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Static spacetimes with prescribed multipole moments: a proof of a conjecture by Geroch

Magnus Herberthson

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In this paper we give sufficient conditions on a sequence of multipole moments for a static spacetime to exist with precisely these moments. The proof is constructive in the sense that a metric having prescribed multipole moments up to a given order can be calculated. Since these sufficient conditions agree with already known necessary conditions, this completes the proof of a long standing conjecture due to Geroch.


PACS

04.20.Jb Exact solutions

04.20.Gz Spacetime topology, causal structure, spinor structure

02.10.De Algebraic structures and number theory

MSC

30C15 Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) (For algebraic theory, see 12D10; for real methods, see 26C10)

46G25 (Spaces of) multilinear mappings, polynomials (See also 46E50, 46G20, 47H60)

30C10 Polynomials

83C05 Einstein's equations (general structure, canonical formalism, Cauchy problems)

83C75 Space-time singularities, cosmic censorship, etc.

Subjects

Mathematical physics

Gravitation and cosmology

Dates

Issue 21 (7 November 2009)

Received 26 June 2009, in final form 24 August 2009

Published 6 October 2009



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