The Kummer tensor density in electrodynamics and in gravity

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, KijklKijkl. This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four TijklTijkl, which is antisymmetric in its first two and its last two indices: Tijkl=−Tjikl=−TijlkTijkl=−Tjikl=−Tijlk. Thus, K∼T3K∼T3, see Eq. (46). (i) If TT is identified with the electromagnetic response tensor of local and linear media, the Kummer tensor density encompasses the generalized Fresnel wave surfaces for propagating light. In the reversible case, the wave surfaces turn out to be Kummer surfaces   as defined in algebraic geometry (Bateman 1910). (ii) If TT is identified with the curvature   tensor RijklRijkl of a Riemann–Cartan spacetime, then K∼R3K∼R3 and, in the special case of general relativity, KK reduces to the Kummer tensor of Zund (1969). This KK is related to the principal null directions   of the curvature. We discuss the properties of the general Kummer tensor density. In particular, we decompose KK irreducibly under the 4-dimensional linear group GL(4,R)GL(4,R) and, subsequently, under the Lorentz group SO(1,3)SO(1,3).