Plebański-Demiański solution of general relativity and its expressions quadratic and cubic in curvature: analogies to electromagnetism
arXiv:1412.1958 · doi:10.1142/S0218271815500790
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 and are evaluated analytically. In the asymptotically flat case, the leading terms of and 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 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.
52 pages, 11 listings of computer algebra code; typos removed, and journal reference added