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Sharp Bounds on the Critical Stability Radius for Relativistic Charged Spheres

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Abstract

In a recent paper by Giuliani and Rothman [17], the problem of finding a lower bound on the radius R of a charged sphere with mass M and charge Q < M is addressed. Such a bound is referred to as the critical stability radius. Equivalently, it can be formulated as the problem of finding an upper bound on M for given radius and charge. This problem has resulted in a number of papers in recent years but neither a transparent nor a general inequality similar to the case without charge, i.e., M ≤ 4R/9, has been found. In this paper we derive the surprisingly transparent inequality

$$\sqrt{M}\leq\frac{\sqrt{R}}{3}+\sqrt{\frac{R}{9}+\frac{Q^2}{3R}}.$$

The inequality is shown to hold for any solution which satisfies p + 2pT ≤ ρ, where p ≥ 0 and pT are the radial- and tangential pressures respectively and ρ ≥ 0 is the energy density. In addition we show that the inequality is sharp, in particular we show that sharpness is attained by infinitely thin shell solutions.

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Authors and Affiliations

  1. Mathematical Sciences, University of Gothenburg, S-41296, Göteborg, Sweden

    Håkan Andréasson

  2. Mathematical Sciences, Chalmers University of Technology, S-41296, Göteborg, Sweden

    Håkan Andréasson

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  1. Håkan Andréasson
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Correspondence to Håkan Andréasson.

Additional information

Communicated by G. W. Gibbons

This work is dedicated to the memory of my father Dan Andréasson (1933–2008).

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Andréasson, H. Sharp Bounds on the Critical Stability Radius for Relativistic Charged Spheres. Commun. Math. Phys. 288, 715–730 (2009). https://doi.org/10.1007/s00220-008-0690-3

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  • DOI: https://doi.org/10.1007/s00220-008-0690-3

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