120 research outputs found
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 , i.e., curvature homogeneous of order 2
but non-homogeneous. This means that tetrad components of are constant, and that essential coordinates first appear as
components of . Covariant derivatives of orders 4,5,6 yield one
additional invariant each, and 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 .Comment: 7 pages, typos corrected, added citation and acknowledgemen
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
Generalization of the Geroch-Held-Penrose formalism to higher dimensions
Geroch, Held and Penrose invented a formalism for studying spacetimes
admitting one or two preferred null directions. This approach is very useful
for studying algebraically special spacetimes and their perturbations. In the
present paper, the formalism is generalized to higher-dimensional spacetimes.
This new formalism leads to equations that are considerably simpler than those
of the higher-dimensional Newman-Penrose formalism employed previously. The
dynamics of p-form test fields is analyzed using the new formalism and some
results concerning algebraically special p-form fields are proved.Comment: 24 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
Exceptional orthogonal polynomials and the Darboux transformation
We adapt the notion of the Darboux transformation to the context of
polynomial Sturm-Liouville problems. As an application, we characterize the
recently described Laguerre polynomials in terms of an isospectral
Darboux transformation. We also show that the shape-invariance of these new
polynomial families is a direct consequence of the permutability property of
the Darboux-Crum transformation.Comment: corrected abstract, added references, minor correction
Black rings with a small electric charge: gyromagnetic ratios and algebraic alignment
We study electromagnetic test fields in the background of vacuum black rings
using Killing vectors as vector potentials. We consider both spacetimes with a
rotating S^1 and with a rotating S^2 and we demonstrate, in particular, that
the gyromagnetic ratio of slightly charged black rings takes the value g=3
(this will in fact apply to a wider class of spacetimes). We also observe that
a S^2-rotating black ring immersed in an external "aligned" magnetic field
completely expels the magnetic flux in the extremal limit. Finally, we discuss
the mutual alignment of principal null directions of the Maxwell 2-form and of
the Weyl tensor, and the algebraic type of exact charged black rings. In
contrast to spherical black holes, charged rings display new distinctive
features and provide us with an explicit example of algebraically general (type
G) spacetimes in higher dimensions. Appendix A contains some global results on
black rings with a rotating 2-sphere. Appendix C shows that g=D-2 in any D>=4
dimensions for test electromagnetic fields generated by a time translation.Comment: 22 pages, 3 figures. v2: new appendix C finds the gyromagnetic ratio
g=D-2 in any dimensions, two new references. To appear in JHE
Quasi-exact solvability beyond the SL(2) algebraization
We present evidence to suggest that the study of one dimensional
quasi-exactly solvable (QES) models in quantum mechanics should be extended
beyond the usual \sla(2) approach. The motivation is twofold: We first show
that certain quasi-exactly solvable potentials constructed with the \sla(2)
Lie algebraic method allow for a new larger portion of the spectrum to be
obtained algebraically. This is done via another algebraization in which the
algebraic hamiltonian cannot be expressed as a polynomial in the generators of
\sla(2). We then show an example of a new quasi-exactly solvable potential
which cannot be obtained within the Lie-algebraic approach.Comment: Submitted to the proceedings of the 2005 Dubna workshop on
superintegrabilit
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
VSI_i spacetimes and the epsilon-property
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.Comment: To appear in JM
Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations
An alternative derivation is presented of the infinitely many exceptional
Wilson and Askey-Wilson polynomials, which were introduced by the present
authors in 2009. Darboux-Crum transformations intertwining the discrete quantum
mechanical systems of the original and the exceptional polynomials play an
important role. Infinitely many continuous Hahn polynomials are derived in the
same manner. The present method provides a simple proof of the shape invariance
of these systems as in the corresponding cases of the exceptional Laguerre and
Jacobi polynomials.Comment: 24 pages. Comments and references added. To appear in J.Phys.
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