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

Jens Boos

Institute for Theoretical Physics, University of Cologne, 50937 Köln, Germany

https://doi.org/10.1142/S0218271815500790Cited by:8 (Source: Crossref)

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 B² - E² 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 (B² + E²)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.

Keywords:

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