Quantum Black Holes: the Event Horizon as a Fuzzy Sphere

DIAS-STP-04-12 Quan um Black Holes: he E en Ho izon as a Fuzzy Sphe e B ian P. Dolan∗ Dep . o Ma hema ical Physics, NUI, Maynoo h, I eland and School o Theo e ical Physics Dublin Ins i u e o Ad anced S udies 10 Bu ling on Rd., Dublin 8, I eland Augus 26, 2005 Abs ac Modeling he e en ho izon o a black hole by a uzzy sphe e i is shown ha in he classical limi , o la ge as ophysical black-holes, he e en ho izon looks locally like a non-commu a i e plane wi h non-commu a i e pa ame e dic a ed by he Planck leng h. Some sugges ions in he li e a u e conce ning black hole mass spec a a e used o de i e a o mula o he mass spec um o quan um black holes in e ms o ou in ege s which de ine he a ea, angula momen um, elec ic and magne ic cha ge o he black hole. We also sugges how he classical bounds on ex emal black holes migh be modi ied in he quan um heo y. 1 In oduc ion Bekens ein’s sugges ion ha he su ace a ea o a black hole is ela ed o en opy and ha he en opy should in ac be p opo ional o he a ea [1], ∗bdolan@ hphys.may.ie 1 was iumphan ly indica ed by Hawking’s calcula ion o he black hole em- pe a u e and en opy as a unc ion o a ea [2]. I he en opy is o be ini e i hen necessa y ha he e be a ini e numbe o deg ees o eedom asso- cia ed wi h he e en ho izon a ea – i should be quan ised [3]. Quan ising he e en ho izon is e y eminiscen o he concep o a “ uzzy sphe e”, S2 F [4], in which poin s a e “smea ed ou ” and he geome y becomes non-local. In his pape we shall in es iga e modeling a black hole e en ho izon wi h a uzzy sphe e and show ha his idea i s nicely wi h many o Bekens ein’s sugges ions o ea ing a black hole as a pa icle, [5] [6] (a poin o iew also s ongly ad oca ed by ’ Hoo [7]). I has been sugges ed ha he a ea o a black hole should ha e a quan ised spec um A=a(N+η)l2 P,(1) wi h N= 1,2..., and a > 0, η > −1 unde e mined cons an s (lP= qGN¯h/c3is he Planck leng h), [3, 6, 8]). This idea has since been de eloped u he in [9] and disc e isa ion o he ho izon has also been pos ula ed by ’ Hoo [10]. I was sugges ed some ime ago ha a black hole e en ho izon migh be modeled by a uzzy sphe e [11]. I is shown in sec ion 2 ha , in a uzzy sphe e model in he classical limi N→ ∞, he neighbou hood o a poin on he e en ho izon locally looks like a non-commu a i e plane wi h non-commu a i i y pa ame e θ=al2 P 4π(2) whe e ais a nume ical cons an o o de one ela ed o he e en ho izon a ea by (1). A ela ion be ween quan isa ion o he e en ho izon a ea and he non-commu a i e plane was sugges ed in [16]. Non-commu a i i y on he e en ho izon was also sugges ed in [17] and a di ec app oach o de i ing non-commu a i i y in black hole physics was ecen ly ini ia ed in [18]. Pa o he cha ac e isa ion o a uzzy sphe e is an i educible ep esen a- ion o SU(2) o dimension N= 2k+1, wi h kei he in eg al o hal -in eg al. Func ions on he uzzy sphe e a e hen ep esen ed by N×Nma ices ac - ing on an N-dimensional Hilbe space. We a gue in he ollowing ha i is na u al o ake he a ea o he e en ho izon o be A= 4π(2k+ 1)l2 P(3) so ha a= 4πand η= 0 abo e. The alue o a= 4π ha is na u al in a uzzy sphe e cons uc ion has also been ound in he semi-classical app oach o [12] and a mini-supe space app oach o black hole quan isa ion in [13] [14]. 2 An equal spaced a ea spec um like ha o (3) was ound in [15] hough he p e ac o was unde e mined, Wi h he alues a= 4πand η= 0 abo e we show ha he mass spec um o black holes sugges ed by Bekens ein [6] is modi ied o gi e: M2 k,j,qe=((2k+ 1 + αq2 e)2+ 4j(j+ 1) 4(2k+ 1) )mP 2,(4) whe e jis in eg al o hal -in eg al and qeis an in ege , ep esen ing angula momen um and elec ic cha ge espec i ely, α=e2/¯hc is he ine s uc u e cons an and mP=qc¯h/GNis he Planck mass ( he e is a modi ica ion o his o mula when magne ic monopoles a e included). The smalles possible mass o a black hole in his scheme is he e o e M=1 2mP,(5) when k=j=qe= 0. Fo gi en jand qe he quan um numbe kis bounded below by (2k+ 1)2≥4j(j+ 1) + α2q4 e.(6) In pa icula , o a ze o cha ge black hole, he classical bound J2≤M4(7) (in uni s wi h GN=c= 1) is eplaced by J2≤M4−π2l4 P A2¯h2.(8) The layou o he pape is as ollows. In sec ion 2 he quan isa ion o he a ea a ising om he uzzy sphe e hypo hesis is discussed o Schwa zschild black holes and he p ojec ion o he non-commu a i e plane is explained. Sec ion 3 analyses non-ze o angula momen um and he associa ed bounds on he mass while sec ion 4 does he same o cha ged and o a ing holes. The ela ion o en opy is discussed in sec ion 5 and he esul s a e summa ised in sec ion 6 2 Schwa zschild Black Holes The 2-dimensional sphe e is a symplec ic mani old — a phase-space in physics language, albei a compac one. This phase-space can be quan ised o gi e 3 S2 F. The concep o a poin on S2 Fis no de ined bu ins ead he poin s a e smea ed ou in o a ini e numbe o phase-space ‘cells’, hence he name ‘ uzzy’, [4]. Fo any in ege , N= 2k+ 1 wi h klabelling SU(2) ep e- sen a ions ei he in eg al o hal -in eg al, S2 Fhas Ncells and ope a o s on phase-space a e N×Nma ices ac ing on a N-dimensional Hilbe space, [19]. Visually S2 Fmigh be iewed as being like he su ace o Jupi e , wi h he bel s being uni cells, bu his is no essen ial since, as in any quan um phase space, only he a ea o he undamen al cells, no hei shape, is ixed. I we pic u e he e en ho izon o a black hole as a uzzy sphe e hen he o al a ea o he e en ho izon is na u ally a mul iple o he a ea o a undamen al uni cell. Suppose he uni cells ha e a ea al2 P, wi h aa posi i e dimensionless cons an o o de one. Then he o al a ea o he e en ho izon is A=Nal2 P,(9) and, since N= 2k+ 1, we conclude ha η= 0 in equa ion (1). Fo a non- o a ing black hole wi h ze o cha ge (9) immedia ely implies ha he Schwa zschild adius RSis also quan ised R2 S=A/4π=Nal2 P 4π.(10) To a oid messy ac o s o 4πi is con enien o de ine ¯ A=A/4πand ¯a= a/4πso R2 S=¯ A=N¯al2 P.(11) The mass o he hole can now be exp essed as M=RSc2 2GN =√N¯alPc2 2GN =√N¯amP 2.(12) The hypo hesis ha he e en ho izon is a uzzy-sphe e hus immedia ely leads us o conclude ha black hole masses a e quan ised M2=N¯a 4mP 2(13) wi h Na posi i e in ege . Fo as ophysical black holes Nis so la ge ha he quan um na u e o he mass would be unobse able, bu in he inal s ages o black hole e apo a ion he black hole would go h ough a se ies o disc e e s a es un il he inal s a e is eached, wi h N= 1 (i.e. k= 0) and esidual mass M0=√¯amP/2. Thus in his pic u e e apo a ing black holes do no disappea bu mus necessa ily lea e behind a esidual hole o he o de o he Planck mass. As ema ked 4 in [5] he si ua ion is eminiscen o he Boh model o he a om in which o bi ing elec ons can only occupy a disc e e se o o bi s, dic a ed by he Boh -Somme eld cons ain Hpdq = 2πN¯hon he o bi als, and decaying elec ons mus inally lodge in he g ound s a e hus ende ing a oms s able. Non-commu ing co-o dina es on he uzzy-sphe e can be ep esen ed glob- ally by h ee N×Nma ices Xi,i= 1,2,3, sa is ying XiXi=R2 S1,(14) whe e 1is he N×Nuni ma ix, wi h Xip opo ional o he gene a o s Lio SU(2) in he i educible N×N ep esen a ion, [Li,Lj] = iijkLk,LiLi=k(k+ 1)1.(15) F om his we deduce ha Xi=λkLi⇒[Xi,Xj] = iλkijkXk,(16) wi h λk:= √¯alPs2k+ 1 k(k+ 1).(17) A i s glance i appea s ha , in he la ge Nlimi , he Xiin equa ion (16) become commu a i e and he commu a i e sphe e is eco e ed, since λk→0 in he limi , bu upon mo e ca e ul conside a ion his is no in ac co ec .1Heu is ically his can be seen by ocusing on a egion nea he sou h pole o a la ge black-hole, in he limi o la ge k. A he sou h pole X1and X2a e ans e se o he su ace and X3is no mal o i , wi h X3≈ −Rsand [X1,X2] = iλkX3=iλ2 kL3.(18) In a basis in which L3=   k... −k   (19) is diagonal he eigen alue X3≈ −Rsco esponds o he minimum eigen alue −ko L3so [X1,X2]≈ −iλ2 kk(20) and, as k→ ∞, [X1,X2] = −2i¯al2 P.(21) 1An ea lie e sion o his pape con ained an e o on his poin and I am g a e ul o Al S e n and Eli Hawkins o b inging his o my a en ion. 5 Hence, in an in ini esimal egion a ound he pole, he e en ho izon looks like a non-commu a i e plane in he in ini e klimi . This obse a ion can be pu on a mo e o mal oo ing using he analysis o [20] (see also [21]) in which is shown ha he k→ ∞ limi o (16) de- sc ibes a non-commu a i e plane unde s e eog aphic p ojec ion. This is seen by de ining X±=X1±iX2and pe o ming he analogue o s e eog aphic p ojec ion o uzzy co-o dina es: Z=X−(1−X3/RS)−1,Z†= (1−X3/RS)−1X+.(22) Then, o la ge k, [Z,Z†] = 2λkRS(1−X3/RS)−2+o(1/k).(23) Now, al hough he ope a o X3/RShas eigen alues be ween −1 o +1 in- clusi e, only a e y small ange abo e −1 is necessa y o co e he whole Z-plane. To see his obse e ha 1 2(ZZ†+Z†Z) = R2 S 1+X3 RS! 1−X3 RS!−1 +o(1/k).(24) W i ing X3/RS=−1+T/R2 S, whe e T/R2 Shas eigen alues be ween 0 and 2, his eads 1 2(ZZ†+Z†Z) = 1 2T 1−T 2R2 S!−1 +o(1/k).(25) Now, in he k→ ∞ limi , we can co e he whole o he Z-plane by p o- jec ing all ope a o s on o he subspace spanned by o eigen ec o s o Twi h eigen alues in he ange 0 o ¯a√k l2 P. Hence, o k→ ∞,T/R2 S→0 in (25) and we can eplace X3 RSwi h −1in (23) o gi e ZZ†+Z†Z=Tand [Z,Z†] = θ(26) wi h non-commu a i i y pa ame e θ= lim k→∞ λkRS 2= lim k→∞ ¯al2 P 2 (2k+ 1) qk(k+ 1) = ¯al2 P.(27) The in e es ing conclusion o his analysis is ha , e en o la ge as ophysical black-holes, he e is a es ige o non-commu a i i y a he Planck leng h. I he assump ions made he e a e co ec he e en ho izon o a black-hole is a physical example o a sys em in which Connes’ non-commu a i e geome y mani es s i sel in he con inuum. 6 3 Ro a ing Black Holes Now conside a o a ing black hole wi h angula momen um J2=j(j+ 1)¯h2 and ze o cha ge. The e en ho izon is s ill opologically a sphe e, hough no me ically a ound sphe e i s ill has a uzzy desc ip ion. The classical o mula o he mass as a unc ion o angula momen um and a ea ( he Ch is odoulou-Ru ini mass [22]) is 2 M2=1 4¯ A+J2 ¯ A,(28) o ¯ A 2=M2+√M4−J2(29) ( he posi i e squa e oo is aken he e because Ais he a ea o he ou e ho izon). F om he abo e o mula comes he bound J2≤M4,(30) o he wise ¯ Abecomes complex. Using (28) and (29) his is equi alen o J2≤1 4¯ A2.(31) Classically he maximum allowed angula momen um is when (30) is sa - u a ed: J2 max =M4=1 4¯ A2.(32) Conside he quan um e sion o (32). Using J2 max =jmax(jmax + 1)¯h2, oge he wi h he ansa z (9), gi es jmax +1 22 = ¯a2k+1 22 +1 4.(33) Quan um mechanically he bound migh no be sa u a ed so all we can sa ely say is ha jmax +1 22 ≤¯a2k+1 22 +1 4.(34) Suppose ha he bound is sa u a ed in he limi o la ge k, and hence la ge jmax, so ha lim k→∞ J2 max M4= 1 ⇔lim k→∞ 4J2 max ¯ A2= 1 ⇔lim k→∞ j2 max k2= ¯a2.(35) 2He e we use uni s in which GN=c2= 1 o keep he o mula clean, bu ¯hwill be e ained so as o highligh quan um phenomena. Hence l2 P=mP 2= ¯h. 7 Now he uzzy sphe e is associa ed wi h a Hilbe space whose maximum angula momen um is k, so i seems e y na u al o ake jmax =k, in which case ¯a= 1. Then (32) mus be modi ied o ead J2 max/¯h2=jmax(jmax + 1) = 1 4(¯ A2/¯h2−1) (36) wi h ¯ A= (2k+ 1)¯h. (37) No e ha a k= 0 black hole necessa ily has j= 0 and is he e o e a boson wi h spin ze o. I is possible ha he e is a co ela ion be ween kand j, e en away om ex emali y, and ha in eg al jimplies in eg al kand hal -in eg al jimplies hal -in eg al k. Indeed he a ea spec um ound in [13] o non- o a ing black holes equi es in eg al kwhen j= 0 o a hole ca ying ze o cha ge, hal - in eg al konly appea o cha ged black holes in hei analysis. The spec um ound in [14] o ze o cha ge equi es ha jand ka e bo h in eg al. The uzzy sphe e app oach he e does no impose any such es ic ions. While a co ela ion be ween in eg al kand j, equi ing ha hey be ei he bo h in eg al o bo h hal -in eg al, seems plausible we ha e no ound a p oo ha i is necessa y. The ac ha he di e ence be ween he quan um bound (36) and he classical bound (32) is independen o Ais a di ec consequence o he choice ¯a= 1. Equa ion (28) now eads M2=(k(k+ 1) + j(j+ 1) + 1 4 (2k+ 1) )¯h. (38) The mass o a black hole o a gi en a ea ( ixed k) wi h maximum allowed angula momen um is now M2(Jmax) = 1 4(8k(k+ 1) + 1 2k+ 1 )¯h. (39) In he quan um heo y equa ion (32) is hen eplaced wi h J2 max =M4(Jmax)−¯h4 16 ¯ A2=1 4(¯ A2−¯h2),(40) so (30) is ne e sa u a ed o ini e k. In e ms o jand k he bound is (2k+ 1)2>4j(j+ 1).(41) 8 4 Cha ged Black Holes Including elec ic cha ge Qe he classical Ch is odoulou-Ru ini o mula eads M2=1 ¯ A1 4(¯ A+Q2 e)2+J2(42) o , i magne ic monopoles wi h cha ge Qma e also included, M2=1 ¯ A1 4(¯ A+Q2)2+J2(43) whe e Q2=Q2 e+Q2 m.(44) Wi h ¯ A= (2k+ 1)¯hand Qequan ised in mul iples o he elec ic cha ge e he quan um e sion o (43) becomes M2=    h2k+ 1 + αq2 e+α−1(qm/2)2i2+ 4j(j+ 1) 4(2k+ 1)     ¯h, (45) wi h qeand qmin ege s (we use uni s wi h 4π0= 1 so ha he ine s uc u e cons an is α=e2/¯hwhen c= 1, he ac o o α−1/4 mul iplying q2 mallows o he Di ac quan isa ion condi ion, QeQm= N¯h/2 whe e Nis an in ege ). Thus, as sugges ed in [6], he black hole mass is cha ac e ised by ou disc e e numbe s: kand j, which can each be ei he in eg al o hal -in eg al, and qe and qmwhich a e bo h in ege s. This pa icle pic u e o black holes has also been a cen al heme in he wo k o ’ Hoo , [7] [10]. 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