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 $SO(1,3)$.
Categories: gr-qc hep-th