This function calculates any of the ten invariants listed above of the Riemann tensor, as designated by their corresponding flags.  For detailed definitions and descriptions of these invariants, refer to the paper listed in the References section of this page.

Simplification :

tensor[invars] has two simplifiers, `tensor/invars/simp` and `tensor/invars/Msimp`.

`tensor/invars/simp` is applied once after the invariant has been formally constructed.

Due to lengths of the actual formulas for the invariants, when calculating r3, m2, m3, m4, and m5, an extra simplifier, `tensor/invars/Msimp`, is employed.  `tensor/invars/Msimp` is used to simplify the sum of every 15 terms in the formulas for the five invariants mentioned above.  And then `tensor/invars/simp` is applied on top of `tensor/invars/Msimp` to put the 15-term segments together.

Note: that if the user finds it unnecessary, one of these simplifiers can actually be defined to perform no action.

This function is part of the tensor package, and can be used in the form invars(..) only after performing the command with(tensor), or with(tensor, invars).  The function can always be accessed in the long form tensor[invars].

$\mathrm{with}\left(\mathrm{tensor}\right)\:$

$\mathrm{coord}≔\left[t\,r\,\mathrm{\theta }\,\mathrm{\phi }\right]\:$

$\mathrm{g\_compts}≔\mathrm{array}\left(\mathrm{symmetric}\,1..4\,1..4\right)\:$

$$\begin{gathered} \mathbf{for}i\mathbf{to}4\mathbf{do} \\ \mathbf{for}j\mathbf{from}i+1\mathbf{to}4\mathbf{do} \\ \mathrm{g\_compts}\left[i,j\right]≔0 \\ \mathbf{end}\mathbf{do} \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$$\mathrm{g\_compts}\left[1,1\right]≔a\left(r\right)\:$$\mathrm{g\_compts}\left[2,2\right]≔-b\left(r\right)\:$$\mathrm{g\_compts}\left[3,3\right]≔-r^2\:$$\mathrm{g\_compts}\left[4,4\right]≔-r^2{\sin \left(\mathrm{\theta }\right)}^2\:$$g≔\mathrm{table}\left(\left['\mathrm{index\_char}'=\left[-1\,-1\right]\,'\mathrm{compts}'=\mathrm{op}\left(\mathrm{g\_compts}\right)\right]\right)$

$g≔table\left(\left[\mathrm{index\_char}=\left[\mathrm{-1}\,\mathrm{-1}\right]\,\mathrm{compts}=\left[\begin{array}{cccc}a\left(r\right)& 0& 0& 0\\ 0& -b\left(r\right)& 0& 0\\ 0& 0& -r^2& 0\\ 0& 0& 0& -r^2{\sin \left(\mathrm{\theta }\right)}^2\end{array}\right]\right]\right)$

$\mathrm{h\_compts}≔\mathrm{array}\left(\mathrm{sparse}\,1..4\,1..4\right)\:$

$\mathrm{h\_compts}\left[1,1\right]≔\frac{1}{2}{2}^{\frac{1}{2}}{a\left(r\right)}^{\frac{1}{2}}\:$

$\mathrm{h\_compts}\left[1,2\right]≔\frac{1}{2}{2}^{\frac{1}{2}}{b\left(r\right)}^{\frac{1}{2}}\:$

$\mathrm{h\_compts}\left[2,1\right]≔\frac{1}{2}{2}^{\frac{1}{2}}{a\left(r\right)}^{\frac{1}{2}}\:$

$\mathrm{h\_compts}\left[2,2\right]≔-\frac{1}{2}{2}^{\frac{1}{2}}{b\left(r\right)}^{\frac{1}{2}}\:$

$\mathrm{h\_compts}\left[3,3\right]≔\frac{1}{2}{2}^{\frac{1}{2}}r\:$

$\mathrm{h\_compts}\left[3,4\right]≔\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}r\sin \left(\mathrm{\theta }\right)\:$

$\mathrm{h\_compts}\left[4,3\right]≔\frac{1}{2}{2}^{\frac{1}{2}}r\:$

$\mathrm{h\_compts}\left[4,4\right]≔-\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}r\sin \left(\mathrm{\theta }\right)\:$

$h≔\mathrm{create}\left(\left[1\,-1\right]\,\mathrm{op}\left(\mathrm{h\_compts}\right)\right)$

$h≔table\left(\left[\mathrm{index\_char}=\left[1\,\mathrm{-1}\right]\,\mathrm{compts}=\left[\begin{array}{cccc}\frac{\sqrt{2}\sqrt{a\left(r\right)}}{2}& \frac{\sqrt{2}\sqrt{b\left(r\right)}}{2}& 0& 0\\ \frac{\sqrt{2}\sqrt{a\left(r\right)}}{2}& -\frac{\sqrt{2}\sqrt{b\left(r\right)}}{2}& 0& 0\\ 0& 0& \frac{\sqrt{2}r}{2}& \frac{\mathrm{I}}{2}\sqrt{2}r\sin \left(\mathrm{\theta }\right)\\ 0& 0& \frac{\sqrt{2}r}{2}& -\frac{\mathrm{I}}{2}\sqrt{2}r\sin \left(\mathrm{\theta }\right)\end{array}\right]\right]\right)$

$\mathrm{SPN}≔\mathrm{npspin}\left(\mathrm{coord}\,h\,'G'\,'\mathrm{any}'\right)\:$

$\mathrm{Curve}≔\mathrm{npcurve}\left(\mathrm{SPN}\,\mathrm{any}\right)\:$

`tensor/invars/simp`:=proc(x) x end proc:

$\mathrm{R1}≔\mathrm{invars}\left('\mathrm{r1}'\,\mathrm{Curve}\right)$

$\mathrm{R1}≔\frac{{\left(\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)b\left(r\right)+\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right)\right)}^2}{8{b\left(r\right)}^4{r}^2{a\left(r\right)}^2}+\frac{{\left(2\left(\frac{{\ⅆ}^2}{\ⅆr^2}a\left(r\right)\right)b\left(r\right)a\left(r\right)r^2-b\left(r\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^2{r}^2-\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)r^2+4{a\left(r\right)}^2{b\left(r\right)}^2-4{a\left(r\right)}^{2}b\left(r\right)\right)}^2}{64{a\left(r\right)}^4{b\left(r\right)}^4{r}^4}$

`tensor/invars/simp`:=proc(x) simplify(factor(x)) end proc:

$\mathrm{R1\_}≔\mathrm{invars}\left('\mathrm{r1}'\,\mathrm{Curve}\right)$

$\mathrm{R1\_}≔\frac{4{\left(\frac{{\ⅆ}^2}{\ⅆr^2}a\left(r\right)\right)}^2{b\left(r\right)}^2{a\left(r\right)}^2{r}^4-4\left(b\left(r\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^2{r}^2+\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)r^2-4{a\left(r\right)}^{2}b\left(r\right)\left(b\left(r\right)-1\right)\right)r^{2}b\left(r\right)a\left(r\right)\left(\frac{{\ⅆ}^2}{\ⅆr^2}a\left(r\right)\right)+{b\left(r\right)}^2{\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^4{r}^4+2b\left(r\right)\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^3{r}^4+r^2{a\left(r\right)}^2\left(r^2{\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)}^2-8{b\left(r\right)}^3+16{b\left(r\right)}^2\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^2-8r^2{a\left(r\right)}^{3}b\left(r\right)\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)\left(b\left(r\right)-3\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)+8{a\left(r\right)}^4\left(r^2{\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)}^2+2{b\left(r\right)}^2{\left(b\left(r\right)-1\right)}^2\right)}{64{a\left(r\right)}^4{b\left(r\right)}^4{r}^4}$

$\mathrm{simplify}\left(\mathrm{R1}-\mathrm{R1\_}\right)$

$0$

`tensor/invars/Msimp`:=proc(x) x end proc:

$\mathrm{M3}≔\mathrm{invars}\left('\mathrm{m3}'\,\mathrm{Curve}\right)$

$\mathrm{M3}≔\frac{{\left(\left(\frac{{\ⅆ}^2}{\ⅆr^2}a\left(r\right)\right)b\left(r\right)a\left(r\right)r^2-\frac{b\left(r\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^2{r}^2}{2}-\frac{ra\left(r\right)\left(r\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)+2b\left(r\right)\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}{2}+{a\left(r\right)}^2\left(r\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)-2{b\left(r\right)}^2+2b\left(r\right)\right)\right)}^2\left({\left(\frac{{\ⅆ}^2}{\ⅆr^2}a\left(r\right)\right)}^2{b\left(r\right)}^2{a\left(r\right)}^2{r}^4-\left(b\left(r\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^2{r}^2+\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)r^2-4{a\left(r\right)}^{2}b\left(r\right)\left(b\left(r\right)-1\right)\right)r^{2}b\left(r\right)a\left(r\right)\left(\frac{{\ⅆ}^2}{\ⅆr^2}a\left(r\right)\right)+\frac{{b\left(r\right)}^2{\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^4{r}^4}{4}+\frac{b\left(r\right)\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^3{r}^4}{2}+\frac{r^2{a\left(r\right)}^2\left(r^2{\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)}^2-8{b\left(r\right)}^3+10{b\left(r\right)}^2\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^2}{4}-2\left(b\left(r\right)-\frac{3}{2}\right)r^{2}b\left(r\right)\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right){a\left(r\right)}^3\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)+\frac{{a\left(r\right)}^4\left(r^2{\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)}^2+8{b\left(r\right)}^2{\left(b\left(r\right)-1\right)}^2\right)}{2}\right)}{576{a\left(r\right)}^8{b\left(r\right)}^8{r}^8}$

`tensor/invars/simp`:=proc(x) x end proc:

$\mathrm{M3\_}≔\mathrm{invars}\left('\mathrm{m3}'\,\mathrm{Curve}\right)$

$\mathrm{M3\_}≔\frac{{\left(2\left(\frac{{\ⅆ}^2}{\ⅆr^2}a\left(r\right)\right)b\left(r\right)a\left(r\right)r^2-b\left(r\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^2{r}^2-\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)r^2-4{a\left(r\right)}^2{b\left(r\right)}^2-2b\left(r\right)a\left(r\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)r+2\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right){a\left(r\right)}^{2}r+4{a\left(r\right)}^{2}b\left(r\right)\right)}^2{\left(\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)b\left(r\right)+\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right)\right)}^2}{4608{a\left(r\right)}^6{b\left(r\right)}^8{r}^6}+\frac{{\left(2\left(\frac{{\ⅆ}^2}{\ⅆr^2}a\left(r\right)\right)b\left(r\right)a\left(r\right)r^2-b\left(r\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^2{r}^2-\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)r^2-4{a\left(r\right)}^2{b\left(r\right)}^2-2b\left(r\right)a\left(r\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)r+2\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right){a\left(r\right)}^{2}r+4{a\left(r\right)}^{2}b\left(r\right)\right)}^2{\left(2\left(\frac{{\ⅆ}^2}{\ⅆr^2}a\left(r\right)\right)b\left(r\right)a\left(r\right)r^2-b\left(r\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^2{r}^2-\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)r^2+4{a\left(r\right)}^2{b\left(r\right)}^2-4{a\left(r\right)}^{2}b\left(r\right)\right)}^2}{9216{a\left(r\right)}^8{b\left(r\right)}^8{r}^8}$

$\mathrm{simplify}\left(\mathrm{M3}-\mathrm{M3\_}\right)$

$0$

$\mathrm{M3\_\_}≔\mathrm{invars}\left('\mathrm{m3}'\,\mathrm{Curve}\,\left[\left[r\,\mathrm{rBAR}\right]\,\left[\mathrm{\theta }\,\mathrm{thetaBAR}\right]\right]\right)$

$\mathrm{M3\_\_}≔\frac{\left(2\left(\frac{{\ⅆ}^2}{\ⅆr^2}a\left(r\right)\right)b\left(r\right)a\left(r\right)r^2-b\left(r\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^2{r}^2-\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)r^2-4{a\left(r\right)}^2{b\left(r\right)}^2-2b\left(r\right)a\left(r\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)r+2\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right){a\left(r\right)}^{2}r+4{a\left(r\right)}^{2}b\left(r\right)\right)\left(2\left(\frac{{\ⅆ}^2}{\ⅆ{\mathrm{rBAR}}^2}a\left(\mathrm{rBAR}\right)\right)b\left(\mathrm{rBAR}\right)a\left(\mathrm{rBAR}\right){\mathrm{rBAR}}^2-b\left(\mathrm{rBAR}\right){\left(\frac{\ⅆ}{\ⅆ\mathrm{rBAR}}a\left(\mathrm{rBAR}\right)\right)}^2{\mathrm{rBAR}}^2-\left(\frac{\ⅆ}{\ⅆ\mathrm{rBAR}}b\left(\mathrm{rBAR}\right)\right)a\left(\mathrm{rBAR}\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{rBAR}}a\left(\mathrm{rBAR}\right)\right){\mathrm{rBAR}}^2-4{a\left(\mathrm{rBAR}\right)}^2{b\left(\mathrm{rBAR}\right)}^2-2b\left(\mathrm{rBAR}\right)a\left(\mathrm{rBAR}\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{rBAR}}a\left(\mathrm{rBAR}\right)\right)\mathrm{rBAR}+2\left(\frac{\ⅆ}{\ⅆ\mathrm{rBAR}}b\left(\mathrm{rBAR}\right)\right){a\left(\mathrm{rBAR}\right)}^2\mathrm{rBAR}+4{a\left(\mathrm{rBAR}\right)}^{2}b\left(\mathrm{rBAR}\right)\right){\left(\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)b\left(r\right)+\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right)\right)}^2}{4608{a\left(r\right)}^4{b\left(r\right)}^6{r}^4{a\left(\mathrm{rBAR}\right)}^2{b\left(\mathrm{rBAR}\right)}^2{\mathrm{rBAR}}^2}+\frac{\left(2\left(\frac{{\ⅆ}^2}{\ⅆr^2}a\left(r\right)\right)b\left(r\right)a\left(r\right)r^2-b\left(r\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^2{r}^2-\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)r^2-4{a\left(r\right)}^2{b\left(r\right)}^2-2b\left(r\right)a\left(r\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)r+2\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right){a\left(r\right)}^{2}r+4{a\left(r\right)}^{2}b\left(r\right)\right)\left(2\left(\frac{{\ⅆ}^2}{\ⅆ{\mathrm{rBAR}}^2}a\left(\mathrm{rBAR}\right)\right)b\left(\mathrm{rBAR}\right)a\left(\mathrm{rBAR}\right){\mathrm{rBAR}}^2-b\left(\mathrm{rBAR}\right){\left(\frac{\ⅆ}{\ⅆ\mathrm{rBAR}}a\left(\mathrm{rBAR}\right)\right)}^2{\mathrm{rBAR}}^2-\left(\frac{\ⅆ}{\ⅆ\mathrm{rBAR}}b\left(\mathrm{rBAR}\right)\right)a\left(\mathrm{rBAR}\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{rBAR}}a\left(\mathrm{rBAR}\right)\right){\mathrm{rBAR}}^2-4{a\left(\mathrm{rBAR}\right)}^2{b\left(\mathrm{rBAR}\right)}^2-2b\left(\mathrm{rBAR}\right)a\left(\mathrm{rBAR}\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{rBAR}}a\left(\mathrm{rBAR}\right)\right)\mathrm{rBAR}+2\left(\frac{\ⅆ}{\ⅆ\mathrm{rBAR}}b\left(\mathrm{rBAR}\right)\right){a\left(\mathrm{rBAR}\right)}^2\mathrm{rBAR}+4{a\left(\mathrm{rBAR}\right)}^{2}b\left(\mathrm{rBAR}\right)\right){\left(2\left(\frac{{\ⅆ}^2}{\ⅆr^2}a\left(r\right)\right)b\left(r\right)a\left(r\right)r^2-b\left(r\right){\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)}^2{r}^2-\left(\frac{\ⅆ}{\ⅆr}b\left(r\right)\right)a\left(r\right)\left(\frac{\ⅆ}{\ⅆr}a\left(r\right)\right)r^2+4{a\left(r\right)}^2{b\left(r\right)}^2-4{a\left(r\right)}^{2}b\left(r\right)\right)}^2}{9216{a\left(r\right)}^6{b\left(r\right)}^6{r}^6{a\left(\mathrm{rBAR}\right)}^2{b\left(\mathrm{rBAR}\right)}^2{\mathrm{rBAR}}^2}$

Carminati, J., and McLenaghan, R.G. "Algebraic invariants of the Riemann tensor in a four-dimensional Lorentzian space." Journal of Mathematical Physics, Vol. 32 No. 11. (Nov. 1991).