Dilaton gravity, charged dust, and (quasi-) black holes

K. A. Bronnikov†

Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya Street, Moscow 119361, Russia;Institute of Gravitation and Cosmology, PFUR, 6 Miklukho-Maklaya Street, Moscow 117198, Russia;

and I. Kant Baltic Federal University, Alexander Nevsky Street 14, Kaliningrad 236041, Russia

J. C. Fabris‡ and R. Silveira*

Departamento de Física, Universidade Federal do Espírito Santo, Avenida Fernando Ferrari 514,Campus de Goiabeiras, CEP 29075-910 Vitória, Espirito Santo, Brazil

O. B. Zaslavskii§

Department of Physics and Technology, Kharkov V. N. Karazin National University,4 Svoboda Square, Kharkov 61077, Ukraine and Institute of Mathematics and Mechanics,

Kazan Federal University, 18 Kremlyovskaya Street, Kazan 420008, Russia(Received 21 December 2013; published 22 May 2014)

We consider Einstein-Maxwell-dilaton gravity with charged dust and interaction of the formPðχÞFμνFμν, where PðχÞ is an arbitrary function of the dilaton field χ that can be normal or phantom.For any regular PðχÞ, static configurations are possible with arbitrary functions g00 ¼ expð2γðxiÞÞ (i ¼ 1,2, 3) and χ ¼ χðγÞ, without any assumption of spatial symmetry. The classical Majumdar-Papapetrousystem is restored by putting χ ¼ const. Among possible solutions are black-hole (BH) and quasi-black-hole (QBH) ones. Some general results on BH and QBH properties are deduced and confirmed byexamples. It is found, in particular, that asymptotically flat BHs and QBHs can exist with positive energydensities of matter and both scalar and electromagnetic fields.

DOI: 10.1103/PhysRevD.89.107501 PACS numbers: 04.70.Dy, 04.40.Nr, 04.70.Bw

An important type of static charged dust configurationsis represented by the Majumdar-Papapetrou (MP) solution[1,2]; it comprises an equilibrium between gravitationalattraction and electric repulsion without any spatial sym-metry assumption: equilibrium is established for any spatialshape of the charged dust cloud provided the charge tomass density ratio takes everywhere the proper value,ρe=ρm ¼ �1 in natural units (c ¼ G ¼ 1).The MP system was recently revived in a new context,

that of the so-called quasi-black holes (QBHs) [3–9]. Usingthe fact that in this solution the force balance implies acharge-to-mass ratio similar to that in the vacuum extremalReissner-Nordstrom solution, a configuration has beenproposed where such a starlike object has a size very closeto the horizon radius. Such a system looks, for a distantexternal observer, quite similar to a true BH, though anevent horizon has not been formed.We here extend this treatment to include a dilatonic

scalar field, which can be partly motivated by studies instring theory. Along with general observations on possibleequilibrium configurations [to be called dilatonic MP(DMP) systems], we consider BHs and QBHs supportedby certain electric and scalar charge distributions. Inparticular, we try to find phantom-free configurations,

i.e., those able to exist with positive-definite energydensities of matter and both fields.This problem has been considered in a Ph.D. thesis of

one of the co-authors of this paper, Robson Silveira, whodied in 2009 before completing his study. He obtainedsome initial results indicating that such scalar QBHsare really possible and described some of their mainproperties. Our goal here is to briefly report on a moregeneral analysis strongly developing his findings. A moredetailed presentation can be found in Ref. [10].Consider the Lagrangian (c ¼ G ¼ 1)

L¼ 1

16π½Rþ 2εð∂χÞ2 −F2PðχÞ�þLmþAμjμþ Jχ; (1)

where ε ¼ �1 (ε ¼ 1 for a normal scalar field χ), Lm isthe Lagrangian of matter, J is the scalar charge density,F2 ≡ FαβFαβ (Fμν ¼ ∂μAν − ∂νAμ, the electromagneticfield), jμ ¼ ρeuμ is the 4-current, and uμ is the 4-velocity.We do not fix the sign of PðχÞ to provide correspondencewith [11,12]. Following the ideas of the MP solution, weconsider a static equilibrium with the metric

ds2 ¼ e2γdt2 − e−2γhikdxidxk; (2)

and assume only the electric components F0i ¼ −Fi0 ¼ ϕi

to be nonzero among Fμν; γ, hik, ϕ, χ are functions of xi,i ¼ 1, 2, 3; hik is the Euclidean metric, in general, incurvilinear coordinates. We use the notations γi ¼ ∂iγ,

*Deceased.†kb20@yandex.ru‡fabris@pq.cnpq.br§zaslav@ukr.net

PHYSICAL REVIEW D 89, 107501 (2014)

1550-7998=2014=89(10)=107501(5) 107501-1 © 2014 American Physical Society

ϕi ¼ ∂iϕ, etc; spatial indices are raised and lowered withthe metric hik and its inverse hik. Also, uμ ¼ δμ0e

−γ .The equations for χ and ϕ and the relevant combinations

of the Einstein equations can be written in the followingform:

2εe2γΔχ þ Pχϕiϕi ¼ −8πJ; (3)

∇iðe−2γPϕiÞ ¼ 4πρee−3γ; (4)

e2γðγiγk þ εχiχkÞ ¼ Pϕiϕ

k; (5)

e2γðΔγ − γiγi − εχiχiÞ ¼ 4πρm; (6)

where∇i and the Laplace operatorΔ ¼ ∇i∇i are defined interms of the metric hik. Equation (5) does not contain thedensities; hence it holds both in vacuum and in matter;Eq. (6) is a convenient expression for ρm in terms ofγðxÞ and χðxÞ. The Einstein equations also lead to theequilibrium condition

ρmγi − ρeϕie−γ ¼ Jχi: (7)

The tensor equation (5) implies that γ, χ, and ϕ arefunctionally related, and if γ ≠ const, we can put ϕ ¼ ϕðγÞ,χ ¼ χðγÞ; Eq. (5) then reduces to

e2γð1þ εχ2γÞ ¼ Pϕ2γ : (8)

Hence we have the following arbitrariness: for any PðχÞand any three-dimensional (3D) profile γðxiÞ, even morethan that, for an arbitrary scalar field distribution χ ¼ χðγÞ,we findϕðγÞ from (5), and the remaining field equations (3),(4), and (6) give us the mass, electric, and scalar chargedistributions that support this field configuration.In what follows wewill try to obtain examples of BH and

QBH configurations in the simplest case of sphericalsymmetry, and of special interest can be those where allkinds of matter are “normal,” i.e., P > 0, ε ¼ þ1,and ρm ≥ 0.The classical MP system is reproduced if we put

χ ¼ const, PðχÞ≡ 1, and we necessarily obtainjρej ¼ ρm. On the contrary, putting ϕ ¼ const, we obtainMP-like systems with an arbitrary function γðxiÞ, existingonly with a phantom χ field, as follows from Eq. (8).In the case of spherical symmetry, the metric (2) reads

ds2 ¼ e2γdt2 − e−2γðdx2 þ x2dΩ2Þ; (9)

where x is a radial coordinate and dΩ2 is the line elementon a unit sphere. The usual spherical (areal) radius isrðxÞ ¼ xe−γ . Our set of equations takes the form

2εx−2e2γðx2χ0Þ0 þ Pχϕ02 ¼ −8πJðxÞ; (10)

x−2ðPe−2γx2ϕ0Þ0 ¼ 4πρee−3γ; (11)

e2γðγ00 þ 2γ0=x − γ02 − εχ02Þ ¼ 4πρm; (12)

γ02 þ εχ02 ¼ e−2γPϕ02; (13)

ρmγ0 − ρeϕ

0e−γ ¼ Jχ0; (14)

where the prime denotes d=dx. The above arbitrarinesstransforms here into the freedom of choosing the functionsγðxÞ and χðxÞ even if the coupling function PðχÞ has beenprescribed from the outset. All other quantities are thenfound from Eqs. (10–14).It is of interest how to choose the arbitrary functions in

order to obtain a starlike configuration with a regular centeror a BH. It is also of interest to seek phantom-freeconfigurations such that ε ¼ þ1 and ρm ≥ 0.A regular center is obtained in the metric (9) at x ¼ 0 if

and only if γðxÞ ¼ γc þOðx2Þ, γc ¼ const. Using a Taylorexpansion for e2γ ≡ AðxÞ at small x, one can show thatρm > 0 near the center requires that g00 ¼ AðxÞ shouldhave there a minimum.Near a horizon we must have e2γ ∼ ðx − xhorÞn, where

n ∈ N is the order of the horizon. From (9) it is clear that ahorizon of finite radius rhor ¼ xe−γjx¼xhor is only possiblewith xhor ¼ 0 and n ¼ 2 (a double or extremal horizon).Thus at small x we can write AðxÞ ¼ 1

2A2x2 þ 1

6A3x3 þ � � �,

Ai ¼ const, A2 > 0. Assuming that χ and χ0 are finite at thehorizon, we obtain ρm ∼ x2, but it can be of any signwithout a direct correlation with ε. From the field equationsit follows that ρe ∼ x or possibly ρe ¼ oðxÞ, while Jgenerically tends there to a finite limit. Thus such con-figurations, being in general perfectly regular and smooth,still contain an anomaly: the density ratios ρe=ρm and J=ρmare infinite at the horizon.For dust balls of finite size placed in vacuum, the external

domain is described by the corresponding “vacuum”Einstein-Maxwell-dilaton (EMD) solution; however, suchsolutions to the field equations are only known for somespecial choices of PðχÞ, e.g., P ¼ e2λχ [13–15]. Therefore,instead, we consider asymptotically flatmatter distributionswith a smoothly decaying density. At large x we can take

AðxÞ ¼ 1 −2mx

þ q2�x2

þ � � � ; χðxÞ ¼ χ∞ þ χ1xþ � � � ;

(15)

and Eq. (12) then yields

4πρm ¼ 1

x4ð−3m2 þ q2� − εχ21Þ þ oðx−4Þ: (16)

BRIEF REPORTS PHYSICAL REVIEW D 89, 107501 (2014)

107501-2

This clearly shows that large charges q are necessary forobtaining ρm > 0 if ε ¼ þ1. [Note that the extremeReissner-Nordström solution with the charge q ¼ m cor-responds in the notation (15) to q2� ¼ 3m2.] The densities ρeand J also behave in general as 1=x4 at large x.Integral charges.—The field at flat spatial infinity is

characterized by integral charges: the electric charge q suchthat the electric field strength is ϕ0 ¼ q=x2 þ oð1=x2Þ, thescalar charge D such that χ0 ¼ D=x2 þ oð1=x2Þ, and themass m corresponding to the Schwarzschild asymptoticeγ ≈ 1 −m=x; hence γ0 ≈m=x2 (note that x ≈ r at large x).A relation between these three quantities directly followsfrom Eq. (13). Indeed, multiply (13) by x4 and take the limitx → ∞ to obtain

m2 − q2 þ εD2 ¼ 0; (17)

since eγ → 1 and P → 1 (assuming that a weak electro-magnetic field should be Maxwell). This generalizes asimilar relation (2.12) from [12], written there for vacuumEMD systems with PðχÞ ∼ e2λχ .Thus, as compared to the MP system where q ¼ �m, a

balance in the DMP system requires m2 > q2 if ε ¼ −1(both electric and phantom scalar fields are repulsive), butm2 < q2 with a canonical, attractive scalar field.Equation (17) is valid for all asymptotically flat

(islandlike) EMD systems since they are approximatelyspherically symmetric in the asymptotic region.Quasi-black holes.—By definition, in some region r ≤

r�ðcÞ of a QBH it holds that eγ ∼ c, where c is a smallparameter, and the limit c → 0 usually corresponds to aBH. The most general static, spherically symmetric QBH inour problem setting is a system with the metric (9) and aregular center, and at small x we can write

e2γ ≡ Aðx; cÞ ¼ A0ðcÞ þ1

2A2ðcÞx2 þ � � � ; (18)

where A0ðcÞ → 0 as c → 0 while A2ð0Þ is finite. Withoutloss of generality we can assume

e2γ ¼ x2 þ c2

f2ðx; cÞ ; (19)

where f is a smooth function that has a well-definednonzero limit c → 0. The value c ¼ 0 in (19) correspondsto an extreme BH metric with a horizon at x ¼ 0. Inparticular, taking fðx; 0Þ ¼ xþm, we obtain the extremeReissner-Nordström metric. At small enough c and x≲ c,e2γ ¼ Oðc2Þ is arbitrarily small.Let us stress that, given (19), the region, where the

“redshift function” eγ is small, is itself not small at all.Indeed, suppose fðx; cÞ ¼ Oð1Þ, and c ≪ 1. Then theradius rðcÞ of the sphere x ¼ c (which belongs to the

high redshift region) is fðc; cÞ= ffiffiffi2

p ¼ Oð1Þ; the distancefrom the center to this sphere,

Rc0 e

−γdx, is also Oð1Þ.Example 1. Let us choose the metric function

eγ ¼ zmþ 2z− y

; y≔ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ a2

p; z≔

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ c2

p;

(20)

with certain positive constantsm, a, c. At small and large xwe have

x → 0∶ e2γ ¼ c2

ðm− aþ 2cÞ2 þ x2m− aþ c2=aðm− aþ 2cÞ3 þOðx4Þ;

(21)

x → ∞∶ e2γ ¼ 1 −2mx

þ 3m2 þ a2 − c2

x2þOðx−3Þ: (22)

The system has a regular center and is asymptotically flat,and m is the Schwarzschild mass. Assuming

c < a < m; (23)

we can be sure that ρm > 0 near the center since eγ has aminimum there (see above). For ρm there is a bulkyexpression leading to ρm > 0 for proper choices of thedilaton field profile χðxÞ with ε ¼ þ1 under the condition(23). It is the case, for instance, if we assume

χ0 ¼ b=y2; b ¼ const > 0 (24)

with sufficiently small b.The expressions for the electric and scalar charge

densities are bulky, but their particular form can addnothing to our understanding of the situation; it is onlyimportant that they are finite and regular.The limit c → 0 leads to an extreme BH metric,

eγ ¼ xmþ 2x − y

; y ≔ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ a2

p: (25)

We thus obtain an asymptotically flat BH without phan-toms. With (24) for χ and ε ¼ þ1, we obtain from (25)

4πρm ¼ x2½ða2 þ b2Þy − b2ð2xþmÞ�y4ð2x − yþmÞ3 : (26)

We have ρm > 0 at all x > 0 in a certain region of theparameter space. Thus, putting m ¼ 1 (fixing the units)and a ¼ 0.5 (for example), we find that ρm > 0 for0 < b < b0 ≈ 0.369.The expressions for ρe and J are cumbersome; it is only

important that, for a generic choice of PðχÞ, they areeverywhere finite and regular and behave at the horizon asdescribed above.

BRIEF REPORTS PHYSICAL REVIEW D 89, 107501 (2014)

107501-3

Example 2. Our framework allows for describing poly-centric systems, with any number of mass concentrations.For instance, one can consider the metric (2) in Cartesiancoordinates xi ¼ ðx; y; zÞ (so that hik ¼ δik) and choose

e−γðxiÞ ≡ fðxiÞ ¼ 1

n

Xna¼1

faðXaÞ; (27)

where fa are functions of Xa ≔ jxi − xiaj, xia being the(fixed) coordinates of the ath center. As fa, one can takeany functions providing asymptotically flat sphericallysymmetric solutions, e.g., BHs or QBHs. A completesolution is obtained after choosing the function χðγÞ, orequivalently χðfÞ, which should be regular at all relevantvalues of f and decay sufficiently rapidly at spatial infinity,as f → 1.What follows is an example of a system of two QBHs: let

fðxiÞ ¼ m1 þ z12z1

þm2 þ z22z2

; (28)

χðfÞ ¼ 1

2bðf − 1Þ2; z1 ≔ ðj~x − ~x1j2 þ c21Þ1=2;

z2 ≔ ðj~x − ~x2j2 þ c22Þ1=2;~x1 ¼ ð0; 0; aÞ; ~x2 ¼ ð0; 0;−aÞ; (29)

with constants m1 > 0, m2 > 0, a > 0, b ≥ 0, c1 ≥ 0, andc2 ≥ 0. The electric potential ϕ and all densities are foundfrom Eqs. (8), (3), (4), and (6). In particular, for the massdensity we obtain

4πρm ¼ 1

f2ðxiÞ�

3m1c21z21ðm1 þ z1Þ3

þ 3m2c122

z22ðm2 þ z2Þ3

− εb2ðf − 1Þ2fifi�: (30)

The special case b ¼ 0 corresponds to a bicentric MPconfiguration. If c1 or c2 is zero, the corresponding“center” is a BH, while at small nonzero ca it is a QBH.Figure 1 shows the 3D behavior of the metric function

e2γ ≡ f−2ðxiÞ and the mass density ρmðxiÞ for the chosenexample of a system of two QBHs for the specifiedparameter values. Evidently, the density is everywherepositive in both cases in Fig. 1 [middle (a MP system)and right (a DMP system with a canonical scalar field)],although inclusion of a scalar field makes it smaller.In conclusion, let us enumerate the main results.(1) It has been shown that, with the Lagrangian (1),

static configurations are possible with arbitraryfunctions g00 ¼ e2γðxiÞ (i ¼ 1, 2, 3) and χ ¼ χðγÞ,for any regular coupling function PðχÞ, without anyassumption of spatial symmetry.

(2) There are purely scalar analogs of MP systems, butonly with phantom scalar fields.

(3) There is a universal balance condition, (17), betweenthe Schwarzschild mass and the electric and scalarcharges, valid for any asymptotically flat DMPsystems, including those with horizons and/orsingularities. It generalizes the results previouslyobtained for special cases (e.g., [12]).

(4) In the case of spherical symmetry, the existenceconditions have been formulated for BH and QBHconfigurations with smooth matter, electric charge,and scalar charge density distributions. It turns out thathorizons inDMPsystems are secondorder (extremal),in agreement with the general properties of QBHs [8].

(5) Examples of phantom-free spherically symmetricBH and QBH solutions have been obtained, and anexample of a phantom-free system of two QBHs.

ACKNOWLEDGMENTS

We thank CNPq (Brazil) and FAPES (Brazil) for partialfinancial support.

2

0

2

x

5

05

z

0.2

0.4

0.6

20

2

x

50

5

z

0.00

0.05

0.10

2

02

x

50

5

z

0.00

0.05

0.10

FIG. 1 (color online). Plots for Example 2, sections y ¼ 0 of different 3D profiles for a system of two identical QBHs. Left: the metricfunction e2γðx;y;zÞ form1 ¼ m2 ¼ 1, a ¼ 1.5, c1 ¼ c2 ¼ 0.2. Middle: the density ρmðx; y; zÞ for the same parameters and b ¼ 0, i.e., for apure MP system. Right: the same for ε ¼ þ1 and b ¼ 0.07, i.e., for a DMP system with the specified χðfÞ.

BRIEF REPORTS PHYSICAL REVIEW D 89, 107501 (2014)

107501-4

[1] S. D. Majumdar, Phys. Rev. 72, 390 (1947).[2] A. Papapetrou, Proc. R. Irish Acad., Sect. A 51, 191

(1947).[3] J. P. S. Lemos and E. J. Weinberg, Phys. Rev. D 69, 104004

(2004).[4] J. P. S. Lemos and O. B. Zaslavskii, Phys. Rev. D 76,

084030 (2007).[5] J. P. S. Lemos and O. B. Zaslavskii, Phys. Rev. D 82,

024029 (2010).[6] J. P. S. Lemos and O. B. Zaslavskii, Phys. Lett. B 695, 37

(2011).[7] J. P. S. Lemos and V. T. Zanchin, Phys. Rev. D 81, 124016

(2010); arXiv:1004.3574[8] J. P. S. Lemos, Sci. Proc. Kazan State Univ. 153, 215 (2011).

[9] R. Meinel and M. Hütten, Classical Quantum Gravity 28,225010 (2011).

[10] K. A. Bronnikov, J. C. Fabris, R. Silveira, and O. B.Zaslavskii, arXiv:1312.4891.

[11] G. Clément, J. C. Fabris, and M. E. Rodrigues, Phys. Rev. D79, 064021 (2009).

[12] M. Azreg-Ainou, G. Clément, J. C. Fabris, and M. E.Rodrigues, Phys. Rev. D 83, 124001 (2011);arXiv:1102.4093.

[13] K. A. Bronnikov and G. N. Shikin, Izv. Vuzov SSSR Fiz. 9,25 (1977); Russ. Phys. J. 20, 1138 (1977).

[14] G.W.Gibbons andK.Maeda, Nucl. Phys.B298, 741 (1988).[15] D. Garfinkle, G. T. Horowitz, and A. Strominger, Phys. Rev.

D 43, 3140 (1991); 45, 3888(E) (1992).

BRIEF REPORTS PHYSICAL REVIEW D 89, 107501 (2014)

107501-5