Kerr black holes as elementary particles
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Abstract
<jats:title>A<jats:sc>bstract</jats:sc> </jats:title><jats:p>Long ago, Newman and Janis showed that a complex deformation <jats:italic>z → z</jats:italic> + <jats:italic>ia</jats:italic> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \sqrt{\mathrm{Kerr}} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mtext>Kerr</mml:mtext> </mml:msqrt> </mml:math></jats:alternatives></jats:inline-formula>. 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 large- spin 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \sqrt{\mathrm{Kerr}} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mtext>Kerr</mml:mtext> </mml:msqrt> </mml:math></jats:alternatives></jats:inline-formula> 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.</jats:p>