Let $g$ be metric on a 4-dimensional manifold. If $g$satisfies the Rainich conditions, then there exists a non-null electromagnetic field $F \($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 $T^{\mathrm{ij}} \= {g_{\mathrm{hk}}F}^{\mathrm{ih}}{F}^{\mathrm{jk}} -\frac{1}{4} g^{\mathrm{ij}} {F_{\mathrm{hk}}F}^{\mathrm{hk}}$ is the electromagnetic energy-momentum tensor. Note that the Rainich conditions require that the Ricci tensor is non-null, that is, $N\= R^{\mathrm{ij}}{R}_{\mathrm{ij}}\ne 0.$ 

The electromagnetic field $F$is constructed as follows. First, define

The command RainichElectromagneticField returns the electromagnetic 2-form $F$.

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);

$\mathrm{true}$

F := RainichElectromagneticField(g);

$F:=\frac{2\cos \left(\mathrm{\_C1}\right)\left(\mathrm{csgn}\left(\frac{1}{t^2}\right)\+1\right)\mathrm{dt}\bigwedge \mathrm{dx}}{\sqrt{6\mathrm{csgn}\left(\frac{1}{t^2}\right)\+6}}\+\frac{\sin \left(\mathrm{\_C1}\right)\sqrt{3}\mathrm{csgn}\left(\frac{1}{t^2}\right)\left(\mathrm{csgn}\left(\frac{1}{t^2}\right)\+1\right)\mathrm{dy}\bigwedge \mathrm{dz}}{\sqrt{6\mathrm{csgn}\left(\frac{1}{t^2}\right)\+6}}$

simplify(F) assuming t::real;

$\frac{2}{3}\cos \left(\mathrm{\_C1}\right)\sqrt{3}\mathrm{dt}\bigwedge \mathrm{dx}\+\sin \left(\mathrm{\_C1}\right)\mathrm{dy}\bigwedge \mathrm{dz}$

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, alpha);

$\mathrm{true}$

F:= RainichElectromagneticField(g, R, CR, alpha) assuming t::real;

$F:=\frac{2}{3}\cos \left(\mathrm{\_C1}\right)\sqrt{3}\mathrm{dt}\bigwedge \mathrm{dx}\+\sin \left(\mathrm{\_C1}\right)\mathrm{dy}\bigwedge \mathrm{dz}$

T := EnergyMomentumTensor("Electromagnetic", g, F);

$T:=\frac{1}{2}\frac{\mathrm{D\_t}\mathrm{D\_t}}{t^2}-\frac{3}{8}\frac{\mathrm{D\_x}\mathrm{D\_x}}{t^4}\+\frac{1}{2}\frac{{\ⅇ}^{2x}\mathrm{D\_y}\mathrm{D\_y}}{t^3}\+\frac{1}{2}\frac{{\ⅇ}^{-2x}\mathrm{D\_z}\mathrm{D\_z}}{t^3}$

E := EinsteinTensor(g);

$E:=\frac{1}{2}\frac{\mathrm{D\_t}\mathrm{D\_t}}{t^2}-\frac{3}{8}\frac{\mathrm{D\_x}\mathrm{D\_x}}{t^4}\+\frac{1}{2}\frac{{\ⅇ}^{2x}\mathrm{D\_y}\mathrm{D\_y}}{t^3}\+\frac{1}{2}\frac{{\ⅇ}^{-2x}\mathrm{D\_z}\mathrm{D\_z}}{t^3}$

E &minus T;

$0\mathrm{D\_t}\mathrm{D\_t}$

MatterFieldEquations("Electromagnetic", g, F);

$0\mathrm{D\_t}\,0\mathrm{dt}\bigwedge \mathrm{dx}\bigwedge \mathrm{dy}$

DGsetup([t, y, phi, v], M2);

$\mathrm{frame\; name:\; M2}$

g2 := evalDG(t^(-2)*(dt &t dt + dy &t dy) + t^2 * dphi &t dphi - (dv + 2*y*dphi) &t (dv + 2*y*dphi));

$\mathrm{g2}:=\frac{\mathrm{dt}\mathrm{dt}}{t^2}\+\frac{\mathrm{dy}\mathrm{dy}}{t^2}-\left(4y^2-t^2\right)\mathrm{dphi}\mathrm{dphi}-2y\mathrm{dphi}\mathrm{dv}-2y\mathrm{dv}\mathrm{dphi}-\mathrm{dv}\mathrm{dv}$

R2 := RicciTensor(g2);

$\mathrm{R2}:=-\frac{2\mathrm{dt}\mathrm{dt}}{t^2}\+\frac{2\mathrm{dy}\mathrm{dy}}{t^2}\+\left(8y^2\+2t^2\right)\mathrm{dphi}\mathrm{dphi}\+4y\mathrm{dphi}\mathrm{dv}\+4y\mathrm{dv}\mathrm{dphi}\+2\mathrm{dv}\mathrm{dv}$

C2 := Christoffel(g2);

$\mathrm{C2}:=-\frac{\mathrm{dt}\mathrm{D\_t}\mathrm{dt}}{t}-\frac{\mathrm{dt}\mathrm{D\_y}\mathrm{dy}}{t}\+\frac{\mathrm{dt}\mathrm{D\_phi}\mathrm{dphi}}{t}-\frac{2y\mathrm{dt}\mathrm{D\_v}\mathrm{dphi}}{t}\+\frac{\mathrm{dy}\mathrm{D\_t}\mathrm{dy}}{t}-\frac{\mathrm{dy}\mathrm{D\_y}\mathrm{dt}}{t}-\frac{2y\mathrm{dy}\mathrm{D\_phi}\mathrm{dphi}}{t^2}-\frac{\mathrm{dy}\mathrm{D\_phi}\mathrm{dv}}{t^2}\+\frac{\left(4y^2\+t^2\right)\mathrm{dy}\mathrm{D\_v}\mathrm{dphi}}{t^2}\+\frac{2y\mathrm{dy}\mathrm{D\_v}\mathrm{dv}}{t^2}-t^3\mathrm{dphi}\mathrm{D\_t}\mathrm{dphi}\+4t^{2}y\mathrm{dphi}\mathrm{D\_y}\mathrm{dphi}\+t^2\mathrm{dphi}\mathrm{D\_y}\mathrm{dv}\+\frac{\mathrm{dphi}\mathrm{D\_phi}\mathrm{dt}}{t}-\frac{2y\mathrm{dphi}\mathrm{D\_phi}\mathrm{dy}}{t^2}-\frac{2y\mathrm{dphi}\mathrm{D\_v}\mathrm{dt}}{t}\+\frac{\left(4y^2\+t^2\right)\mathrm{dphi}\mathrm{D\_v}\mathrm{dy}}{t^2}\+t^2\mathrm{dv}\mathrm{D\_y}\mathrm{dphi}-\frac{\mathrm{dv}\mathrm{D\_phi}\mathrm{dy}}{t^2}\+\frac{2y\mathrm{dv}\mathrm{D\_v}\mathrm{dy}}{t^2}$

CR2 := CovariantDerivative(R2, C2);

$\mathrm{CR2}:=\frac{4\mathrm{dt}\mathrm{dy}\mathrm{dy}}{t^3}-4t\mathrm{dt}\mathrm{dphi}\mathrm{dphi}\+\frac{4\mathrm{dy}\mathrm{dt}\mathrm{dy}}{t^3}-8y\mathrm{dy}\mathrm{dphi}\mathrm{dphi}-4\mathrm{dy}\mathrm{dv}\mathrm{dphi}-4t\mathrm{dphi}\mathrm{dt}\mathrm{dphi}-8y\mathrm{dphi}\mathrm{dy}\mathrm{dphi}\+16y\mathrm{dphi}\mathrm{dphi}\mathrm{dy}\+4\mathrm{dphi}\mathrm{dv}\mathrm{dy}-4\mathrm{dv}\mathrm{dy}\mathrm{dphi}\+4\mathrm{dv}\mathrm{dphi}\mathrm{dy}$

RainichConditions(g2, R2, CR2, alpha2) assuming t > 0;

$\mathrm{true}$

alpha2;

$\frac{2\mathrm{dt}}{t}$

F2 := RainichElectromagneticField(g2, R2, CR2, alpha2):

F2 := simplify(F2) assuming t > 0, y > 0;

$\mathrm{F2}:=\frac{4\cos \left(2\ln \left(t\right)\+\mathrm{\_C1}\right)y\mathrm{dt}\bigwedge \mathrm{dphi}}{t}\+\frac{2\cos \left(2\ln \left(t\right)\+\mathrm{\_C1}\right)\mathrm{dt}\bigwedge \mathrm{dv}}{t}\+2\sin \left(2\ln \left(t\right)\+\mathrm{\_C1}\right)\mathrm{dy}\bigwedge \mathrm{dphi}$

T2 := EnergyMomentumTensor("Electromagnetic", g2, F2);

$\mathrm{T2}:=-2t^2\mathrm{D\_t}\mathrm{D\_t}\+2t^2\mathrm{D\_y}\mathrm{D\_y}\+\frac{2\mathrm{D\_phi}\mathrm{D\_phi}}{t^2}-\frac{4y\mathrm{D\_phi}\mathrm{D\_v}}{t^2}-\frac{4y\mathrm{D\_v}\mathrm{D\_phi}}{t^2}\+\frac{2\left(4y^2\+t^2\right)\mathrm{D\_v}\mathrm{D\_v}}{t^2}$

E2 := EinsteinTensor(g2);

$\mathrm{E2}:=-2t^2\mathrm{D\_t}\mathrm{D\_t}\+2t^2\mathrm{D\_y}\mathrm{D\_y}\+\frac{2\mathrm{D\_phi}\mathrm{D\_phi}}{t^2}-\frac{4y\mathrm{D\_phi}\mathrm{D\_v}}{t^2}-\frac{4y\mathrm{D\_v}\mathrm{D\_phi}}{t^2}\+\frac{2\left(4y^2\+t^2\right)\mathrm{D\_v}\mathrm{D\_v}}{t^2}$

E2 &minus T2;

$0\mathrm{D\_t}\mathrm{D\_t}$

MatterFieldEquations("Electromagnetic", g2, F2);

$0\mathrm{D\_t}\,0\mathrm{dt}\bigwedge \mathrm{dy}\bigwedge \mathrm{dphi}$