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 ${\mathbf{B}}^2-{\mathbf{E}}^2$ and $\mathbf{E}\cdot \mathbf{B}$ are evaluated analytically. In the asymptotically flat case, the leading terms of $\mathbf{E}$ and $\mathbf{B}$ 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 $({\mathbf{B}}^2+{\mathbf{E}}^2{)}^2$ 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.

arXiv comment52 pages, 11 listings of computer algebra code; typos removed, and journal reference added

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