Algebraic invariants of the Riemann tensor in a four-dimensional Lorentzian space

Abstract

Abstract. In a recent comment Sneddon discussed the set of fourteen algebraic invariants of the Riemann curvature tensor in four dimensions. The focus was rectification of an error (in the form of lack of independence) in an earlier construction and the presentation of a corrected set suitable for application. Several authors who have worked on this problem were mentioned. The comment, however, did not mention the work of Narlikar and Karmarkar who presented a set of invariants well before the earliest work cited in the comment. The original publication by Narlikar and Karmarkar may not be readily available so we list and make a few comments on their set.Recently, Sneddon (1986) discussed briefly the form in which several authors have presented the fourteen independent algebraic invariants (Haskins 1902)$ of the Riemann curvature tensor in four dimensions. Among the authors mentioned were GChCniau and Debever (1956a, b, c) with implied priority of publication. This accords with common usage in which these scalars are almost universally referred to as the 'GChCniau-Debever' scalars. The fact is that Narlikar and Karmarkar (1948) published such a set substantially earlier.

A minimal set of invariants as a systematic approach to higher order gravity models

“…There has been some pioneering work by [25,33,34,35,36,37,38] and then a renewed interest by several recent papers [26,27,39,40]. This long series of studies resulted in theorems about minimal sets of invariants and classifications of spacetime manifolds.…”

“…This long series of studies resulted in theorems about minimal sets of invariants and classifications of spacetime manifolds. It was shown for example in [26,27] that there are at most 14 independent real algebraic curvature invariants in a 4-dimensional Lorentzian space. The number of independent invariants depends on the symmetries of the spacetime as delineated by the Petrov classification [29,30,31,32] and also the algebraic type of the Ricci tensor as for example described by the Segre classification [28,32]).…”

“…It is worth clarifying that Petrov and Segre classifications depend on the symmetries of the space-time metric under consideration and were originally derived without reference to source fluids. Indeed, most papers as for example [25,27,35,36,39] derived and discussed minimum sets of invariants in terms of the algebraic Segre and Petrov types without reference to source fluids, while [26] considered the type of source fluid in their discussion as obtained via the Einstein field equations. Since we study here modified gravity models where the field equations are not anymore the Einstein equations, we thus consider the minimum sets from the point of view of the algebraic Segre and Petrov classifications of the space-time manifold.…”

“…Since we study here modified gravity models where the field equations are not anymore the Einstein equations, we thus consider the minimum sets from the point of view of the algebraic Segre and Petrov classifications of the space-time manifold. Now, as it was derived in original papers on curvature invariants, the elements of the basis can be built from contractions of the trace-free Ricci tensor plus those of the Weyl tensor above, C αβγδ , as given by equation (4) and the complex conjugate of its self-dual tensor, i.e.C αβγδ [26,27].…”

“…We explore an idea based on theorems from the theory of invariants in general relativity [25,26,27] in order to develop a systematic framework that allows one to substantially reduce this very large class of model to a systematic list (taxonomy) of models. The idea is that curvature invariants are not independent from each other and, for a given algebraic type of the Ricci tensor (see for example the Segre classification [28,32]) and a given Petrov type of the Weyl tensor (i.e.…”