Let  be a space-time metric on a 4-dimensional manifold. The Rainich conditions are necessary and locally sufficient conditions for there to exist a non-null electromagnetic fielda non-null 2-form satisfying the source-free Maxwell equations) such that the Einstein equations hold. Here is the Einstein tensor and  is the electromagnetic energy-momentum tensor.  The Rainich conditions apply only to those metrics for which the Ricci tensor is non-null, that is,

There are 2 algebraic Rainich conditions and 1 differential condition

Space-times which satisfy these Rainich conditions are called electro-vac space-times.

If the Rainich conditions hold, then an electromagnetic fieldwhich solves the Einstein-Maxwell equations can be found. See RainichElectromagneticField.

The command RainichConditions returns true or false. With output = "tensor", the 3 tensors defined by the left-hand sides of the equations C1, C2, C3 are returned. If the argument alpha is present, then the value of the 1-form in C3 is assigned to alpha.

For subsequent computations with RainichElectromagneticField it is more efficient to first calculate/simplify the Ricci tensor and its covariant derivative and then to use the second calling sequence.

with(DifferentialGeometry): with(Tensor):

DGsetup([t, x, y, z], M):

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);

RainichConditions(g);

R := RicciTensor(g);

C := Christoffel(g);

CR := CovariantDerivative(R, C);

RainichConditions(g, R, CR);

RainichConditions(g, R, CR, 'alpha');

alpha;

DGsetup([t, x, y, z], M):

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);

C1, C2, C3 := RainichConditions(g, output = "tensor"):

C2, C3;

Eq := Tools:-DGinfo(C1, "CoefficientSet") union Tools:-DGinfo(C2, "CoefficientSet"):

nops(Eq);

Eq[1];

solution := pdsolve(Eq);