Let $g$ 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 field$\($a non-null 2-form satisfying the source-free Maxwell equations) such that the Einstein equations $G^{\mathrm{ii}}\= T^{\mathrm{ij}}$hold. Here $G^{\mathrm{ii} }\=R^{\mathrm{ij}} -\frac{1}{2}{g}^{\mathrm{ij}}R$is the Einstein tensor and $T^{\mathrm{ij}}$ is the electromagnetic energy-momentum tensor. The Rainich conditions apply only to those metrics $g$for which the Ricci tensor is non-null, that is, $N\= R^{\mathrm{ij}}{R}_{\mathrm{ij}}\ne 0.$There are 2 algebraic Rainich conditions and 1 differential condition

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);

$g:=-\mathrm{dt}\mathrm{dt}\+\frac{4}{3}{t}^2\mathrm{dx}\mathrm{dx}\+t{\ⅇ}^{-2x}\mathrm{dy}\mathrm{dy}\+t{\ⅇ}^{2x}\mathrm{dz}\mathrm{dz}$

RainichConditions(g);

$\mathrm{true}$

R := RicciTensor(g);

$R:=\frac{1}{2}\frac{\mathrm{dt}\mathrm{dt}}{t^2}-\frac{2}{3}\mathrm{dx}\mathrm{dx}\+\frac{1}{2}\frac{{\ⅇ}^{-2x}\mathrm{dy}\mathrm{dy}}{t}\+\frac{1}{2}\frac{{\ⅇ}^{2x}\mathrm{dz}\mathrm{dz}}{t}$

C := Christoffel(g);

$C:=\frac{\mathrm{dt}\mathrm{D\_x}\mathrm{dx}}{t}\+\frac{1}{2}\frac{\mathrm{dt}\mathrm{D\_y}\mathrm{dy}}{t}\+\frac{1}{2}\frac{\mathrm{dt}\mathrm{D\_z}\mathrm{dz}}{t}\+\frac{4}{3}t\mathrm{dx}\mathrm{D\_t}\mathrm{dx}\+\frac{\mathrm{dx}\mathrm{D\_x}\mathrm{dt}}{t}-\mathrm{dx}\mathrm{D\_y}\mathrm{dy}\+\mathrm{dx}\mathrm{D\_z}\mathrm{dz}\+\frac{1}{2}{\ⅇ}^{-2x}\mathrm{dy}\mathrm{D\_t}\mathrm{dy}\+\frac{3}{4}\frac{{\ⅇ}^{-2x}\mathrm{dy}\mathrm{D\_x}\mathrm{dy}}{t}\+\frac{1}{2}\frac{\mathrm{dy}\mathrm{D\_y}\mathrm{dt}}{t}-\mathrm{dy}\mathrm{D\_y}\mathrm{dx}\+\frac{1}{2}{\ⅇ}^{2x}\mathrm{dz}\mathrm{D\_t}\mathrm{dz}-\frac{3}{4}\frac{{\ⅇ}^{2x}\mathrm{dz}\mathrm{D\_x}\mathrm{dz}}{t}\+\frac{1}{2}\frac{\mathrm{dz}\mathrm{D\_z}\mathrm{dt}}{t}\+\mathrm{dz}\mathrm{D\_z}\mathrm{dx}$

CR := CovariantDerivative(R, C);

$\mathrm{CR}:=-\frac{\mathrm{dt}\mathrm{dt}\mathrm{dt}}{t^3}-\frac{1}{2}\frac{{\ⅇ}^{-2x}\mathrm{dt}\mathrm{dy}\mathrm{dy}}{t^2}-\frac{1}{2}\frac{{\ⅇ}^{2x}\mathrm{dt}\mathrm{dz}\mathrm{dz}}{t^2}\+\frac{4}{3}\frac{\mathrm{dx}\mathrm{dx}\mathrm{dt}}{t}\+\frac{{\ⅇ}^{-2x}\mathrm{dx}\mathrm{dy}\mathrm{dy}}{t}-\frac{{\ⅇ}^{2x}\mathrm{dx}\mathrm{dz}\mathrm{dz}}{t}-\frac{1}{2}\frac{{\ⅇ}^{-2x}\mathrm{dy}\mathrm{dt}\mathrm{dy}}{t^2}\+\frac{{\ⅇ}^{-2x}\mathrm{dy}\mathrm{dx}\mathrm{dy}}{t}-\frac{{\ⅇ}^{-2x}\mathrm{dy}\mathrm{dy}\mathrm{dt}}{t^2}-\frac{1}{2}\frac{{\ⅇ}^{2x}\mathrm{dz}\mathrm{dt}\mathrm{dz}}{t^2}-\frac{{\ⅇ}^{2x}\mathrm{dz}\mathrm{dx}\mathrm{dz}}{t}-\frac{{\ⅇ}^{2x}\mathrm{dz}\mathrm{dz}\mathrm{dt}}{t^2}$

RainichConditions(g, R, CR);

$\mathrm{true}$

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

$\mathrm{true}$

alpha;

$0\mathrm{dt}$

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);

$g:=-\frac{z^2\mathrm{dt}\mathrm{dt}}{x^2}\+\frac{A\left(x\right)\mathrm{dx}\mathrm{dx}}{x^2}\+\frac{B\left(x\right)\mathrm{dy}\mathrm{dy}}{x^2}\+\frac{\mathrm{dz}\mathrm{dz}}{x^2{z}^2}$

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

C2, C3;

$-\frac{1}{2}\frac{24A\left(x\right){B\left(x\right)}^2\+4x^2{A\left(x\right)}^2{B\left(x\right)}^2-6xB\left(x\right)A\left(x\right)\left(\frac{\ⅆ}{\ⅆx}B\left(x\right)\right)\+6x\left(\frac{\ⅆ}{\ⅆx}A\left(x\right)\right){B\left(x\right)}^2-{\left(\frac{\ⅆ}{\ⅆx}B\left(x\right)\right)}^2{x}^{2}A\left(x\right)-\left(\frac{\ⅆ}{\ⅆx}B\left(x\right)\right)x^2\left(\frac{\ⅆ}{\ⅆx}A\left(x\right)\right)B\left(x\right)\+2B\left(x\right)x^{2}A\left(x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}B\left(x\right)\right)}{{A\left(x\right)}^2{B\left(x\right)}^2}\,0\mathrm{dt}\bigwedge \mathrm{dx}$

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

nops(Eq);

$5$

Eq[1];

$-\frac{1}{2}\frac{24A\left(x\right){B\left(x\right)}^2\+4x^2{A\left(x\right)}^2{B\left(x\right)}^2-6xB\left(x\right)A\left(x\right)\left(\frac{\ⅆ}{\ⅆx}B\left(x\right)\right)\+6x\left(\frac{\ⅆ}{\ⅆx}A\left(x\right)\right){B\left(x\right)}^2-{\left(\frac{\ⅆ}{\ⅆx}B\left(x\right)\right)}^2{x}^{2}A\left(x\right)-\left(\frac{\ⅆ}{\ⅆx}B\left(x\right)\right)x^2\left(\frac{\ⅆ}{\ⅆx}A\left(x\right)\right)B\left(x\right)\+2B\left(x\right)x^{2}A\left(x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}B\left(x\right)\right)}{{A\left(x\right)}^2{B\left(x\right)}^2}$

solution := pdsolve(Eq);

$\mathrm{solution}:=\left\{A\left(x\right)\=-\frac{\mathrm{\_C1}}{x^2\left(\mathrm{\_C1}\+\mathrm{\_C2}x\+\mathrm{\_C3}{x}^2\right)}\,B\left(x\right)\=\mathrm{\_C1}{x}^2\+\mathrm{\_C2}{x}^3\+\mathrm{\_C3}{x}^4\right\}$