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Type D conformal initial data

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Abstract

For a vacuum initial data set of the Einstein field equations it is possible to carry out a conformal rescaling or conformal compactification of the data giving rise to an initial data set for the Friedrich vacuum conformal equations. When will the data development with respect to the conformal equations of this set be a conformal extension of a type D solution? In this work we provide a set of necessary and sufficient conditions on a set of initial data for the conformal equations that guarantees that the data development of the conformal equations has a subset that is conformal to a vacuum type D solution of the Einstein’s equations. In particular we find the conditions under which this vacuum solution corresponds to the Kerr solution. Using our results we are able to show that there are no obstructions to extend the Petrov type of the physical spacetime to the unphysical spacetime if the conformal data are hyperboloidal.

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Notes

  1. The reasoning of [24] was formulated for the case with \(\lambda =0\) but it still holds when \(\lambda \ne 0\).

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Acknowledgements

We thank Dr. Valiente Kroon for his assistance with the computations dealing with the initial value formulation of the vacuum conformal equations, for a careful reading of the manuscript and for many comments and suggestions that improved it. We thank the financial support from Grant 14-37086G and the consecutive Grant 19-01850S of the Czech Science Foundation. Partial support from the projects IT956-16 (“Eusko Jaurlaritza”, Spain), FIS2014-57956-P (“Ministerio de Economía y Competitividad”, Spain), PTDC/MAT-ANA /1275/2014 (“Fundação para a Ciência e a Tecnologia”, Portugal) and the Mobility Fund of the Charles University is also gratefully acknowledged.

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  1. Alfonso García-Parrado

    Present address: Departamento de Matemáticas, Campus de Rabanales, Universidad de Córdoba, 14071, Córdoba, Spain

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  1. Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, V Holešovičkách 2, 180 00, Prague 8, Czech Republic

    Alfonso García-Parrado

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Correspondence to Alfonso García-Parrado.

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García-Parrado, A. Type D conformal initial data. Gen Relativ Gravit 52, 39 (2020). https://doi.org/10.1007/s10714-020-02687-x

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