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Speaker |
Boos, Jens |
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| Talk Title |
Plebański-Demiański solution of general relativity and its expressions quadratic and cubic in curvature: analogies to electromagnetism |
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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 B^2−E^2 and E\cdot B are evaluated analytically. In the asymptotically flat case, the leading terms of E and B can be interpreted as gravitatoelectric 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 Podolsk\'y (GP) allowing for a more straightforward physical interpretation of the free parameters. |
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