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Title: 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, K{sup ijkl}. This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four T{sup ijkl}, which is antisymmetric in its first two and its last two indices: T{sup ijkl}=−T{sup jikl}=−T{sup ijlk}. Thus, K∼T{sup 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 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 T is identified with the curvature tensor R{sup ijkl} of a Riemann–Cartan spacetime, then K∼R{sup 3} and, in the special case of general relativity, K reduces to the Kummer tensor of Zund (1969). This K is related to the 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 GL(4,R) and, subsequently, under the Lorentz group SO(1,3)
Authors:
 [1] ;  [2] ;  [3] ;  [4] ;  [5] ;  [6]
  1. University of Appl. Sciences, 40474 Düsseldorf (Germany)
  2. Inst. Physics, Carl-von-Ossietzky-Univ., 26111 Oldenburg (Germany)
  3. Inst. Mathematics, Hebrew University of Jerusalem (Israel)
  4. (Israel)
  5. Inst. Theor. Physics, University of Cologne, 50923 Köln (Germany)
  6. (United States)
Publication Date:
OSTI Identifier:
22403415
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics (New York); Journal Volume: 349; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ELECTRODYNAMICS; GAUGE INVARIANCE; GENERAL RELATIVITY THEORY; GEOMETRY; GRAVITATION; LORENTZ GROUPS; SPACE-TIME; TENSORS