 1/f[r[]], r[]^2,r[]^2 Sin[\[Theta][]]^2}], {-B, -B}];
g[-\[Mu], -\[Nu]]
SetCMetric[g, B, SignatureOfMetric -> {3, 1, 0}]
MetricCompute[g, B, All]

This part is pretty straightforward. Than I want to use CTensors to construct the geodesic equation, put I really don't know what to do.

A need to write something like x[a]''-Christoffel[a,-b,-c]x[b]'x[c]' ==0

  1. How do I define the velocity vector to work with the components?
  2. I tried a projector u = diag[{-1,0,0,0}] acting on X={X0,X1,X2,X3}, u[a]CD[-a] should give me the projected time derivative.
  3. u[a]CD[-a]X[b] should be the velocity V and V[a]CD[-a]@V[a] whould, in principle give me the geodesics.

It does not work. The output is enter image description here

I appreciate any help. Thanks in advance.

Luciano Barosi
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1 Answers1

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Rather than starting with the "vector of coordinates", which is not really a vector, it is better to start with the velocity components, which do form a vector.

Define the same structures you had:

<< xAct`xCoba`

DefManifold[M4, 4, {\[Alpha], \[Beta], \[Rho], \[Sigma], \[Mu], \[Nu], \[Lambda]}]

DefConstantSymbol[M]

DefChart[B, M4, {0, 1, 2, 3}, {t[], r[], \[Theta][], \[Phi][]}]

f[expr_] := (1 - 2 M/expr)

g = CTensor[DiagonalMatrix[{-f[r[]], 1/f[r[]], r[]^2, r[]^2 Sin[\[Theta][]]^2}], {-B, -B}]

You can then find the LeviCivita connection of this metric with

cd = LC[g]

which can be used like any other covariant derivative in xAct. Try for example

cd[-\[Alpha]][g[-\[Mu], -\[Nu]]]

Its Christoffel, from the PDB of the chart you introduced, will be

Christoffel[cd, PDB][\[Alpha], -\[Mu], -\[Nu]]

To get the geodesic equation, you just need to introduce the velocity vector. To facilitate integration later, let us use scalar components for both the position and the velocity:

DefScalarFunction /@ {xt, xr, x\[Theta], x\[Phi], vt, vr, v\[Theta], v\[Phi]};

and a parameter tau for the trajectory:

DefParameter[tau]

Then we have this list of coordinates (not a vector):

x = {xt[tau], xr[tau], x\[Theta][tau], x\[Phi][tau]};

and the velocity vector in the coordinated basis B:

v = CTensor[{vt[tau], vr[tau], v\[Theta][tau], v\[Phi][tau]}, {B}]

You need the relation between x and v:

Thread[ParamD[tau][x] == First[v]]

and the actual geodesic equation, evaluating the Christoffel tensor on the trajectory:

ParamD[tau][v[\[Alpha]]] + Christoffel[cd, PDB][\[Alpha], -\[Mu], -\[Nu]] v[\[Mu]] v[\[Nu]] /. Thread[ScalarsOfChart[B] -> x] // Simplify
jose
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