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with(DifferentialGeometry): with(Tensor):
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Example 1.
We define a space-time metric and check that the Rainich conditions hold.
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DGsetup([t, x, y, z], M):
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g := evalDG(4/3*t^2* dx &t dx + t*(exp(-2*x)* dy &t dy + exp(2*x)*dz &t dz) - dt &t dt);
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| (2.1) |
1. First calling sequence.
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RainichConditions(g);
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2. To use the 2nd calling sequence first calculate the Ricci tensor and its covariant derivative.
| (2.3) |
| (2.4) |
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CR := CovariantDerivative(R, C);
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| (2.5) |
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RainichConditions(g, R, CR);
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3. Let's see the value of the 1-form equation C3.
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RainichConditions(g, R, CR, 'alpha');
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Example 2
We consider a metric depending upon 2 arbitrary functions and determine those functions for which the Rainich conditions hold.
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DGsetup([t, x, y, z], M):
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g := (1/x^2) &mult evalDG(A(x)*dx &t dx + B(x)*dy &t dy + 1/z^2*dz &t dz - z^2*dt &t dt);
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| (2.9) |
Here are the Rainich conditions. The first condition is too complicated to display here, but the 2nd and 3rd are simple.
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C1, C2, C3 := RainichConditions(g, output = "tensor"):
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| (2.10) |
To impose the Rainich conditions, we set the coefficients of the tensors and to zero. The command DGinfo/"CoefficientSet" gives us these coefficients. Again, they are too long to display here.
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Eq := Tools:-DGinfo(C1, "CoefficientSet") union Tools:-DGinfo(C2, "CoefficientSet"):
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We see that there are a total of 5 scalar conditions on .
Here is one of the Rainich conditions.
| (2.12) |
We use pdsolve to solve all the Rainich conditions.
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solution := pdsolve(Eq);
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| (2.13) |
For these values of the metric defines an electro-vac space-time.