The Kummer tensor density in electrodynamics and in gravity

Peter Baekler; Alberto Favaro; Yakov Itin; Friedrich W. Hehl

doi:10.1016/j.aop.2014.06.007

paper

The Kummer tensor density in electrodynamics and in gravity

arXiv:1403.3467 · doi:10.1016/j.aop.2014.06.007

Abstract

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, $K^{ij k l}$ . This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four $T^{ij k l}$ , which is antisymmetric in its first two and its last two indices: $T^{ij k l} = - T^{j ik l} = - T^{ij l k}$ . Thus, $K \sim T^{3}$ , see Eq.(46). (i) If $T$ 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 $T$ is identified with the {\it curvature} tensor $R^{ij k l}$ of a Riemann-Cartan spacetime, then $K \sim R^{3}$ and, in the special case of general relativity, $K$ reduces to the Kummer tensor of Zund (1969). This $K$ 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 $K$ irreducibly under the 4-dimensional linear group $G L (4, R)$ and, subsequently, under the Lorentz group $S O (1, 3)$ .

54 pages, 6 figures, written in LaTex; improved version in accordance with the referee report

The Kummer tensor density in electrodynamics and in gravity

Abstract

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