RainichElectromagneticField - Maple Help

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Tensor[RainichElectromagneticField] - from a given metric satisfying the Rainich conditions, calculate an electromagnetic field which solves the Einstein-Maxwell equations

Calling Sequences

RainichElectromagneticField(g, ${\mathbf{α}}{\mathbf{)}}$

RainichElectromagneticField(g, R, CR, ${\mathbf{α}}$, option)

Parameters

g       - a metric tensor on a 4-dimensional manifold

R       - the Ricci tensor of g

CR      -  a rank 3 tensor, the covariant derivative of the Ricci tensor

alpha   - (optional) 1-form

Description

 • Let $g$ be metric on a 4-dimensional manifold. If satisfies the Rainich conditions, then there exists a non-null electromagnetic field a 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. Note that the Rainich conditions require that the Ricci tensor is non-null, that is,
 • The electromagnetic field is constructed as follows. First, define

and

Define a rank 2 skew-symmetric tensor ${f}_{\mathrm{ij}}$ by ${f}_{\mathrm{ij}}{f}_{\mathrm{hk}}$. For example,  if then Let  Let be the 1-form defined by Find a function such that The Rainich electromagnetic field is  The electromagnetic field may have complex values if the metric is not of Lorentz signature.

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

Examples

 > with(DifferentialGeometry): with(Tensor):

Example 1.

We define a space-time metric and check that the Rainich conditions hold. Then we find the Rainich electromagnetic field.

 M > DGsetup([t, x, y, z], M):
 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);
 ${g}{:=}{-}{\mathrm{dt}}{}{\mathrm{dt}}{+}\frac{{4}}{{3}}{}{{t}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{t}{}{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{t}{}{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.1)

1. First calling sequences.

 M > RainichConditions(g);
 ${\mathrm{true}}$ (2.2)
 M > F := RainichElectromagneticField(g);
 ${F}{:=}\frac{{2}{}{\mathrm{cos}}{}\left({\mathrm{_C1}}\right){}\left({\mathrm{csgn}}{}\left(\frac{{1}}{{{t}}^{{2}}}\right){+}{1}\right){}{\mathrm{dt}}{}{\bigwedge }{}{\mathrm{dx}}}{\sqrt{{6}{}{\mathrm{csgn}}{}\left(\frac{{1}}{{{t}}^{{2}}}\right){+}{6}}}{+}\frac{{\mathrm{sin}}{}\left({\mathrm{_C1}}\right){}\sqrt{{3}}{}{\mathrm{csgn}}{}\left(\frac{{1}}{{{t}}^{{2}}}\right){}\left({\mathrm{csgn}}{}\left(\frac{{1}}{{{t}}^{{2}}}\right){+}{1}\right){}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}}{\sqrt{{6}{}{\mathrm{csgn}}{}\left(\frac{{1}}{{{t}}^{{2}}}\right){+}{6}}}$ (2.3)

We can simplify this output with the assuming command.

 M > simplify(F) assuming t::real;
 $\frac{{2}}{{3}}{}{\mathrm{cos}}{}\left({\mathrm{_C1}}\right){}\sqrt{{3}}{}{\mathrm{dt}}{}{\bigwedge }{}{\mathrm{dx}}{+}{\mathrm{sin}}{}\left({\mathrm{_C1}}\right){}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}$ (2.4)

Note that because the first calling sequence for either RainichCondition or RainichElectromagneticField requires coordinate differentiation (to calculate the Ricci tensor and its covariant derivative), assumptions such as assuming t::real cannot be applied directly to these commands. For this reason and for efficiency, it is better to use the second calling sequences.

2. Second calling sequences. First calculate the Ricci tensor and its covariant derivative.

 > R := RicciTensor(g);
 ${R}{:=}\frac{{1}}{{2}}{}\frac{{\mathrm{dt}}{}{\mathrm{dt}}}{{{t}}^{{2}}}{-}\frac{{2}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{t}}{+}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{dz}}}{{t}}$ (2.5)
 M > C := Christoffel(g);
 ${C}{:=}\frac{{\mathrm{dt}}{}{\mathrm{D_x}}{}{\mathrm{dx}}}{{t}}{+}\frac{{1}}{{2}}{}\frac{{\mathrm{dt}}{}{\mathrm{D_y}}{}{\mathrm{dy}}}{{t}}{+}\frac{{1}}{{2}}{}\frac{{\mathrm{dt}}{}{\mathrm{D_z}}{}{\mathrm{dz}}}{{t}}{+}\frac{{4}}{{3}}{}{t}{}{\mathrm{dx}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{+}\frac{{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dt}}}{{t}}{-}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{+}{\mathrm{dx}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dy}}{}{\mathrm{D_t}}{}{\mathrm{dy}}{+}\frac{{3}}{{4}}{}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dy}}{}{\mathrm{D_x}}{}{\mathrm{dy}}}{{t}}{+}\frac{{1}}{{2}}{}\frac{{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dt}}}{{t}}{-}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{+}\frac{{1}}{{2}}{}{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{D_t}}{}{\mathrm{dz}}{-}\frac{{3}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{D_x}}{}{\mathrm{dz}}}{{t}}{+}\frac{{1}}{{2}}{}\frac{{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dt}}}{{t}}{+}{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dx}}$ (2.6)
 M > CR := CovariantDerivative(R, C);
 ${\mathrm{CR}}{:=}{-}\frac{{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dt}}}{{{t}}^{{3}}}{-}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{{t}}^{{2}}}{-}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dz}}}{{{t}}^{{2}}}{+}\frac{{4}}{{3}}{}\frac{{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dt}}}{{t}}{+}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{t}}{-}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dz}}}{{t}}{-}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dy}}}{{{t}}^{{2}}}{+}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{t}}{-}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dt}}}{{{t}}^{{2}}}{-}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}}{{{t}}^{{2}}}{-}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dz}}}{{t}}{-}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dt}}}{{{t}}^{{2}}}$ (2.7)
 M > RainichConditions(g, R, CR, alpha);
 ${\mathrm{true}}$ (2.8)

Here is the Rainich electromagnetic field tensor. The constant $\mathrm{_C1}$ reflects the non-uniqueness of theta.

 M > F:= RainichElectromagneticField(g, R, CR, alpha) assuming t::real;
 ${F}{:=}\frac{{2}}{{3}}{}{\mathrm{cos}}{}\left({\mathrm{_C1}}\right){}\sqrt{{3}}{}{\mathrm{dt}}{}{\bigwedge }{}{\mathrm{dx}}{+}{\mathrm{sin}}{}\left({\mathrm{_C1}}\right){}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}$ (2.9)

We check that the Einstein equations are satisfied (See EinsteinTensor, EnergyMomentumTensor).

 M > T := EnergyMomentumTensor("Electromagnetic", g, F);
 ${T}{:=}\frac{{1}}{{2}}{}\frac{{\mathrm{D_t}}{}{\mathrm{D_t}}}{{{t}}^{{2}}}{-}\frac{{3}}{{8}}{}\frac{{\mathrm{D_x}}{}{\mathrm{D_x}}}{{{t}}^{{4}}}{+}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{D_y}}{}{\mathrm{D_y}}}{{{t}}^{{3}}}{+}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{D_z}}{}{\mathrm{D_z}}}{{{t}}^{{3}}}$ (2.10)
 M > E := EinsteinTensor(g);
 ${E}{:=}\frac{{1}}{{2}}{}\frac{{\mathrm{D_t}}{}{\mathrm{D_t}}}{{{t}}^{{2}}}{-}\frac{{3}}{{8}}{}\frac{{\mathrm{D_x}}{}{\mathrm{D_x}}}{{{t}}^{{4}}}{+}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{D_y}}{}{\mathrm{D_y}}}{{{t}}^{{3}}}{+}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{D_z}}{}{\mathrm{D_z}}}{{{t}}^{{3}}}$ (2.11)
 M > E &minus T;
 ${0}{}{\mathrm{D_t}}{}{\mathrm{D_t}}$ (2.12)

We check that the Maxwell equations (see MatterFieldEquations)  are satisfied.

 M > MatterFieldEquations("Electromagnetic", g, F);
 ${0}{}{\mathrm{D_t}}{,}{0}{}{\mathrm{dt}}{}{\bigwedge }{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}$ (2.13)

Example 2.

We present an example where the 1-form is non-zero.

 M > DGsetup([t, y, phi, v], M2);
 ${\mathrm{frame name: M2}}$ (2.14)
 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));
 ${\mathrm{g2}}{:=}\frac{{\mathrm{dt}}{}{\mathrm{dt}}}{{{t}}^{{2}}}{+}\frac{{\mathrm{dy}}{}{\mathrm{dy}}}{{{t}}^{{2}}}{-}\left({4}{}{{y}}^{{2}}{-}{{t}}^{{2}}\right){}{\mathrm{dphi}}{}{\mathrm{dphi}}{-}{2}{}{y}{}{\mathrm{dphi}}{}{\mathrm{dv}}{-}{2}{}{y}{}{\mathrm{dv}}{}{\mathrm{dphi}}{-}{\mathrm{dv}}{}{\mathrm{dv}}$ (2.15)
 > R2 := RicciTensor(g2);
 ${\mathrm{R2}}{:=}{-}\frac{{2}{}{\mathrm{dt}}{}{\mathrm{dt}}}{{{t}}^{{2}}}{+}\frac{{2}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{{t}}^{{2}}}{+}\left({8}{}{{y}}^{{2}}{+}{2}{}{{t}}^{{2}}\right){}{\mathrm{dphi}}{}{\mathrm{dphi}}{+}{4}{}{y}{}{\mathrm{dphi}}{}{\mathrm{dv}}{+}{4}{}{y}{}{\mathrm{dv}}{}{\mathrm{dphi}}{+}{2}{}{\mathrm{dv}}{}{\mathrm{dv}}$ (2.16)
 M > C2 := Christoffel(g2);
 ${\mathrm{C2}}{:=}{-}\frac{{\mathrm{dt}}{}{\mathrm{D_t}}{}{\mathrm{dt}}}{{t}}{-}\frac{{\mathrm{dt}}{}{\mathrm{D_y}}{}{\mathrm{dy}}}{{t}}{+}\frac{{\mathrm{dt}}{}{\mathrm{D_phi}}{}{\mathrm{dphi}}}{{t}}{-}\frac{{2}{}{y}{}{\mathrm{dt}}{}{\mathrm{D_v}}{}{\mathrm{dphi}}}{{t}}{+}\frac{{\mathrm{dy}}{}{\mathrm{D_t}}{}{\mathrm{dy}}}{{t}}{-}\frac{{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dt}}}{{t}}{-}\frac{{2}{}{y}{}{\mathrm{dy}}{}{\mathrm{D_phi}}{}{\mathrm{dphi}}}{{{t}}^{{2}}}{-}\frac{{\mathrm{dy}}{}{\mathrm{D_phi}}{}{\mathrm{dv}}}{{{t}}^{{2}}}{+}\frac{\left({4}{}{{y}}^{{2}}{+}{{t}}^{{2}}\right){}{\mathrm{dy}}{}{\mathrm{D_v}}{}{\mathrm{dphi}}}{{{t}}^{{2}}}{+}\frac{{2}{}{y}{}{\mathrm{dy}}{}{\mathrm{D_v}}{}{\mathrm{dv}}}{{{t}}^{{2}}}{-}{{t}}^{{3}}{}{\mathrm{dphi}}{}{\mathrm{D_t}}{}{\mathrm{dphi}}{+}{4}{}{{t}}^{{2}}{}{y}{}{\mathrm{dphi}}{}{\mathrm{D_y}}{}{\mathrm{dphi}}{+}{{t}}^{{2}}{}{\mathrm{dphi}}{}{\mathrm{D_y}}{}{\mathrm{dv}}{+}\frac{{\mathrm{dphi}}{}{\mathrm{D_phi}}{}{\mathrm{dt}}}{{t}}{-}\frac{{2}{}{y}{}{\mathrm{dphi}}{}{\mathrm{D_phi}}{}{\mathrm{dy}}}{{{t}}^{{2}}}{-}\frac{{2}{}{y}{}{\mathrm{dphi}}{}{\mathrm{D_v}}{}{\mathrm{dt}}}{{t}}{+}\frac{\left({4}{}{{y}}^{{2}}{+}{{t}}^{{2}}\right){}{\mathrm{dphi}}{}{\mathrm{D_v}}{}{\mathrm{dy}}}{{{t}}^{{2}}}{+}{{t}}^{{2}}{}{\mathrm{dv}}{}{\mathrm{D_y}}{}{\mathrm{dphi}}{-}\frac{{\mathrm{dv}}{}{\mathrm{D_phi}}{}{\mathrm{dy}}}{{{t}}^{{2}}}{+}\frac{{2}{}{y}{}{\mathrm{dv}}{}{\mathrm{D_v}}{}{\mathrm{dy}}}{{{t}}^{{2}}}$ (2.17)
 M > CR2 := CovariantDerivative(R2, C2);
 ${\mathrm{CR2}}{:=}\frac{{4}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{{t}}^{{3}}}{-}{4}{}{t}{}{\mathrm{dt}}{}{\mathrm{dphi}}{}{\mathrm{dphi}}{+}\frac{{4}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dy}}}{{{t}}^{{3}}}{-}{8}{}{y}{}{\mathrm{dy}}{}{\mathrm{dphi}}{}{\mathrm{dphi}}{-}{4}{}{\mathrm{dy}}{}{\mathrm{dv}}{}{\mathrm{dphi}}{-}{4}{}{t}{}{\mathrm{dphi}}{}{\mathrm{dt}}{}{\mathrm{dphi}}{-}{8}{}{y}{}{\mathrm{dphi}}{}{\mathrm{dy}}{}{\mathrm{dphi}}{+}{16}{}{y}{}{\mathrm{dphi}}{}{\mathrm{dphi}}{}{\mathrm{dy}}{+}{4}{}{\mathrm{dphi}}{}{\mathrm{dv}}{}{\mathrm{dy}}{-}{4}{}{\mathrm{dv}}{}{\mathrm{dy}}{}{\mathrm{dphi}}{+}{4}{}{\mathrm{dv}}{}{\mathrm{dphi}}{}{\mathrm{dy}}$ (2.18)
 M > RainichConditions(g2, R2, CR2, alpha2) assuming t > 0;
 ${\mathrm{true}}$ (2.19)
 M > alpha2;
 $\frac{{2}{}{\mathrm{dt}}}{{t}}$ (2.20)
 M2 > F2 := RainichElectromagneticField(g2, R2, CR2, alpha2):
 M2 > F2 := simplify(F2) assuming t > 0, y > 0;
 ${\mathrm{F2}}{:=}\frac{{4}{}{\mathrm{cos}}{}\left({2}{}{\mathrm{ln}}{}\left({t}\right){+}{\mathrm{_C1}}\right){}{y}{}{\mathrm{dt}}{}{\bigwedge }{}{\mathrm{dphi}}}{{t}}{+}\frac{{2}{}{\mathrm{cos}}{}\left({2}{}{\mathrm{ln}}{}\left({t}\right){+}{\mathrm{_C1}}\right){}{\mathrm{dt}}{}{\bigwedge }{}{\mathrm{dv}}}{{t}}{+}{2}{}{\mathrm{sin}}{}\left({2}{}{\mathrm{ln}}{}\left({t}\right){+}{\mathrm{_C1}}\right){}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dphi}}$ (2.21)

We check that the Einstein equations are satisfied (See EinsteinTensor, EnergyMomentumTensor).

 M2 > T2 := EnergyMomentumTensor("Electromagnetic", g2, F2);
 ${\mathrm{T2}}{:=}{-}{2}{}{{t}}^{{2}}{}{\mathrm{D_t}}{}{\mathrm{D_t}}{+}{2}{}{{t}}^{{2}}{}{\mathrm{D_y}}{}{\mathrm{D_y}}{+}\frac{{2}{}{\mathrm{D_phi}}{}{\mathrm{D_phi}}}{{{t}}^{{2}}}{-}\frac{{4}{}{y}{}{\mathrm{D_phi}}{}{\mathrm{D_v}}}{{{t}}^{{2}}}{-}\frac{{4}{}{y}{}{\mathrm{D_v}}{}{\mathrm{D_phi}}}{{{t}}^{{2}}}{+}\frac{{2}{}\left({4}{}{{y}}^{{2}}{+}{{t}}^{{2}}\right){}{\mathrm{D_v}}{}{\mathrm{D_v}}}{{{t}}^{{2}}}$ (2.22)
 M2 > E2 := EinsteinTensor(g2);
 ${\mathrm{E2}}{:=}{-}{2}{}{{t}}^{{2}}{}{\mathrm{D_t}}{}{\mathrm{D_t}}{+}{2}{}{{t}}^{{2}}{}{\mathrm{D_y}}{}{\mathrm{D_y}}{+}\frac{{2}{}{\mathrm{D_phi}}{}{\mathrm{D_phi}}}{{{t}}^{{2}}}{-}\frac{{4}{}{y}{}{\mathrm{D_phi}}{}{\mathrm{D_v}}}{{{t}}^{{2}}}{-}\frac{{4}{}{y}{}{\mathrm{D_v}}{}{\mathrm{D_phi}}}{{{t}}^{{2}}}{+}\frac{{2}{}\left({4}{}{{y}}^{{2}}{+}{{t}}^{{2}}\right){}{\mathrm{D_v}}{}{\mathrm{D_v}}}{{{t}}^{{2}}}$ (2.23)
 M2 > E2 &minus T2;
 ${0}{}{\mathrm{D_t}}{}{\mathrm{D_t}}$ (2.24)

We check that the Maxwell equations (see MatterFieldEquations)  are satisfied.

 M2 > MatterFieldEquations("Electromagnetic", g2, F2);
 ${0}{}{\mathrm{D_t}}{,}{0}{}{\mathrm{dt}}{}{\bigwedge }{}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dphi}}$ (2.25)