RainichConditions - Maple Help

Tensor[RainichConditions] - check that a metric tensor satisfies the Rainich conditions

Calling Sequences

RainichConditions${(}$g, option)

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

Parameters

g      - a metric tensor

R      - the Ricci tensor

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

alpha  - (optional) an unevaluated name

option - the keyword argument output = "tensor"

Description

 • 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 hold. Here is the Einstein tensor and ${T}^{\mathrm{ij}}$ 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

C1:     C2: C3: d= 0, where

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.

Examples

 > with(DifferentialGeometry): with(Tensor):

Example 1.

We define a space-time metric and check that the Rainich conditions hold.

 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 sequence.

 M > RainichConditions(g);
 ${\mathrm{true}}$ (2.2)

2. To use the 2nd calling sequence first calculate the Ricci tensor and its covariant derivative.

 > R := RicciTensor(g);
 ${R}{:=}\frac{{1}}{{2}}{}\frac{{\mathrm{dt}}{}{\mathrm{dt}}}{{{t}}^{{2}}}{-}\frac{{2}}{{3}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{t}}{+}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{dz}}}{{t}}$ (2.3)
 M > C := Christoffel(g);
 ${C}{:=}\frac{{\mathrm{dt}}{}{\mathrm{D_x}}{}{\mathrm{dx}}}{{t}}{+}\frac{{1}}{{2}}{}\frac{{\mathrm{dt}}{}{\mathrm{D_y}}{}{\mathrm{dy}}}{{t}}{+}\frac{{1}}{{2}}{}\frac{{\mathrm{dt}}{}{\mathrm{D_z}}{}{\mathrm{dz}}}{{t}}{+}\frac{{4}}{{3}}{}{t}{}{\mathrm{dx}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{+}\frac{{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dt}}}{{t}}{-}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{+}{\mathrm{dx}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dy}}{}{\mathrm{D_t}}{}{\mathrm{dy}}{+}\frac{{3}}{{4}}{}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dy}}{}{\mathrm{D_x}}{}{\mathrm{dy}}}{{t}}{+}\frac{{1}}{{2}}{}\frac{{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dt}}}{{t}}{-}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{+}\frac{{1}}{{2}}{}{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{D_t}}{}{\mathrm{dz}}{-}\frac{{3}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{D_x}}{}{\mathrm{dz}}}{{t}}{+}\frac{{1}}{{2}}{}\frac{{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dt}}}{{t}}{+}{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dx}}$ (2.4)
 M > CR := CovariantDerivative(R, C);
 ${\mathrm{CR}}{:=}{-}\frac{{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dt}}}{{{t}}^{{3}}}{-}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{{t}}^{{2}}}{-}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dz}}}{{{t}}^{{2}}}{+}\frac{{4}}{{3}}{}\frac{{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dt}}}{{t}}{+}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{t}}{-}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dz}}}{{t}}{-}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dy}}}{{{t}}^{{2}}}{+}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}}{{t}}{-}\frac{{{ⅇ}}^{{-}{2}{}{x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dt}}}{{{t}}^{{2}}}{-}\frac{{1}}{{2}}{}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}}{{{t}}^{{2}}}{-}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dz}}}{{t}}{-}\frac{{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dt}}}{{{t}}^{{2}}}$ (2.5)
 M > RainichConditions(g, R, CR);
 ${\mathrm{true}}$ (2.6)

3. Let's see the value of the 1-form equation C3.

 M > RainichConditions(g, R, CR, 'alpha');
 ${\mathrm{true}}$ (2.7)
 M > alpha;
 ${0}{}{\mathrm{dt}}$ (2.8)

Example 2

We consider a metric depending upon 2 arbitrary functions and determine those functions for which the Rainich conditions hold.

 M > DGsetup([t, x, y, z], M):
 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);
 ${g}{:=}{-}\frac{{{z}}^{{2}}{}{\mathrm{dt}}{}{\mathrm{dt}}}{{{x}}^{{2}}}{+}\frac{{A}{}\left({x}\right){}{\mathrm{dx}}{}{\mathrm{dx}}}{{{x}}^{{2}}}{+}\frac{{B}{}\left({x}\right){}{\mathrm{dy}}{}{\mathrm{dy}}}{{{x}}^{{2}}}{+}\frac{{\mathrm{dz}}{}{\mathrm{dz}}}{{{x}}^{{2}}{}{{z}}^{{2}}}$ (2.9)

Here are the Rainich conditions. The first condition is too complicated to display here, but the 2nd and 3rd are simple.

 M > C1, C2, C3 := RainichConditions(g, output = "tensor"):
 M > C2, C3;
 ${-}\frac{{1}}{{2}}{}\frac{{24}{}{A}{}\left({x}\right){}{{B}{}\left({x}\right)}^{{2}}{+}{4}{}{{x}}^{{2}}{}{{A}{}\left({x}\right)}^{{2}}{}{{B}{}\left({x}\right)}^{{2}}{-}{6}{}{x}{}{B}{}\left({x}\right){}{A}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{B}{}\left({x}\right)\right){+}{6}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{A}{}\left({x}\right)\right){}{{B}{}\left({x}\right)}^{{2}}{-}{\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{B}{}\left({x}\right)\right)}^{{2}}{}{{x}}^{{2}}{}{A}{}\left({x}\right){-}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{B}{}\left({x}\right)\right){}{{x}}^{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{A}{}\left({x}\right)\right){}{B}{}\left({x}\right){+}{2}{}{B}{}\left({x}\right){}{{x}}^{{2}}{}{A}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{B}{}\left({x}\right)\right)}{{{A}{}\left({x}\right)}^{{2}}{}{{B}{}\left({x}\right)}^{{2}}}{,}{0}{}{\mathrm{dt}}{}{\bigwedge }{}{\mathrm{dx}}$ (2.10)

To impose the Rainich conditions, we set the coefficients of the tensors $\mathrm{C1}$ and $\mathrm{C2}$ to zero. The command DGinfo/"CoefficientSet" gives us these coefficients. Again, they are too long to display here.

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

We see that there are a total of 5 scalar conditions on .

 M > nops(Eq);
 ${5}$ (2.11)

Here is one of the Rainich conditions.

 M > Eq[1];
 ${-}\frac{{1}}{{2}}{}\frac{{24}{}{A}{}\left({x}\right){}{{B}{}\left({x}\right)}^{{2}}{+}{4}{}{{x}}^{{2}}{}{{A}{}\left({x}\right)}^{{2}}{}{{B}{}\left({x}\right)}^{{2}}{-}{6}{}{x}{}{B}{}\left({x}\right){}{A}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{B}{}\left({x}\right)\right){+}{6}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{A}{}\left({x}\right)\right){}{{B}{}\left({x}\right)}^{{2}}{-}{\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{B}{}\left({x}\right)\right)}^{{2}}{}{{x}}^{{2}}{}{A}{}\left({x}\right){-}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{B}{}\left({x}\right)\right){}{{x}}^{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{A}{}\left({x}\right)\right){}{B}{}\left({x}\right){+}{2}{}{B}{}\left({x}\right){}{{x}}^{{2}}{}{A}{}\left({x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{B}{}\left({x}\right)\right)}{{{A}{}\left({x}\right)}^{{2}}{}{{B}{}\left({x}\right)}^{{2}}}$ (2.12)

We use pdsolve to solve all the Rainich conditions.

 M > solution := pdsolve(Eq);
 ${\mathrm{solution}}{:=}\left\{{A}{}\left({x}\right){=}{-}\frac{{\mathrm{_C1}}}{{{x}}^{{2}}{}\left({\mathrm{_C1}}{+}{\mathrm{_C2}}{}{x}{+}{\mathrm{_C3}}{}{{x}}^{{2}}\right)}{,}{B}{}\left({x}\right){=}{\mathrm{_C1}}{}{{x}}^{{2}}{+}{\mathrm{_C2}}{}{{x}}^{{3}}{+}{\mathrm{_C3}}{}{{x}}^{{4}}\right\}$ (2.13)

For these values of the metric defines an electro-vac space-time.