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Revisiting the Birkhoff theorem from a dual null point of view

A. Maciel, M. Le Delliou, J. P. Mimoso

Abstract
The Birkhoff theorem is a well-known result in general relativity, and it is used in many applications. However, its most general version, due to Bona, is almost unknown and presented in a form less accessible to the relativist and cosmologist community. Moreover, many wield it mistakenly as a simple transposition of Newton's iron sphere theorem. In the present work, we propose a modern, dual null, presentationuseful in many explorations, including black holes-of the theorem that renders accessible most of the results of Bona's version. In addition, we discuss the fluid contents admissible for the application of the theorem, beyond a vacuum, and we demonstrate how the formalism greatly simplifies solving the dynamical equations and allows one to express the solution as a power expansion in r. We present a family of solutions that share the properties predicted by the Birkhoff theorem and discuss the existence of trapped and antitrapped regions. The formalism manifestly shows how the type of region-trapped or untrapped-determines the character of the Killing vector.

Keywords
BLACK-HOLE DYNAMICS; GENERAL-RELATIVITY; GRAVITY

Physical Review D
Volume 98, Issue 2, Page 11
2018 July

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Instituto de Astrofísica e Ciências do Espaço Universidade do Porto Faculdade de Ciências da Universidade de Lisboa
Fundação para a Ciência e a Tecnologia COMPETE 2020 PORTUGAL 2020 União Europeia