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On scalar curvature invariants in three dimensional spacetimes

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Abstract

We wish to construct a minimal set of algebraically independent scalar curvature invariants formed by the contraction of the Riemann (Ricci) tensor and its covariant derivatives up to some order of differentiation in three dimensional (3D) Lorentzian spacetimes. In order to do this we utilize the Cartan–Karlhede equivalence algorithm since, in general, all Cartan invariants are related to scalar polynomial curvature invariants. As an example we apply the algorithm to the class of 3D Szekeres cosmological spacetimes with comoving dust and cosmological constant \(\Lambda \). In this case, we find that there are at most twelve algebraically independent Cartan invariants, including \(\Lambda \). We present these Cartan invariants, and we relate them to twelve independent scalar polynomial curvature invariants (two, four and six, respectively, zeroth, first, and second order scalar polynomial curvature invariants).

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Notes

  1. Typically the maximum order of differentiation is written as \(q=p+1\) where p denotes the iteration of the CKA where the number of functionally independent invariants reaches a maximum and the dimension of the isotropy group of the nth covariant derivative of the curvature tensor for \(n> p\) reaches its minimum.

  2. The question of the maximal order of covariant derivative required for the invariant classification of a n-dimensional pseudo-Riemannian manifold \((\eta , M)\) is relevant in determining the worst case scenarios for implementing the Cartan–Karlhede equivalence algorithm. Cartan established the theoretical upper-bound to be \(q \le n(n+1)/2\); that is, the dimension of \(O(\eta , M)\). Karlhede improved the estimation of the upper bound to be \(q \le n+N_0+1 < \dim O(\eta , M)\). For the 3D Lorentzian manifolds, this implies the upper-bound is at most five as \(\dim O(\eta , M)= \dim O(1,2)=6\).

  3. These types correspond to the Segre type \(\{(11),1\}, \{1(1,1)\}\) and \( \{(21)\}\) for the traceless Ricci tensors, respectively.

  4. Here we denote the isotropy group of the n-th covariant derivative of the curvature tensor as \(H_n\).

  5. We will choose to write the spin coefficients only in terms of \(\gamma \), \(\sigma -\lambda \), \(\pi \), \(\sigma +\lambda \).

  6. What we are calling the normal forms of the Ricci tensor, Sousa et al. [23] refer to as canonical forms.

  7. We will call invariants constructed from combinations of Cartan invariants of different orders extended Cartan invariants.

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Acknowledgments

We would like to thank Malcolm MacCallum and Robert Milson for helpful comments. This research was supported by the Natural Sciences and Engineering Research Council of Canada and by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

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Correspondence to N. K. Musoke.

Appendices

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(ty), (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.

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Musoke, N.K., McNutt, D.D., Coley, A.A. et al. On scalar curvature invariants in three dimensional spacetimes. Gen Relativ Gravit 48, 27 (2016). https://doi.org/10.1007/s10714-016-2022-9

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