Appendix: Second order invariants
In this appendix we demonstrate some of the expressions for the second order polynomial scalar curvature invariants which were too long to include in the main text.
$$\begin{aligned} I_{8,d,2}= & {} R^{;pq} R_{;pq}\\= & {} 55296\,{{ {\Psi }_2}}^{2}{{ \gamma }} ^{2}{{ \pi }}^{2}+6912\,{{ {\Psi }_2} }^{2}{{ \gamma }}^{2}{{ (\sigma -\lambda )} }^{2}-13824\,{{ {\Psi }_2}}^{2}{{ \gamma }}^{2}{ (\sigma -\lambda )}\,{ (\sigma +\lambda )}\\&+\,6912\,{{ {\Psi }_2}}^{2}{{ \gamma }}^{2}{{ (\sigma +\lambda )}}^{2}+ 18432\,{{ {\Psi }_2}}^{2}{ \gamma }\, { (\sigma -\lambda )}\,{{ \pi }}^{2}-2304\, {{ {\Psi }_2}}^{2}{ \gamma }\,{{ (\sigma -\lambda )}}^{3}\\&+\,4608\,{{ {\Psi }_2}}^{ 2}{ \gamma }\,{{ (\sigma -\lambda )}}^{2}{ (\sigma +\lambda )}-2304\,{{ {\Psi }_2}}^{2 }{ \gamma }\,{ (\sigma -\lambda )}\,{{ (\sigma +\lambda )}}^{2}\\&-\,4608\,{{ {\Psi }_2}}^{ 2}{{ (\sigma -\lambda )}}^{2}{{ \pi }}^{2}\\&+\, 1728\,{{ {\Psi }_2}}^{2}{{ (\sigma -\lambda )}}^{4}-3456\,{{ {\Psi }_2}}^{ 2}{{ (\sigma -\lambda )}}^{3}{ (\sigma +\lambda )} \\&+\,1728\,{{ {\Psi }_2}}^{2}{{ (\sigma -\lambda )}}^{2}{{ (\sigma +\lambda )}}^{2}\\&+\, 2304\,\Lambda \,{ \gamma }\,{{ {\Psi }_2}}^{2}{ (\sigma -\lambda )}-2304\, \Lambda \,{ \gamma }\,{{ {\Psi }_2}}^{ 2}{ (\sigma +\lambda )}-1152\,\Lambda \,{{ {\Psi }_2}}^{2}{{ (\sigma -\lambda )}}^{2}\\&+\,1152 \,\Lambda \,{{ {\Psi }_2}}^{2}{ (\sigma -\lambda )}\,{ (\sigma +\lambda )}-9216\,{{ {\Psi }_2}}^{2}{ \gamma }\,{ \pi }\,{ \delta (\sigma -\lambda )}\\&-\,4608\,{{ {\Psi }_2}}^{2}{ (\sigma -\lambda )}\,{ \pi }\,{ \delta (\sigma -\lambda )}\\&+\,18432\,{ {\Psi }_2}\,{{ \gamma }}^{3}{ \Delta {\Psi }_2}+9216\,{ {\Psi }_2}\,{{ \gamma }}^{2}{ \Delta {\Psi }_2}\,{ (\sigma +\lambda )}-36864\,{ {\Psi }_2}\,{ \gamma }\,{ \Delta {\Psi }_2}\,{{ \pi }}^{2} \end{aligned}$$
$$\begin{aligned}&-\,9216\,{ {\Psi }_2}\,{ \gamma }\,{ (\sigma -\lambda )}\,{ \pi }\,{ \delta {\Psi }_2}-9216\,{ {\Psi }_2}\,{ \gamma }\,{ (\sigma +\lambda )}\,{ \pi }\,{ \delta {\Psi }_2}\nonumber \\&+\,4608\,{ {\Psi }_2}\,{ \gamma }\,{{ (\sigma -\lambda )}}^{2}{ \Delta {\Psi }_2}\nonumber \\&+\,2304\,{ {\Psi }_2}\,{ \gamma }\,{ (\sigma -\lambda )}\,{ \Delta {\Psi }_2}\,{ (\sigma +\lambda )}-2304\,{ {\Psi }_2}\,{ \gamma }\,{ \Delta {\Psi }_2}\,{{ (\sigma +\lambda )}}^{2}\nonumber \\&-\, 9216\,{ {\Psi }_2}\,{{ (\sigma -\lambda )}}^ {2}{ \pi }\,{ \delta {\Psi }_2}+4608\,{ {\Psi }_2}\,{ (\sigma -\lambda )}\,{ (\sigma +\lambda )}\,{ \pi }\,{ \delta {\Psi }_2}\nonumber \\&+\,2304\,{ {\Psi }_2}\,{{ (\sigma -\lambda )}}^{3}{ \Delta {\Psi }_2}\nonumber \\&-\,1152\,{ {\Psi }_2}\,{{ (\sigma -\lambda )}}^ {2}{ \Delta {\Psi }_2}\,{ (\sigma +\lambda )}- 1152\,{ {\Psi }_2}\,{ (\sigma -\lambda )}\,{ \Delta {\Psi }_2}\,{{ (\sigma +\lambda )}}^{2 }\nonumber \\&+\,288\,{\Lambda }^{2}{{ {\Psi }_2}}^{2}\nonumber \\&-\,3456\,\Lambda \, { \gamma }\,{ {\Psi }_2}\,{ \Delta {\Psi }_2}+1152\,\Lambda \,{ {\Psi }_2}\,{ \delta {\Psi }_2}\,{ \pi }-1152\,\Lambda \,{ {\Psi }_2}\,{ (\sigma -\lambda )}\,{ \Delta {\Psi }_2}\nonumber \\&+\,9216\,{ {\Psi }_2}\,{{ \gamma }}^{2}{ \Delta \Delta {\Psi }_2}+18432\,{ {\Psi }_2} \,{ \gamma }\,{ \pi }\,{ \Delta \delta {\Psi }_2}+9216\,{ {\Psi }_2}\, { \gamma }\,{ (\sigma -\lambda )}\,{ \Delta \Delta {\Psi }_2}\nonumber \\&+\,2304\,{ {\Psi }_2}\, { \gamma }\,{ (\sigma -\lambda )}\,{ \delta \delta {\Psi }_2}-4608\,{ {\Psi }_2}\, { \gamma }\,{ (\sigma +\lambda )}\,{ \Delta \Delta {\Psi }_2}\nonumber \\&-\,2304\,{ {\Psi }_2}\, { \gamma }\,{ (\sigma +\lambda )}\,{ \delta \delta {\Psi }_2}+4608\,{ {\Psi }_2}\, { \Delta {\Psi }_2}\,{ \pi }\,{ \delta (\sigma -\lambda )}\nonumber \\&+\,1152\,{ {\Psi }_2}\,{{ (\sigma -\lambda )}}^{2}{ \delta \delta {\Psi }_2}\nonumber \\&+\,2304\,{ {\Psi }_2}\, { (\sigma -\lambda )}\,{ (\sigma +\lambda )}\,{ \Delta \Delta {\Psi }_2}-1152\,{ {\Psi }_2}\,{ (\sigma -\lambda )}\,{ (\sigma +\lambda )}\,{ \delta \delta {\Psi }_2}\nonumber \\&+\, 2304\,{{ \gamma }}^{2}{{ \Delta {\Psi }_2}}^{2}-4608\,{ \gamma }\,{ \Delta {\Psi }_2}\,{ \pi }\,{ \delta {\Psi }_2}+4608\,{ \gamma }\,{ (\sigma -\lambda )}\,{{ \Delta {\Psi }_2}}^{2}\nonumber \\&+\, 4608\,{{ \Delta {\Psi }_2}}^{2}{{ \pi } }^{2}\nonumber \\&+\,2304\,{{ \pi }}^{2}{{ \delta {\Psi }_2}}^{2}+4608\,{ (\sigma -\lambda )}\,{ \Delta {\Psi }_2}\,{ \pi }\,{ \delta {\Psi }_2}+1152\,{{ (\sigma -\lambda )}}^{ 2}{{ \Delta {\Psi }_2}}^{2}\nonumber \\&+\,576\,{{ \Delta {\Psi }_2}}^{2}{{ (\sigma +\lambda )}}^{2} +576\,\Lambda \,{ {\Psi }_2}\,{ \Delta \Delta {\Psi }_2}-2304\,{ {\Psi }_2}\, { \Delta \delta {\Psi }_2}\,{ \delta (\sigma -\lambda )}\nonumber \\&-\,6912\,{ \gamma }\,{ \Delta {\Psi }_2}\,{ \Delta \Delta {\Psi }_2}-4608\,{ \gamma }\,{ \delta {\Psi }_2}\,{ \Delta \delta {\Psi }_2}-4608\,{ \Delta {\Psi }_2}\,{ \pi }\,{ \Delta \delta {\Psi }_2}\nonumber \\&+\,2304\,{ \pi }\,{ \delta {\Psi }_2}\,{ \Delta \Delta {\Psi }_2}\!-\!2304\,{ (\sigma \!-\!\lambda ) }\,{ \Delta {\Psi }_2}\,{ \Delta \Delta {\Psi }_2}\!+\!1152\,{ \Delta {\Psi }_2}\,{ (\sigma +\lambda )}\,{ \delta \delta {\Psi }_2}\nonumber \\&-\,4608\,{ (\sigma -\lambda )}\,{ \delta {\Psi }_2}\,{ \Delta \delta {\Psi }_2}+576\,{{ \Delta \Delta {\Psi }_2}}^{2}+1152\,{{ \Delta \delta {\Psi }_2}}^{2}+576\,{{ \delta \delta {\Psi }_2}}^{2}\nonumber \\ \end{aligned}$$
(51)
$$\begin{aligned} I_{8,d,3}= & {} R^{;pq} \Box R_{pq}\\= & {} -\,4608\,{{ \gamma }}^{3}{{{\Psi }_2}}^{2}{(\sigma -\lambda )}-4608\,{{\gamma }}^{3}{{{\Psi }_2}}^{2}{(\sigma +\lambda )} -9216\,{{{\Psi }_2}}^{2}{{\gamma }} ^{2}{{\pi }}^{2}\\&+\,5760\,{{{\Psi }_2}}^{2}{{ \gamma }}^{2}{{(\sigma -\lambda )}}^{2}\\&-\,18432\,{{{\Psi }_2}}^{2}{{\gamma }}^{2}{ (\sigma -\lambda )}\,{(\sigma +\lambda )}+8064\,{{{\Psi }_2}}^{2}{{\gamma }}^{2}{{(\sigma +\lambda )}}^{2}\\&+\, 9216\,{{{\Psi }_2}}^{2}{\gamma }\,{(\sigma -\lambda )}\,{{\pi }}^{2}\\&-\,1152\,{{{\Psi }_2}}^{2}{\gamma }\,{{(\sigma -\lambda )}}^{2}{(\sigma +\lambda )}-1152\,{{{\Psi }_2}}^{2}{\gamma }\,{(\sigma -\lambda )}\,{{(\sigma +\lambda )}}^{2}\\&-\,2304 \,{{{\Psi }_2}}^{2}{{(\sigma -\lambda )}}^ {2}{{\pi }}^{2}\\&+\,1440\,{{{\Psi }_2}} ^{2}{{(\sigma -\lambda )}}^{4}-1152\,{{{\Psi }_2}}^{2}{{(\sigma -\lambda )}}^{3}{(\sigma +\lambda )}\\&-\,288\,{{{\Psi }_2}}^{2}{{(\sigma -\lambda )}}^{2}{{(\sigma +\lambda )}}^{2}\\&+\,2304\,\Lambda \,{{\gamma }}^{2}{{{\Psi }_2}}^{2}+3744\,\Lambda \,{\gamma }\,{{{\Psi }_2}}^{2}{(\sigma -\lambda )}- 2592\,\Lambda \,{\gamma }\,{{{\Psi }_2}}^{2}{(\sigma +\lambda )}\\&-\,144\,\Lambda \,{{{\Psi }_2}}^{2}{{(\sigma -\lambda )}}^{2}\\&+\,432\,\Lambda \,{{{\Psi }_2}}^{2}{(\sigma -\lambda )}\,{(\sigma +\lambda )}+4608\,{{{\Psi }_2}}^{2}{\gamma }\,{\pi }\,{\delta (\sigma -\lambda )}\\&-\,2304\,{{{\Psi }_2}}^{2}{(\sigma -\lambda )}\,{\pi }\,{\delta (\sigma -\lambda )}\\&+\,9216\,{{\Psi }_2}\,{{\gamma }}^{3}{\Delta {\Psi }_2}+13824 \,{{\gamma }}^{2}{\pi }\,{\delta {\Psi }_2}\,{{\Psi }_2}-9216\,{\Delta {\Psi }_2}\,{{\gamma }}^{2}{{\Psi }_2}\,{(\sigma -\lambda )}\\&+\,13824\,{{\Psi }_2}\,{{\gamma }}^{2}{\Delta {\Psi }_2}\,{(\sigma +\lambda )}+16128\,{{\Psi }_2}\,{\gamma }\,{(\sigma -\lambda )}\,{\pi }\,{\delta {\Psi }_2}\\&-\,6912\,{{\Psi }_2}\,{\gamma }\,{(\sigma +\lambda )}\,{\pi }\,{\delta {\Psi }_2}\\&-\,4608\,{{\Psi }_2}\,{\gamma }\,{{(\sigma -\lambda )}}^{2}{\Delta {\Psi }_2}+11520 \,{{\Psi }_2}\,{\gamma }\,{(\sigma -\lambda )}\,{\Delta {\Psi }_2}\,{(\sigma +\lambda )} \end{aligned}$$
$$\begin{aligned}&-\,3456\,{{\Psi }_2}\,{\gamma }\,{\Delta {\Psi }_2}\,{{(\sigma +\lambda )}}^{2}-4608\,{{\Psi }_2}\,{{(\sigma -\lambda )}}^{2}{\pi }\,{\delta {\Psi }_2}\nonumber \\&+\,1152\,{{\Psi }_2}\,{(\sigma -\lambda )}\,{(\sigma +\lambda )}\,{\pi }\,{\delta {\Psi }_2}\nonumber \\&+\,576\,{{\Psi }_2}\,{(\sigma -\lambda )}\,{\Delta {\Psi }_2}\,{{(\sigma +\lambda )}}^{2}+144\,{\Lambda }^{2}{{{\Psi }_2}}^{2}-2592\,\Lambda \,{\gamma }\,{{\Psi }_2}\,{\Delta {\Psi }_2}\nonumber \\&+\,864\,\Lambda \,{{\Psi }_2}\,{\delta {\Psi }_2}\,{\pi }\nonumber \\&-\,864\,\Lambda \,{{\Psi }_2}\,{(\sigma -\lambda )} \,{\Delta {\Psi }_2}\!+\!432\,\Lambda \,{{\Psi }_2}\,{\Delta {\Psi }_2}\,{(\sigma +\lambda )}-576\,{{\delta (\sigma -\lambda )}}^{2}{{{\Psi }_2}}^{2}\nonumber \\&+\,4608\,{{\Psi }_2}\,{{\gamma }}^{2}{\Delta \Delta {\Psi }_2}+2304\,{\delta \delta {\Psi }_2}\,{{\gamma }}^{2}{{\Psi }_2}+4608\,{{\Psi }_2}\,{\gamma }\,{(\sigma -\lambda )}\,{\Delta \Delta {\Psi }_2}\nonumber \\&+\,4608\,{{\Psi }_2}\,{\gamma }\,{(\sigma -\lambda )}\,{\delta \delta {\Psi }_2}\nonumber \\&-\,3456\,{{\Psi }_2}\,{\gamma }\,{(\sigma +\lambda )}\,{\Delta \Delta {\Psi }_2}-3456\,{{\Psi }_2}\,{\gamma }\,{(\sigma +\lambda )}\,{\delta \delta {\Psi }_2}\nonumber \\&-\,1152\,{\delta (\sigma -\lambda )}\,{\gamma }\,{\delta {\Psi }_2}\,{{\Psi }_2}\nonumber \\&+\,576\,{{\Psi }_2}\,{{(\sigma -\lambda )}}^{2}{\delta \delta {\Psi }_2}+576\,{{\Psi }_2}\,{(\sigma -\lambda )}\,{(\sigma +\lambda )}\,{\Delta \Delta {\Psi }_2}\nonumber \\&+\,576\,{{\Psi }_2}\,{(\sigma -\lambda )}\,{(\sigma +\lambda )}\,{\delta \delta {\Psi }_2}-2304\,{\delta (\sigma -\lambda )}\,{\delta {\Psi }_2}\,{{\Psi }_2}\,{(\sigma -\lambda )}\nonumber \\&+\,1152\,{{\gamma }}^{2}{{\Delta {\Psi }_2}}^{2} \nonumber \\&-\,6912\,{\gamma }\,{\Delta {\Psi }_2}\,{\pi }\,{\delta {\Psi }_2}+4608\,{\gamma }\,{(\sigma -\lambda )}\,{{\Delta {\Psi }_2}}^{2}-3456\,{{\Delta {\Psi }_2}}^{2}{\gamma }\,{(\sigma +\lambda )}\nonumber \\&-\,2304\,{\gamma }\,{{\delta {\Psi }_2}}^{2}{(\sigma -\lambda )}+1152\,{{\pi }}^{2}{{\delta {\Psi }_2}}^{2}-2304\,{(\sigma -\lambda )}\,{\Delta {\Psi }_2}\,{\pi }\,{\delta {\Psi }_2}\nonumber \\&+\,1152\,{\Delta {\Psi }_2}\,{\pi }\,{\delta {\Psi }_2}\,{(\sigma +\lambda )}+1152\,{{(\sigma -\lambda )}}^{2}{{\Delta {\Psi }_2}}^{2}\nonumber \\&-\,1152\,{{\Delta {\Psi }_2}}^{2}{(\sigma -\lambda )}\,{(\sigma +\lambda )}\nonumber \\&+\,288\,{{\Delta {\Psi }_2}}^{2}{{(\sigma +\lambda )}}^{2}-2304\,{{\delta {\Psi }_2}}^{2}{{(\sigma -\lambda )}}^{2}\nonumber \\&+\,432\,\Lambda \,{{\Psi }_2}\,{\Delta \Delta {\Psi }_2}+432\,\Lambda \,{{\Psi }_2}\,{\delta \delta {\Psi }_2}\nonumber \\&-\,3456\,{\gamma }\,{\Delta {\Psi }_2}\,{\Delta \Delta {\Psi }_2}-3456\,{\delta \delta {\Psi }_2}\,{\Delta {\Psi }_2}\,{\gamma }\nonumber \\&+\,1152\,{\pi }\,{\delta {\Psi }_2}\,{\Delta \Delta {\Psi }_2}+1152\,{\delta \delta {\Psi }_2}\,{\pi }\,{\delta {\Psi }_2}\nonumber \\&-\,1152\,{ (\sigma -\lambda )}\,{\Delta {\Psi }_2}\,{\Delta \Delta {\Psi }_2}-1152\,{\delta \delta {\Psi }_2}\,{\Delta {\Psi }_2}\,{(\sigma -\lambda )}\nonumber \\&+\,576\,{\Delta {\Psi }_2} \,{\Delta \Delta {\Psi }_2}\,{(\sigma +\lambda )}\nonumber \\&+\,576\,{\Delta {\Psi }_2}\,{(\sigma +\lambda )}\,{\delta \delta {\Psi }_2} \!+\!288\,{{\Delta \Delta {\Psi }_2}}^{2}+576\,{\delta \delta {\Psi }_2}\,{\Delta \Delta {\Psi }_2}\!+\!288\,{{\delta \delta {\Psi }_2}}^{2}\nonumber \\ \end{aligned}$$
(52)
Appendix 2: A special case
It is illustrative to demonstrate the above relationships by examining a special case of the 3D Szekeres spacetimes. We first motivate our choice of form for the arbitrary function S(y) and those appearing in \(\hat{\nu }(x,y) = -\ln (A(y) x^2 + B(y) x + C(y))\).
1.1
S(y)
We desire a special case in which the Cartan invariants take on a simpler (but nondegenerate) form. They can then be more easily compared with the polynomial scalar curvature invariants explicitly.
\(F_{,x}=0\) for our Szekeres metrics, so (as in (7)) equations (87)-(88) of Barrow et al. become
$$\begin{aligned} \hat{\kappa } \hat{\rho } = \frac{E(x,y; \Lambda )}{R(R_{,y} + R \nu _{,y})} \end{aligned}$$
(53)
where
$$\begin{aligned} E(x,y; \Lambda ) = e^{-2\nu }[\nu _{,xy} \nu _{,x} - \nu _{,xxy} ] + \frac{1}{2} e^{-2\nu }(K e^{2\nu })_{,y} \end{aligned}$$
(54)
and
$$\begin{aligned} K(x,y,t) = \dot{R}^2 - \Lambda R^2 - 2 (SF_{,y})^2. \end{aligned}$$
(55)
Defining S(y) to be such that K vanishes, that is,
$$\begin{aligned} S^2 = \frac{\dot{R}^2 - \Lambda R^2}{ 2 {F_{,y}}^2} = \frac{-\Lambda + F^2}{ 2 {F_{,y}}^2}, \end{aligned}$$
(56)
we have
$$\begin{aligned} \Psi _2 = \frac{1}{12} \hat{\kappa } \hat{\rho } = \frac{1}{12} \frac{e^{-2\nu }[\nu _{,xy} \nu _{,x} - \nu _{,xxy} ]}{R(R_{,y} + R \nu _{,y})}, \end{aligned}$$
(57)
which is simpler. Note that due to a differential identity of R(t, y), (56) is in fact not a function of t. Also note that we must assume \(-\Lambda + F^2 \gneq 0\).
1.2
\(\hat{\nu }(x,y)\)
We have
$$\begin{aligned} \hat{\nu } = -\ln (A(y) x^2 + B(y) x + C(y)). \end{aligned}$$
(58)
Choosing only \(B(y)=0\) and \(C(y)=c\) constant simplifies the derivatives of \(\hat{\nu }\) without causing the \(\Psi _2\) above to vanish. This implies that only two arbitrary functions remain: A(y) and F(y).
1.3 Extended Cartan Invariants
After making these choices, the Cartan invariants take the forms \(\Lambda \) constant and:
$$\begin{aligned} \Psi _2= & {} 1/6\,{\frac{A_{{y}}c \left( A{x}^{2}-c \right) }{R \left( -R_{{y}}A{x }^{2}+RA_{{y}}{x}^{2}-R_{{y}}c \right) }} \end{aligned}$$
(59)
$$\begin{aligned} \gamma= & {} -1/4\,{\frac{R_{{t}}}{R}} \end{aligned}$$
(60)
$$\begin{aligned} \frac{\sigma -\lambda }{2}= & {} -1/4\,{\frac{-R_{{t,y}}A{x}^{2}+R_{{t}}A_{{y}}{x}^{2}-R_{{t,y}}c}{-R_ {{y}}A{x}^{2}+RA_{{y}}{x}^{2}-R_{{y}}c}} \end{aligned}$$
(61)
$$\begin{aligned} \delta \Psi _2= & {} \frac{-\sqrt{-\Lambda \,{R}^{2}+{R_{{t}}}^{2}}c \left( A{x}^{2}+c \right) }{6\, \left( -R_{{y}}A{x}^{2}+RA_{{y}}{x}^{2}-R_{{y}}c \right) ^{3}{R}^{ 2} } \Bigl ( A_{{y}}{A}^{2}R_{{y,y }}R{x}^{4}\nonumber \\&-\,{A}^{2}R_{{y}}A_{{y,y}}R{x}^{4} +A_{{y}}{A}^{2}{R_{{y}}}^{2} {x}^{4} -2\,{A_{{y}}}^{2}AR_{{y}}R{x}^{4}+{A_{{y}}}^{3}{R}^{2}{x}^{4}\nonumber \\&-\,A_{{y}}R_{{y,y}}R{c}^{2}+R_{{y}}A_{{y,y}}R{c}^{2}-A_{{y}}{R_{{y}}}^{2}{ c}^{2} \Bigr ) \end{aligned}$$
(62)
$$\begin{aligned} \Delta \Psi _2= & {} \frac{-A_{{y}}c }{12\,{R}^{2} \left( -R_{{y}}A{x}^{2}+RA_{{y}}{x}^{2}-R_{{y}}c \right) ^{2}} \Bigl ( -R_{{t,y}}R{A}^{2}{x}^{4}\nonumber \\&-\,R_{{t}}R_{{y}}{A}^{2}{x}^{4 }+2\,A_{{y}}R_{{t}}RA{x}^{4}+4\,R_{{y}}{A}^{2}c{x}^{3}-2\,A_{{y}}RAc{x }^{3}\nonumber \\&-\,2\,A_{{y}}R_{{t}}Rc{x}^{2}+4\,R_{{y}}A{c}^{2}x-2\,A_{{y}}R{c}^{2 }x+R_{{t,y}}R{c}^{2}+R_{{t}}R_{{y}}{c}^{2} \Bigr )\qquad \end{aligned}$$
(63)
$$\begin{aligned} \pi= & {} 1/4\,{\frac{\sqrt{-\Lambda \,{R}^{2}+{R_{{t}}}^{2}}}{R}} \end{aligned}$$
(64)
$$\begin{aligned} \frac{\sigma +\lambda }{2}= & {} 1/2\,{\frac{A_{{y}}xc}{-R_{{y}}A{x}^{2}+RA_{{y}}{x}^{2}-R_{{y}}c}} \end{aligned}$$
(65)
$$\begin{aligned} \delta (\sigma -\lambda )= & {} \frac{-\sqrt{-\Lambda \,{R}^{2}+{R_{{t}}}^{2}}\left( A{x}^{2}+c \right) }{4\, \left( -R_{{y}}A{x}^{2}+RA_{{y}}{x}^{2}-R_{{y}}c \right) ^{3}} \Bigl ( -{A}^{2}R_{{y}}R_{{y,t ,y}}{x}^{4}\nonumber \\&+\,R_{{t,y}}{A}^{2}R_{{y,y}}{x}^{4}+RA_{{y}}AR_{{y,t,y}}{x}^{ 4}-R_{{t,y}}RAA_{{y,y}}{x}^{4}\nonumber \\&-\,R_{{t}}A_{{y}}AR_{{y,y}}{x}^{4}+R_{{t}} AR_{{y}}A_{{y,y}}{x}^{4}-2\,AR_{{y}}R_{{y,t,y}}c{x}^{2}\nonumber \\&+\,2\,R_{{t,y}}AR _{{y,y}}c{x}^{2}+RA_{{y}}R_{{y,t,y}}c{x}^{2}-R_{{t,y}}RA_{{y,y}}c{x}^{ 2}\nonumber \\&-\,R_{{t}}A_{{y}}R_{{y,y}}c{x}^{2}+R_{{t}}R_{{y}}A_{{y,y}}c{x}^{2}-R_{ {y}}R_{{y,t,y}}{c}^{2}+R_{{t,y}}R_{{y,y}}{c}^{2} \Bigr ) \end{aligned}$$
(66)
$$\begin{aligned} \delta \delta \Psi _2= & {} \text {196 terms} \end{aligned}$$
(67)
$$\begin{aligned} \Delta \Delta \Psi _2= & {} \frac{1}{24\,{R}^{3} \left( -R_{{y}}A{x}^{2}+RA_{{y}}{x}^{2}-R_{{y}}c \right) ^{3}} \Bigl ( -A_{{y}}c \left( 4\,{A}^{4}{R_{{y}}}^{2}c{x}^{6}\right. \nonumber \\&\left. -2\,{R}^{2}{A}^{3}{R_{ {t,y}}}^{2}{x}^{6}-6\,A_{{y}}R{A}^{3}R_{{y}}c{x}^{6}-2\,R{A}^{3}R_{{t} }R_{{y}}R_{{t,y}}{x}^{6}\right. \nonumber \\&\left. +R_{{t,t}}R{A}^{3}{R_{{y}}}^{2}{x}^{6}-2\,{A}^ {3}{R_{{t}}}^{2}{R_{{y}}}^{2}{x}^{6}+2\,{A_{{y}}}^{2}{R}^{2}{A}^{2}c{x }^{6}\right. \nonumber \\&\left. +6\,A_{{y}}{R}^{2}{A}^{2}R_{{t}}R_{{t,y}}{x}^{6}-3\,A_{{y}}R_{{t, t}}{R}^{2}{A}^{2}R_{{y}}{x}^{6}+6\,A_{{y}}R{A}^{2}{R_{{t}}}^{2}R_{{y}} {x}^{6}\right. \nonumber \\&\left. +2\,{A_{{y}}}^{2}R_{{t,t}}{R}^{3}A{x}^{6}-6\,{A_{{y}}}^{2}{R}^{ 2}A{R_{{t}}}^{2}{x}^{6}+8\,R{A}^{3}R_{{y}}R_{{t,y}}c{x}^{5}\right. \nonumber \\&\left. +12\,{A}^{3 }R_{{t}}{R_{{y}}}^{2}c{x}^{5}-30\,A_{{y}}R{A}^{2}R_{{t}}R_{{y}}c{x}^{5 }+10\,{A_{{y}}}^{2}{R}^{2}AR_{{t}}c{x}^{5}\right. \nonumber \\&\left. +4\,{A}^{3}{R_{{y}}}^{2}{c}^ {2}{x}^{4}-2\,{R}^{2}{A}^{2}{R_{{t,y}}}^{2}c{x}^{4}-18\,A_{{y}}R{A}^{2 }R_{{y}}{c}^{2}{x}^{4}\right. \nonumber \\&\left. -2\,R{A}^{2}R_{{t}}R_{{y}}R_{{t,y}}c{x}^{4}+R_{{ t,t}}R{A}^{2}{R_{{y}}}^{2}c{x}^{4}-2\,{A}^{2}{R_{{t}}}^{2}{R_{{y}}}^{2 }c{x}^{4}\right. \nonumber \\&\left. +8\,{A_{{y}}}^{2}{R}^{2}A{c}^{2}{x}^{4}-2\,{A_{{y}}}^{2}R_{{t ,t}}{R}^{3}c{x}^{4}+6\,{A_{{y}}}^{2}{R}^{2}{R_{{t}}}^{2}c{x}^{4}\right. \nonumber \\&\left. +16\,R {A}^{2}R_{{y}}R_{{t,y}}{c}^{2}{x}^{3}+24\,{A}^{2}R_{{t}}{R_{{y}}}^{2}{ c}^{2}{x}^{3}-8\,A_{{y}}{R}^{2}AR_{{t,y}}{c}^{2}{x}^{3}\right. \nonumber \\&\left. -32\,A_{{y}}RAR _{{t}}R_{{y}}{c}^{2}{x}^{3}+10\,{A_{{y}}}^{2}{R}^{2}R_{{t}}{c}^{2}{x}^ {3}-4\,{A}^{2}{R_{{y}}}^{2}{c}^{3}{x}^{2}\right. \nonumber \\&\left. +2\,{R}^{2}A{R_{{t,y}}}^{2}{c }^{2}{x}^{2}-10\,A_{{y}}RAR_{{y}}{c}^{3}{x}^{2}+2\,RAR_{{t}}R_{{y}}R_{ {t,y}}{c}^{2}{x}^{2}\right. \nonumber \\&\left. -R_{{t,t}}RA{R_{{y}}}^{2}{c}^{2}{x}^{2}+2\,A{R_{{t }}}^{2}{R_{{y}}}^{2}{c}^{2}{x}^{2}+6\,{A_{{y}}}^{2}{R}^{2}{c}^{3}{x}^{ 2}\right. \nonumber \\&\left. -6\,A_{{y}}{R}^{2}R_{{t}}R_{{t,y}}{c}^{2}{x}^{2}+3\,A_{{y}}R_{{t,t}} {R}^{2}R_{{y}}{c}^{2}{x}^{2}-6\,A_{{y}}R{R_{{t}}}^{2}R_{{y}}{c}^{2}{x} ^{2}\right. \nonumber \\&\left. +8\,RAR_{{y}}R_{{t,y}}{c}^{3}x+12\,AR_{{t}}{R_{{y}}}^{2}{c}^{3}x-8 \,A_{{y}}{R}^{2}R_{{t,y}}{c}^{3}x-2\,A_{{y}}RR_{{t}}R_{{y}}{c}^{3}x\right. \nonumber \\&\left. -4 \,A{R_{{y}}}^{2}{c}^{4}+2\,{R}^{2}{R_{{t,y}}}^{2}{c}^{3}+2\,A_{{y}}RR_ {{y}}{c}^{4}+2\,RR_{{t}}R_{{y}}R_{{t,y}}{c}^{3}\right. \nonumber \\&\left. -R_{{t,t}}R{R_{{y}}}^{2 }{c}^{3}+2\,{R_{{t}}}^{2}{R_{{y}}}^{2}{c}^{3}\right) \Bigr ) \end{aligned}$$
(68)
$$\begin{aligned} \Delta \delta \Psi _2=\text {214 terms} \end{aligned}$$
(69)
1.4 Scalar polynomial invariants
Likewise, the explicit polynomial scalar invariants are simplified. Let us simply present the two 0th order invariants from Section 3.1:
$$\begin{aligned} R= & {} \frac{1}{R \left( -R_{{y}}A{x}^{2}+RA_{{y}}{x}^{2}-R_{{y}}c \right) } \Bigl (-\Lambda \,RAR_{{y}}{x}^{2}+\Lambda \,{R}^{2}A_{{y}}{x}^{2}\nonumber \\&+\,2\,A{x}^{2}A _{{y}}c-R_{{t,t}}AR_{{y}}{x}^{2}+2\,A_{{y}}{x}^{2}RR_{{t,t}}-\Lambda \, RR_{{y}}c-2\,A_{{y}}{c}^{2}\nonumber \\&-\,R_{{t,t}}R_{{y}}c \Bigr ) \end{aligned}$$
(70)
$$\begin{aligned} R^{ab} R_{ab}= & {} \frac{1}{2\,{R}^{2} \left( -R_{{y}}A{x}^{2}+A_{{y}}{x}^{2}R-R_{{y}}c \right) ^{2}} \Bigl ( {\Lambda }^{2}{R}^{2}{A}^{2}{R_{{y}}}^{2}{x}^{4}-2\,{\Lambda }^{2}{R}^{3 }A_{{y}}AR_{{y}}{x}^{4}\nonumber \\&+\,{\Lambda }^{2}{R}^{4}{A_{{y}}}^{2}{x}^{4}-4\, \Lambda \,RA_{{y}}{A}^{2}R_{{y}}c{x}^{4}+\Lambda \,R{A}^{2}{R_{{y}}}^{2} R_{{t,t}}{x}^{4}\nonumber \\&+\,4\,\Lambda \,{R}^{2}{A_{{y}}}^{2}Ac{x}^{4}-3\,\Lambda \,{R}^{2}A_{{y}}AR_{{y}}R_{{t,t}}{x}^{4}+2\,\Lambda \,{R}^{3}{A_{{y}}}^ {2}R_{{t,t}}{x}^{4}\nonumber \\&+\,4\,{A_{{y}}}^{2}{A}^{2}{c}^{2}{x}^{4}-2\,A_{{y}}{A }^{2}R_{{y}}R_{{t,t}}c{x}^{4}+{A}^{2}{R_{{y}}}^{2}{R_{{t,t}}}^{2}{x}^{ 4}\!+\!2\,{\Lambda }^{2}{R}^{2}A{R_{{y}}}^{2}c{x}^{2}\nonumber \\&+\,4\,R{A_{{y}}}^{2}AR_{ {t,t}}c{x}^{4}-3\,RA_{{y}}AR_{{y}}{R_{{t,t}}}^{2}{x}^{4}-2\,{\Lambda }^ {2}{R}^{3}A_{{y}}R_{{y}}c{x}^{2}\nonumber \\&+\,3\,{R}^{2}{A_{{y}}}^{2}{R_{{t,t}}}^{2 }{x}^{4}+2\,\Lambda \,RA{R_{{y}}}^{2}R_{{t,t}}c{x}^{2}-4\,\Lambda \,{R}^ {2}{A_{{y}}}^{2}{c}^{2}{x}^{2}\nonumber \\&-\,3\,\Lambda \,{R}^{2}A_{{y}}R_{{y}}R_{{t, t}}c{x}^{2}-8\,{A_{{y}}}^{2}A{c}^{3}{x}^{2}+2\,A{R_{{y}}}^{2}{R_{{t,t} }}^{2}c{x}^{2}+{\Lambda }^{2}{R}^{2}{R_{{y}}}^{2}{c}^{2}\nonumber \\&-\,4\,R{A_{{y}}}^ {2}R_{{t,t}}{c}^{2}{x}^{2}-3\,RA_{{y}}R_{{y}}{R_{{t,t}}}^{2}c{x}^{2}+4 \,\Lambda \,RA_{{y}}R_{{y}}{c}^{3}\nonumber \\&+\,\Lambda \,R{R_{{y}}}^{2}R_{{t,t}}{c}^ {2}+4\,{A_{{y}}}^{2}{c}^{4}+2\,A_{{y}}R_{{y}}R_{{t,t}}{c}^{3}+{R_{{y}} }^{2}{R_{{t,t}}}^{2}{c}^{2} \Bigr ) \end{aligned}$$
(71)
Using these expressions and those above, we verify that (21) holds in the special case. The higher order polynomial invariants can also be calculated in the special case. However, as one may imagine from the expressions for the 1st and 2nd order Cartan invariants above, the expressions are somewhat unwieldy.