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 sqrt{mathrm{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 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 sqrt{mathrm{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.
Highlights
S particles in the high energy limit
As one can see we have recovered eq (3.7): importantly, we identify the shift in Kerr solution explicitly with the exponentiation of s m for spinning particles in the large spin limit! the shift b → b+ia arises because of the exponential structure of minimally coupled amplitudes, and the Fourier factor eiq·b in expressions for observables in terms of amplitudes
In this paper we demonstrated that the exponentiation induced in taking the large-spin limit of minimally coupled spinning particles, precisely maps to the Newman-Janis complex shift relating the Schwarzschild and Kerr solutions in position space
Summary
An early example of the utility of complexified space-time was the derivation of the Kerr metric from a complex coordinate transformation of the Schwarzschild metric [18]. It is remarkable that φKerr can be obtained from φSch by a complex shift, which is as simple as z → z + ia To see how this connects the Schwarzschild to the Kerr solution, note that the quantity r2 = x2 + y2 + z2 shifts to x2 + y2 + z2 − a2 + 2iaz = r2 − a2 cos θ + 2iar cos θ = (r + ia cos θ), where r is the solution to equation (2.5). The Kerr-Schild form of the metric is convenient for revealing double copy relations between classical solutions of the Yang-Mills and Einstein equations. √ We first study the equivalence between the electromagnetic field of the Kerr solution with the minimally coupled spinning particle, in the infinite spin limit, by computing the impulse induced on a charged particle. We will reproduce the above result from the scattering amplitude involving minimally coupled spinning particles
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