# Janis–Newman Algorithm: Generating Rotating and NUT Charged Black Holes

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivations

#### 1.2. The Janis–Newman Algorithm

#### 1.3. Overview

- all bosonic fields with spin $\le 2$;
- topological horizons;
- charges $m,n,q,p,a$ (with a only for $\Lambda =0$);
- extend to d = 3, 5 dimensions (and proposal for higher).

#### 1.4. Outlook

#### 1.5. Summary

## 2. Algorithm: Main Ideas

#### 2.1. Summary

- Perform a change of coordinates $(t,r)$ to $(u,r)$ and a gauge transformation such that ${g}_{rr}=0$ and ${A}_{r}=0$.
- Take $u,r\in \mathbb{C}$ and replace the functions ${f}_{i}(r)$ inside the real fields by new real-valued functions ${\tilde{f}}_{i}(r,\overline{r})$ (there is a set of “empirical” rules).
- Perform a change of coordinates to simplify the metric (for example to Boyer–Lindquist system). If the transformation is infinitesimal then one should check that it is a valid diffeomorphism, i.e., that it is integrable.

#### 2.2. Algorithm

#### 2.2.1. Seed Metric and Gauge Fields

#### 2.2.2. Janis–Newman Prescription: Newman–Penrose Formalism

#### 2.2.3. Giampieri Prescription

#### 2.2.4. Transforming the Functions

#### 2.2.5. Boyer–Lindquist Coordinates

#### 2.3. Examples

#### 2.3.1. Flat Space

#### 2.3.2. Kerr–Newman

## 3. Extension through Simple Examples

#### 3.1. Magnetic Charges: Dyonic Kerr–Newman

#### 3.2. NUT Charge and Cosmological Constant and Topological Horizon: (Anti-)de Sitter Schwarzschild–NUT

- Embedding Einstein–Maxwell into $N=2$ supergravity with a negative cosmological constant $\Lambda =-3{g}^{2}$, the solution is BPS if [72]$${\kappa}^{\prime}=-1,\phantom{\rule{2.em}{0ex}}n=\pm \frac{1}{2g},$$
- The Euclidean NUT solution is obtained from the Wick rotation$$t=-i\tau ,\phantom{\rule{2.em}{0ex}}n=i\nu .$$

#### 3.3. Complex Scalar Fields

## 4. Complete Algorithm

#### 4.1. Seed Configuration

#### 4.2. Janis–Newman Algorithm

#### 4.2.1. Complex Transformation

#### 4.2.2. Function Transformation

#### 4.2.3. Null Coordinates

#### 4.2.4. Boyer–Lindquist Coordinates

#### Degenerate Schwarzschild Seed

#### Degenerate Isotropic Seed

#### Constant F

#### 4.3. Open Questions

## 5. Examples

^{3}model and STU model): this theory is reviewed in Appendix A.

#### 5.1. Kerr–Newman–NUT

#### 5.2. Charged (a)dS–BBMB–NUT

#### 5.3. Ungauged $N=2$ BPS Solutions

#### 5.3.1. Pure Supergravity

#### 5.3.2. STU Model

#### 5.4. Non-Extremal Rotating Solution in ${T}^{3}$ Model

#### 5.5. SWIP Solutions

#### 5.6. Gauged $N=2$ Non-Extremal Solution

## Acknowledgments

## Conflicts of Interest

## Appendix A. Review of N = 2 Ungauged Supergravity

## References

- Plebański, J.F. A Class of Solutions of Einstein-Maxwell Equations. Ann. Phys.
**1975**, 90, 196–255. [Google Scholar] [CrossRef] - Plebański, J.F.; Demiański, M. Rotating, Charged, and Uniformly Accelerating Mass in General Relativity. Ann. Phys.
**1976**, 98, 98–127. [Google Scholar] [CrossRef] - Newman, E.T.; Janis, A.I. Note on the Kerr Spinning-Particle Metric. J. Math. Phys.
**1965**, 6, 915–917. [Google Scholar] [CrossRef] - Newman, E.T.; Couch, E.; Chinnapared, K.; Exton, A.; Prakash, A.; Torrence, R. Metric of a Rotating, Charged Mass. J. Math. Phys.
**1965**, 6, 918–919. [Google Scholar] [CrossRef] - Giampieri, G. Introducing Angular Momentum into a Black Hole Using Complex Variables. 1990; (unpublished). [Google Scholar]
- Nawarajan, D.; Visser, M. Cartesian Kerr-Schild Variation on the Newman-Janis Ansatz. ArXive, 2016; arXiv:1601.03532. [Google Scholar]
- Demiański, M. New Kerr-like Space-Time. Phys. Lett. A
**1972**, 42, 157–159. [Google Scholar] [CrossRef] - Drake, S.P.; Szekeres, P. Uniqueness of the Newman-Janis Algorithm in Generating the Kerr-Newman Metric. Gen. Relativ. Gravit.
**2000**, 32, 445–457. [Google Scholar] [CrossRef] - Azreg-Aïnou, M. From Static to Rotating to Conformal Static Solutions: Rotating Imperfect Fluid Wormholes with(out) Electric or Magnetic Field. Eur. Phys. J. C
**2014**, 74, 2865. [Google Scholar] [CrossRef] - Talbot, C.J. Newman-Penrose Approach to Twisting Degenerate Metrics. Commun. Math. Phys.
**1969**, 13, 45–61. [Google Scholar] [CrossRef] - Gürses, M.; Gürsey, F. Lorentz Covariant Treatment of the Kerr–Schild Geometry. J. Math. Phys.
**1975**, 16, 2385–2390. [Google Scholar] [CrossRef] - Schiffer, M.M.; Adler, R.J.; Mark, J.; Sheffield, C. Kerr Geometry as Complexified Schwarzschild Geometry. J. Math. Phys.
**1973**, 14, 52–56. [Google Scholar] [CrossRef] - Newman, E.T. Complex Coordinate Transformations and the Schwarzschild-Kerr Metrics. J. Math. Phys.
**1973**, 14, 774–776. [Google Scholar] [CrossRef] - Newman, E.T.; Winicour, J. A Curiosity Concerning Angular Momentum. J. Math. Phys.
**1974**, 15, 1113–1115. [Google Scholar] [CrossRef] - Newman, E.T. Heaven and Its Properties. Gen. Relativ. Gravit.
**1976**, 7, 107–111. [Google Scholar] [CrossRef] - Ferraro, R. Untangling the Newman-Janis Algorithm. Gen. Relativ. Gravit.
**2014**, 46, 1705. [Google Scholar] [CrossRef] - Adamo, T.; Newman, E.T. The Kerr-Newman Metric: A Review. Scholarpedia
**2014**, 9, 31791. [Google Scholar] - Ernst, F.J. New Formulation of the Axially Symmetric Gravitational Field Problem. Phys. Rev.
**1968**, 167, 1175–1178. [Google Scholar] [CrossRef] - Ernst, F.J. New Formulation of the Axially Symmetric Gravitational Field Problem. II. Phys. Rev.
**1968**, 168, 1415–1417. [Google Scholar] [CrossRef] - Quevedo, H. Complex Transformations of the Curvature Tensor. Gen. Relativ. Gravit.
**1992**, 24, 693–703. [Google Scholar] [CrossRef] - Quevedo, H. Determination of the Metric from the Curvature. Gen. Relativ. Gravit.
**1992**, 24, 799–819. [Google Scholar] [CrossRef] - Xu, D.Y. Exact Solutions of Einstein and Einstein-Maxwell Equations in Higher-Dimensional Spacetime. Class. Quantum Gravity
**1988**, 5, 871. [Google Scholar] - Kim, H. Notes on Spinning AdS_3 Black Hole Solution. Availiable online: http://cds.cern.ch/record/327099/files/9706008.pdf (accessed on 2 March 2017).
- Kim, H. Spinning BTZ Black Hole versus Kerr Black Hole: A Closer Look. Phys. Rev. D
**1999**, 59, 064002. [Google Scholar] [CrossRef] - Yazadjiev, S. Newman-Janis Method and Rotating Dilaton-Axion Black Hole. Gen. Relativ. Gravit.
**2000**, 32, 2345–2352. [Google Scholar] [CrossRef] - Herrera, L.; Jiménez, J. The Complexification of a Nonrotating Sphere: An Extension of the Newman–Janis Algorithm. J. Math. Phys.
**1982**, 23, 2339–2345. [Google Scholar] [CrossRef] - Drake, S.P.; Turolla, R. The Application of the Newman-Janis Algorithm in Obtaining Interior Solutions of the Kerr Metric. Class. Quantum Gravity
**1997**, 14, 1883–1897. [Google Scholar] [CrossRef] - Glass, E.N.; Krisch, J.P. Kottler-Lambda-Kerr Spacetime. ArXiv, 2004; arXiv:gr-qc/0405143. [Google Scholar]
- Ibohal, N. Rotating Metrics Admitting Non-Perfect Fluids in General Relativity. Gen. Relativ. Gravit.
**2005**, 37, 19–51. [Google Scholar] [CrossRef] - Azreg-Aïnou, M. Generating Rotating Regular Black Hole Solutions without Complexification. Phys. Rev. D
**2014**, 90, 064041. [Google Scholar] [CrossRef] - Carter, B. Hamilton-Jacobi and Schrödinger Separable Solutions of Einstein’s Equations. Commun. Math. Phys.
**1968**, 10, 280–310. [Google Scholar] - Gibbons, G.W.; Hawking, S.W. Cosmological Event Horizons, Thermodynamics, and Particle Creation. Phys. Rev. D
**1977**, 15, 2738–2751. [Google Scholar] [CrossRef] - Klemm, D.; Moretti, V.; Vanzo, L. Rotating Topological Black Holes. Phys. Rev. D
**1998**, 57. [Google Scholar] [CrossRef] - De Urreta, E.J.G.; Socolovsky, M. Extended Newman-Janis Algorithm and Rotating and Kerr-Newman de Sitter (Anti de Sitter) Metrics. ArXiv, 2015; arXiv:1504.01728. [Google Scholar]
- Mallett, R. Metric of a Rotating Radiating Charged Mass in a de Sitter Space. Phys. Lett. A
**1988**, 126, 226–228. [Google Scholar] [CrossRef] - Viaggiu, S. Interior Kerr Solutions with the Newman-Janis Algorithm Starting with Static Physically Reasonable Space-Times. Int. J. Mod. Phys. D
**2006**, 15, 1441–1453. [Google Scholar] [CrossRef] - Whisker, R. Braneworld Black Holes. Ph.D. Thesis, University of Durham, Durham, UK, 2008. [Google Scholar]
- Lessner, G. The “complex Trick” in Five-Dimensional Relativity. Gen. Relativ. Gravit.
**2008**, 40, 2177–2184. [Google Scholar] [CrossRef] - Capozziello, S.; De Laurentis, M.; Stabile, A. Axially Symmetric Solutions in f(R)-Gravity. Class. Quantum Gravity
**2010**, 27, 165008. [Google Scholar] [CrossRef] - Caravelli, F.; Modesto, L. Spinning Loop Black Holes. Class. Quantum Gravity
**2010**, 27, 245022. [Google Scholar] [CrossRef] - Dadhich, N.; Ghosh, S.G. Rotating Black Hole in Einstein and Pure Lovelock Gravity. ArXiv, 2013; arXiv:1307.6166. [Google Scholar]
- Ghosh, S.G.; Papnoi, U. Spinning Higher Dimensional Einstein-Yang-Mills Black Holes. Eur. Phys. J. C
**2013**, 74, 3016. [Google Scholar] [CrossRef] - Ghosh, S.G. Rotating Black Hole and Quintessence. Eur. Phys. J. C
**2016**, 76, 222. [Google Scholar] [CrossRef] - Azreg-Aïnou, M. Comment on “Spinning Loop Black holes”. Class. Quantum Gravity
**2011**, 28, 148001. [Google Scholar] [CrossRef] - Xu, D. Radiating Metric, Retarded Time Coordinates of Kerr-Newman-de Sitter Black Holes and Related Energy-Momentum Tensor. Sci. China Ser. A Math.
**1998**, 41, 663–672. [Google Scholar] [CrossRef] - Demiański, M.; Newman, E.T. Combined Kerr-NUT Solution of the Einstein Field Equations. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys.
**1966**, 14, 653–657. [Google Scholar] - Patel, L.K. Radiating Demianski-Type Space-Times. Indian J. Pure Appl. Math.
**1978**, 9, 1019. [Google Scholar] - Krori, K.D.; Chaudhury, T.; Bhattacharjee, R. Charged Demianski Metric. J. Math. Phys.
**1981**, 22, 2235–2236. [Google Scholar] [CrossRef] - Patel, L.K.; Akabari, R.P.; Dave, U.K. Radiating Demianski-Type Metrics and the Einstein-Maxwell Fields. J. Austral. Math. Soc. Ser. B
**1988**, 30, 120–126. [Google Scholar] [CrossRef] - Pirogov, Y.F. Towards the Rotating Scalar-Vacuum Black Holes. ArXiv, 2013; arXiv:1306.4866. [Google Scholar]
- Hansen, D.; Yunes, N. Applicability of the Newman-Janis Algorithm to Black Hole Solutions of Modified Gravity Theories. Phys. Rev. D
**2013**, 88, 104020. [Google Scholar] [CrossRef] - Horne, J.H.; Horowitz, G.T. Rotating Dilaton Black Holes. Phys. Rev. D
**1992**, 46, 1340–1346. [Google Scholar] [CrossRef] - Cirilo-Lombardo, D. The Newman-Janis Algorithm, Rotating Solutions and Einstein-Born-Infeld Black Holes. Class. Quantum Gravity
**2006**, 21, 1407–1417. [Google Scholar] [CrossRef] - D’Inverno, R. Introducing Einstein’s Relativity; Clarendon Press: Oxford, UK, 1992. [Google Scholar]
- Reed, J.F. Some Imaginary Tetrad-Transformations of Einstein Spaces. Ph.D. Thesis, Rice University, Houston, TX, USA, 1974. [Google Scholar]
- Erbin, H. Janis-Newman Algorithm: Simplifications and Gauge Field Transformation. Gen. Relativ. Gravit.
**2015**, 47, 19. [Google Scholar] [CrossRef] - Erbin, H.; Heurtier, L. Five-Dimensional Janis-Newman Algorithm. Class. Quantum Gravity
**2015**, 32, 16. [Google Scholar] - Erbin, H. Deciphering and Generalizing Demianski-Janis-Newman Algorithm. Gen. Relativ. Gravit.
**2016**, 48, 56. [Google Scholar] [CrossRef] - Erbin, H.; Heurtier, L. Supergravity, Complex Parameters and the Janis-Newman Algorithm. Class. Quantum Gravity
**2015**, 32, 165005. [Google Scholar] [CrossRef] - Erbin, H. Black Holes in N = 2 Supergravity. Ph.D. Thesis, Université Pierre et Marie Curie, Paris, France, 2015. [Google Scholar]
- Keane, A.J. An Extension of the Newman-Janis Algorithm. Class. Quantum Gravity
**2014**, 31, 155003. [Google Scholar] [CrossRef] - Sen, A. Rotating Charged Black Hole Solution in Heterotic String Theory. Phys. Rev. Lett.
**1992**, 69, 1006–1009. [Google Scholar] [CrossRef] [PubMed] - Perry, M.J. Black Holes Are Coloured. Phys. Lett. B
**1977**, 71, 234–236. [Google Scholar] [CrossRef] - Behrndt, K.; Lüst, D.; Sabra, W.A. Stationary Solutions of N = 2 Supergravity. Nucl. Phys. B
**1998**, 510, 264–288. [Google Scholar] [CrossRef] - Bergshoeff, E.; Kallosh, R.; Ortín, T. Stationary Axion/Dilaton Solutions and Supersymmetry. Nucl. Phys. B
**1996**, 478, 156–180. [Google Scholar] [CrossRef] [Green Version] - Bardoux, Y.; Caldarelli, M.M.; Charmousis, C. Integrability in Conformally Coupled Gravity: Taub-NUT Spacetimes and Rotating Black Holes. J. High Energy Phys.
**2014**, 2014, 039. [Google Scholar] [CrossRef] - Myers, R.; Perry, M. Black Holes in Higher Dimensional Space-Times. Ann. Phys.
**1986**, 172, 304–347. [Google Scholar] [CrossRef] - Breckenridge, J.C.; Myers, R.C.; Peet, A.W.; Vafa, C. D-Branes and Spinning Black Holes. Phys. Lett. B
**1997**, 391, 93–98. [Google Scholar] [CrossRef] - Gnecchi, A.; Hristov, K.; Klemm, D.; Toldo, C.; Vaughan, O. Rotating Black Holes in 4d Gauged Supergravity. J. High Energy Phys.
**2014**, 2014, 127. [Google Scholar] [CrossRef] - Visser, M. The Kerr Spacetime: A Brief Introduction. In The Kerr Spacetime. Rotating Black Holes in General Relativity; Wiltshire, D.L., Visser, M., Scott, S.M., Eds.; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Carroll, S.M. Spacetime and Geometry: An Introduction to General Relativity; Addison Wesley: Boston, MA, USA, 2004. [Google Scholar]
- Alonso-Alberca, N.; Meessen, P.; Ortín, T. Supersymmetry of Topological Kerr-Newmann-Taub-NUT-aDS Spacetimes. Class. Quantum Gravity
**2000**, 17, 2783–2797. [Google Scholar] [CrossRef] - Griffiths, J.B.; Podolsky, J. A New Look at the Plebanski-Demianski Family of Solutions. Int. J. Mod. Phys. D
**2006**, 15, 335–369. [Google Scholar] [CrossRef] - Chamblin, A.; Emparan, R.; Johnson, C.V.; Myers, R.C. Large N Phases, Gravitational Instantons and the Nuts and Bolts of AdS Holography. Phys. Rev. D
**1999**, 59, 064010. [Google Scholar] [CrossRef] - Johnson, C.V. Thermodynamic Volumes for AdS-Taub-NUT and AdS-Taub-Bolt. Class. Quantum Gravity
**2014**, 31, 235003. [Google Scholar] [CrossRef] - Krasiński, A. Inhomogeneous Cosmological Models; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Bekenstein, J.D. Exact Solutions of Einstein-Conformal Scalar Equations. Ann. Phys.
**1974**, 82, 535–547. [Google Scholar] [CrossRef] - Bocharova, N.M.; Bronnikov, K.A.; Melnikov, V.N. An Exact Solution of the System of Einstein Equations and Mass-Free Scalar Field. Vestn. Mosk. Univ. Fiz. Astro.
**1970**, 6, 706. [Google Scholar] - Hristov, K.; Looyestijn, H.; Vandoren, S. BPS Black Holes in N = 2 D = 4 Gauged Supergravities. J. High Energy Phys.
**2010**, 2010, 103. [Google Scholar] [CrossRef] - Chow, D.D.K.; Compère, G. Black Holes in N = 8 Supergravity from SO(4,4) Hidden Symmetries. Phys. Rev. D
**2014**, 90, 025029. [Google Scholar] [CrossRef] - Ortín, T. Gravity and Strings; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Freedman, D.Z.; Van Proeyen, A. Supergravity; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Andrianopoli, L.; Bertolini, M.; Ceresole, A.; D’Auria, R.; Ferrara, S.; Fré, P. General Matter Coupled N = 2 Supergravity. Nucl. Phys. B
**1996**, 476, 397–417. [Google Scholar] [CrossRef] - Andrianopoli, L.; Bertolini, M.; Ceresole, A.; D’Auria, R.; Ferrara, S.; Fré, P.; Magri, T. N = 2 Supergravity and N = 2 Super Yang-Mills Theory on General Scalar Manifolds: Symplectic Covariance, Gaugings and the Momentum Map. J. Geom. Phys.
**1997**, 23, 111–189. [Google Scholar] [CrossRef] [Green Version]

^{1}For simplifying, we will say that we complexify the functions inside the metric when we perform this transformation, even if in practice we “realify” them.^{2}It has not been proved that the KS condition is necessary, but all known examples seem to fit in this category.^{4}There are some errors in the introduction of [51]: they report incorrectly that the result from [50] implies that Sen’s black hole cannot be derived from the JN algorithm, as was done by Yazadjiev [25]. But this black hole corresponds to $\alpha =1$ and as reported above there is no problem in this case (see [52] for comparison). Moreover they argue that several works published before 2013 did not take into account the results of Pirogov [50], published in 2013.^{5}It may be possible to circumvent the result of [53] by using the results described in this review since several tools were not known by its author.^{6}Another solution has been proposed by Keane [61] but it is applicable only to the Newman–Penrose coefficients of the field strength. Our proposal requires less computations and yields directly the gauge field from which all relevant quantities can easily be derived.^{7}Available at http://www.lpthe.jussieu.fr/~erbin/.^{8}Derived by D. Klemm and M. Rabbiosi, unpublished work.^{9}We leave aside the case of the plane ${\mathbb{R}}^{2}$ with $\kappa =0$. The formulas can easily be extended to this case.^{10}We stress that at this stage these formula do not satisfy Einstein equations, they are just proxies to simplify later computations.^{11}Due to the convention of [66] there is no κ in the transformations.^{12}We omit the tilde that is present in [64] to avoid the confusion with the quantities that are transformed by the JNA. No confusion is possible since the index position will always indicate which function we are using.^{13}They correspond to singular solutions, but we are not concerned with regularity here.^{14}This model can be obtained from the STU model by setting the sections pairwise equal ${X}^{2}={X}^{0}$ and ${X}^{3}={X}^{1}$ [80]. It is also a truncation of pure $N=4$ supergravity.^{16}The original derivation is due to D. Klemm and M. Rabbiosi and has not been published. I am grateful to them for allowing me to reproduce it here.

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**MDPI and ACS Style**

Erbin, H.
Janis–Newman Algorithm: Generating Rotating and NUT Charged Black Holes. *Universe* **2017**, *3*, 19.
https://doi.org/10.3390/universe3010019

**AMA Style**

Erbin H.
Janis–Newman Algorithm: Generating Rotating and NUT Charged Black Holes. *Universe*. 2017; 3(1):19.
https://doi.org/10.3390/universe3010019

**Chicago/Turabian Style**

Erbin, Harold.
2017. "Janis–Newman Algorithm: Generating Rotating and NUT Charged Black Holes" *Universe* 3, no. 1: 19.
https://doi.org/10.3390/universe3010019