Electric multipole fields of higher-dimensional massive bodies

Matthew S. Fox
Phys. Rev. D 102, 044008 – Published 5 August 2020

Abstract

It was shown in a recent paper [J. Math. Phys. (N.Y.) 60, 102502 (2019)] that slowly lowering an electric charge into a Schwarzschild-Tangherlini (ST) black hole endows the final state with electric multipole fields, which implies that the final-state geometry is not Reissner-Nordström-Tangherlini in nature. This conclusion departs from the four-dimensional case in which the no-hair theorem (NHT) requires the final state to be a Reissner-Nordström black hole. To better understand this discrepancy clearly requires a deeper understanding of the origin of the multipole hair in the higher-dimensional case. In this paper, we advance the conjecture that charged, static, and asymptotically flat higher-dimensional black holes can acquire electric multipole hair only after they form. This supposition derives from studying the asymptotic behavior of the field of a multipole charge onto which a massive and hyperspherical shell with an exterior ST geometry is collapsing. In the mathematical limit as the shell approaches its ST radius, we find that the multipole fields (except the monopole) vanish. This implies that the only information of an arbitrary (but finite) charge distribution inside the collapsing shell that is available to an asymptotic observer is the total electric charge. Our results yield considerable insight into how higher-dimensional black holes acquire electric multipole hair, and also imply that, in four dimensions, the fadeaway of multipole moments during gravitational collapse is not strictly because of the NHT.

  • Received 29 March 2020
  • Accepted 27 July 2020

DOI:https://doi.org/10.1103/PhysRevD.102.044008

© 2020 American Physical Society

Physics Subject Headings (PhySH)

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Matthew S. Fox*

  • Department of Physics, Harvey Mudd College, Claremont, California 91711, USA

  • *msfox@g.hmc.edu

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Issue

Vol. 102, Iss. 4 — 15 August 2020

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