General Electric-Magnetic decomposition of fields, positivity and Rainich-like conditions

Abstract

We show how to generalize the classical electric-magnetic decomposition of the Maxwell or the Weyl tensors to arbitrary fields described by tensors of any rank in general $n$-dimensional spacetimes of Lorentzian signature. The properties and applications of this decomposition are reviewed. In particular, the definition of tensors quadratic in the original fields and with important positivity properties is given. These tensors are usually called "super-energy" (s-e) tensors, they include the traditional energy-momentum, Bel and Bel-Robinson tensors, and satisfy the so-called Dominant Property, which is a straightforward generalization of the classical dominant energy condition satisfied by well-behaved energy-momentum tensors. We prove that, in fact, any tensor satisfying the dominant property can be decomposed as a finite sum of the s-e tensors. Some remarks about the conservation laws derivable from s-e tensors, with some explicit examples, are presented. Finally, we will show how our results can be used to provide adequate generalizations of the Rainich conditions in general dimension and for any physical field.

20 pages, no figures, plenary talks given at the Spanish Relativity Meeting 2000, to be publisehd in the Proceedings of ERE-00 Some minor errors corrected, one formula and acknowledgements added

Cited by in corpus (20)

• The Kiselev black hole is neither perfect fluid, nor is it quintessence
• Algebraic classification of higher dimensional spacetimes based on null alignment
• Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension
• Essential core of the Hawking--Ellis types
• Twisted Self-Duality for Linearized Gravity in D dimensions
• Dynamical symmetry breaking in geometrodynamics
• Covariant gravitational dynamics in 3+1+1 dimensions
• Equations for general shells
• Some remarks on the Bel-Robinson Tensor
• Generalized Rainich conditions, generalized stress-energy conditions, and the Hawking-Ellis classification
• Area deficits and the Bel-Robinson tensor
• The universal 'energy' operator
• Algebraic classification of the Weyl tensor in higher dimensions based on its "superenergy" tensor
• Symmetric hyperbolic systems for a large class of fields in arbitrary dimension
• Hawking-Ellis type III spacetime geometry
• Photon traps in nonlinear electrodynamics
• On conformally recurrent manifolds of dimension greater than 4
• Remarks on the algebraic structure of (2, 2) double forms
• Electric and magnetic Weyl tensors in higher dimensions
• Gravity from thermodynamics in vacuum: Lorentz invariance and the role of Bel-Robinson tensor