We examine the issue of magnetic charge quantization in the presence of black
holes. It is pointed out that quantization of magnetic charge can lead to the
mass quantization for magnetically charged black holes. We also discuss some
implications for the experimental searches of magnetically charged black holes.Comment: RevTeX, 11 pages, Invited paper for the first editorial volume of the
  book series "Contemporary Fundamental Physics" by the Nova Science Publisher

Ignatiev, A. Yu.

Joshi, G. C.

Wali, Kameshwar C.

English

arXiv

We examine the issue of magnetic charge quantization in the presence of black holes. It is pointed out that quantization of magnetic charge can lead to the mass quantization for magnetically charged black holes. We also discuss some implications for the experimental searches of magnetically charged black holes

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Syracuse University SURFACE Physics College of Arts and Sciences 11-13-1998 Black Holes with Magnetic Charge and Quantized Mass A. Yu. Ignatiev The University of Melbourne G. C. Joshi The University of Melbourne Kameshwar C. Wali Syracuse University Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation arXiv:hep-ph/9811320v1 This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:hep-ph/9811320v1  13 Nov 1998UM-P-98/46, RCHEP-98/12Black holes with magnetic charge and quantized massA.Yu.Ignatiev, G.C.Joshi and Kameshwar C.WaliResearch Centre for High Energy Physics, School of Physics, University of Melbourne, Parkville3052, Victoria, AustraliaDepartment of Physics, Syracuse University, Syracuse, NY 13210(Invited paper for the first editorial volume of the book series ”Contemporary FundamentalPhysics” by the Nova Science Publishers)AbstractWe examine the issue of magnetic charge quantization in the presence ofblack holes. It is pointed out that quantization of magnetic charge can leadto the mass quantization for magnetically charged black holes. We also discusssome implications for the experimental searches of magnetically charged blackholes.Typeset using REVTEX1The purpose of this note is to draw attention to some aspects of the interplay betweengravity and the quantization of magnetic charge. The issue we are going to address is notonly of fundamental theoretical importance but also can be relevant for the experimentalsearches of monopoles.Let us view the problem of magnetic charge from two perspectives. First, the magneticcharge gRN of the extreme Reissner-Nordstrom black hole is related to its mass accordingto the equation,gRN =√GM, (1)where M is the mass of the black hole. This relation follows from an exact solution ofEinstein-Maxwell equations [1,2] , and is a purely classical result with no appeal to quantummechanics. Similarly, an extension to coupled Einstein- Yang-Mills-Higgs system leads tonon-abelian and abelian black holes with magnetic charges [3]. References to earlier andrelated work are contained in this paper. More recently, generalizations to slowly rotatingnon-Abelian black holes have been reported by M.S. Volkov and N. Straumann [4]. Second,the Dirac quantization condition [5] reads:gD =nh̄c2e. (2)We use the Gauss system of units where e2/h̄c = α.Now, what is the relationship between the Eqs.(1) and (2)? Should we equate theleft-hand sides of the above formulas? Or are they totally independent of each other?Surprisingly, this supposedly simple question, to the best of our knowledge, does not havea straightforward answer. What we really obtain here is a dilemma: on the one hand,we can insist that the Dirac quantization condition derived in flat space-time, should bevalid also in curved space-time in the presence of a black hole; on the other hand, we mayconjecture that there may exist some mechanism of evading the Dirac quantization conditionin the regions with strong gravitational fields. Both these alternatives have interestingconsequences. Which of them is realized in Nature seems to be an open question at present.For some particular cases it has been claimed that the Dirac quantization condition remainsintact in the presence of gravity [6]. However, there is no general proof that we know of,establishing the validity of the Dirac quantization condition in arbitrary curved space-time.Neither there are any models that would definitely establish a possibility of its violation.Therefore, we examine qualitatively each scenario and discuss the consequences.If we assume that the Dirac quantization condition is valid in the presence of gravitythen we can equate the left-hand sides of Eqs.(1) and (2) to obtain the equationM =nh̄c√G2e= nMP l2√α. (3)This says that the masses of Reissner-Nordstrom black holes are quantized and theminimum possible mass for the extreme magnetically charged black holes is given byMmin = 5.85MP l, (4)with an equi-spaced mass spectrum analogous to the energy spectrum of a quantum simpleharmonic oscillator. Whether such black holes are the final stages of more massive black2holes undergoing Hawking radiation and reaching a stable state is an interesting question.Anyway, theory aside, the range of magnetic monopole masses around the point 5.85MP lshould be considered seriously in the monopole search experiments.Now, let us consider the other alternative, namely, the possible violation of the Diracquantization condition in the presence of gravity. Not knowing the precise nature of itsviolation assume that the magnetic charge of a Reissner-Nordstrom black hole obeys only oneconstraint, that of Eq.(1), being unrestricted by Eq.(2). Thus, we have magnetic monopoleswith ”dequantized” magnetic charge determined by the mass [7]. Thus there is no restrictionon the range of allowed magnetic charges. However, let us confine our attention to chargesbetween zero and one Dirac unit, g = 68.5e and note that very small magnetic chargescorrespond to the black holes with the mass much less than the Planck mass for whichthe Compton wavelength h̄Mcis longer than the Schwarzschild radius 2GMc2. Therefore, theeffects of quantum gravity should be quite important for such black holes. For this reason itwould be safer to leave very small magnetic charges from our consideration. A particularlyinteresting case from the point of view of symmetry between electricity and magnetism, isg = e. This equality of magnetic charge of the black hole and the electric charge of theelectron would occur for the black hole massM =√αMP l = 0.08MP l (5)Another special case of interest is the black hole with the Planck mass for which the magneticcharge isgP l =e√α= 11.7e, (6)so gP l = 1 in dimensionless units (h̄ = c = 1). Black holes with the Planck mass have beencalled maximons in [8].In this context, it is interesting to recall that the reported monopole observations ofEhrenhaft [9] corresponded to the values of magnetic charge less than the standard Diraccharge of 68.5e, and hence they were given less credibility on the ground that they violatedDirac quantization condition.Are there any arguments in favour of the violation of the Dirac quantization conditionby the gravitational effects? Originally, the Dirac quantization condition appeared as thenecessary condition for the unobservability of the Dirac string. The Dirac string is the lineof a singularity in the vector potential describing the monopole. It starts at the monopolelocation and stretches to infinity. Analytically it can be written, for example, asADr = ADθ = 0, (7a)ADϕ =grtan θ2. (7b)The magnetic field corresponding to that vector potential is also singular and has the form[10] (here, n is the unit vector along the string, i.e. opposite the z-axis):∇×AD = gr3r + n4πgθ(−z)δ(x)δ(y). (8)Note that to obtain the above equation rigorously one has to treat all quantities involvedin the sense of distributions rather than regular functions [11]. Therefore, the string makes3a singular contribution to the energy momentum tensor of the electromagnetic field. Anatural question then arises: can such a string be observed by its gravitational effects? Thegravitational field of a gauge string has been studied in [12] (A gauge string has a finitemass µ per unit length whereas a Dirac string has infinite line density). It has been foundthat the spacetime around such a string is locally flat, but globally it is similar to a cone.It means that the range of the polar angle φ (measured around the string) is not 2π but0 ≤ φ ≤ 2π(1 − 4Gµ). This angular deficit has been called a conical singularity. Thepresence of conical singularity leads to several observable effects: double images of sourceslocated behind the string, anisotropy of the background radiation, and the creation of sheetsof matter in the wake of the string. These results strongly suggest that the Dirac stringmay become observable by its gravitational effects regardless of the value of the magneticcharge. 1 Thus we are led to conjecture that in the presence of gravitation it may becomeirrelevant whether the quantization condition for the Dirac monopole is fulfilled or violated.(Of course, an alternative interpretation of the same result would be to claim that the wholeconcept of the Dirac monopole is inconsistent with gravity.)Let us now turn to the situation with the t’Hooft-Polyakov type of monopoles. For thesemonopoles there exist two very different pictures: one uses a gauge with the “hedgehog”configuration of the scalar field; this gauge is completely free of any singularity lines (non-singular gauge). The other approach is very close to the picture of a Dirac monopole becauseof the presence of a string similar to the Dirac string (string gauge). These two pictures arebelieved to be physically equivalent because there exist a (singular) gauge transformationfrom one to the other. Therefore, if we choose to work in the string gauge we can againrequire the string to be unobservable and thus obtain the quantization condition.Now, we would like to think of the monopole coupled to gravity. One of the fundamentalquestions is: will the two gauges remain physically equivalent? If they are then we can useagain the same line of argument as above. Consequently, we can again conjecture that thequantization condition may become irrelevant in the presence of gravity. On the other hand,the non-singular gauge and the string gauge may become non-equivalent once the gravity isswitched on, in which case we do not know the form the quantization condition takes. Inother words, at the present state of our knowledge, arbitrary magnetic charges unconstrainedby the Dirac quantization condition seem not to be ruled out by sound theoretical arguments.This should be kept in mind in experimental searches.We now turn to the analysis of the experimental constraints on the flux of magneticallycharged black holes (“black poles” for short). The most stringent constraints come from1Note that there exists some evidence that the string may become visible even in flat space. In[13] the motion of a monopole in the magnetic field of an external electric current was analyzed.Roughly speaking, the physical results depend on how many times the monopole string winds uparound the current line. Another configuration where the strings turn out to be a problem hasbeen considered in [14]: a charged particle in the magnetic field of a Dirac monopole line. Furtherproblems related to the Dirac quantization condition and existence of strings have been raisedrecently in the context of quantum field theory with monopoles, see e.g. [15–17].It is not clearwhether or not the stringless formalisms of Wu-Yang and others [18–20] based on the fibre bundletheory would help in solving these problems.4the results of non- accelerator experiments devoted to the search of superheavy magneticmonopoles arising in Grand Unified Theories of strong, weak, and electromagnetic (but notincluding gravitational) interactions. These searches can be divided into two main classes:the first uses the effect of current induction caused by the monopole magnetic field andthe second is based upon the ionizing properties of monopoles. The induction experimentsgive the most direct upper limits on the monopole flux because they are independent of theunknown characteristics of the monopoles such as its mass and velocity. The best upperbounds on the monopole flux F from induction experiments are:F < 4.4 × 10−12cm−2sr−1s−1 [21], (9)F < 7.2 × 10−13cm−2sr−1s−1 [22] (10)F < 5 × 10−14cm−2sr−1s−1 [23] (11)These constraints can be taken over for the case of quantized black holes regardless of theirmass and velocity. As for the dequantized black holes, however, the situation is different.The reason is that the magnetic flux through a superconducting loop is quantized and canchange only in multiples of magnetic flux quantum φ0 = 12(4π× h̄c2e). Therefore, it is unclearto what extent the superconductive induction experiments are sensitive to the dequantizedmagnetic charges. A sufficient condition for a magnetic charge g to be detected in theinduction experiment reads4πg =φ0nl, (12)where nl is a geometric factor which is specific for the loop configuration adopted in aparticular experiment. For instance, for the original Cabrera’s detector nl = 4 since thereare 4 superconducting loops connected in a series. An interesting problem is how to designan experimental setup (or use the existing setups) in such a way that it would be capableof detecting both quantized and dequantized magnetic charges (including the charges notsatisfying Eq. (12)).Let us now turn to the experiments that use the ionizing properties of monopoles. Weconsider first the case of quantized black holes, assuming that the black holes with n ≥ 1are stable. The ionization experiments are much more sensitive to the theoretical inputthan the induction experiments. In particular, their sensitivity crucially depends on themonopole’s velocity: the monopole would be undetectable if its velocity were below thethreshold, somewhere between 10−3c and 10−4c. To find out the typical monopole velocity,we need to find the greater of the two characteristic velocities: the first is the velocity withwhich the monopole enters the Galaxy (∼ 10−3c) and the second is the typical monopolevelocity due to its acceleration by the galactic magnetic field, given byvn ≃ 3 × 10−3c(1016GeVMn)1/2. (13)The quantized magnetically charged black holes with masses M >∼ 6 × 1019 GeV can beviewed as ultraheavy monopoles (as contrasted with superheavy GUT monopoles with the5mass of the order of 1016 GeV). It is readily seen that the total spectrum of the quantizedblack poles can be divided into two parts: 1)“low-lying”, low charge black poles, correspond-ing to n <∼ 600 and 2)“very heavy ”, heavily charged black poles” with n >∼ 600. For thelow-lying black poles the galactic magnetic field is not strong enough to accelerate them,so their typical velocity is expected to be of the order of 10−3c. Therefore the astrophys-ical bounds on their flux obtained from the condition of survival of the galactic magneticfield (the Parker bound and its refinements) are inapplicable for them. On the contrary, forthe superheavy black poles their dynamics is dominated by the galactic magnetic field, somost of them are soon ejected out of the galaxy similar to the case of the ordinary GUTmonopoles. Thus, the low-lying black poles with the mass in the interval (6 − 3600)MP land, consequently, the magnetic charge in the range (1−600)g1 seem to be more interestingexperimentally. Although the Parker-type bounds do not hold for these poles, their flux cannevertheless be constrained at a similar level, F <∼ 10−15cm−2sr−1s−1, from the requirementthat the galactic density of the poles should not be unacceptably high. The best availablelimits from the ionization detectors on the ultraheavy monopole flux with velocities of theorder of 10−3c have the same order of magnitude. For instance,F < 5.6 × 10−15cm−2sr−1s−1 [24], F < 2.7 × 10−15cm−2sr−1s−1 [25] (14)Furthermore, if one takes into account the possibility of monopole catalysis of nucleon decay(Rubakov-Callan effect) [26]) then a somewhat stronger limit can be obtained [27] (it is,however, sensitive to the assumed catalysis cross-section):F < 5 × 10−16cm−2sr−1s−1. (15)To summarize, we have examined the issue of magnetic charge quantization in the presenceof black holes. It was pointed out that quantization of magnetic charge can lead to the massquantization for magnetically charged black holes. We have also discussed some implicationsfor the experimental searches of magnetically charged black holes.This work was supported in part by a grant from National Science Foundation, Division ofInternational Program (U.S.-Australia Cooperative Research). One of the authors (K.C.W)would also like to acknowledge the hospitality at the Institut des Hautes Etudes Scientifiquesduring the course of this work.6REFERENCES[1] S.D.Majumdar, Phys.Rev. 72, 390 (1947); A.Papapetrou, Proc.R.Irish Acad. Sect.A 51,191 (147)[2] J.B.Hartle and S.W.Hawking, Commun.Math.Phys. 26, 87 (1979)[3] For instance, see B.S.Balakrishna and K.C.Wali, Phys.Rev.D46, R522 (1992) N.A.Vietand K.C.Wali, Phys.Rev.D 51, 1664 (1995)[4] M.S.Volkov and N.Straumann, ZU-TH/97 hep-th/9704026[5] P.A.M.Dirac, Proc.Roy.Soc. (London) A133, 60 (1931)[6] R.J.Adler, Phys.Rev. D14, 392 (1976); A.Inomata and M.Trinkala, Phys. Rev. D18,1861 (1978); J.L.Friedman, S.Mayer and L.Parker, Phys.Rev. D17, 1957 (1978)[7] E.B.Bogomolnyi, Yad.Phys.24, 449 (1976); M.K.Prasad and C.H.Sommerfield,Phys.Rev.Lett. 35, 760 (1975)[8] M.A.Markov, Soviet Physics JETP 24, 584 (1967)[9] Ehrenhaft, Z.Phys.31, 478 (1930); V.F.Mikhailov, Phys.Lett. 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Black holes with magnetic charge and quantized mass

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