Kerr black holes as elementary particles
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Springer
Received: October 3, 2019 Accepted: December 14, 2019 Published: January 8, 2020
Kerr black holes as elementary particles
a
School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, U.S.A. b Department of Physics and Astronomy, National Taiwan University, Taipei 10617, Taiwan c Physics Division, National Center for Theoretical Sciences, National Tsing-Hua University, No.101, section 2, Kuang-Fu Road, Hsinchu, Taiwan d Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3JZ, Scotland, U.K.
E-mail: [email protected], [email protected], [email protected] Abstract: Long ago, Newman and Janis showed that a complex deformation z → z + ia of the Schwarzschild solution produces the Kerr solution. The underlying explanation for this relationship has remained obscure. The complex deformation has an electromagnetic counterpart: by shifting the Coloumb potential, we obtain the EM field of a certain rotating √ charge distribution which we term Kerr. In this note, we identify the origin of this shift as arising from the exponentiation of spin operators for the recently defined “minimally coupled” three-particle amplitudes of spinning particles coupled to gravity, in the largespin limit. We demonstrate this by studying the impulse imparted to a test particle in the background of the heavy spinning particle. We first consider the electromagnetic case, √ where the impulse due to Kerr is reproduced by a charged spinning particle; the shift of the Coloumb potential is matched to the exponentiated spin-factor appearing in the amplitude. The known impulse due to the Kerr black hole is then trivially derived from the gravitationally coupled spinning particle via the double copy. Keywords: Black Holes, Scattering Amplitudes ArXiv ePrint: 1906.10100
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)046
JHEP01(2020)046
Nima Arkani-Hamed,a Yu-tin Huangb,c and Donal O’Connelld
Contents 1
2 Complexifying Schwarzschild and the double copy
3
√ 3 From Kerr to spinning particles √ 3.1 Impulse from Kerr 3.2 Impulse from x 3.3 Impulse for Kerr black hole
4 4 5 8
4 Discussions and conclusions
8
1
Introduction
The no hair theorem states that black holes are characterized by only their mass, charge and angular momentum, implying that externally the black hole behaves as a point particle. For a long time this point of view has been utilized to derive the spin-independent part of the two-body classical potential for inspiralling black holes [1–7], from the scattering amplitudes of gravitationally coupled scalars. (See [8–11] for some recent results, and [12] for a more comprehensive review.) Of course any massive object with spin, viewed from sufficiently long distances, can be effectively treated as a point particle. From the perspective of on-shell scattering amplitudes, the most important first issue is to determine the three-particle amplitude, coupling the massive particles to gravitons
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