A Riemann invariant is a scalar invariant constructed from the curvature tensor of a metric $g$. These are second-order invariants in the sense that no covariant derivatives of the curvature are used. There are many lists of Riemann invariants listed in the literature. At present the command RiemannInvariants will compute those found in the papers:

J. Carminati and R. G. McLenaghan, Algebraic invariants of the Riemann tensor in a four dimensional Lorentzian spacetime, J. Math. Physics 32(11), 3135--314.

Zakhary and C. B. G. McIntosh, A complete set of Riemann Invariants, General Relativity and Gravitation 29(5) 1997, 539 -- 581.

The command RiemannInvariants with return a table $T$ which contains the Riemann invariants as denoted in these papers.

If author = "ZakharyMcIntosh" and invariants = "AllComplex", then the indices of $T$ are ["I", "J", "R", "I6", "I7", "I8", "K", "L", "M", "M1", "M2"].

If author = "ZakharyMcIntosh" and invariants = "AllReal", then the indices of $T$ are [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

If author = "CarminatiMcLenaghan", then the indices of $T$ are ["r1", "r2", "r3", "w1", "w2", "m1", "m2", "m3", "m4", "m5"].

The default option is author = "ZakharyMcIntosh" and invariants = "AllComplex".

The invariants of ZakharyMcIntosh are calculated using either the spinor formalism or the Newman-Penrose formalism. Use either the 2nd, 3rd or 4th calling sequences. See Examples 1--4.

The invariants of CarminatiMcLenaghan are calculated using either the tensor or spinor formalism. Use the 1st or 4th calling sequence. See Examples 5--7.

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

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

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

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

$g≔\mathrm{evalDG}\left(x^4\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$g≔x^4\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{Inv1}≔\mathrm{RiemannInvariants}\left(g\right)$

$\mathrm{Inv1}≔table\left(\left[\right]\right)$

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

$\mathrm{OTetrad}≔\left[\frac{1}{x^2}\mathrm{D\_t}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]$

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

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

$\mathrm{Inv2}≔\mathrm{RiemannInvariants}\left(\mathrm{NTetrad}\right)$

$\mathrm{Inv2}≔table\left(\left[''I''=\frac{1}{9x^4}\&comma;''L''=\frac{2}{9x^8}\&comma;''K''=-\frac{2}{3x^6}\&comma;''I6''=\frac{1}{3x^4}\&comma;''R''=''R''\&comma;''M''=0\&comma;''I8''=0\&comma;''I7''=0\&comma;''J''=-\frac{1}{27x^6}\&comma;''M2''=-\frac{4}{27x^{10}}\&comma;''M1''=\frac{4}{9x^8}\right]\right)$

$\mathrm{RiemannInvariants}\left(\mathrm{NTetrad}\&comma;\mathrm{invariants}=\left[''L''\right]\right)$

$table\left(\left[''L''=\frac{2}{9x^8}\right]\right)$

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

$\mathrm{RicciScalars},\mathrm{WeylScalars}≔table\left(\left[''Lambda''=-\frac{1}{6x^2}\&comma;''Phi11''=0\&comma;''Phi12''=0\&comma;''Phi20''=-\frac{1}{2x^2}\&comma;''Phi22''=\frac{1}{2x^2}\&comma;''Phi21''=0\&comma;''Phi00''=\frac{1}{2x^2}\&comma;''Phi01''=0\&comma;''Phi02''=-\frac{1}{2x^2}\&comma;''Phi10''=0\right]\right),table\left(\left[''Psi1''=0\&comma;''PsiBar4''=\frac{1}{2x^2}\&comma;''Psi0''=\frac{1}{2x^2}\&comma;''PsiBar2''=-\frac{1}{6x^2}\&comma;''PsiBar3''=0\&comma;''Psi2''=-\frac{1}{6x^2}\&comma;''PsiBar1''=0\&comma;''PsiBar0''=\frac{1}{2x^2}\&comma;''Psi4''=\frac{1}{2x^2}\&comma;''Psi3''=0\right]\right)$

$\mathrm{Inv3}≔\mathrm{RiemannInvariants}\left(\mathrm{RicciScalars}\&comma;\mathrm{WeylScalars}\right)$

$\mathrm{Inv3}≔table\left(\left[''I''=\frac{1}{9x^4}\&comma;''L''=\frac{2}{9x^8}\&comma;''K''=-\frac{2}{3x^6}\&comma;''I6''=\frac{1}{3x^4}\&comma;''R''=''R''\&comma;''M''=0\&comma;''I8''=0\&comma;''I7''=0\&comma;''J''=-\frac{1}{27x^6}\&comma;''M2''=-\frac{4}{27x^{10}}\&comma;''M1''=\frac{4}{9x^8}\right]\right)$

$\mathrm{DGsetup}\left(\left[t\&comma;x\&comma;y\&comma;z\right]\&comma;\left[\mathrm{z1}\&comma;\mathrm{z2}\&comma;\mathrm{w1}\&comma;\mathrm{w2}\right]\&comma;N\right)$

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

$\mathrm{g1}≔\mathrm{evalDG}\left(x^4\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g1}≔x^4\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{OTetrad}≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D\_t}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]\&comma;\mathrm{g1}\&comma;\mathrm{signature}=\left[1\&comma;-1\&comma;-1\&comma;-1\right]\right)\mathbf{assuming}0<x$

$\mathrm{OTetrad}≔\left[\frac{1}{x^2}\mathrm{D\_t}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]$

$S≔\mathrm{SolderForm}\left(\mathrm{OTetrad}\right)$

$S≔\frac{x^2\sqrt{2}}{2}\mathrm{dt}\mathrm{D\_z1}\mathrm{D\_w1}+\frac{x^2\sqrt{2}}{2}\mathrm{dt}\mathrm{D\_z2}\mathrm{D\_w2}+\frac{\sqrt{2}}{2}\mathrm{dx}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\sqrt{2}}{2}\mathrm{dx}\mathrm{D\_z2}\mathrm{D\_w1}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_z2}\mathrm{D\_w1}+\frac{\sqrt{2}}{2}\mathrm{dz}\mathrm{D\_z1}\mathrm{D\_w1}-\frac{\sqrt{2}}{2}\mathrm{dz}\mathrm{D\_z2}\mathrm{D\_w2}$

$\mathrm{RicciSpin}≔\mathrm{RicciSpinor}\left(S\right)$

$\mathrm{RicciSpin}≔-\frac{1}{2x^2}\mathrm{dz1}\mathrm{dz1}\mathrm{dw1}\mathrm{dw1}+\frac{1}{2x^2}\mathrm{dz1}\mathrm{dz1}\mathrm{dw2}\mathrm{dw2}+\frac{1}{2x^2}\mathrm{dz2}\mathrm{dz2}\mathrm{dw1}\mathrm{dw1}-\frac{1}{2x^2}\mathrm{dz2}\mathrm{dz2}\mathrm{dw2}\mathrm{dw2}$

$\mathrm{WeylSpin}≔\mathrm{WeylSpinor}\left(S\right)$

$\mathrm{WeylSpin}≔-\frac{1}{2x^2}\mathrm{dz1}\mathrm{dz1}\mathrm{dz1}\mathrm{dz1}+\frac{1}{6x^2}\mathrm{dz1}\mathrm{dz1}\mathrm{dz2}\mathrm{dz2}+\frac{1}{6x^2}\mathrm{dz1}\mathrm{dz2}\mathrm{dz1}\mathrm{dz2}+\frac{1}{6x^2}\mathrm{dz1}\mathrm{dz2}\mathrm{dz2}\mathrm{dz1}+\frac{1}{6x^2}\mathrm{dz2}\mathrm{dz1}\mathrm{dz1}\mathrm{dz2}+\frac{1}{6x^2}\mathrm{dz2}\mathrm{dz1}\mathrm{dz2}\mathrm{dz1}+\frac{1}{6x^2}\mathrm{dz2}\mathrm{dz2}\mathrm{dz1}\mathrm{dz1}-\frac{1}{2x^2}\mathrm{dz2}\mathrm{dz2}\mathrm{dz2}\mathrm{dz2}$

$\mathrm{barWeylSpin}≔\mathrm{ConjugateSpinor}\left(\mathrm{WeylSpin}\right)$

$\mathrm{barWeylSpin}≔-\frac{1}{2x^2}\mathrm{dw1}\mathrm{dw1}\mathrm{dw1}\mathrm{dw1}+\frac{1}{6x^2}\mathrm{dw1}\mathrm{dw1}\mathrm{dw2}\mathrm{dw2}+\frac{1}{6x^2}\mathrm{dw1}\mathrm{dw2}\mathrm{dw1}\mathrm{dw2}+\frac{1}{6x^2}\mathrm{dw1}\mathrm{dw2}\mathrm{dw2}\mathrm{dw1}+\frac{1}{6x^2}\mathrm{dw2}\mathrm{dw1}\mathrm{dw1}\mathrm{dw2}+\frac{1}{6x^2}\mathrm{dw2}\mathrm{dw1}\mathrm{dw2}\mathrm{dw1}+\frac{1}{6x^2}\mathrm{dw2}\mathrm{dw2}\mathrm{dw1}\mathrm{dw1}-\frac{1}{2x^2}\mathrm{dw2}\mathrm{dw2}\mathrm{dw2}\mathrm{dw2}$

$\mathrm{Inv4}≔\mathrm{RiemannInvariants}\left(\mathrm{RicciSpin}\&comma;\mathrm{WeylSpin}\&comma;\mathrm{barWeylSpin}\right)$

$\mathrm{Inv4}≔table\left(\left[''I''=\frac{1}{9x^4}\&comma;''L''=\frac{2}{9x^8}\&comma;''K''=\frac{2}{3x^6}\&comma;''I6''=\frac{1}{3x^4}\&comma;''R''=''R''\&comma;''M''=0\&comma;''I8''=0\&comma;''I7''=0\&comma;''J''=\frac{1}{27x^6}\&comma;''M2''=\frac{4}{27x^{10}}\&comma;''M1''=\frac{4}{9x^8}\right]\right)$

$\mathrm{ChangeFrame}\left(M\right)$

$N$

$\mathrm{Inv5}≔\mathrm{RiemannInvariants}\left(g\&comma;\mathrm{author}=''CarminatiMcLenaghan''\right)$

$\mathrm{Inv5}≔table\left(\left[''m4''=0\&comma;''r2''=0\&comma;''m3''=\frac{4}{9x^8}\&comma;''r3''=\frac{1}{4x^8}\&comma;''m2''=\frac{4}{9x^8}\&comma;''w1''=\frac{2}{3x^4}\&comma;''m1''=\frac{2}{3x^6}\&comma;''w2''=\frac{2}{9x^6}\&comma;''r1''=\frac{1}{x^4}\&comma;''m5''=\frac{8}{27x^{10}}\right]\right)$

$C≔\mathrm{CurvatureTensor}\left(g\right)$

$C≔-\frac{2}{x^2}\mathrm{D\_t}\mathrm{dx}\mathrm{dt}\mathrm{dx}+\frac{2}{x^2}\mathrm{D\_t}\mathrm{dx}\mathrm{dx}\mathrm{dt}-2x^2\mathrm{D\_x}\mathrm{dt}\mathrm{dt}\mathrm{dx}+2x^2\mathrm{D\_x}\mathrm{dt}\mathrm{dx}\mathrm{dt}$

$\mathrm{Inv5}≔\mathrm{RiemannInvariants}\left(g\&comma;C\&comma;\mathrm{author}=''CarminatiMcLenaghan''\right)$

$\mathrm{Inv5}≔table\left(\left[''m4''=0\&comma;''r2''=0\&comma;''m3''=\frac{4}{9x^8}\&comma;''r3''=\frac{1}{4x^8}\&comma;''m2''=\frac{4}{9x^8}\&comma;''w1''=\frac{2}{3x^4}\&comma;''m1''=\frac{2}{3x^6}\&comma;''w2''=\frac{2}{9x^6}\&comma;''r1''=\frac{1}{x^4}\&comma;''m5''=\frac{8}{27x^{10}}\right]\right)$

$\mathrm{RiemannInvariants}\left(\mathrm{NTetrad}\&comma;\mathrm{author}=''CarminatiMcLenaghan''\right)$

$table\left(\left[\right]\right)$

$\mathrm{Inv6}≔\mathrm{RiemannInvariants}\left(\mathrm{RicciSpin}\&comma;\mathrm{WeylSpin}\&comma;\mathrm{barWeylSpin}\&comma;\mathrm{author}=''CarminatiMcLenaghan''\right)$

$\mathrm{Inv6}≔table\left(\left[''m4''=0\&comma;''r2''=0\&comma;''m3''=\frac{4}{9x^8}\&comma;''r3''=\frac{1}{4x^8}\&comma;''m2''=\frac{4}{9x^8}\&comma;''w1''=\frac{2}{3x^4}\&comma;''m1''=\frac{2}{3x^6}\&comma;''w2''=\frac{2}{9x^6}\&comma;''r1''=\frac{1}{x^4}\&comma;''m5''=\frac{4\left({\left|x\right|}^{12}+x^{12}\right)}{27x^{10}{\left|x\right|}^{12}}\right]\right)$

$\mathrm{map}\left(\mathrm{simplify}\&comma;\mathrm{Inv6}\right)\mathbf{assuming}x::\mathrm{real}$

$table\left(\left[''m4''=0\&comma;''r2''=0\&comma;''m3''=\frac{4}{9x^8}\&comma;''r3''=\frac{1}{4x^8}\&comma;''m2''=\frac{4}{9x^8}\&comma;''w1''=\frac{2}{3x^4}\&comma;''m1''=\frac{2}{3x^6}\&comma;''w2''=\frac{2}{9x^6}\&comma;''r1''=\frac{1}{x^4}\&comma;''m5''=\frac{8}{27x^{10}}\right]\right)$