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]. The gene al o m o he spec um (45) was de i ed
by Bekens ein [6], he new ing edien he e is ha some o he cons an s di e
as a consequence o he hypo hesis ha he e en ho izon is modeled by a
uzzy sphe e.
Demanding ha ¯
Ain (43) is eal gi es he classical bound
M4−Q2M2−J2≥0 (46)
De ining
∆2:= M4−Q2M2−J2(47)
(43) can be used o exp ess ∆2in e ms o he a ea
∆2=(¯
A2−Q4−4J2)2
16 ¯
A2.(48)
9
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