Plebański–Demiański solution of general relativity and its expressions quadratic and cubic in curvature: Analogies to electromagnetism

Abstract

Analogies between gravitation and electromagnetism have been known since the 1950s.Here, we examine a fairly general type D solution-the exact seven parameter solution of Plebański-Demiański (PD)-to demonstrate these analogies for a physically meaningful spacetime:The two quadratic curvature invariants B 2 − E 2 and E ⋅ B are evaluated analytically.In the asymptotically flat case, the leading terms of E and B can be interpreted as gravitoelectric mass and gravitoelectric current of the PD solution, respectively, if there are no gravitomagnetic monopoles present.Furthermore, the square of the Bel-Robinson tensor reads (B 2 + E 2 ) 2 for the PD solution, reminiscent of the square of the energy density in electrodynamics. By analogy to the energy-momentum 3-form of the electromagnetic field, we provide an alternative way to derive the recently introduced Bel-Robinson 3-form, from which the Bel-Robinson tensor can be calculated.We also determine the Kummer tensor, a tensor cubic in curvature, for a general type D solution for the first time, and calculate the pieces of its irreducible decomposition.The calculations are carried out in two coordinate systems: in the original polynomial PD coordinates, and in a modified Boyer-Lindquist-like version introduced by Griffiths and Podolský (GP) allowing for a more straightforward physical interpretation of the free parameters. file: 05 elm inv v9.tex, Jul

Transferring energy in general relativity

“…As described in [16,34], for Coulomb-like gravitational fields that are described by a Petrov type D spacetime (of which LRS-II is a subclass), the symmetric 2 index square root of the Bell-Robinson tensor T G ab can be written as…”

Nonlocality and gravitoelectromagnetic duality

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Gravity-Induced Four-Fermion Contact Interaction Implies Gravitational Intermediate W and Z Type Gauge Bosons

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