In this paper we determine the class of four-dimensional Lorentzian manifolds that can be completely characterized by the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. We introduce the notion of an I-non-degenerate spacetime metric, which implies that the spacetime metric is locally determined by its curvature invariants. By determining an appropriate set of projection operators from the Riemann tensor and its covariant derivatives, we are able to prove a number of results (both in the algebraically general and in algebraically special cases) of when a spacetime metric is I-nondegenerate. This enables us to prove our main theorem that a spacetime metric is either I-non-degenerate or a Kundt metric. Therefore, a metric that is not characterized by its curvature invariants must be of Kundt form. We then discuss the inverse question of what properties of the underlying spacetime can be determined from a given a set of scalar polynomial invariants, and some partial results are presented. We also discuss the notions of strong and weak non-degeneracy.

We study Lorentzian spacetimes for which all scalar invariants constructed from the Riemann tensor and its covariant derivatives are constant (CSI spacetimes). We obtain a number of general results in arbitrary dimensions. We study and construct warped product CSI spacetimes and higher-dimensional Kundt CSI spacetimes. We show how these spacetimes can be constructed from locally homogeneous spaces and V SI spacetimes. The results suggest a number of conjectures. In particular, it is plausible that for CSI spacetimes that are not locally homogeneous the Weyl type is II, III, N or O, with any boost weight zero components being constant. We then consider the four-dimensional CSI spacetimes in more detail. We show that there are severe constraints on these spacetimes, and we argue that it is plausible that they are either locally homogeneous or that the spacetime necessarily belongs to the Kundt class of CSI spacetimes, all of which are constructed. The four-dimensional results lend support to the conjectures in higher dimensions.Date: November 7, 2018.Moreover, if a spacetime is V SI k or CSI k for all k, we will simply call the spacetime V SI or CSI, respectively. The set of all locally homogeneous spacetimes will be denoted by H. Clearly V SI ⊂ CSI and H ⊂ CSI. Definition 1.3 (CSI R spacetimes). Let us denote by CSI R all reducible CSI spacetimes that can be built from V SI and H by (i) warped products (ii) fibered products, and (iii) tensor sums (defined more precisely later).Definition 1.4 (CSI F spacetimes). Let us denote by CSI F those spacetimes for which there exists a frame with a null vector ℓ such that all components of the Riemann tensor and its covariants derivatives in this frame have the property that (i) all positive boost weight components (with respect to ℓ) are zero and (ii) all zero boost weight components are constant.Note that CSI R ⊂ CSI and CSI F ⊂ CSI. (There are similar definitions for CSI F,k etc. [8]).Definition 1.5 (CSI K spacetimes). Finally, let us denote by CSI K , those CSI spacetimes that belong to the (higher-dimensional) Kundt class (defined later); the so-called Kundt CSI spacetimes.In particular, we shall study the relationship between CSI R , CSI F , CSI K and especially with CSI\H. We note that by construction CSI R is at least of Weyl type II (i.e., of type II, III, N or O [9]), and by definition CSI F and CSI K are at least of Weyl type II (more precisely, at least of Riemann type II). In 4D, CSI R , CSI F and CSI K are closely related, and it is plausible that CSI\H is at least of Weyl type II (see section 7).

Kundt spacetimes are of great importance in general relativity in four dimensions and have a number of physical applications in higher dimensions in the context of string theory. The degenerate Kundt spacetimes have many special and unique mathematical properties, including their invariant curvature structure and their holonomy structure. We provide a rigorous geometrical kinematical definition of the general Kundt spacetime in four dimensions; essentially a Kundt spacetime is defined as one admitting a null vector that is geodesic, expansion-free, shear-free and twist-free. A Kundt spacetime is said to be degenerate if the preferred kinematic and curvature null frames are all aligned. The degenerate Kundt spacetimes are the only spacetimes in four dimensions that are not I-non-degenerate, so that they are not determined by their scalar polynomial curvature invariants. We first discuss the nonaligned Kundt spacetimes, and then turn our attention to the degenerate Kundt spacetimes. The degenerate Kundt spacetimes are classified algebraically by the Riemann tensor and its covariant derivatives in the aligned kinematic frame; as an example, we classify Riemann type D degenerate Kundt spacetimes in which ∇(Riem), ∇ (2) (Riem) are also of type D. We discuss other local characteristics of the degenerate Kundt spacetimes. Finally, we discuss degenerate Kundt spacetimes in higher dimensions.

We show that in theories of gravity that add quadratic curvature invariants to the EinsteinHilbert action there exist expanding vacuum cosmologies with positive cosmological constant which do not approach the de Sitter universe. Exact solutions are found which inflate anisotropically. This behaviour is driven by the Ricci curvature invariant and has no counterpart in the general relativistic limit. These examples show that the cosmic no-hair theorem does not hold in these higher-order extensions of general relativity and raises new questions about the ubiquity of inflation in the very early universe and the thermodynamics of gravitational fields.PACS numbers: 95.30. Sf, 98.80.Jk, 04.80.Cc, 98.80.Bp, 98.80.Ft, 95.10.Eg

‡Mathematical Institute, Academy of Sciences, ˇ Zitná 25, 115 67 Prague 1, Czech Republic. We investigate Lorentzian spacetimes where all zeroth and first order curvature invariants vanish and discuss how this class differs from the one where all curvature invariants vanish (VSI). We show that for VSI spacetimes all components of the Riemann tensor and its derivatives up to some fixed order can be made arbitrarily small. We discuss this in more detail by way of examples.