Algebraic properties are explored for the curvature tensors of Riemannian manifolds, using the irreducible decomposition of curvature tensors. Our method provides a powerful tool to analyze the irreducible basis as well as an algorithm to determine the linear dependence of arbitrary Riemann polynomials. We completely specify 13 independent basis elements for the quartic scalars and explicitly find 13 linear relations among 26 scalar invariants. Our method provides several completely new results, including some clues to identify 23 independent basis elements from 90 quintic scalars, that are difficult to find otherwise.
The ’t Hooft symbols, dubbed by physicists, were first introduced in [45, 46] as quantities (dubbed “tensor rotors”) connecting a self-dual bivector to a vector in, so-called, “rotor space” and an anti-self-dual bivector to a complex conjugated rotor. See also [47]. The ’t Hooft symbols have played an important role for instanton physics in Yang-Mills gauge theory [51].
There may be a subtlety for pseudo-tensors since the Wick rotation of the Levi-Civita tensor will introduce an extra imaginary factor \(i=\sqrt{-1}\). However, since the pseudo-tensor contains only an odd number of the Levi-Civita tensors, such a subtlety can be easily dealt with.
Therefore, the result for an Einstein manifold can easily be converted to the result for the Weyl tensor of a general Riemannian manifold using the relation (3.17) which can be written as \(W_{abcd} = R^E_{abcd} - \frac{1}{12} R (\delta _{ac} \delta _{bd} - \delta _{ad} \delta _{bc})\) where \(R^E_{abcd}\) is the Riemann tensor given by (3.25). See, for example, Eqs. (3.21) and (3.22).
Our approach can also be applied to pseudo-tensors. For example, we get
However, pseudo-tensors also have as many basis elements as normal tensors, and systematically classifying them as in Ref. [35] is a new problem. We leave this issue as a problem addressed in the future. But, it would be possible to know how to generate a pseudo-scalar basis that is parity odd, as we will discuss later.
We found several errors in Ref. [50]. These are corrected in “Appendix D”. Rectification of such errors played a crucial role in finding the correct independent basis.
There is a subtle element: \(Q_5^{\text {odd}} = \text {Tr} \left( B^T B A_+ B A_- + B B^T A_- B^T A_+ \right) \). This is parity even in itself but the odd power of the matrix B. If this combination were allowed at the quintic order, a similar term \( \text {Tr} \left( B^T B A_+ B + B^T A_- B^T B \right) \) would also be allowed at the quartic order. But this term did not appear at the quartic order. So we rule out \(Q_5^{\text {odd}}\). The result in Table 3 alludes that a term of odd powers of B is only possible in the form of \(\textrm{det}B\) because \(\textrm{det}B = \textrm{det}B^T\).
Therefore, Eqs. (D.6) and (5.10) can be rewritten as the identity
for any symmetric \(3 \times 3\) matrix \(A_\pm \) with \(R = 8 \text {Tr} A_+ = 8 \text {Tr} A_-.\)
Note that the result in [35] was missing the parity odd elements. We have discussed above how these can be restored.
Using the matrix notation (4.26), \(\textrm{det}f^{ij}_{(+-)}\) can be written as \(\textrm{det}B = \frac{1}{3} \text {Tr} B^3 - \frac{1}{2} \text {Tr} B \, \text {Tr} B^2 + \frac{1}{6} \big (\text {Tr} B \big )^3\). This expression appeared in Eq. (4) of Ref. [30] and was used to reduce independent invariants.
The expression for these 14 basis elements is quite complicated and not illuminating, so will not be displayed here. But we can e-mail the result to those who request it.
We used Mathematica (www.wolfram.com) and the add-on package MathSymbolica (www.mathsymbolica.com) for calculation of the quartic Riemann monomials, \(A \sim Z\), and the linear relations, \((a) \sim (l)\). We verified the results by numerical tests using randomly generated numbers for the coefficients \(f^{ij}_{(\pm \pm )}\) and \(f^{ij}_{(\pm {{\mp }})}\), and hence, we believe that our results are error-free.
Eq. (19) in Ref. [50] was derived using Eq. (18), but Eq. (18) contains typographic errors. The rectified version must read as \(- 2 R^2 R_{abcd}^2 \rightarrow + 2 R^2 R_{abcd}^2\) and \(-\frac{64}{3} R R_{ab} R_{bc} R_{ca} \rightarrow + \frac{64}{3} R R_{ab} R_{bc} R_{ca}\).
References
Gautreau, R., Anderson, J.L.: Phys. Lett. A 25, 291 (1967)
McIntosh, C.G.B., Foyster, J.M., Lun, A.W.-C.: The classification of the Ricci and Plebański tensors in general relativity using Newman-Penrose formalism. J. Math. Phys. 22, 2620 (1981)
Ehler, J., Kundt, W.: Exact solutions of the gravitational field equations. In: Witten, L. (ed.) Gravitation: An Introduction to Current Research. John Wiley & Sons, New York (1962)
Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations, 2nd edn. Cambridge University Press, Cambridge (2003)
Gibbons, G.W.: Quantum field theory in curved spacetime. In: Hawking, S.W., Isreal, W. (eds.) General Relativity: An Einstein centenary survey. Cambridge University Press, Cambridge (1979)
DeWitt, B.S.: Quantum gravity: the new synthesis. In: Hawking, S.W., Israel, W. (eds.) General Relativity: An Einstein centenary survey. Cambridge University Press, Cambridge (1979)
Debever, R.: Étude géométrique du tenseur de Riemann-Christoffel des espaces de Riemann à quatre dimensions. Bull. Acad. R. Belg. Cl. Sc. XLII, 313, 608 (1956)
Fulling, S.A., King, R.C., Wybourne, B.G., Cummins, C.J.: Normal forms for tensor polynomials: I. The Riemann tensor. Class. Quantum Grav. 9, 1151 (1992)
Carminati, J., Zakhary, E., McLenaghan, R.G.: On the problem of algebraic completeness for the invariants of the Riemann tensor: II. J. Math. Phys. 43, 1474 (2002)
Jack, I., Parker, L.: Linear independence of renormalization counterterms in curved space-times of arbitrary dimensionality. J. Math. Phys. 28, 1137 (1987)
This research was performed using Mathematica (www.wolfram.com) and the add-on package MathSymbolica (www.mathsymbolica.com). This work was supported by the National Research Foundation of Korea (NRF) with grant number NRF-2018R1D1A1B0705011314 (HSY). We acknowledge the hospitality at APCTP where part of this work was done.
Author information
Authors and Affiliations
School of Electrical Engineering and Computer Science, Gwangju Institute of Science and Technology, Gwangju, 61005, Korea
Youngjoo Chung
Division of Liberal Arts and Sciences, Gwangju Institute of Science and Technology, Gwangju, 61005, Korea
Chi-Ok Hwang
Department of Physics and Photon Science, Gwangju Institute of Science and Technology, Gwangju, 61005, Korea
Here we will analytically prove the identities (4.1) and (4.2) using the decomposition (3.25). In order to simplify the calculation, it is convenient to choose pairs with the most contractions and perform the calculation with them. So we will show (4.1) by choosing the following pair:
and a similar formula holds for the matrix \([{\bar{\eta }}^{i_1} \eta ^{j_1} {\bar{\eta }}^{i_2} \eta ^{j_2}]_{c_1 e_2}\). Now it is easy to see that Eq. (B.1) identically vanishes when recalling Eq. (3.12) and the fact that \([\eta ^{i} {\bar{\eta }}^{j} ]_{c_1 e_2}\) is a symmetric matrix by Eq. (A.7). Although it is straightforward, the proof of Eq. (4.2) by hand requires a little algebra:
where we have used the identity (3.13). Combining all these terms leads to the identity (4.2).
Appendix C: Cubic identities for a general Riemannian manifold
The identity (4.22) becomes trivial for an Einstein manifold which satisfies the equation \(R_{ab} = \frac{1}{4} R \delta _{ab}\) and the identity (4.23) reduces to
Therefore, in order to prove that the identities (4.22) and (4.23) are the only linear relations existent in the cubic terms, we will consider a general Riemannian manifold and represent the eight cubic monomials using the decomposition (3.11):
One can see that the above 7 monomials are expressed in terms of only 5 types of terms besides the term \(R^3\), listed below. All the cubic monomials are symmetric under the interchange (3.23) since they are scalars. The six independent basis elements of the cubic Riemann monomials are given by
where \(\textrm{det}f^{ij}_{(+-)}\) is the determinant of the \(3 \times 3\) matrix \(f^{ij}_{(+-)}\).Footnote 9 The first three basis elements in (C.2) correspond to the quadratic ones (4.3) multiplied by R. The last basis in (C.2) consists of three terms which are self-symmetric under the parity transformation (3.23). Thus they could form independent basis separately, but they must appear with that combination because they come from the product of Ricci tensors, \(R_{ab} R_{bc} R_{ca}\). It is also clear from the fact that such combination forms \(\textrm{det}f^{ij}_{(+-)}\). This proves that the cubic scalars have only six independent basis elements. Replacing the 5 types of terms by the cubic monomials leads to the identities (4.22) and (4.23). There is no more linear relation. Our approach identifies the independent basis of Riemann monomials without any ambiguity.
Now we list the result for the expansion of second-rank cubic monomials in Table 1. For notational simplicity, we will use the matrix notation in (4.26):
Note that the second-rank cubic tensors in Table 1 are parity even, so they must be invariant under the parity transformation (4.27). One can quickly check it by noting that the parity transformation acts on \(P: \big ( \eta ^{i_1} {\bar{\eta }}^{i_2} \big )_{ab} \rightarrow \big ( {\bar{\eta }}^{i_1} \eta ^{i_2} \big )_{ab} = \big ( \eta ^{i_2} {\bar{\eta }}^{i_1} \big )_{ab}\). The second-rank cubic monomials in Table 1 are symmetric with respect to \((a \leftrightarrow b)\) except in G and J which contain an anti-symmetric part as was shown above. Although the expansion is much more complicated compared to the scalar case, it turns out that terms that would appear in the scalar case reappear with the factor \(\frac{1}{4} \delta _{ab}\), and novel terms that do not appear in the scalar case (or should disappear in the scalar case) appear with the factor \((\eta ^i {\bar{\eta }}^j)_{ab}\). Indeed, \(\delta _{ab}\) and \((\eta ^i {\bar{\eta }}^j)_{ab}\) are the only symmetric tensors constructed from the product of ’t Hooft symbols. The latter is trace-free due to Eq. (A.5). In particular, \((\eta ^i {\bar{\eta }}^i)_{ab}\) is a diagonal matrix given by
It is natural to expect that there exist more than two linear relations between the second-rank cubic tensors in Table 1 since contracting the free indices a and b has to reproduce the eight cubic scalars which obey the syzygy relations (4.22) and (4.23). However we found that there exist 14 linearly independent second-rank cubic tensors using a computer algorithm.Footnote 10 This means that Eqs. (C.4) and (C.5) exhaust all possible linear relations for the second-rank cubic tensors in Table 1. It also implies that Eqs. (C.4) and (C.5) have to reproduce the syzygies (4.22) and (4.23) when contracting a and b. It can be checked by eliminating \(R_{a e c g} R_{b f d g} R_{c d e f}\) and \(R_{a c} R_{b e d f} R_{c e d f}\) from Eqs. (C.4) and (C.5), respectively, and then combining them together to obtain
$$\begin{aligned}{} & {} R^2 R_{a b} - 2 R R_{a c} R_{b c} + 4 R_{a c} R_{b d} R_{c d} - 4 R_{a b} R_{c d}^2 - 2 R R_{c d} R_{a c b d} + 4 R_{c e} R_{d e} R_{a c b d} \nonumber \\{} & {} \quad + 4 R_{a c} R_{d e} R_{b e c d} + R_{a b} R_{c d e f}^2 + 4 R_{e f} R_{a c b d} R_{c e d f} \nonumber \\{} & {} \quad - 2 R_{a c} R_{b e d f} R_{c e d f} + 2 R_{a e b g} R_{c d e f} R_{c d f g} = 0, \end{aligned}$$
(C.6)
$$\begin{aligned}{} & {} 3 R^2 R_{a b} - 4 R R_{a c} R_{b c} + 8 R_{a c} R_{b d} R_{c d} - 12 R_{a b} R_{c d}^2 - 4 R R_{c d} R_{a c b d} + 8 R_{c e} R_{d e} R_{a c b d} \nonumber \\{} & {} \quad - 2 R R_{a e c d} R_{b e c d} + 8 R_{e f} R_{a c d e} R_{b c d f}\nonumber \\{} & {} \quad + 4 R_{e f} R_{a e c d} R_{b f c d} + 3 R_{a b} R_{c d e f}^2 + 8 R_{e f} R_{a c b d} R_{c e d f} \nonumber \\{} & {} \quad - 4 R_{a g c d} R_{b g e f} R_{c d e f} + 8 R_{a e c g} R_{b f d g} R_{c d e f} + 4 R_{a e b g} R_{c d e f} R_{c d f g} = 0. \end{aligned}$$
(C.7)
Combining the above equations after contracting a and b reproduces the cubic identities (4.22) and (4.23).
Appendix D: Quartic identities for a general Riemannian manifold
We present the expansion of the 26 quartic monomials in Table 2 using the decomposition (3.11) for a general Riemannian manifold:
Since (D.1) is odd under the parity transformation (3.23), its vanishing can be easily understood from this property. This means that such terms can be written as \(- \frac{1}{4} R \, \textrm{det}f^{ij}_{(+-)}\). It is interesting to compare this expression for \(\textrm{det}f^{ij}_{(+-)}\) with the result in Eq. (C.2). Using the identity (D.1), it is straightforward to identity the basis elements for the quartic monomials in Table 2 which superficially look independent. The result is summarized in Table 5. The first 6 basis elements, from I to VI, are simply coming from the cubic ones (C.2) multiplied by R. The remaining 8 elements are newly generated in the quartic order.
We will show that the 26 quartic monomials in Table 2 can be represented by using only 13 linearly independent basis elements. This means that the 14 basis elements listed in Table 5 are not completely independent. We will explain later how one linear relation among the quartic scalars arises. This implies that there must be 13 linear relations among the quartic monomials. A. Harvey found 6 such relations using the fact [50] that an \((n+1)\) index object anti-symmetrized on an n-dimensional manifold vanishes identically. However his results contain several errors and we will correct them here. We find that there are 7 more linear relations in addition to the 6 relations found by Harvey. We can find such linear relations by replacing the quartic basis elements in Table 5 in favor of the corresponding quartic monomials in Table 2. We first list the 12 linear relations between the Riemann monomials in Table 2.
$$\begin{aligned}{} & {} (a): \; -R^4 + 9 R^2 R_{ab} R_{ab} - 14 R R_{ab} R_{bc} R_{ca} - 6 \big ( R_{ab} R_{ab} \big )^2 + 12 R_{ab} R_{bc} R_{cd} R_{da} \\{} & {} \quad - 6 R R_{ab} R_{cd} R_{acbd} + 12 R_{ab} R_{ce} R_{ed} R_{acbd} = 0, \\{} & {} (b): \; - \frac{R^4}{4} + 2 R^2 R_{a b} R_{a b} -2 R R_{a b} R_{b c} R_{c a} - 2 R R_{a b} R_{c d} R_{a c b d}\\{} & {} \quad - \frac{1}{4} R^2 R_{a b c d} R_{a b c d} + R R_{a b} R_{a c d e} R_{b c d e} = 0, \\{} & {} (c): \; - 2 R^4 + 15 R^2 R_{a b} R_{a b} - 16 R R_{a b} R_{b c} R_{c a} \\{} & {} \quad -12 R R_{a b} R_{c d} R_{a c b d} -3 R_{a b} R_{a b} R_{c d e f} R_{c d e f} \\{} & {} \quad + 12 R_{a b} R_{b c} R_{a d e f} R_{c d e f} = 0, \\{} & {} (d): \; -\frac{11}{2} R^4 + \frac{87}{2} R^2 R_{a b} R_{a b} - 56 R R_{a b} R_{b c} R_{c a} -18 \big (R_{ab} R_{ab} \big )^2 + 36 R_{a b} R_{b c} R_{c d} R_{d a} \\{} & {} \quad - 18 R R_{a b} R_{c d} R_{a c b d} - \frac{3}{2} R^2 R_{a b c d} R_{a b c d} + \frac{3}{2} R_{a b} R_{a b} R_{c d e f} R_{c d e f} + 12 R_{a b} R_{c d} R_{a e c f} R_{b e d f} \\{} & {} \quad - 12 R_{a b} R_{c d} R_{a e b f} R_{c e d f} + 6 R_{a b} R_{c d} R_{a c e f} R_{b d e f} = 0, \\{} & {} (e): \; \frac{5}{4} R^4 - 9 R^2 R_{a b} R_{a b} + 8 R R_{a b} R_{b c} R_{c a} + 6 R R_{a b} R_{c d} R_{a c b d} + \frac{3}{4} R^2 R_{a b c d} R_{a b c d} \\{} & {} \quad - R R_{a b c d} R_{c d e f} R_{e f a b} + 2 R R_{a b c d} R_{a e c f} R_{b e d f} = 0, \\{} & {} (f): \; -2 R^4 + 15 R^2 R_{a b} R_{a b} -28 R R_{a b} R_{b c} R_{c a} +24 R_{a b} R_{b c} R_{c d} R_{d a} -3 R_{a b} R_{a b} R_{c d e f} R_{c d e f} \\{} & {} \quad - 24 R_{a b} R_{c d} R_{a e b f} R_{c e d f} + 12 R_{a b} R_{a c b d} R_{c e f g} R_{d e f g} = 0, \end{aligned}$$
$$\begin{aligned}{} & {} (g): \; -\frac{R^4}{2} + 3 R^2 R_{a b} R_{a b} - 8 R R_{a b} R_{b c} R_{c a} + 4 R R_{a b} R_{c d} R_{a c b d}\\{} & {} \quad + 8 R_{a b} R_{b c} R_{c d} R_{d a} + \frac{1}{2} R^2 R_{a b c d} R_{a b c d}\\{} & {} \quad -8 R_{a b} R_{c d} R_{a e b f} R_{c e d f} - R_{a b} R_{a b} R_{c d e f} R_{c d e f} \\{} & {} \quad - 4 R_{a b} R_{c d} R_{a c e f} R_{b d e f} - R R_{a b c d} R_{c d e f} R_{e f a b} \\{} & {} \quad + 4 R_{a b} R_{c d e f} R_{c d g a} R_{e f g b} = 0, \\{} & {} (h): \; 4 R^4 - 30 R^2 R_{a b} R_{a b} +32 R R_{a b} R_{b c} R_{c a} + 6 \left( R_{a b} R_{a b}\right) ^2 -12 R_{a b} R_{b c} R_{c d} R_{d a} \\{} & {} \quad + 18 R R_{a b} R_{c d} R_{a c b d} + \frac{3}{2} R^2 R_{a b c d} R_{a b c d} - 6 R_{a b} R_{c d} R_{a c e f} R_{b d e f} \\{} & {} \quad -\frac{3}{2} R R_{a b c d} R_{c d e f} R_{e f a b} + 12 R_{a b} R_{c e d f} R_{a d c g} R_{b f e g} = 0, \end{aligned}$$
$$\begin{aligned}{} & {} (i): \;-5 R^4 + 36 R^2 R_{a b} R_{a b} - 64 R R_{a b} R_{b c} R_{c a} \\{} & {} \quad + 48 R_{a b} R_{b c} R_{c d} R_{d a} - 48 R_{a b} R_{c d} R_{a e b f} R_{c e d f} \\{} & {} \quad -3 \left( R_{a b c d} R_{a b c d}\right) ^2 + 12 R_{a b c d} R_{a b c e} R_{d h f g } R_{e h f g} = 0, \\{} & {} (j): \; R^4 - 6 R^2 R_{a b} R_{a b} + 16 R R_{a b} R_{b c} R_{c a} \\{} & {} \quad - 16 R_{a b} R_{b c} R_{c d} R_{d a} - 8 R R_{a b} R_{c d} R_{a c b d} - \frac{1}{2} R^2 R_{a b c d} R_{a b c d} \\{} & {} \quad +4 R_{a b} R_{c d} R_{a c e f} R_{b d e f} +16 R_{a b} R_{c d} R_{a e b f} R_{c e d f} \\{} & {} \quad + R R_{a b c d} R_{c d e f} R_{e f a b} + \frac{1}{2} \left( R_{a b c d} R_{a b c d}\right) ^2 \\{} & {} \quad + 4 R_{a b c d} R_{a b e f} R_{c g e h} R_{d g f h} - R_{a b c d} R_{c d e f} R_{e f g h} R_{g h a b} - 2 R_{a b c d} R_{a b e f} R_{c e g h} R_{g h d f} = 0, \\{} & {} (k): \; -\frac{23}{6} R^4 + 28 R^2 R_{a b} R_{a b} -\frac{128}{3} R R_{a b} R_{b c} R_{c a} -8 \left( R_{a b} R_{a b}\right) ^2 + 32 R_{a b} R_{b c} R_{c d} R_{d a} \\{} & {} \quad - R^2 R_{a b c d} R_{a b c d} + 4 R_{a b} R_{a b} R_{c d e f} R_{c d e f} - 32 R_{a b} R_{c d} R_{a e b f} R_{c e d f} -\frac{3}{2} \left( R_{a b c d} R_{a b c d}\right) {}^2 \\{} & {} \quad + 8 R_{a c b d} R_{c e d f} R_{e g f h} R_{a g b h} + R_{a b c d} R_{c d e f} R_{e f g h} R_{g h a b} = 0, \\{} & {} (l): \; \frac{17}{12} R^4 - 12 R^2 R_{a b} R_{a b} - \frac{128}{3} R R_{a b} R_{b c} R_{c a} - 6 \left( R_{a b} R_{a b}\right) ^2 + 84 R_{a b} R_{b c} R_{c d} R_{d a} \\{} & {} \quad + 72 R R_{a b} R_{c d} R_{a c b d} + \frac{5}{2} R^2 R_{a b c d} R_{a b c d} -20 R_{a b} R_{c d} R_{a c e f} R_{b d e f} + 8 R_{a b} R_{a b} R_{c d e f} R_{c d e f} \\{} & {} \quad - 104 R_{a b} R_{c d} R_{a e b f} R_{c e d f} - 6 R R_{a b c d} R_{c d e f} R_{e f a b} -\frac{13}{4} \left( R_{a b c d} R_{a b c d}\right) ^2 \\{} & {} \quad + \frac{13}{2} R_{a b c d} R_{c d e f} R_{e f g h} R_{g h a b} - 3 R_{a b c d} R_{a b e f} R_{c e g h} R_{g h d f} + 32 R_{a c b d} R_{a e b f} R_{c h e g } R_{d h f g} = 0. \end{aligned}$$
Our results \((a), \, (b)\) and (c) precisely reproduce Eqs. (11), (9), and (10) in Ref. [50], respectively.Footnote 11 Unfortunately, Eqs. (15), (17), (18), and (19) in Ref. [50] contain errors too.Footnote 12 For example, Eq. (15) in Ref. [50] should read as
$$\begin{aligned}{} & {} -\frac{5}{4} R^4 + \frac{39}{4} R^2 R_{a b} R_{a b} - 12 R R_{a b} R_{b c} R_{c a} - 3 \big (R_{ab} R_{ab} \big )^2 + 6 R_{a b} R_{b c} R_{c d} R_{d a} \nonumber \\{} & {} \quad - 5 R R_{a b} R_{c d} R_{a c b d} - \frac{1}{4} R^2 R_{a b c d} R_{a b c d} + \frac{1}{4} R_{a b} R_{a b} R_{c d e f} R_{c d e f} + 2 R_{a b} R_{b c} R_{a d e f} R_{c d e f} \nonumber \\{} & {} \quad + 2 R_{a b} R_{c d} R_{a e c f} R_{b e d f} - 2 R_{a b} R_{c d} R_{a e b f} R_{c e d f} + R_{a b} R_{c d} R_{a c e f} R_{b d e f} = 0. \end{aligned}$$
(D.2)
In particular, Eq. (15) in Ref. [50] was missing the term, \(- \frac{1}{4} R^2 R_{a b c d} R_{a b c d}\). The correct equation (D.2) can simply be obtained by adding (c) and (d) and dividing by 6. Since Eq. (17) in Ref. [50] used the incorrect equation (15), it is incorrect too. The correct forms of Eqs. (17) and (19) are given by, respectively,Footnote 13
$$\begin{aligned}{} & {} \text {(17)}: \; R^4 - 6 R^2 R_{a b} R_{a b} + 5 R R_{a b} R_{b c} R_{c a} + 6 R_{a b} R_{c e} R_{e d} R_{a c b d} + \frac{3}{2} R R_{a b} R_{c d e a} R_{c d e b} \nonumber \\{} & {} \quad + 3 R_{a b} R_{b c a d} R_{e f g c} R_{e f g d} - 3 R_{a b} R_{c d e f} R_{e f g a} R_{g b c d}\nonumber \\{} & {} \quad + 6 R_{a b} R_{c e d f} R_{e g f a} R_{g c b d} = 0, \end{aligned}$$
(D.3)
$$\begin{aligned}{} & {} \text {(19)}: \; - \frac{5}{48} R^4 + \frac{1}{2} R^2 R^2_{a b} -\frac{1}{3} R R_{a b} R_{b c} R_{c a} + \frac{1}{16} R_{abcd}^4 - R_{a b} R_{c d e f} R_{e f g a} R_{g b c d} \nonumber \\{} & {} \quad \; + 2 R_{ab} R_{c e d f} R_{e g f a} R_{g c b d} + R_{a b} R_{b c a d} R_{e f g c} R_{g d e f} + R_{a c b d} R_{c e d f} R_{e g f h} R_{a g b h} \nonumber \\{} & {} \quad \; - 2 R_{a b c d} R_{a b e f} R_{c g e h} R_{d g f h} - 2 R_{a c b d} R_{a e b f} R_{c h e g} R_{d h f g}\nonumber \\{} & {} \quad + \frac{1}{8} R_{a b c d} R_{c d e f} R_{e f g h} R_{g h a b} \nonumber \\{} & {} \quad \; + R_{a b c d} R_{a b e f} R_{c e g h} R_{g h d f} - R_{a b c d} R_{a b c e} R_{d h f g} R_{f g e h} = 0. \end{aligned}$$
(D.4)
Eq. (D.3) can be reproduced from our results by considering the combination, \(\frac{1}{4} (f) - \frac{3}{4} (g) + \frac{1}{2} (h) + \frac{1}{2} (a) + \frac{3}{2}(b)\). Using 12 linear equations \((a) \sim (l)\), Eq. (D.4) can be further reduced as
$$\begin{aligned}{} & {} - \frac{1}{4} R^4 + R^2 R_{a b} R_{a b} + R_{ab}^4 - 2 R_{a b} R_{b c} R_{c d} R_{d a} \nonumber \\{} & {} \quad - 2 R_{a b} R_{c d} R_{e f a c} R_{e f b d} + 2 R_{a c b d} R_{c e d f} R_{e g f h} R_{g a h b} \nonumber \\{} & {} \quad - 4 R_{a c b d} R_{e a f b} R_{f g d h} R_{g e h c} + \frac{3}{2} R_{a b c d} R_{a b e f} R_{c e g h} R_{g h d f} \nonumber \\{} & {} \quad - R_{a b c d} R_{a b c e} R_{f g h d} R_{h e f g} = 0. \end{aligned}$$
(D.5)
Using the notation (4.26), Eqs. (D.4) and (D.5) are equally written as
The reason why the matrix B does not appear in (D.6) is that Eqs. (D.4) and (D.5) originated from a quartic polynomial of Weyl tensors. The validity of the linear relation (D.6) may be understood by the fact that the four elements, \(III, \, IV, \, X\) and XIV, in Table 5 can be shown to be implicitly connected by this relationship. We have confirmed that Eq. (D.4) cannot be derived as a linear combination of the 12 equations, \((a) \sim (l)\), by checking all possible cases out of 26 terms. This means that Eq. (D.4) must be regarded as a new linear relation in addition to the 12 relations, \((a) \sim (l)\). It is quite revealing to see that Eq. (D.6) is an identity for general symmetric \(3 \times 3\) matrices \(A_\pm \) satisfying the property \(\text {Tr} A_+ = \text {Tr} A_- = \frac{R}{8}\). It can be most easily checked in a diagonalized frame such that \(A_\pm = \text {diag} (a^1_\pm , a^2_\pm , a^3_\pm )\) and \(a^1_+ + a^2_+ + a^3_+ = a^1_- + a^2_- + a^3_-\). Since Eq. (D.6) provides one more linear relation among the quartic basis elements in Table 5, the number of linearly independent basis elements for quartic scalars is 13.
Note added. The above linear relations, (D.4), (D.5), and (D.6), have been derived from the identity \({R^{ab}}_{[ab} {R^{cd}}_{cd} {R^{ef}}_{ef} {R^{gh}}_{gh]} = 0\) whose expansion is very complicated [50]. But we found that Eq. (D.6) can be rewritten as \(P- 6\,S + 3X = R R_{abcd} R_{cedf} R_{eafb} - 6 R_{ab} R_{cedf} R_{egfa} R_{gcbd} + 3 R_{abcd} R_{efab} R_{gche} R_{gdhf} =0\) using only the three members in Table 2.
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