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In the Kerr geometry, we calculate various surfaces of constant curvature invariants. These extend well beyond the Kerr horizon, and we argue that they might be of observational significance in connection with non-minimally coupled matter fields. Moreover, we demonstrate that the principal null directions of the Kerr geometry can be obtained by projections involving either the Bel--Robinson or the Kummer tensor. We conjecture that this is also possible in more general settings., Comment: 4 pages, 10 figures. Essay written for the Gravity Research Foundation 2017 Awards for Essays on Gravitation

The propagation of light through bianisotropic materials is studied in the geometrical optics approximation. For that purpose, we use the quartic general dispersion equation specified by the Tamm-Rubilar tensor, which is cubic in the electromagnetic response tensor of the medium. A collection of different and remarkable Fresnel (wave) surfaces is gathered, and unified via the projective geometry of Kummer surfaces., Comment: 3 pages, 5 figures

It is known that the Fresnel wave surfaces of transparent biaxial media have 4 singular points, located on two special directions. We show that, in more general media, the number of singularities can exceed 4. In fact, a highly symmetric linear material is proposed whose Fresnel surface exhibits 16 singular points. Because, for every linear material, the dispersion equation is quartic, we conclude that 16 is the maximum number of isolated singularities. The identity of Fresnel and Kummer surfaces, which holds true for media with a certain symmetry (zero skewon piece), provides an elegant interpretation of the results. We describe a metamaterial realization for our linear medium with 16 singular points. It is found that an appropriate combination of metal bars, split-ring resonators, and magnetized particles can generate the correct permittivity, permeability, and magnetoelectric moduli. Lastly, we discuss the arrangement of the singularities in terms of Kummer's (16,6)-configuration of points and planes. An investigation parallel to ours, but in linear elasticity, is suggested for future research., Comment: 7 pages, 3 figures

Guided by results in the premetric electrodynamics of local and linear media, we introduce on 4-dimensional spacetime the new abstract notion of a Kummer tensor density of rank four, ${\cal K}^{ijkl}$. This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four ${\cal T}^{ijkl}$, which is antisymmetric in its first two and its last two indices: ${\cal T}^{ijkl} = - {\cal T}^{jikl} = - {\cal T}^{ijlk}$. Thus, ${\cal K}\sim {\cal T}^3$, see Eq.(46). (i) If $\cal T$ is identified with the electromagnetic response tensor of local and linear media, the Kummer tensor density encompasses the generalized {\it Fresnel wave surfaces} for propagating light. In the reversible case, the wave surfaces turn out to be {\it Kummer surfaces} as defined in algebraic geometry (Bateman 1910). (ii) If $\cal T$ is identified with the {\it curvature} tensor $R^{ijkl}$ of a Riemann-Cartan spacetime, then ${\cal K}\sim R^3$ and, in the special case of general relativity, ${\cal K}$ reduces to the Kummer tensor of Zund (1969). This $\cal K$ is related to the {\it principal null directions} of the curvature. We discuss the properties of the general Kummer tensor density. In particular, we decompose $\cal K$ irreducibly under the 4-dimensional linear group $GL(4,R)$ and, subsequently, under the Lorentz group $SO(1,3)$., Comment: 54 pages, 6 figures, written in LaTex; improved version in accordance with the referee report

Geometrical optics describes, with good accuracy, the propagation of high-frequency plane waves through an electromagnetic medium. Under such approximation, the behaviour of the electromagnetic fields is characterised by just three quantities: the temporal frequency $\omega$, the spatial wave (co)vector $k$, and the polarisation (co)vector $a$. Numerous key properties of a given optical medium are determined by the Fresnel surface, which is the visual counterpart of the equation relating $\omega$ and $k$. For instance, the propagation of electromagnetic waves in a uniaxial crystal, such as calcite, is represented by two light-cones. Kummer, whilst analysing quadratic line complexes as models for light rays in an optical apparatus, discovered in the framework of projective geometry a quartic surface that is linked to the Fresnel one. Given an arbitrary dispersionless linear (meta)material or vacuum, we aim to establish whether the resulting Fresnel surface is equivalent to, or is more general than, a Kummer surface., Comment: Invited lecture at "Electromagnetic Spacetimes", Wolfgang Pauli Institute, Vienna, 19--23 Nov 2012

Based on a recent work by Schuller et al., a geometric representation of all skewonless, non-birefringent, linear media is obtained. The derived constitutive law is based on a "core", encoding the optical metric up to a constant. All further corrections are provided by two (anti-)selfdual bivectors, and an "axion". The bivectors are found to vanish if the optical metric has signature (3,1) - that is, if the Fresnel equation is hyperbolic. We propose applications of this result in the context of transformation optics and premetric electrodynamics., Comment: 12 pages, REVTeX; v2: typos; v3: relevant changes in text, reorganization of manuscript; v4: new section added, final version, to appear in Annalen der Physik