Pleba\'nski-Demia\'nski solution of general relativity and its expressions quadratic and cubic in curvature: analogies to electromagnetism

TL;DR

This paper explores the Plebański-Demiański solution in general relativity, analyzing its curvature invariants and analogies to electromagnetism, including new calculations of the Bel–Robinson and Kummer tensors in different coordinate systems.

Contribution

It provides the first calculation of the Kummer tensor for a general type D solution and offers new insights into the electromagnetic analogies of the PD spacetime.

Findings

Analytic expressions for quadratic curvature invariants in the PD solution.

Interpretation of gravitoelectric fields as mass and current in asymptotically flat spacetime.

Calculation of the Kummer tensor and its decomposition for type D solutions.

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 Pleba\'nski--Demia\'nski (PD)---to demonstrate these analogies for a physically meaningful spacetime: The two quadratic curvature invariants $\bf{B}^2-\bf{E}^2$ and $\bf{E}\cdot\bf{B}$ are evaluated analytically. In the asymptotically flat case, the leading terms of $\bf{E}$ and $\bf{B}$ 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 Podolsk\'y (GP) allowing for a more straightforward physical interpretation of the free parameters.