# Black holes with Lambert W function horizons

## Abstract

We consider Einstein gravity with a negative cosmological constant endowed with distinct matter sources. The different models analyzed here share the following two properties: (1) they admit static symmetric solutions with planar base manifold characterized by their mass and some additional Noetherian charges, and (2) the contribution of these latter in the metric has a slower falloff to zero than the mass term, and this slowness is of logarithmic order. Under these hypothesis, it is shown that, for suitable bounds between the mass and the additional Noetherian charges, the solutions can represent black holes with two horizons whose locations are given in term of the real branches of the Lambert W functions. We present various examples of such black hole solutions with electric, dyonic or axionic charges with AdS and Lifshitz asymptotics. As an illustrative example, we construct a purely AdS magnetic black hole in five dimensions with a matter source given by three different Maxwell invariants.

## 1 Introduction

The AdS/CFT correspondence has been proved to be extremely useful for getting a better understanding of strongly coupled systems by studying classical gravity, and more specifically black holes. In particular, the gauge/gravity duality can be a powerful tool for analyzing finite temperature systems in presence of a background magnetic field. In such cases, from the dictionary of the correspondence, the black holes must be endowed with a magnetic charge corresponding to the external magnetic field of the CFT. In light of this constatation, it is clear that dyonic black holes are of great importance in order to study the charge transport at quantum critical point, particulary for strongly coupled CFTs in presence of an external magnetic field. For example, four-dimensional dyonic black holes have been proved to be relevant for a better comprehension of planar condensed matter phenomena such as the quantum Hall effect [1], the superconductivity-superfluidity [2] or the Nernst effect [3]. The study of dyonic black holes is not only interesting in four dimensions, but also in higher dimensions where their holographic applications have been discussed in the current literature. For example, it has been shown that large dyonic AdS black holes are dual to stationary solutions of a charged fluid in presence of an external magnetic field [4]. In this last reference, the AdS/CFT correspondence was used conversely and stationary solutions of the Navier-Stokes equations were constructed corresponding to an hypothetical five-dimensional AdS dyonic rotating black string with nonvanishing momentum along the string. We can also mention that magnetic/dyonic black holes present some interest from a purely gravity point of view. Indeed, there is a wide range of contexts in which magnetic/dyonic solutions are currently studied including in particular supergravity models [5, 6], Einstein-Yang-Mills theory [7] or nonlinear electrodynamics [8]. Nevertheless, in spite of partial results, the problem of finding magnetic solutions in higher dimension is an highly nontrivial problem. For example, it is easy to demonstrate that under suitable hypothesis, magnetic solutions in odd dimensions \(D\ge 5\) for the Einstein–Maxwell or for the Lovelock–Maxwell theories do not exist [9, 10]. This observation is in contrast with the four-dimensional situation where static dyonic configuration can be easily constructed thanks to the electromagnetic duality which rotates the electric field into the magnetic field. In the same register, one may also suspect the lack of electromagnetic duality and of the conformal invariance in dimension \(D>4\) to explain the difficulty for constructing the higher-dimensional extension of the Kerr–Newmann solution.

The purpose of the present paper is twofold. Firstly, we would like to present a simple dyonic extension of the five-dimensional Reissner–Nordstrom solution with planar horizon. The solution will be magnetically charged by considering an electromagnetic source composed by *at least* three different Maxwell gauge fields. Each of these *U*(1) gauge fields will be sustained by one of the three different coordinates of the planar base manifold. Interestingly enough, the magnetic contribution in the metric has an asymptotically logarithmic falloff of the form \(\frac{\ln r}{r^2}\). Nevertheless, in spite of this slowly behavior, the thermodynamics analysis yields finite quantities even for the magnetic charge. Since we are working in five dimensions, we extend as well this dyonic solution to the case of Einstein–Gauss–Bonnet gravity. We can also mention that the causal structure of the dyonic solution can not be done analytically. Nevertheless from different simulations, one can observe that the solution has a Reissner–Nordstrom like behavior. Indeed, depending on the election of the integration constants, the solution can be a black hole with inner and outer horizons or an extremal black hole or the solution can have a naked singularity located at the origin. On the other hand, we notice that the horizon structure of the purely magnetic solution can be treated analytically. More precisely, we will show that, as for the Reissner–Nordstrom solution, the absence of naked singularity can be guaranteed for a suitable bound relation between the mass and the magnetic charge. Moreover, in this case, the location of the inner and outer horizons are expressed analytically in term of the real branches of so-called Lambert W function. This latter is defined to be the multivalued inverse of the complex function \(f(\omega )=\omega e^{\omega }\) which has an infinite countable number of branches but only two of them are real-valued, see Ref. [11] for a nice review. The Lambert W functions have a wide range of applications as for example in combinatoric with the tree functions that are used in the enumeration of trees [12] or for equations with delay that have applications for biological, chemical or physical phenomena, see e.g. [13] or in the AdS/CFT correspondence as in the expression of the large-spin expansion of the energy of the Gubser–Klebanov–Polyakov string theory [14]. Just to conclude this parenthesis about the Lambert W function, we also mention that this function can be used in the case of the Schwarzschild metric as going from the Eddington–Finkelstein coordinates to the standard Schwarzschild coordinates

The plan of the paper is organized as follows. In the next section, we present our toy model for dyonic solutions which consists on the five-dimensional Einstein–Gauss–Bonnet action with three different Abelian gauge fields. For this model, we derive a dyonic black hole configuration as well as its GR limit. A particular attention will be devoted to the purely magnetic GR solution for which a bound relation between the mass and the magnetic charge ensures the existence of an event horizon covering the naked singularity. In this case, the inner and outer horizons are expressed in term of the two real branches of the Lambert W functions. We will establish that this mass bound is essentially due to the fact that the magnetic charge has a slower falloff of logarithmic order to zero than the mass term in the metric function. Starting from this observation, we will present in Sect. 3 various examples of black holes sharing this same feature with electric, axionic or magnetic charges and with AdS and Lifshitz asymptotics. In Sect. 4, we extend the previous solutions to general dyonic configurations with axionic charges. Finally, the last section is devoted to our conclusion and an appendix is provided where some useful properties of the Lambert W functions are given.

## 2 Five-dimensional dyonic black hole solution

*U*(1) gauge fields \({\mathcal {A}}_I\) for \(I=\{1, 2, 3\}\) and \(\alpha \) represents the Gauss–Bonnet coupling constant. The field equations obtained by varying this action read

The causal structure of the dyonic solution is quite involved and can not be treated analytically as in the case of the four-dimensional Reissner–Nordstrom dyonic solution. Nevertheless, it is quite simple to see that the GR solution (2.4) with \(\Lambda <0\) and without magnetic charge, has a Reissner–Nordstrom like behavior in the sense that for \({\mathcal {M}}\ge { 3^{\frac{5}{3}} \vert {\mathcal {Q}}_{e}\vert ^{\frac{4}{3}}\left( -\Lambda \right) ^{\frac{1}{3}}}\big / {4^{\frac{4}{3}} \vert \Sigma _3 \vert ^{\frac{1}{3}}}\), the solution describes a (extremal) black hole while the case \({\mathcal {M}}<{ 3^{\frac{5}{3}} \vert {\mathcal {Q}}_{e}\vert ^{\frac{4}{3}}\left( -\Lambda \right) ^{\frac{1}{3}}}\big / {4^{\frac{4}{3}} \vert \Sigma _3 \vert ^{\frac{1}{3}}}\) will yield a naked singularity. The dyonic GR solution has also a similar behavior which can be appreciated only by means of some simulations reported in the graphics below. In the next subsection, we will see that in the purely magnetic case, the causal structure of the solution can be analyzed analytically (Fig. 1).

### 2.1 Purely magnetic GR solution

*F*, it is useful to define

*h*has a global minimum at \(x=\frac{3{\mathcal {Q}}_m^2}{-2\Lambda \vert \Sigma _3\vert ^2}\). The equation for the zeros of the function

*h*that will give as well the location of the horizons for the metric function

*F*through (2.8) is of the form (6.2). Hence, its corresponding discriminant as defined in Eq. (6.3) is given by

*F*has an inner (Cauchy) horizon \(r_{-}\) and an outer (event) horizon \(r_{+}\) whose locations are expressed in term of the two real branches of the Lambert W functions, \(W_{0}\) and \(W_{-1}\) as

## 3 Other examples of black holes with Lambert W function horizons

*A*, \({\mathcal {A}}\) and the axionic fields \(\psi _j\) to a dilaton field \(\phi \).

### 3.1 Electrically charged AdS black holes for nonlinear Maxwell theory in odd dimension

*q*is of the form \(q=\frac{D-1}{2}\). As shown in Ref. [32], there exists a purely electric solution with logarithmic falloff, and this solution, in order to be real, must be restricted to odd dimension \(D=2k+1\) with \(k\ge 1\). Hence the Maxwell nonlinearity is \(q=k\) and the metric function and the electric potential are given by

*F*can be put in the form (6.2) by substituting \(x=r^{2k}\) and in this case, the discriminant (6.3) is given by

*k*. Indeed, for even

*k*or equivalently for odd dimensions \(D=5\,\,\text{ mod }\,\, 4\), the discriminant is positive, and hence the solution is a black hole for any value of the mass \({\mathcal {M}}\), and there is a single horizon located at

*k*or equivalently for odd dimensions \(D=3\,\,\text{ mod }\,\,4\), the solution will be a black hole provided that the mass satisfies the following bound relation with the electric charge

### 3.2 Five-dimensional AdS dyonic black holes and particular stealth configuration

*F*is given by

- (i)For \(A<0\), that is forthe solution has a single horizon located at$$\begin{aligned} \vert {\mathcal {Q}}_e\vert >\frac{2^{\frac{5}{4}}\vert {\mathcal {Q}}_m\vert ^{\frac{3}{2}}}{\vert \Sigma _3\vert ^{\frac{1}{2}}}, \end{aligned}$$but the solution does not satisfy the dominant energy conditions neither the weak energy conditions since the energy density \(\mu =\frac{3A}{2r^4}\) is always negative.$$\begin{aligned} r_h=e^{-\frac{W_{0}(\Delta )}{4}-\frac{2{\mathcal {M}}}{3\vert \Sigma _3\vert A}}, \end{aligned}$$
- (ii)For \(A>0\), that is forthe solution represents a dyonic AdS black hole only if$$\begin{aligned} \vert {\mathcal {Q}}_e\vert <\frac{2^{\frac{5}{4}}\vert {\mathcal {Q}}_m\vert ^{\frac{3}{2}}}{\vert \Sigma _3\vert ^{\frac{1}{2}}}, \end{aligned}$$and in this case, the solution is shown to satisfy the dominant energy conditions (2.6).$$\begin{aligned} {\mathcal {M}} \ge \frac{3\vert \Sigma _3\vert A}{8}\left[ 1-\ln \left( \frac{3A}{-2\Lambda }\right) \right] , \end{aligned}$$
- (iii)Finally, for \(A=0\) that is forthe solution represents a black hole stealth dyonic configuration on the Schwarzschild AdS background where the horizon is located at$$\begin{aligned} \vert {\mathcal {Q}}_e\vert =\frac{2^{\frac{5}{4}}\vert {\mathcal {Q}}_m\vert ^{\frac{3}{2}}}{\vert \Sigma _3\vert ^{\frac{1}{2}}}, \end{aligned}$$$$\begin{aligned} r_h=\left( \frac{4{\mathcal {M}}}{-\Lambda \vert \Sigma _3\vert }\right) ^{\frac{1}{4}}. \end{aligned}$$

### 3.3 Purely magnetic Lifshitz black hole with dynamical exponent \(z=D-4\)

We now turn to derive examples with a different asymptotic behavior characterized by an anisotropy scale between the time and the space, the so-called Lifshitz asymptotic. This anisotropy is reflected by a dynamical exponent denoted usually by *z* and defined such that the case \(z=1\) corresponds to the AdS isotropic case. Note that Lifshitz black holes have been insensitively studied during the last decade, see for examples Refs. [36, 37, 38, 39, 40].

### 3.4 Axionic Lifshitz black hole with dynamical exponent \(z=D-2\)

^{1}

*E*is a constant. The perturbed Maxwell current given by \(J=\sqrt{-g}e^{\lambda \phi }F^{rx_1}\) is a conserved quantity along the radial coordinate. A straightforward computation along the same lines as those in [27, 28] yields a DC conductivity \(\sigma _{\tiny {\text{ DC }}}\) given by

Just to conclude this section, we would like to mention that the Lifshitz solutions presented here can be extended to the so-called hyperscaling violation black holes by adding an extra gauge field as done in Refs. [41, 42, 43, 44]. In these last references, the thermoelectric DC conductivities of the hyperscaling violation black holes were also computed.

## 4 More general dyonic-axionic solutions in arbitrary dimension

*n*extra gauge fields that will sustain the magnetic charge, \({\mathcal {F}}_{(I)\mu \nu }=\partial _{\mu } {\mathcal {A}}_{(I)\nu }-\partial _{\nu } {\mathcal {A}}_{(I)\mu }\) for \(I=1,\ldots n\) with \(n=1\) in even dimension and \(n=3\) in odd dimension. The field equations read

*z*of the field equations is given by

## 5 Conclusion

*F*(6.3) is given by

*i*. Indeed, in this case, the consistency of the Einstein equations \(T_t^t+(D-2)T_i^i+(D-1)\Lambda =0\) yields to a nonhomogeneous Euler’s differential equation of second-order

We have also presented two other examples with Lifshitz asymptotics with fixed values of the dynamical exponent with a magnetic charge and an axionic charge. The emergence of such asymptotic solutions is essentially due to the presence of dilatonic fields. Note that there also exist Lifshitz black holes with a logarithmic falloff in the case of higher-order gravity [47]. Finally, for completeness, we have extended the previous solutions to accommodate a dyonic as well an axionic charge in arbitrary dimension.

## Footnotes

- 1.
For simplicity, we only consider perturbations along one of the planar coordinate \(x_1\).

## Notes

### Acknowledgements

MB is supported by grant Conicyt/ Programa Fondecyt de Iniciación en Investigación No. 11170037.

## References

- 1.S.A. Hartnoll, P. Kovtun, Phys. Rev. D
**76**, 066001 (2007)ADSCrossRefGoogle Scholar - 2.S.A. Hartnoll, C.P. Herzog, G.T. Horowitz, Phys. Rev. Lett.
**101**, 031601 (2008)ADSCrossRefGoogle Scholar - 3.S.A. Hartnoll, P.K. Kovtun, M. Muller, S. Sachdev, Phys. Rev. B
**76**, 144502 (2007)ADSCrossRefGoogle Scholar - 4.M.M. Caldarelli, O.J.C. Dias, D. Klemm, JHEP
**0903**, 025 (2009)ADSCrossRefGoogle Scholar - 5.A.H. Chamseddine, W.A. Sabra, Phys. Lett. B
**485**, 301 (2000)ADSMathSciNetCrossRefGoogle Scholar - 6.D.D.K. Chow, G. Compere, Phys. Rev. D
**89**(6), 065003 (2014)ADSCrossRefGoogle Scholar - 7.B.C. Nolan, E. Winstanley, Class. Quant. Gravit.
**29**, 235024 (2012)ADSCrossRefGoogle Scholar - 8.K.A. Bronnikov, Gravit. Cosmol.
**23**(4), 343 (2017)ADSMathSciNetCrossRefGoogle Scholar - 9.M. Ortaggio, J. Podolsky, M. Zofka, Class. Quant. Gravit.
**25**, 025006 (2008)ADSCrossRefGoogle Scholar - 10.H. Maeda, M. Hassaine, C. Martinez, JHEP
**1008**, 123 (2010)ADSCrossRefGoogle Scholar - 11.R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, Adv. Comput. Math.
**5**(4), 329–359 (1996)MathSciNetCrossRefGoogle Scholar - 12.M. Josuat-Verges, Ramanujan J.
**38**(1), 1–15 (2015)MathSciNetCrossRefGoogle Scholar - 13.P.B. Brito, M.F. Fabiao, A.G. St. Aubyn, Numer. Funct. Anal. Optim.
**32**:11, 1116–1126 (2011)Google Scholar - 14.E. Floratos, G. Georgiou, G. Linardopoulos, JHEP
**1403**, 018 (2014)ADSCrossRefGoogle Scholar - 15.R.E. Arias, I. Salazar Landea, JHEP
**1712**, 087 (2017)ADSCrossRefGoogle Scholar - 16.M. Bravo-Gaete, M. Hassaine, Phys. Rev. D
**97**(2), 024020 (2018)ADSMathSciNetCrossRefGoogle Scholar - 17.M. Cvetic, S. Nojiri, S.D. Odintsov, Nucl. Phys. B
**628**, 295 (2002)ADSCrossRefGoogle Scholar - 18.D.G. Boulware, S. Deser, Phys. Rev. Lett.
**55**, 2656 (1985)ADSCrossRefGoogle Scholar - 19.M. Hassaine, C. Martinez, Phys. Rev. D
**75**, 027502 (2007)ADSMathSciNetCrossRefGoogle Scholar - 20.M. Hassaine, C. Martinez, Class. Quant. Gravit.
**25**, 195023 (2008)ADSCrossRefGoogle Scholar - 21.S.H. Hendi, S. Panahiyan, H. Mohammadpour, Eur. Phys. J. C
**72**, 2184 (2012)ADSCrossRefGoogle Scholar - 22.S.H. Hendi, Adv. High Energy Phys.
**2014**, 697863 (2014)Google Scholar - 23.H.A. Gonzalez, M. Hassaine, C. Martinez, Phys. Rev. D
**80**, 104008 (2009)ADSCrossRefGoogle Scholar - 24.A. Rincon, E. Contreras, P. Bargueno, B. Koch, G. Panotopoulos, Eur. Phys. J. C
**78**(8), 641 (2018)ADSCrossRefGoogle Scholar - 25.M. Ghanaatian, F. Naeimipour, A. Bazrafshan, M. Eftekharian, Phys. Rev. D
**99**(2), 024006 (2019)ADSCrossRefGoogle Scholar - 26.T. Andrade, B. Withers, JHEP
**1405**, 101 (2014)ADSCrossRefGoogle Scholar - 27.A. Donos, J.P. Gauntlett, JHEP
**1404**, 040 (2014)ADSCrossRefGoogle Scholar - 28.A. Donos, J.P. Gauntlett, JHEP
**1411**, 081 (2014)ADSCrossRefGoogle Scholar - 29.A. Cisterna, C. Erices, X.M. Kuang, M. Rinaldi, Phys. Rev. D
**97**(12), 124052 (2018)ADSMathSciNetCrossRefGoogle Scholar - 30.A. Cisterna, L. Guajardo, M. Hassaine, arXiv:1901.00514 [hep-th]
- 31.A. Cisterna, M. Hassaine, J. Oliva, M. Rinaldi, Phys. Rev. D
**96**(12), 124033 (2017)ADSMathSciNetCrossRefGoogle Scholar - 32.H. Maeda, M. Hassaine, C. Martinez, Phys. Rev. D
**79**, 044012 (2009)ADSCrossRefGoogle Scholar - 33.E. Ayon-Beato, C. Martinez, J. Zanelli, Gen. Relat. Gravit.
**38**, 145 (2006)ADSCrossRefGoogle Scholar - 34.E. Babichev, C. Charmousis, M. Hassaine, JCAP
**1505**, 031 (2015)ADSCrossRefGoogle Scholar - 35.A. Cisterna, M. Hassaine, J. Oliva, M. Rinaldi, Phys. Rev. D
**94**(10), 104039 (2016)ADSMathSciNetCrossRefGoogle Scholar - 36.K. Balasubramanian, J. McGreevy, Phys. Rev. D
**80**, 104039 (2009)ADSMathSciNetCrossRefGoogle Scholar - 37.G. Bertoldi, B.A. Burrington, A. Peet, Phys. Rev. D
**80**, 126003 (2009)ADSMathSciNetCrossRefGoogle Scholar - 38.E. Ayon-Beato, A. Garbarz, G. Giribet, M. Hassaine, Phys. Rev. D
**80**, 104029 (2009)ADSMathSciNetCrossRefGoogle Scholar - 39.H. Lu, Y. Pang, C.N. Pope, J.F. Vazquez-Poritz, Phys. Rev. D
**86**, 044011 (2012)ADSCrossRefGoogle Scholar - 40.E. Babichev, C. Charmousis, M. Hassaine, JHEP
**1705**, 114 (2017)ADSCrossRefGoogle Scholar - 41.X.H. Ge, Y. Tian, S.Y. Wu, S.F. Wu, S.F. Wu, JHEP
**1611**, 128 (2016)ADSCrossRefGoogle Scholar - 42.S. Cremonini, H.S. Liu, H. Lu, C.N. Pope, JHEP
**1704**, 009 (2017)ADSCrossRefGoogle Scholar - 43.N. Bhatnagar, S. Siwach, Int. J. Mod. Phys. A
**33**(04), 1850028 (2018)ADSCrossRefGoogle Scholar - 44.S. Cremonini, M. Cvetic, I. Papadimitriou, JHEP
**1804**, 099 (2018)ADSCrossRefGoogle Scholar - 45.J. Tarrio, S. Vandoren, JHEP
**1109**, 017 (2011)ADSCrossRefGoogle Scholar - 46.G.W. Gibbons, S.W. Hawking, Phys. Rev. D
**15**, 2752 (1977)ADSCrossRefGoogle Scholar - 47.E. Ayon-Beato, A. Garbarz, G. Giribet, M. Hassaine, JHEP
**1004**, 030 (2010)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}