# Kerr black holes as elementary particles

## 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.

- Authors:

- Inst. for Advanced Study, Princeton, NJ (United States)
- National Taiwan Univ., Taipei (Taiwan); National Tsing Hua Univ., Hsinchu (Taiwan)
- Univ. of Edinburgh, Scotland (United Kingdom)

- Publication Date:

- Research Org.:
- Inst. for Advanced Study, Princeton, NJ (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC); Simons Foundation; Science and Technology Facilities Council (STFC)

- OSTI Identifier:
- 1596096

- Grant/Contract Number:
- [SC0009988; 106-2628-M-002-012-MY3]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- [Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2020; Journal Issue: 1]; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Black Holes; Scattering Amplitudes

### Citation Formats

```
Arkani-Hamed, Nima, Huang, Yu-tin, and O’Connell, Donal. Kerr black holes as elementary particles. United States: N. p., 2020.
Web. doi:10.1007/JHEP01(2020)046.
```

```
Arkani-Hamed, Nima, Huang, Yu-tin, & O’Connell, Donal. Kerr black holes as elementary particles. United States. doi:10.1007/JHEP01(2020)046.
```

```
Arkani-Hamed, Nima, Huang, Yu-tin, and O’Connell, Donal. Wed .
"Kerr black holes as elementary particles". United States. doi:10.1007/JHEP01(2020)046. https://www.osti.gov/servlets/purl/1596096.
```

```
@article{osti_1596096,
```

title = {Kerr black holes as elementary particles},

author = {Arkani-Hamed, Nima and Huang, Yu-tin and O’Connell, Donal},

abstractNote = {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.},

doi = {10.1007/JHEP01(2020)046},

journal = {Journal of High Energy Physics (Online)},

number = [1],

volume = [2020],

place = {United States},

year = {2020},

month = {1}

}

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