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, ${\mathcal{K}}^{ijkl}$. This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four ${\mathcal{T}}^{ijkl}$, which is antisymmetric in its first two and its last two indices: ${\mathcal{T}}^{ijkl}=-{\mathcal{T}}^{jikl}=-{\mathcal{T}}^{ijlk}$. Thus, $\mathcal{K}\sim {\mathcal{T}}^3$, see Eq.(46). (i) If $\mathcal{T}$ 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 $\mathcal{T}$ is identified with the {\it curvature} tensor $R^{ijkl}$ of a Riemann-Cartan spacetime, then $\mathcal{K}\sim R^3$ and, in the special case of general relativity, $\mathcal{K}$ 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 $\mathcal{K}$ irreducibly under the 4-dimensional linear group $GL(4,R)$ and, subsequently, under the Lorentz group $SO(1,3)$.