Quantum Black Holes: the Event Horizon as a Fuzzy Sphere

DIAS-STP-04-12 Quantum Black Holes: the Event Horizon as a Fuzzy Sphere Brian P. Dolan∗ Dept. of Mathematical Physics, NUI, Maynooth, Ireland and School of Theoretical Physics Dublin Institute for Advanced Studies 10 Burlington Rd., Dublin 8, Ireland August 26, 2005 Abstract Modeling the event horizon of a black hole by a fuzzy sphere it is shown that in the classical limit, for large astrophysical black-holes, the event horizon looks locally like a non-commutative plane with non-commutative parameter dictated by the Planck length. Some suggestions in the literature concerning black hole mass spectra are used to derive a formula for the mass spectrum of quantum black holes in terms of four integers which define the area, angular momentum, electric and magnetic charge of the black hole. We also suggest how the classical bounds on extremal black holes might be modified in the quantum theory. 1 Introduction Bekenstein’s suggestion that the surface area of a black hole is related to entropy and that the entropy should in fact be proportional to the area [1], ∗[email protected] 1 was triumphantly vindicated by Hawking’s calculation of the black hole temperature and entropy as a function of area [2]. If the entropy is to be finite it then necessary that there be a finite number of degrees of freedom associated with the event horizon area – it should be quantised [3]. Quantising the event horizon is very reminiscent of the concept of a “fuzzy sphere”, S2 F [4], in which points are “smeared out” and the geometry becomes non-local. In this paper we shall investigate modeling a black hole event horizon with a fuzzy sphere and show that this idea fits nicely with many of Bekenstein’s suggestions of treating a black hole as a particle, [5] [6] (a point of view also strongly advocated by ’t Hooft [7]). It has been suggested that the area of a black hole should have a quantised spectrum A=a(N+η)l2 P,(1) with N= 1,2..., and a > 0, η > −1 undetermined constants (lP= qGN¯h/c3is the Planck length), [3, 6, 8]). This idea has since been developed further in [9] and discretisation of the horizon has also been postulated by ’t Hooft [10]. It was suggested some time ago that a black hole event horizon might be modeled by a fuzzy sphere [11]. It is shown in section 2 that, in a fuzzy sphere model in the classical limit N→ ∞, the neighbourhood of a point on the event horizon locally looks like a non-commutative plane with non-commutativity parameter θ=al2 P 4π(2) where ais a numerical constant of order one related to the event horizon area by (1). A relation between quantisation of the event horizon area and the non-commutative plane was suggested in [16]. Non-commutativity on the event horizon was also suggested in [17] and a direct approach to deriving non-commutativity in black hole physics was recently initiated in [18]. Part of the characterisation of a fuzzy sphere is an irreducible representation of SU(2) of dimension N= 2k+1, with keither integral or half-integral. Functions on the fuzzy sphere are then represented by N×Nmatrices acting on an N-dimensional Hilbert space. We argue in the following that it is natural to take the area of the event horizon to be A= 4π(2k+ 1)l2 P(3) so that a= 4πand η= 0 above. The value of a= 4πthat is natural in a fuzzy sphere construction has also been found in the semi-classical approach of [12] and a mini-superspace approach to black hole quantisation in [13] [14]. 2 An equal spaced area spectrum like that of (3) was found in [15] though the prefactor was undetermined, With the values a= 4πand η= 0 above we show that the mass spectrum for black holes suggested by Bekenstein [6] is modified to give: M2 k,j,qe=((2k+ 1 + αq2 e)2+ 4j(j+ 1) 4(2k+ 1) )mP 2,(4) where jis integral or half-integral and qeis an integer, representing angular momentum and electric charge respectively, α=e2/¯hc is the fine structure constant and mP=qc¯h/GNis the Planck mass (there is a modification of this formula when magnetic monopoles are included). The smallest possible mass for a black hole in this scheme is therefore M=1 2mP,(5) when k=j=qe= 0. For given jand qethe quantum number kis bounded below by (2k+ 1)2≥4j(j+ 1) + α2q4 e.(6) In particular, for a zero charge black hole, the classical bound J2≤M4(7) (in units with GN=c= 1) is replaced by J2≤M4−π2l4 P A2¯h2.(8) The layout of the paper is as follows. In section 2 the quantisation of the area arising from the fuzzy sphere hypothesis is discussed for Schwarzschild black holes and the projection to the non-commutative plane is explained. Section 3 analyses non-zero angular momentum and the associated bounds on the mass while section 4 does the same for charged and rotating holes. The relation to entropy is discussed in section 5 and the results are summarised in section 6 2 Schwarzschild Black Holes The 2-dimensional sphere is a symplectic manifold — a phase-space in physics language, albeit a compact one. This phase-space can be quantised to give 3 S2 F. The concept of a point on S2 Fis not defined but instead the points are smeared out into a finite number of phase-space ‘cells’, hence the name ‘fuzzy’, [4]. For any integer, N= 2k+ 1 with klabelling SU(2) representations either integral or half-integral, S2 Fhas Ncells and operators on phase-space are N×Nmatrices acting on a N-dimensional Hilbert space, [19]. Visually S2 Fmight be viewed as being like the surface of Jupiter, with the belts being unit cells, but this is not essential since, as in any quantum phase space, only the area of the fundamental cells, not their shape, is fixed. If we picture the event horizon of a black hole as a fuzzy sphere then the total area of the event horizon is naturally a multiple of the area of a fundamental unit cell. Suppose the unit cells have area al2 P, with aa positive dimensionless constant of order one. Then the total area of the event horizon is A=Nal2 P,(9) and, since N= 2k+ 1, we conclude that η= 0 in equation (1). For a non-rotating black hole with zero charge (9) immediately implies that the Schwarzschild radius RSis also quantised R2 S=A/4π=Nal2 P 4π.(10) To avoid messy factors of 4πit is convenient to define ¯ A=A/4πand ¯a= a/4πso R2 S=¯ A=N¯al2 P.(11) The mass of the hole can now be expressed as M=RSc2 2GN =√N¯alPc2 2GN =√N¯amP 2.(12) The hypothesis that the event horizon is a fuzzy-sphere thus immediately leads us to conclude that black hole masses are quantised M2=N¯a 4mP 2(13) with Na positive integer. For astrophysical black holes Nis so large that the quantum nature of the mass would be unobservable, but in the final stages of black hole evaporation the black hole would go through a series of discrete states until the final state is reached, with N= 1 (i.e. k= 0) and residual mass M0=√¯amP/2. Thus in this picture evaporating black holes do not disappear but must necessarily leave behind a residual hole of the order of the Planck mass. As remarked 4 in [5] the situation is reminiscent of the Bohr model of the atom in which orbiting electrons can only occupy a discrete set of orbits, dictated by the Bohr-Sommerfeld constraint Hpdq = 2πN¯hon the orbitals, and decaying electrons must finally lodge in the ground state thus rendering atoms stable. Non-commuting co-ordinates on the fuzzy-sphere can be represented globally by three N×Nmatrices Xi,i= 1,2,3, satisfying XiXi=R2 S1,(14) where 1is the N×Nunit matrix, with Xiproportional to the generators Liof SU(2) in the irreducible N×Nrepresentation, [Li,Lj] = iijkLk,LiLi=k(k+ 1)1.(15) From this we deduce that Xi=λkLi⇒[Xi,Xj] = iλkijkXk,(16) with λk:= √¯alPs2k+ 1 k(k+ 1).(17) At first glance it appears that, in the large Nlimit, the Xiin equation (16) become commutative and the commutative sphere is recovered, since λk→0 in the limit, but upon more careful consideration this is not in fact correct.1Heuristically this can be seen by focusing on a region near the south pole of a large black-hole, in the limit of large k. At the south pole X1and X2are transverse to the surface and X3is normal to it, with X3≈ −Rsand [X1,X2] = iλkX3=iλ2 kL3.(18) In a basis in which L3=   k... −k   (19) is diagonal the eigenvalue X3≈ −Rscorresponds to the minimum eigenvalue −kof L3so [X1,X2]≈ −iλ2 kk(20) and, as k→ ∞, [X1,X2] = −2i¯al2 P.(21) 1An earlier version of this paper contained an error on this point and I am grateful to Al Stern and Eli Hawkins for bringing this to my attention. 5 Hence, in an infinitesimal region around the pole, the event horizon looks like a non-commutative plane in the infinite klimit. This observation can be put on a more formal footing using the analysis of [20] (see also [21]) in which is shown that the k→ ∞ limit of (16) describes a non-commutative plane under stereographic projection. This is seen by defining X±=X1±iX2and performing the analogue of stereographic projection for fuzzy co-ordinates: Z=X−(1−X3/RS)−1,Z†= (1−X3/RS)−1X+.(22) Then, for large k, [Z,Z†] = 2λkRS(1−X3/RS)−2+o(1/k).(23) Now, although the operator X3/RShas eigenvalues between −1 to +1 inclusive, only a very small range above −1 is necessary to cover the whole Z-plane. To see this observe that 1 2(ZZ†+Z†Z) = R2 S 1+X3 RS! 1−X3 RS!−1 +o(1/k).(24) Writing X3/RS=−1+T/R2 S, where T/R2 Shas eigenvalues between 0 and 2, this reads 1 2(ZZ†+Z†Z) = 1 2T 1−T 2R2 S!−1 +o(1/k).(25) Now, in the k→ ∞ limit, we can cover the whole of the Z-plane by projecting all operators onto the subspace spanned by of eigenvectors of Twith eigenvalues in the range 0 to ¯a√k l2 P. Hence, for k→ ∞,T/R2 S→0 in (25) and we can replace X3 RSwith −1in (23) to give ZZ†+Z†Z=Tand [Z,Z†] = θ(26) with non-commutativity parameter θ= lim k→∞ λkRS 2= lim k→∞ ¯al2 P 2 (2k+ 1) qk(k+ 1) = ¯al2 P.(27) The interesting conclusion of this analysis is that, even for large astrophysical black-holes, there is a vestige of non-commutativity at the Planck length. If the assumptions made here are correct the event horizon of a black-hole is a physical example of a system in which Connes’ non-commutative geometry manifests itself in the continuum. 6 3 Rotating Black Holes Now consider a rotating black hole with angular momentum J2=j(j+ 1)¯h2 and zero charge. The event horizon is still topologically a sphere, though not metrically a round sphere it still has a fuzzy description. The classical formula for the mass as a function of angular momentum and area (the Christodoulou-Ruffini mass [22]) is 2 M2=1 4¯ A+J2 ¯ A,(28) or ¯ A 2=M2+√M4−J2(29) (the positive square root is taken here because Ais the area of the outer horizon). From the above formula comes the bound J2≤M4,(30) otherwise ¯ Abecomes complex. Using (28) and (29) this is equivalent to J2≤1 4¯ A2.(31) Classically the maximum allowed angular momentum is when (30) is saturated: J2 max =M4=1 4¯ A2.(32) Consider the quantum version of (32). Using J2 max =jmax(jmax + 1)¯h2, together with the ansatz (9), gives jmax +1 22 = ¯a2k+1 22 +1 4.(33) Quantum mechanically the bound might not be saturated so all we can safely say is that jmax +1 22 ≤¯a2k+1 22 +1 4.(34) Suppose that the bound is saturated in the limit of large k, and hence large jmax, so that lim k→∞ J2 max M4= 1 ⇔lim k→∞ 4J2 max ¯ A2= 1 ⇔lim k→∞ j2 max k2= ¯a2.(35) 2Here we use units in which GN=c2= 1 to keep the formula clean, but ¯hwill be retained so as to highlight quantum phenomena. Hence l2 P=mP 2= ¯h. 7 Now the fuzzy sphere is associated with a Hilbert space whose maximum angular momentum is k, so it seems very natural to take jmax =k, in which case ¯a= 1. Then (32) must be modified to read J2 max/¯h2=jmax(jmax + 1) = 1 4(¯ A2/¯h2−1) (36) with ¯ A= (2k+ 1)¯h. (37) Note that a k= 0 black hole necessarily has j= 0 and is therefore a boson with spin zero. It is possible that there is a correlation between kand j, even away from extremality, and that integral jimplies integral kand half-integral jimplies half-integral k. Indeed the area spectrum found in [13] for non-rotating black holes requires integral kwhen j= 0 for a hole carrying zero charge, halfintegral konly appear for charged black holes in their analysis. The spectrum found in [14] for zero charge requires that jand kare both integral. The fuzzy sphere approach here does not impose any such restrictions. While a correlation between integral kand j, requiring that they be either both integral or both half-integral, seems plausible we have not found a proof that it is necessary. The fact that the difference between the quantum bound (36) and the classical bound (32) is independent of Ais a direct consequence of the choice ¯a= 1. Equation (28) now reads M2=(k(k+ 1) + j(j+ 1) + 1 4 (2k+ 1) )¯h. (38) The mass of a black hole of a given area (fixed k) with maximum allowed angular momentum is now M2(Jmax) = 1 4(8k(k+ 1) + 1 2k+ 1 )¯h. (39) In the quantum theory equation (32) is then replaced with J2 max =M4(Jmax)−¯h4 16 ¯ A2=1 4(¯ A2−¯h2),(40) so (30) is never saturated for finite k. In terms of jand kthe bound is (2k+ 1)2>4j(j+ 1).(41) 8 4 Charged Black Holes Including electric charge Qethe classical Christodoulou-Ruffini formula reads M2=1 ¯ A1 4(¯ A+Q2 e)2+J2(42) or, if magnetic monopoles with charge Qmare also included, M2=1 ¯ A1 4(¯ A+Q2)2+J2(43) where Q2=Q2 e+Q2 m.(44) With ¯ A= (2k+ 1)¯hand Qequantised in multiples of the electric charge e the quantum version of (43) becomes M2=    h2k+ 1 + αq2 e+α−1(qm/2)2i2+ 4j(j+ 1) 4(2k+ 1)     ¯h, (45) with qeand qmintegers (we use units with 4π0= 1 so that the fine structure constant is α=e2/¯hwhen c= 1, the factor of α−1/4 multiplying q2 mallows for the Dirac quantisation condition, QeQm=f N¯h/2 where f Nis an integer). Thus, as suggested in [6], the black hole mass is characterised by four discrete numbers: kand j, which can each be either integral or half-integral, and qe and qmwhich are both integers. This particle picture of black holes has also been a central theme in the work of ’t Hooft, [7] [10]. The general form of the spectrum (45) was derived by Bekenstein [6], the new ingredient here is that some of the constants differ as a consequence of the hypothesis that the event horizon is modeled by a fuzzy sphere. Demanding that ¯ Ain (43) is real gives the classical bound M4−Q2M2−J2≥0 (46) Defining ∆2:= M4−Q2M2−J2(47) (43) can be used to express ∆2in terms of the area ∆2=(¯ A2−Q4−4J2)2 16 ¯ A2.(48) 9 [25] J.M. Maldacena and A. Strominger, Phys.Rev.Lett. 77 (1996) 428, hep-th/9603060 [26] G.T. Horowitz, D.A. Lowe and J.M. Maldacena, Phys. Rev. Lett. 77 (1996) 430, hep-th/9603195 [27] R. Kallosh, A. Linde, T. Ortin, A. Peet and A. van Proeyen, Phys. Rev. D46 (1992) 5278; M. Cvetic and D. Youm, Phys. Rev. D53 (1996) 584, hep-th/9507090; M. Cvetic and A. Tseytlin, Phys. Rev. D53 (1996) 5619, hep-th/9512031; erratum ibid. D55 (1997) 3907; R. Kallosh and B. Koll, Phys. Rev. D53 (1996) 5344, hep-th/9602014 [28] A.Yu. Alekseev, A. Recknagel and V. Schomerus, JHEP 9909 (1999) 023, hep-th/9908040; Y. Hikida, M. Nozaki and Y. Sugawara, Nucl.Phys. B617 (2001) 117, hep-th/0101211; K. Hashimoto and K. Krasnov, Phys.Rev. D64 (2001) 046007, hep-th/0101145 [29] S. Iso, Y. Kimura, K. Tanaka and K. Wakatsuki, Nucl. Phys. B604 (2001) 121, hep-th/0101102 [30] L. Smolin, J. Math. Phys. 36 (1995) 6417; C. Rovelli, Phys. Rev. Lett. 77 (1996) 3288; K. Krasnov, Phys. Rev. D55 (1997) 3505; A. Ashtekar, J. Baez, A. Corichi, K. Krasnov, Phys. Rev. Lett. 80 (1998) 904; A. Ashtekar, J. Baez, K. Krasnov, Adv. Theor. Math. Phys. 4(2000) 1 [31] K.A. Meissner, Black Hole Entropy in Loop Quantum Gravity, gr-qc/0407052 [32] P.A. Zizzi, A Minimal Model for Quantum Gravity,gr-qc/0409069 [33] S. Hod, Phys. Rev. Lett. 81 (1998) 4293, gr-qc/9812002 [34] T. Azuma, K. Nagao and J. Nishimura, Perturbative Dynamics of Fuzzy Spheres at large N,hep-th/0410263; P. Castro-Villarreal, R. DelgadilloBlando and B. Ydri, Nuc. Phys. B704 (2005) 111, hep-th/0405201 16