The Segre classification of the Ricci tensor of the metric $g$ is an algebraic classification based upon the eigenvalues-eigenvectors for the trace-free Ricci tensor. These types are designated as [(1,1)(11)], [(1,1)11], [(1,11)1], [(1,111)], [(2,1)1], [(2,11)], [(3,1)], [1,(111)], [1,1(11)], [1,111], [2,(11)], [2,11], [3,1], [ZZ(11)], [ZZ11]. See Stephani, Kramer, et al for details.

Closely related to the Segre classification is the Plebanski-Petrov classification of the trace-free Ricci tensor. Here the trace-free Ricci tensor is used to construct the Plebanski tensor, a rank 4 tensor with the same algebraic symmetries as the Weyl tensor. Accordingly, the Plebanski tensor may be assigned one of the following types "O", "N", "D", "I", "II", "III" -- these are called the Plebanski-Petrov types of the trace-free Ricci tensor. See Stephani, Kramer, et al for details.

The command SegreType returns a sequence PP, S, where PP is the Plebanski-Petrov classification and S is the Segre classification.

See Details for SegreType for a complete description of the algorithm used.

Because the algorithm is based upon the calculation of the Newman-Penrose Ricci scalars, it is best to use one of the last 4 calling sequences. Note that the Ricci scalars can often be simplified through the application of an appropriate null tetrad transformation. The first calling sequence SegreType(g, pt) will automatically generate a null tetrad (if possible) but the result may not be optimal for calculating all the invariants needed to determine the Plebanski-Petrov and the Segre types.

The Segre type of an arbitrary rank 2 symmetric tensor $R$ can be calculated using the third calling sequence. From the tensor $R$, a trace-free tensor $S$ is calculated using the metric $g$. The Newman-Penrose coefficients of $S$ are determined and these are subsequently used to calculate the Segre type.

It is possible that the algebraic classification can vary from point to point on the manifold $M$. The command SegreType returns, if possible, the algebraic classification at a generic point. If the command fails, try the classification at a specific point with the optional argument pt. Constraints on the coordinate values can be imposed with assuming.

The branching performed by the program SegreType can be tracked by setting infolevel[SegreType] = 2.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form SegreType(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-SegreType.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[t\,x\,y\,z\right]\,M\right)$

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

$g≔\mathrm{evalDG}\left(\frac{t^2}{x^2}\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$g≔\frac{t^2}{x^2}\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{SegreType}\left(g\right)$

$''D'',''[(1,1)(11)]''$

$\mathrm{OTetrad}≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D\_t}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]\&comma;g\&comma;\mathrm{signature}=\left[1\&comma;-1\&comma;-1\&comma;-1\right]\right)\mathbf{assuming}0<x,0<t$

$\mathrm{OTetrad}≔\left[\frac{x}{t}\mathrm{D\_t}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]$

$\mathrm{NTetrad}≔\mathrm{NullTetrad}\left(\mathrm{OTetrad}\right)$

$\mathrm{NTetrad}≔\left[\frac{\sqrt{2}x}{2t}\mathrm{D\_t}+\frac{\sqrt{2}}{2}\mathrm{D\_z}\&comma;\frac{\sqrt{2}x}{2t}\mathrm{D\_t}-\frac{\sqrt{2}}{2}\mathrm{D\_z}\&comma;\frac{\sqrt{2}}{2}\mathrm{D\_x}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\&comma;\frac{\sqrt{2}}{2}\mathrm{D\_x}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\right]$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\right)$

$''D'',''[(1,1)(11)]''$

$\mathrm{FD}≔\mathrm{FrameData}\left(\mathrm{NTetrad}\&comma;\mathrm{AF}\right)$

$\mathrm{FD}≔\left[\left[\mathrm{E1}\&comma;\mathrm{E3}\right]=-\frac{\sqrt{2}\mathrm{E1}}{4x}-\frac{\sqrt{2}\mathrm{E2}}{4x}\&comma;\left[\mathrm{E1}\&comma;\mathrm{E4}\right]=-\frac{\sqrt{2}\mathrm{E1}}{4x}-\frac{\sqrt{2}\mathrm{E2}}{4x}\&comma;\left[\mathrm{E2}\&comma;\mathrm{E3}\right]=-\frac{\sqrt{2}\mathrm{E1}}{4x}-\frac{\sqrt{2}\mathrm{E2}}{4x}\&comma;\left[\mathrm{E2}\&comma;\mathrm{E4}\right]=-\frac{\sqrt{2}\mathrm{E1}}{4x}-\frac{\sqrt{2}\mathrm{E2}}{4x}\right]$

$\mathrm{DGsetup}\left(\mathrm{FD}\right)$

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

$\mathrm{SegreType}\left(\mathrm{AF}\right)$

$''D'',''[(1,1)(11)]''$

$''D''\&comma;''[(1,1)(11)]''$

$\mathrm{RS}≔\mathrm{NPCurvatureScalars}\left(\mathrm{NTetrad}\&comma;\mathrm{output}=\left[''AllRicciScalars''\right]\right)$

$\mathrm{RS}≔table\left(\left[''Phi12''=0\&comma;''Phi11''=0\&comma;''Phi10''=0\&comma;''Phi00''=\frac{1}{2x^2}\&comma;''Phi02''=-\frac{1}{2x^2}\&comma;''Phi01''=0\&comma;''Lambda''=-\frac{1}{6x^2}\&comma;''Phi22''=\frac{1}{2x^2}\&comma;''Phi21''=0\&comma;''Phi20''=-\frac{1}{2x^2}\right]\right)$

$\mathrm{SegreType}\left(\mathrm{RS}\right)$

$''D'',''[(1,1)(11)]''$

$\mathrm{DGsetup}\left(\left[t\&comma;x\&comma;y\&comma;z\right]\&comma;M\right)$

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

$g≔\mathrm{evalDG}\left(\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}+y^2\mathrm{dt}\&s\mathrm{dx}\right)$

$g≔\mathrm{dt}\mathrm{dt}+y^2\mathrm{dt}\mathrm{dx}+y^2\mathrm{dx}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{OTetrad}≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D\_t}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]\&comma;g\&comma;\mathrm{signature}=\left[1\&comma;-1\&comma;-1\&comma;-1\right]\right)\mathbf{assuming}0<y$

$\mathrm{OTetrad}≔\left[\mathrm{D\_t}\&comma;-\frac{y^2}{\sqrt{1+y^4}}\mathrm{D\_t}+\frac{1}{\sqrt{1+y^4}}\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]$

$\mathrm{NTetrad}≔\mathrm{NullTetrad}\left(\mathrm{OTetrad}\right)$

$\mathrm{NTetrad}≔\left[\frac{\sqrt{2}}{2}\mathrm{D\_t}+\frac{\sqrt{2}}{2}\mathrm{D\_z}\&comma;\frac{\sqrt{2}}{2}\mathrm{D\_t}-\frac{\sqrt{2}}{2}\mathrm{D\_z}\&comma;-\frac{\sqrt{2}{y}^2}{2\sqrt{1+y^4}}\mathrm{D\_t}+\frac{\sqrt{2}}{2\sqrt{1+y^4}}\mathrm{D\_x}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\&comma;-\frac{\sqrt{2}{y}^2}{2\sqrt{1+y^4}}\mathrm{D\_t}+\frac{\sqrt{2}}{2\sqrt{1+y^4}}\mathrm{D\_x}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\right]$

$\mathrm{NP}≔\mathrm{NPCurvatureScalars}\left(\mathrm{NTetrad}\&comma;\mathrm{output}=\left[''AllRicciScalars''\right]\right)$

$\mathrm{NP}≔table\left(\left[''Phi12''=-\frac{-1+y^4}{4{\left(1+y^4\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\&comma;''Phi11''=\frac{y^2\left(y^4-3\right)}{4{\left(1+y^4\right)}^2}\&comma;''Phi10''=-\frac{-1+y^4}{4{\left(1+y^4\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\&comma;''Phi00''=\frac{y^2}{2\left(1+y^4\right)}\&comma;''Phi02''=0\&comma;''Phi01''=-\frac{-1+y^4}{4{\left(1+y^4\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\&comma;''Lambda''=-\frac{y^2\left(5+y^4\right)}{12{\left(1+y^4\right)}^2}\&comma;''Phi22''=\frac{y^2}{2\left(1+y^4\right)}\&comma;''Phi21''=-\frac{-1+y^4}{4{\left(1+y^4\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\&comma;''Phi20''=0\right]\right)$

$\mathrm{SegreType}\left(\mathrm{NP}\right)\mathbf{assuming}1<y$

$''I'',''[ZZ11]''$

$\mathrm{SegreType}\left(\mathrm{NP}\&comma;\left[t=\mathrm{t0}\&comma;x=\mathrm{x0}\&comma;y=1\&comma;z=\mathrm{z0}\right]\right)$

$''O'',''[(1,11)1]''$

$\mathrm{SegreType}\left(\mathrm{NP}\&comma;\left[t=\mathrm{t0}\&comma;x=\mathrm{x0}\&comma;y=0\&comma;z=\mathrm{z0}\right]\right)$

$''D'',''[ZZ(11)]''$

$\mathrm{DGsetup}\left(\left[t\&comma;x\&comma;y\&comma;z\right]\&comma;M\right)$

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

$\mathrm{NTetrad}≔\mathrm{evalDG}\left(\left[\frac{1}{2y}z\mathrm{D\_t}+\frac{1}{2y}z\mathrm{D\_x}+\frac{1}{2}{2}^{\frac{1}{2}}\mathrm{D\_z}\&comma;\frac{1}{2y}z\mathrm{D\_t}+\frac{1}{2y}z\mathrm{D\_x}-\frac{1}{2}{2}^{\frac{1}{2}}\mathrm{D\_z}\&comma;\frac{1}{2y}z\mathrm{D\_t}-\frac{1}{2y}z\mathrm{D\_x}+\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}\mathrm{D\_y}\&comma;\frac{1}{2y}z\mathrm{D\_t}-\frac{1}{2y}z\mathrm{D\_x}-\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}\mathrm{D\_y}\right]\right)$

$\mathrm{NTetrad}≔\left[\frac{z}{2y}\mathrm{D\_t}+\frac{z}{2y}\mathrm{D\_x}+\frac{\sqrt{2}}{2}\mathrm{D\_z}\&comma;\frac{z}{2y}\mathrm{D\_t}+\frac{z}{2y}\mathrm{D\_x}-\frac{\sqrt{2}}{2}\mathrm{D\_z}\&comma;\frac{z}{2y}\mathrm{D\_t}-\frac{z}{2y}\mathrm{D\_x}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\&comma;\frac{z}{2y}\mathrm{D\_t}-\frac{z}{2y}\mathrm{D\_x}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\right]$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\right)$

$\mathrm{FAIL}$

$\mathrm{infolevel}\left[\mathrm{SegreType}\right]≔2$

${\mathrm{infolevel}}_{\mathrm{SegreType}}≔2$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\right)$

The NP Ricci coefficients are:

   Phi00: -1/4*(y^2-z^2)/z^2/y^2
   Phi01: 1/2*I/z/y
   Phi02: -1/4*(3*y^2+z^2)/z^2/y^2
   Phi10: -1/2*I/z/y
   Phi11: 1/2/z^2
   Phi12: -1/2*I/z/y
   Phi20: -1/4*(3*y^2+z^2)/z^2/y^2
   Phi21: 1/2*I/z/y
   Phi22: -1/4*(y^2-z^2)/z^2/y^2
The NP coefficients of the Chi spinor are:
   Chi0: 1/8*(3*y^4+2*z^2*y^2-z^4)/z^4/y^4
   Chi1: 0
   Chi2: -1/24*(3*y^4+2*z^2*y^2-z^4)/z^4/y^4
   Chi3: 0
   Chi4: 1/8*(3*y^4+2*z^2*y^2-z^4)/z^4/y^4
The Petrov-Plebanski type of the given symmetric tensor is "D"
The NP coefficients of the E spinor are:
   E00: -1/2*(y^2+z^2)/z^4/y^2
   E01: -1/2*I*(y^2+z^2)/z^3/y^3
   E02: 3/4*(y^2+z^2)/z^4/y^2
   E11: -1/2*(y^2+z^2)/z^4/y^2
   E12: 1/2*I*(y^2+z^2)/z^3/y^3
   E22: -1/2*(y^2+z^2)/z^4/y^2
The r1 invariant is:
   1/12*(9*y^4+10*z^2*y^2+z^4)/z^4/y^4
   If r1 = 0, then the Segre invariant is [ZZ(11)]
   If r1 <> 0, and the E spinor is 0, then the Segre invariant is [(1,1)(11)]
The tildeChi0 invariant is:
   -1/4*(y^2+z^2)^2*(3*y^2-2*z^2)/z^8/y^6
   If r1 <> 0, if Chi0 <> 0 and if tildeChi0 = 0, then the Segre invariant is [2,(11)]
The Ip invariant is:
   1/144*(3*y^4+2*z^2*y^2-z^4)^2/z^8/y^8
The Jp invariant is:
   -1/1728*(3*y^4+2*z^2*y^2-z^4)^3/z^12/y^12
The H invariant is:
   1/144*(3*y^4+2*z^2*y^2-z^4)^2/y^10*(y^2+z^2)/z^12
   If H = 0, then the Segre invariant is [2,(11)]
   If H < 0, then the Segre invariant is [ZZ,(11)]
Unable to evaluate the sign of H

$\mathrm{FAIL}$

$\mathrm{solve}\left(3y^4+2y^2{z}^2-z^4\&comma;z\right)$

$\sqrt{3}y,-\sqrt{3}y,\mathrm{I}y,\mathrm{-I}y$

$\mathrm{infolevel}\left[\mathrm{SegreType}\right]≔0$

${\mathrm{infolevel}}_{\mathrm{SegreType}}≔0$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;\left[x=a\&comma;y=b\&comma;z=\mathrm{sqrt}\left(3\right)b\&comma;t=c\right]\right)\mathbf{assuming}b::\mathrm{real}$

$''O'',''[(1,11)1]''$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;\left[x=a\&comma;y=b\&comma;z=b\&comma;t=d\right]\right)\mathbf{assuming}0<b$

$''D'',''[(1,1)11]''$

$g≔\mathrm{evalDG}\left(\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$g≔\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{NTetrad}≔\mathrm{evalDG}\left(\left[\frac{1}{2}{2}^{\frac{1}{2}}\mathrm{D\_t}+\frac{1}{2}{2}^{\frac{1}{2}}\mathrm{D\_z}\&comma;\frac{1}{2}{2}^{\frac{1}{2}}\mathrm{D\_t}-\frac{1}{2}{2}^{\frac{1}{2}}\mathrm{D\_z}\&comma;\frac{1}{2}{2}^{\frac{1}{2}}\mathrm{D\_x}+\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}\mathrm{D\_y}\&comma;\frac{1}{2}{2}^{\frac{1}{2}}\mathrm{D\_x}-\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}\mathrm{D\_y}\right]\right)$

$\mathrm{NTetrad}≔\left[\frac{\sqrt{2}}{2}\mathrm{D\_t}+\frac{\sqrt{2}}{2}\mathrm{D\_z}\&comma;\frac{\sqrt{2}}{2}\mathrm{D\_t}-\frac{\sqrt{2}}{2}\mathrm{D\_z}\&comma;\frac{\sqrt{2}}{2}\mathrm{D\_x}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\&comma;\frac{\sqrt{2}}{2}\mathrm{D\_x}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\right]$

$R≔\mathrm{evalDG}\left(a\mathrm{dt}\&t\mathrm{dt}-a\mathrm{dx}\&t\mathrm{dx}-a\mathrm{dy}\&t\mathrm{dy}-a\mathrm{dz}\&t\mathrm{dz}\right)$

$R≔a\mathrm{dt}\mathrm{dt}-a\mathrm{dx}\mathrm{dx}-a\mathrm{dy}\mathrm{dy}-a\mathrm{dz}\mathrm{dz}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)\mathbf{assuming}a<b$

$''O'',''[(1,111)]''$

$R≔\mathrm{evalDG}\left(a\mathrm{dt}\&t\mathrm{dt}-b\mathrm{dx}\&t\mathrm{dx}-b\mathrm{dy}\&t\mathrm{dy}-b\mathrm{dz}\&t\mathrm{dz}\right)$

$R≔a\mathrm{dt}\mathrm{dt}-b\mathrm{dx}\mathrm{dx}-b\mathrm{dy}\mathrm{dy}-b\mathrm{dz}\mathrm{dz}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)\mathbf{assuming}a<b$

$''O'',''[1,(111)]''$

$R≔\mathrm{evalDG}\left(a\mathrm{dt}\&t\mathrm{dt}-a\mathrm{dx}\&t\mathrm{dx}-a\mathrm{dy}\&t\mathrm{dy}-b\mathrm{dz}\&t\mathrm{dz}\right)$

$R≔a\mathrm{dt}\mathrm{dt}-a\mathrm{dx}\mathrm{dx}-a\mathrm{dy}\mathrm{dy}-b\mathrm{dz}\mathrm{dz}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)\mathbf{assuming}a<b$

$''O'',''[(1,11)1]''$

R := evalDG((a + 1)*dt &t dt + dt &t dx + dx &t dt + (-a + 1)*dx &t dx - a*dy &t dy - a*dz &t dz);

$R≔\left(a+1\right)\mathrm{dt}\mathrm{dt}+\mathrm{dt}\mathrm{dx}+\mathrm{dx}\mathrm{dt}-\left(a-1\right)\mathrm{dx}\mathrm{dx}-a\mathrm{dy}\mathrm{dy}-a\mathrm{dz}\mathrm{dz}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)\mathbf{assuming}a<b$

$''O'',''[(2,11)]''$

$R≔\mathrm{evalDG}\left(\left(a+1\right)\mathrm{dt}\&t\mathrm{dt}+\mathrm{dt}\&t\mathrm{dx}+\mathrm{dx}\&t\mathrm{dt}+\left(-a+1\right)\mathrm{dx}\&t\mathrm{dx}-a\mathrm{dy}\&t\mathrm{dy}-b\mathrm{dz}\&t\mathrm{dz}\right)$

$R≔\left(a+1\right)\mathrm{dt}\mathrm{dt}+\mathrm{dt}\mathrm{dx}+\mathrm{dx}\mathrm{dt}-\left(a-1\right)\mathrm{dx}\mathrm{dx}-a\mathrm{dy}\mathrm{dy}-b\mathrm{dz}\mathrm{dz}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)\mathbf{assuming}a<b$

$''N'',''[(2,1)1]''$

$R≔\mathrm{evalDG}\left(a\mathrm{dt}\&t\mathrm{dt}+\mathrm{dt}\&t\mathrm{dy}-a\mathrm{dx}\&t\mathrm{dx}-\mathrm{dx}\&t\mathrm{dy}+\mathrm{dy}\&t\mathrm{dt}-\mathrm{dy}\&t\mathrm{dx}-a\mathrm{dy}\&t\mathrm{dy}-a\mathrm{dz}\&t\mathrm{dz}\right)$

$R≔a\mathrm{dt}\mathrm{dt}+\mathrm{dt}\mathrm{dy}-a\mathrm{dx}\mathrm{dx}-\mathrm{dx}\mathrm{dy}+\mathrm{dy}\mathrm{dt}-\mathrm{dy}\mathrm{dx}-a\mathrm{dy}\mathrm{dy}-a\mathrm{dz}\mathrm{dz}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)\mathbf{assuming}a<b$

$''N'',''[(3,1)]''$

$R≔\mathrm{evalDG}\left(a\mathrm{dt}\&t\mathrm{dt}-a\mathrm{dx}\&t\mathrm{dx}-b\mathrm{dy}\&t\mathrm{dy}\right)$

$R≔a\mathrm{dt}\mathrm{dt}-a\mathrm{dx}\mathrm{dx}-b\mathrm{dy}\mathrm{dy}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)\mathbf{assuming}0<a,a<b$

$''D'',''[(1,1)11]''$

$R≔\mathrm{evalDG}\left(a\mathrm{dt}\&t\mathrm{dt}-b\mathrm{dx}\&t\mathrm{dx}\right)$

$R≔a\mathrm{dt}\mathrm{dt}-b\mathrm{dx}\mathrm{dx}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)\mathbf{assuming}0<a,a<b$

$''D'',''[1,1(11)]''$

$R≔\mathrm{evalDG}\left(a\mathrm{dt}\&t\mathrm{dt}+b\mathrm{dt}\&t\mathrm{dx}+b\mathrm{dx}\&t\mathrm{dt}-a\mathrm{dx}\&t\mathrm{dx}-c\mathrm{dy}\&t\mathrm{dy}-c\mathrm{dz}\&t\mathrm{dz}\right)$

$R≔a\mathrm{dt}\mathrm{dt}+b\mathrm{dt}\mathrm{dx}+b\mathrm{dx}\mathrm{dt}-a\mathrm{dx}\mathrm{dx}-c\mathrm{dy}\mathrm{dy}-c\mathrm{dz}\mathrm{dz}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)\mathbf{assuming}0<a,0<b,a<c$

$''D'',''[ZZ(11)]''$

$R≔\mathrm{evalDG}\left(a\mathrm{dt}\&t\mathrm{dt}-a\mathrm{dx}\&t\mathrm{dx}-b\mathrm{dy}\&t\mathrm{dy}\right)$

$R≔a\mathrm{dt}\mathrm{dt}-a\mathrm{dx}\mathrm{dx}-b\mathrm{dy}\mathrm{dy}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)\mathbf{assuming}0<a,a<b$

$''D'',''[(1,1)11]''$

$R≔\mathrm{evalDG}\left(\left(a+1\right)\mathrm{dt}\&t\mathrm{dt}+\mathrm{dt}\&t\mathrm{dx}+\mathrm{dx}\&t\mathrm{dt}+\left(-a+1\right)\mathrm{dx}\&t\mathrm{dx}-b\mathrm{dy}\&t\mathrm{dy}-b\mathrm{dz}\&t\mathrm{dz}\right)$

$R≔\left(a+1\right)\mathrm{dt}\mathrm{dt}+\mathrm{dt}\mathrm{dx}+\mathrm{dx}\mathrm{dt}-\left(a-1\right)\mathrm{dx}\mathrm{dx}-b\mathrm{dy}\mathrm{dy}-b\mathrm{dz}\mathrm{dz}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)\mathbf{assuming}a<b$

$''D'',''[2,(11)]''$

$R≔\mathrm{evalDG}\left(\left(a+1\right)\mathrm{dt}\&t\mathrm{dt}+\mathrm{dt}\&t\mathrm{dx}+\mathrm{dx}\&t\mathrm{dt}+\left(-a+1\right)\mathrm{dx}\&t\mathrm{dx}-b\mathrm{dy}\&t\mathrm{dy}-c\mathrm{dz}\&t\mathrm{dz}\right)$

$R≔\left(a+1\right)\mathrm{dt}\mathrm{dt}+\mathrm{dt}\mathrm{dx}+\mathrm{dx}\mathrm{dt}-\left(a-1\right)\mathrm{dx}\mathrm{dx}-b\mathrm{dy}\mathrm{dy}-c\mathrm{dz}\mathrm{dz}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)$

$''II'',''[2,11]''$

$R≔\mathrm{evalDG}\left(a\mathrm{dt}\&t\mathrm{dt}-b\mathrm{dx}\&t\mathrm{dx}-c\mathrm{dy}\&t\mathrm{dy}-d\mathrm{dz}\&t\mathrm{dz}\right)$

$R≔a\mathrm{dt}\mathrm{dt}-b\mathrm{dx}\mathrm{dx}-c\mathrm{dy}\mathrm{dy}-d\mathrm{dz}\mathrm{dz}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)\mathbf{assuming}a<b,b<c,c<d,0<a$

$''I'',''[1,111]''$

$R≔\mathrm{evalDG}\left(a\mathrm{dt}\&t\mathrm{dt}+b\mathrm{dt}\&t\mathrm{dx}+b\mathrm{dx}\&t\mathrm{dt}-a\mathrm{dx}\&t\mathrm{dx}-b\mathrm{dy}\&t\mathrm{dy}+\left(b+2a\right)\mathrm{dz}\&t\mathrm{dz}\right)$

$R≔a\mathrm{dt}\mathrm{dt}+b\mathrm{dt}\mathrm{dx}+b\mathrm{dx}\mathrm{dt}-a\mathrm{dx}\mathrm{dx}-b\mathrm{dy}\mathrm{dy}+\left(b+2a\right)\mathrm{dz}\mathrm{dz}$

$\mathrm{SegreType}\left(\mathrm{NTetrad}\&comma;g\&comma;R\right)\mathbf{assuming}a<b,0<a$

$''I'',''[ZZ11]''$