Bianchi identities in higher dimensions

A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension $n$. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in $n-4$ spacelike dimensions or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers-Perry metric as an explicit example of a vacuum type D spacetime to show that principal geodesic null congruences in vacuum type D spacetimes do not share this property.Comment: 25 pages, v3: Corrections to Appendix B as given in Erratum-ibid.24:1691,2007 are now incorporated (A factor of 2 was missing in certain Bianchi equations.

Electric and magnetic Weyl tensors in higher dimensions

Recent results on purely electric (PE) or magnetic (PM) spacetimes in n dimensions are summarized. These include: Weyl types; diagonalizability; conditions under which direct (or warped) products are PE/PM.Comment: 4 pages; short summary of (parts of) arXiv:1203.3563. Proceedings of "Relativity and Gravitation - 100 Years after Einstein in Prague", Prague, June 25-29, 2012 (http://ae100prg.mff.cuni.cz/

Alignment and algebraically special tensors in Lorentzian geometry

We develop a dimension-independent theory of alignment in Lorentzian geometry, and apply it to the tensor classification problem for the Weyl and Ricci tensors. First, we show that the alignment condition is equivalent to the PND equation. In 4D, this recovers the usual Petrov types. For higher dimensions, we prove that, in general, a Weyl tensor does not possess aligned directions. We then go on to describe a number of additional algebraic types for the various alignment configurations. For the case of second-order symmetric (Ricci) tensors, we perform the classification by considering the geometric properties of the corresponding alignment variety.Comment: 19 pages. Revised presentatio

All metrics have curvature tensors characterised by its invariants as a limit: the \epsilon-property

We prove a generalisation of the $\epsilon$-property, namely that for any dimension and signature, a metric which is not characterised by its polynomial scalar curvature invariants, there is a frame such that the components of the curvature tensors can be arbitrary close to a certain "background". This "background" is defined by its curvature tensors: it is characterised by its curvature tensors and has the same polynomial curvature invariants as the original metric.Comment: 6 page

Vanishing Scalar Invariant Spacetimes in Higher Dimensions

We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher-dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order.Comment: final versio

The type N Karlhede bound is sharp

We present a family of four-dimensional Lorentzian manifolds whose invariant classification requires the seventh covariant derivative of the curvature tensor. The spacetimes in questions are null radiation, type N solutions on an anti-de Sitter background. The large order of the bound is due to the fact that these spacetimes are properly $CH_2$, i.e., curvature homogeneous of order 2 but non-homogeneous. This means that tetrad components of $R, \nabla R, \nabla^{(2)}R$ are constant, and that essential coordinates first appear as components of $\nabla^{(3)}R$. Covariant derivatives of orders 4,5,6 yield one additional invariant each, and $\nabla^{(7)}R$ is needed for invariant classification. Thus, our class proves that the bound of 7 on the order of the covariant derivative, first established by Karlhede, is sharp. Our finding corrects an outstanding assertion that invariant classification of four-dimensional Lorentzian manifolds requires at most $\nabla^{(6)}R$.Comment: 7 pages, typos corrected, added citation and acknowledgemen

Note on the invariant classification of vacuum type D spacetimes

We illustrate the fact that the class of vacuum type D spacetimes which are $\mathcal{I}$-\emph{non-degenerate} are invariantly classified by their scalar polynomial curvature invariants

Quasi-Exact Solvability and the direct approach to invariant subspaces

We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line which are not Lie-algebraic. It is also applied to generate potentials with multiple algebraic sectors. We discuss two illustrative examples of these two applications: an interesting generalization of the Lam\'e potential which posses four algebraic sectors, and a quasi-exactly solvable deformation of the Morse potential which is not Lie-algebraic.Comment: 17 pages, 3 figure