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RESEARCH PRODUCT
Type D vacuum solutions: a new intrinsic approach
Juan Antonio SáezJoan Josep Ferrandosubject
PhysicsRiemann curvature tensorPure mathematicsPhysics and Astronomy (miscellaneous)FOS: Physical sciencesGeneral Relativity and Quantum Cosmology (gr-qc)Invariant (physics)Upper and lower boundsGeneral Relativity and Quantum Cosmologysymbols.namesakeRiemann hypothesisKilling vector fieldGeneral Relativity and Quantum CosmologyHomogeneous spacesymbolsCovariant transformationField equationdescription
We present a new approach to the intrinsic properties of the type D vacuum solutions based on the invariant symmetries that these spacetimes admit. By using tensorial formalism and without explicitly integrating the field equations, we offer a new proof that the upper bound of covariant derivatives of the Riemann tensor required for a Cartan-Karlhede classification is two. Moreover we show that, except for the Ehlers-Kundt's C-metrics, the Riemann derivatives depend on the first order ones, and for the C-metrics they depend on the first order derivatives and on a second order constant invariant. In our analysis the existence of an invariant complex Killing vector plays a central role. It also allows us to easily obtain and to geometrically interpret several known relations. We apply to the vacuum case the intrinsic classification of the type D spacetimes based on the first order differential properties of the 2+2 Weyl principal structure, and we show that only six classes are compatible. We define several natural and suitable subclasses and present an operational algorithm to detect them.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2013-09-18 |