On Spacetimes with Constant Scalar Invariants

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 $VSI$ 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.