Hall Scrambling on Black Hole Horizons

0.77 MB·English

by  Willy Fischler

UTTG-31-14
TCC-031-14
Hall Scrambling on Black Hole Horizons
Willy Fischlera,b, Sandipan Kundua,b,c
aTheory Group, Department of Physics, University of Texas, Austin, TX 78712
bTexas Cosmology Center, University of Texas, Austin, TX 78712
cDepartment of Physics, Cornell University, Ithaca, New York, 14853, USA and
E-mail: fi[email protected], [email protected]
We explore the effect of the electrodynamics θ-angle on the macroscopic properties of black hole
horizons. UsingonlyclassicalEinstein-Maxwell-Chern-Simonstheoryin(3+1)−dimensions,inthe
form of the membrane paradigm, we show that in the presence of the θ-term, a black hole horizon
5
behavesasaHallconductor,foranobserverhoveringoutside. Westudyhowlocalizedperturbations
1
created on the stretched horizon scramble on the horizon by dropping a charged particle. We show
0
that the θ-angle affects the way perturbations scramble on the horizon, in particular, it introduces
2
vorticeswithoutchangingthescramblingtime. ThisHallscramblingofinformationisalsoexpected

g to occur on cosmological horizons.
u
A
I. INTRODUCTION AND A SUMMARY zons (see [9] and references therein). Macroscopic prop-
1
ertiesofblackholehorizonscanalsoprovidecrucialhints
3
aboutdetailsofthemicroscopicphysics[10]. Predictions
Black hole horizons have eluded physicists ever since
] astronomer Karl Schwarzschild first found them in 1915 ofthemembraneparadigmaregenerallyconsideredtobe
h robust since they depend on some very general assump-
t as a simple yet puzzling consequence of the general the-
- tions: (i) the effective number of degrees of freedom be-
p ory of relativity. The description of the near horizon
tween the actual black hole horizon and the stretched
e region of a black hole in terms of quantum field theory
h in curved spacetimes [1, 2] historically has provided us horizon are vanishingly small, (ii) physics outside the
[ black hole, classically must not be affected by the dy-
with a deep insight into the mysterious innards of quan-
namics inside the black hole.
2 tum gravity, e.g. thermodynamic descriptions of black
Quantum chromodynamics (QCD) is an integral cor-
v holes. Itwasbelievedthatquantumfieldtheoryincurved
6 nerstone of our understanding of particle physics. It is
spacetimes provided a good physical description when
1 wellknownthatLorentzandgaugeinvarianceallowQCD
quantum gravitational effects do not play a significant
3 action to have a CP-violating, topological θ sector
1 role, however this idea has recently been challenged [3]. QCD
[13]
0 Various puzzles and paradoxes [4, 5] indicate a possible
1. failure of quantum effective field theory in the near hori- θ g2
0 zon region of a black hole. Lθ = Q6C4πD2 (cid:15)αβµνFαaβFµaν ,
5
String theory, Matrix Theory [6], and the AdS/CFT
1 where Fa is the field strength and a priori parameter
: correspondence [7], which are the only models of quan- αβ
v tum gravity over which we have mathematical control, θQCD can take any value between −π and π. This topo-
i logicaltermcanhavephysicaleffects,forexampleitcon-
X strongly suggest that black hole evolution as seen by an
tributes to the neutron electric dipole moment. Experi-
external observer, obeys the usual rules of unitary quan-
r
a tum mechanics, and verifies the Bekenstein-Hawking en- ments set a rather strong limit: |θQCD|<<10−9, which
indicates that this term is unnaturally small. Similarly,
tropy formula in a large class of special cases. Thus far
Lorentz and gauge invariance also allow the electrody-
none of them give us a comprehensive description of the
namics action to have a CP-violating θ term
physics of the black hole, and in particular, there is no
definitive clue about the microscopic description of the (cid:90) (cid:20) √g θ (cid:21)
near horizon region of a black hole. S = d4x − F Fµν + (cid:15)αβµνF F .
4 µν 8 αβ µν
Ontheotherhand, themembrane paradigm[8,9]pro-
vides a powerful framework to study macroscopic prop- Theelectrodynamicsθ-termisanelusivequantity;itdoes
ertiesofblackholehorizonsbyreplacingthetruemathe- not affect the classical equations of motion because it
maticalhorizonbyastretched horizon, aneffectivetime- is a total derivative. Therefore, it does not contribute
like membrane located roughly one Planck length away forperturbativequantumelectrodynamics(QED),which
from the true horizon. Indeed, in astrophysics the mem- strongly indicates that the effects of the θ term in QED,
braneparadigmhasbeensuccessfullyusedtostudyphe- if any, are non-perturbative, making it difficult to de-
nomena in the vicinity of black holes, outside their hori- tect. However, as Witten showed [14], in the presence
2
of illusive magnetic monopoles, this term can have a and conclude in section VI.
measurablephysicaleffectbecauseitprovidesmonopoles
with electric charges proportional to θ. In this paper,
by coupling this theory to gravity we will show that II. MEMBRANE PARADIGM
the θ-term can affect the electrical properties of the
black hole horizon. In particular, in the context of the Let us begin with a brief discussion of the black hole
membrane paradigm by using only classical Einstein- membrane paradigm [8, 9]. Finiteness of the black hole
Maxwell-Chern-Simons theoryin(3+1)−dimensions,we entropy indicates that at a distance less than the Planck
argue that in the presence of the θ-term, a black hole length from the black hole horizon, the effective number
horizon behaves as a Hall conductor with Hall conduc- of degrees of freedom should be vanishingly small. So,
tance ≈θ(377Ω)−1. it is more natural as well as more convenient to replace
Wewillprovide,inthispaper,evidenceforthisbehav- the true mathematical horizon by a stretched horizon,
ior by analyzing a simple thought experiment, in which an effective time-like membrane M located roughly one
an outside observer drops a charge onto the black hole Planck length away from the true horizon.
andwatcheshowtheperturbationscramblesontheblack Advantagesof having a stretched horizonwill be more
hole horizon. In quantum mechanics, information con- apparentifweconsidersomefieldsintheblackholeback-
tained inside a small subsystem of a bigger system, is ground with an action
said to be scrambled when the small subsystem becomes (cid:90) √
S = dd+1x gL(φ ,∇ φ ) , (1)
entangled with the rest of the system and the informa- tot I µ I
tion can only be recovered by examining nearly all the
where φ with I =1,2,... stands for any fields. It is nec-
degrees of freedom of the system. It is indeed remark- I
essary to impose some boundary conditions on the fields
able, as pointed out by Sekino and Susskind, that black
φ inordertoobtainequationsofmotionbyvaryingthis
holehorizons(andalsodeSitterhorizons)arethefastest I
action. We will impose Dirichlet boundary conditions
scramblers in nature [10, 11]; in particular, for a black
δφ = 0 at the boundary of space-time. The stretched
hole with temperature T and entropy S, the scrambling I
horizon M divides the whole space-time in regions: A
time goes as t T ≈ (cid:126)lnS. This “fast-scrambling” on
s (outsidethemembraneM)andB (insidethemembrane
black hole and de Sitter horizons strongly indicates that
M). Therefore, we can write S =S +S .
the microscopic description of scrambling of information tot A B
Now imagine an observer O who is hovering outside
on static horizons must involve non-local degrees of free-
the horizon of a black hole. Observer O has access only
dom [10, 11]. We will show that the θ-angle affects the
to the region outside the black hole and physics he ob-
waychargesscrambleonthehorizon, inparticular, itin-
serves, classically must not be affected by the dynamics
troduces vortices without changing the scrambling rate
inside the black hole. That means observer O should be
– we will call this phenomenon “Hall scrambling”. Since
abletoobtainthecorrectequationsofmotionbyvarying
Hall scrambling depends only on the Rindler-like char-
someactionS whichisrestrictedonlytothespace-time
acter of the black hole horizon but not on the details O
outside the black hole. Clearly, S (cid:54)= S because the
of the metric, the same conclusion is also true for arbi- O A
boundary terms generated on M from S are in general
trary cosmological horizons. A microscopic description A
non-vanishing. Therefore, we should add some surface
of fast-scrambling should be able to explain the origin of
terms that exactly cancel all these boundary terms. Let
Hall-scrambling in the presence of the θ-angle.
us now rewrite the total action in the following way [15]
It is very tempting to apply the framework of mem-
brane paradigm in the context of holography. In fact the S =(S +S )+(S −S ) , (2)
tot A surf B surf
AdS/CFT correspondence [7] has taught us that the low
such that S +S and S −S are independent
frequency limit of linear response of a strongly coupled A surf B surf
of each other and correct equations of motion can be
quantum field theory is related to that of the membrane
obtained by varying them individually. For the observer
paradigmfluidontheblackholehorizonofthedualgrav-
O, the action S = S +S for fields φ now have
ity theory [16–18]. This raises a deeper question. Can O A surf I
sources residing on the stretched horizon
we then conclude that the same has to be true for holo-
(cid:90) √
graphic models of cosmological spacetimes? S = dd+1x −gL(φ ,∇ φ )
O I µ I
The rest of the paper is organized as follows. We start A
withabriefdiscussionofthemembraneparadigminsec- (cid:88)(cid:90) √
+ ddx −h JI φ (3)
tion II. In section III, we show that in the presence of M I
I M
electrodynamics θ-term, the black hole horizon behaves
where, histhedeterminantoftheinducedmetriconthe
as a Hall conductor. Then in section IV, we introduce
stretched horizon M and sources are
Hall scrambling for Rindler, black hole and cosmological
(cid:20) (cid:21)
horizons. Finally, we make some comments in section V ∂L
JI = n (4)
about its connection with the AdS/CFT correspondence M µ∂(∇ φ )
µ I M
3
where, n is the outward pointing normal vector to the This can be regarded as a portion of Minkowski space,
µ
time-like stretched horizon M with n nµ = 1. The ob- formally known as the Rindler wedge. In particular, un-
µ
serverOwhoishoveringoutsidethehorizon,canactually der the redefinitions
perform real experiments on the stretched horizon M to
t=ρsinhω , x=ρcoshω, (10)
measure the sources JI . It is important to note that
M
one should interpret JMI as external sources such that one arrives to the more familiar metric
δJMI =0.
δφJ ds2 =−dt2+dx2+dy2+dz2. (11)
For the observer O, who follows orbits of the time-like
A. Electromagnetic fields and stretched horizon
Killing vector ξ = ∂ , there is a horizon at the edge of
ω
the Rindler wedge, x = |t|, or equivalently, ρ = 0. We
The action for electromagnetic fields in the curved
will replace the mathematical horizon by the stretched
space-time in (3+1)−dimensions is
horizon located at ρ = (cid:15), where (cid:15) is about one Planck
length.
(cid:90) √ (cid:20) 1 (cid:21)
S = gd4x − F Fµν +j Aµ , (5) Since,4−velocityUµ oftheobserverOissingularnear
4 µν µ
the horizon, the electric and magnetic fields E and B as
measured by O (eνµαβ is the Levi-Civita tensor)
where,asusualF =∂ A −∂ A andg =−det(g ).
µν µ ν ν µ µν
jµ isaconservedcurrent,i.e.,∇µjµ =0. Ourconvention Eµ =FµνU , Bµ = 1eνµαβF U (12)
of the metric is that the Minkowski metric has signature ν 2 αβ ν
(−+++). The equation of motion obtained from action
can be singular in general. In order to understand the
(5) is
behavior of E and B on the horizon let us consider a
∇ Fµν =−jν . (6) freely falling observer (FFO) P:
µ
FFO P : x=a, y =z =0 . (13)
Field strength tensor F also obeys ∂ F =0.
µν [µ νλ]
We will also assume that the conserved current jµ is
A freely falling observer does not see the coordinate sin-
contained inside the membrane M and hence the ob-
gularity and hence electric and magnetic fields E and
P
server O who has access only to the region outside the
B as measured by P should be non-singular. Relating
stretched horizon does not see the current jµ. However, P
E and B with E and B , we obtain
the observer will see a surface current Jµ on the mem- P P
M
brane. Let us start with the action for the observer O Eρ| ,Bρ| =O(1) ,
M M
S =−1(cid:90) √gd4xF Fµν +(cid:90) √−hd3xJ Aµ . Ey|M+Bz|M =O((cid:15)) , (14)
O 4 µν M;µ Ez| −By| =O((cid:15)) .
A M M M
(7)
Note that the action is invariant under the gauge trans- It is important to note that the above relations can be
formation: A →A +∂ αonlyifJ nµ =0,wheren thought of as ingoing boundary conditions for the elec-
µ µ µ M;µ µ
istheoutgoingunitnormalvectoronM. Inorderforthe tromagneticradiationsanditisaconsequenceofthefact
observer O to recover the vacuum Maxwell’s equations, that black holes behave as perfect absorbers.
the boundary terms on M should cancel out and hence Therefore, from equation (8), we obtain,
from equation (4) we obtain
Jy =Ey| , Jz =Ez| . (15)
M M M M
Jµ =n Fµν| . (8)
M ν M The black hole horizon behaves like an Ohmic conductor
It is obvious that J nµ = 0 and hence action (7) is with conductivity
M;µ
invariant under gauge transformations.
σ =1 . (16)
That is, the surface resistivity of the black hole is r =
B. Black hole horizon and electrical conductivity 1/σ ≈377Ω [8, 9].
We will end this section with a brief discussion on
AgainconsiderafiducialobserverO hoveringjustout- Ohmic dissipation in the stretched horizon because
side a (3+1)−dimensional black hole. For such an ob- of the electromagnetic fields on the horizon. For a
server, the near horizon geometry is a good approxima- SchwarzschildblackholeofmassM,thefirstlawofther-
tion and the metric takes the Rindler form modynamics states
ds2 =−ρ2dω2+dρ2+dy2+dz2. (9) TdS =dM (17)
4
where T is the temperature and S is the entropy of the A. Hall conductivity
blackhole. Presenceofelectromagneticfieldsonthehori-
zon can increase the mass and hence the entropy of the Followingthediscussionofsection(IIB),equations(14)
black hole [9] and (24), now lead to
dM dS (cid:90)
=T =− S(cid:126) .dA(cid:126) . (18) (cid:18)Jy (cid:19) (cid:18)σ −θ(cid:19)(cid:18)Ey (cid:19)
dt dt H M = M . (25)
M Jz θ σ Ez
M M
S(cid:126) is the renormalized Poynting flux
H
Therefore, surface Hall conductance of the black hole
S(cid:126) =(cid:15)2(E×B) . (19) horizon is
H M
S(cid:126)H is the amount of red-shifted energy (as measured at σzy =−σyz =θ ≈θ(377Ω)−1 (26)
infinity) crossing a unit area per unit time at infinity.
Equation (14), leads to the famous result of the Joule and in principle one can find out the value of the θ-angle
heating law for black hole horizon bymeasuringtheHallconductivityoftheblackholehori-
dM dS (cid:90) zon.
=T = (cid:15)2rJ(cid:126)2 dA . (20) From equation (18), it is obvious that the presence of
dt dt M
M theθ-termdoesnotcontributetotheincreaseofentropy
of the black hole. However, it can be easily checked that
the stretched horizon now does not follow the standard
III. ELECTRODYNAMICS θ-TERM AND THE
MEMBRANE PARADIGM Joule heating law (20). Instead it obeys
dM dS (cid:90) (cid:18) σ (cid:19)
The gauge invariance of electrodynamics allows for a =T = (cid:15)2 J(cid:126)2 dA . (27)
dt dt θ2+σ2 M
θ-term in the action M
(cid:90) √ (cid:20) 1 (cid:21)
S = gd4x − F Fµν +j Aµ
4 µν µ IV. HALL SCRAMBLING OF CHARGES ON THE
θ (cid:90) STRETCHED HORIZON
+ d4x(cid:15)αβµνF F (21)
8 αβ µν
Scrambling is the process by which a localized pertur-
where one can write
bation of a system spreads out into the whole system.
θ θ√ In quantum mechanics, information contained inside a
(cid:15)αβµνF F = gF ∗Fµν . (22)
8 αβ µν 4 µν small subsystem of a bigger system, is said to be scram-
bled when the small subsystem becomes entangled with
OurconventionoftheLevi-Civitatensordensity(cid:15)αβµν is
therestofthesystem. Andscramblingtimet isdefined
the following: (cid:15)0123 =1, (cid:15) =−g. s
0123
asthetimeittakesforalocalizedperturbationtobecome
The electrodynamics θ-term is a total derivative and
fullyscrambledsuchthattheinformationitcontainscan
hence it does not affect the classical equations of mo-
only be recovered by examining nearly all the degrees
tion. Therefore, this term does not contribute even for
of freedom. In a local quantum field theory, scrambling
perturbative quantum electrodynamics, which strongly
time t is expected to be at least as long as the diffusion
indicates that the effects of the QED θ-term, if any, are s
time. Consequently, for a strongly correlated quantum
non-perturbative. However, by coupling this theory to
fluid in d-spatial dimensions and at temperature T, the
gravitywewillshowthattheelectrodynamicsθ-termcan
scrambling time satisfies
affect the electrical properties of the stretched horizon.
Let us again imagine a stretched horizon M that di-
t T ≥c(cid:126)S2/d , (28)
videsthewholespace-timeintworegionsandanobserver s
O who has access only to the region outside the horizon.
wherecissomedimensionlessconstantandS isthetotal
The action for the observer O is
entropy. In [10, 11], it has been argued that this is a
(cid:90) √ (cid:20) 1 θ (cid:21) universal bound on the scrambling time. Hence, it is
S = gd4x − F Fµν + F ∗Fµν
O 4 µν 4 µν indeed remarkable that information scrambles on black
A
(cid:90) √ hole and de Sitter horizons exponentially fast
+ −hd3xJ Aµ . (23)
M;µ
M t T ≈(cid:126)lnS (29)
s
Therefore from equation (4) the membrane surface cur-
rent is given by, violating the bound (28). This unusual process is fa-
mously known as “fast-scrambling” and it strongly indi-
JMµ =(nνFµν −θ nν ∗Fµν)|M . (24) cates that the microscopic description of scrambling of
5
informationonstatichorizonsmustinvolvenon-localde- Fµν in the Rindler frame
R
grees offreedom [10,11]. Non-localityis indeedessential
Jµ =(Fµρ−θ∗Fµρ)| . (31)
for fast scrambling [20–24] and in fact it is well known M R R M
thatnon-localinteractionscanincreasethelevelofentan-
At any given time the Rindler coordinates are related to
glement among different degrees of freedom of a theory
the Minkowski coordinates by a boost along the x-axis.
[25–27].
In Minkowski coordinates:
In this section, we will focus on scrambling of point-
charges on the horizon in the presence of the electrody- F01 = Q(x−a) =−F10 , (32)
namicsθ-term. Wewillarguethattheθ-angleaffectsthe M 4π[(x−a)2+y2+z2]3/2 M
waychargesscrambleonthehorizon, inparticular, itin-
Qy
troduces vortices without changing the scrambling rate FM02 = 4π[(x−a)2+y2+z2]3/2 =−FM20 , (33)
– we will call this phenomenon “Hall scrambling”. Let
Qz
us note that a microscopic description of fast-scrambling F03 = =−F30 (34)
shouldbeabletoexplaintheoriginofHall-scramblingin M 4π[(x−a)2+y2+z2]3/2 M
the presence of the θ-angle.
and all the other components are zero. Now one can
compute Fµν by performing the change of coordinates.
R
A. Rindler coordinates That leads to:
Q((cid:15)coshω−a)
Jω = , (35)
Let us again consider an accelerated observer in M 4π(cid:15)[((cid:15)coshω−a)2+y2+z2]3/2
Minkowski space. For such an observer, the metric takes
Jρ =0 , (36)
the Rindler form (9). We again replace the mathemat- M
ical horizon by the stretched horizon at ρ = (cid:15), where (cid:15) Q(ysinhω−θzcoshω)
Jy = , (37)
is about one Planck length. We will consider a single M 4π[((cid:15)coshω−a)2+y2+z2]3/2
charge which is stationary in the Minkowski frame [12]
Q(zsinhω+θycoshω)
Jz = . (38)
Charge Q: x=a, y =z =0 . (30) M 4π[((cid:15)coshω−a)2+y2+z2]3/2
However, in the Rindler frame, the charge is falling into We are mainly interested in the current density on the
the horizon. Figure 1 represents schematically this situ- stretched horizon after the point charge Q crosses the
ation. stretched horizon. An observer hovering just outside the
stretched horizon will measure a surface charge density
t
10 ρ (y,z) = (cid:15)J0 on the horizon. Note that the surface
H M
ω=∞ charge density ρ (y,z) does not receive any correction
H
in the presence of the θ-term.
The point charge crosses the horizon at the Rindler
5 time ω =ω , where
ρ=0 1
a
coshω = . (39)
1 (cid:15)
a
0 x In the limit ω →ω (but ω <ω ), we obtain
1 1
Q
ρ (y,z)=− δ(y)δ(z) . (40)
H 2
(cid:45)5
In the limit ω →ω (but ω >ω )
1 1
Q
ρ = δ(y)δ(z) . (41)
ω=−∞ H 2
(cid:45)10
(cid:45)10 (cid:45)5 0 5 10 Note that there is a discontinuity at ω = ω . The total
1
charge on the horizon is given by (ω >ω ):
1
FIG. 1. Rindler coordinates plotted on a Minkowski dia- (cid:90) Q
gram. The dashed lines correspond to the Rindler horizons. Q = dydzρ (y,z)= . (42)
Constant-ρ and constant-ω lines are shown in red and black, H H 2
respectively. The blue line corresponds to the worldline for a
AnobserverO hoveringnearthethehorizonwillmea-
free falling charge.
sure the current Jµ and electric field E (12):
M
To compute the surface current, we need to compute Ey =(coshω)F02| , Ez =(coshω)F03| . (43)
M M M M
6
Finally equations (37,38) can be rewritten in the follow- V
ing way:
(cid:18)Jy (cid:19) (cid:18)tanhω −θ (cid:19)(cid:18)Ey(cid:19)
M = . (44)
Jz θ tanhω Ez
M
Therefore, in this case, conductivity of the stretched
horizon, as measured by the observer O is time-
II
dependent
σ =tanhω ≈1−2e−2ω . (45) III (cid:73)
U
Late time behavior: Scrambling
IV
In the limit ω >>ω we obtain:
1
(cid:15)Qe−2ω
ρ (y,z)= , (46)
H π[(cid:15)2+r2]3/2
⊥
(y−θz)Qe−2ω
Jy = , (47)
M π[(cid:15)2+r2]3/2
⊥
(z+θy)Qe−2ω FIG. 2. Maximal analytic extension of Schwarzschild so-
Jz = , (48)
M π[(cid:15)2+r2]3/2 lution in Kruskal-Szekeres coordinates. Region I is the
⊥ Schwarzschild patch, where constant r-lines are shown in
black and constant t-lines are shown in Blue. The solid red
where r2 = 4e−2ω(y2 + z2). And in this limit the
⊥ linesrepresentthesingularity. Theworldlineofafreefalling
conductivities of the stretched horizon are constants:
charge from V =0 is shown in brown.
σ = σ ≈ 1, σ = −σ = θ and the horizon be-
yy zz zy yz
haves like an ideal Hall conductor.
Before proceeding further, a few comments are in or- event horizon is located at V = ±U and the curvature
der. From equation (46), it is clear that the presence singularity is at V2−U2 = 1. In Schwarzschild coordi-
of the θ-term does not change the scrambling time. On nates, the metric is
the other hand, equations (47) and (48) indicate that (cid:16) r (cid:17) dr2
the θ-angle affects the way the charge scramble on the ds2 =− 1− 2Gm dt2+ (cid:0)1− r (cid:1)
horizon. Let us first represent {y,z}-plane in the polar 2Gm
+r2(cid:0)dθ¯2+sin2θ¯dφ2(cid:1) . (51)
coordinates: {b,φ}, where y = bcosφ and z = bsinφ,
as usual. When θ = 0, φ-component of the horizon cur- For r >2Gm, coordinates {U,V} and {r,T} are related
rent JMφ = 0. Whereas, for θ (cid:54)= 0, JMφ = θJMb and in the following way
∇(cid:126) ×J(cid:126)M (cid:54)= 0 which clearly indicate the presence of vor- (cid:16) r (cid:17)1/2 (cid:18) t (cid:19)
tices; this is a direct consequence of the non-zero Hall V = −1 e4Grm sinh , (52)
2Gm 4Gm
conductivity.
(cid:16) r (cid:17)1/2 (cid:18) t (cid:19)
U = −1 e4Grm cosh . (53)
2Gm 4Gm
B. Schwarzschild black hole
Following the standard procerdure, we will replace the
mathematical horizon by the stretched horizon at r =
The metric of a Schwarzschild black hole in Kruskal-
2Gm+δ, where δ <<2Gm.
Szekeres coordinates is
We will consider a large black hole and restrict the
ds2 = 32Gr3m3e−2Grm (cid:0)−dV2+dU2(cid:1) icnhaargesms taollbaengnuealarrthreeghioonrizaornb,itir.aer.il|yr/c2eGntmer−ed1|at<<θ¯ 1=,
+r2(cid:0)dθ¯2+sin2θ¯dφ2(cid:1) , (49) 0. In that case, we can replace the angular part of
both Kruskal-Szekeres and Schwarzschild coordinates by
where r is the radial Schwarzschild coordinate Cartesian coordinates
V2−U2 =(cid:16)1− r (cid:17)e2Grm (50) r2(cid:0)dθ¯2+sin2θ¯dφ2(cid:1)≈dy2+dz2 , (54)
2Gm
where,
and we are using the symbol θ¯for angular coordinate to
y =2mGθ¯cosφ , z =2mGθ¯sinφ . (55)
differentiate it from the θ-angle of the action (21). The
7
Similarly, the Schwarzschild metric leads to
ds2 ≈−ρ2dω2+dρ2+dy2+dz2 . (59)
Therefore, we can use the results of the Rindler case to
analyze smearing of charges on the Schwarzschild hori-
zon. AgainwehaveachargeQandinitialconditionsare
given at V =0:
Q: U =U , θ¯=0 . (60)
0
(a) Inthenearhorizonapproximation,wecanmapthisprob-
lem to the problem of scrambling in Rindler spacetime
with
√
a=4GmeU , (cid:15)=2 2Gmδ . (61)
0
Therefore, the Schwarzschild observer O can see the
charge Q for t < t . At Schwarzschild time t = t , the
1 1
Schwarzschild observer will see that the charge density
on the stretched horizon is localized at a point , where
√
(cid:18) (cid:19)
t eU 2Gm
cosh 1 = 0√ . (62)
4Gm δ
(b)
Finally using (46-48), in the late time limit t >> t , (in
1
the small angle approximation) we obtain:
√
2 2GmδQe−t/2Gm
ρ (θ¯,φ,t)= , (63)
H π[8Gmδ+r2]3/2
⊥
2mGθ¯Qe−t/2Gm
Jθ¯ (θ¯,φ,t)= , (64)
M π[8Gmδ+r2]3/2
⊥
Jφ (θ¯,φ,t)=θJθ¯ (θ¯,φ,t) , (65)
M M
where r = 4mGθ¯e−t/4mG. So, an observer hover-
⊥
(c) ing outside the horizon will see Hall scrambling on the
Schwarzschild horizon as expected (see figure 3). Note
FIG. 3. A schematic diagram of Hall scrambling on a black that the Hall scrambling depends only on the Rindler-
hole horizon. Lines represent surface electric current on the like character of the horizon but not on the details of
blackholehorizonafterapositivechargeisdroppedatpoint
the Schwarzschild metric. We can define the scrambling
P,asobservedbyanoutsideobserverfor(a)θ=0,(b)θ>0,
time t in the following way: it is the time at which the
(c) θ<0. s
charge density ρ (θ¯,φ,t )≈Q/4πr2 , where r =2Gm
H s H H
is the radius of the horizon. Applying that for θ¯=0, we
obtain,
We will now define two sets of coordinates to describe (cid:18) (cid:19) (cid:18) (cid:19)
2Gm m
the near horizon region of the Schwarzschild black hole: t ≈2Gmln ≈4Gmln , (66)
s δ 2πM
P
(cid:114)
2Gm t wherewehaveused(cid:15)∼Plancklength. SotheHallscram-
ρ=4Gm 1− , ω = . (56)
r 4Gm bling does not affect the scrambling rate.
X =4GmeU, T =4GmeV . (57) It is important to note that there is a crucial differ-
ence between the discussions of Rindler spacetime and
Where e is the Euler’s number. Therefore, in terms of the Schwarzschild black hole. In a Schwarzschild black
{T,X,y,z}, the Kruskal-Szekeres metric becomes (near hole, all freely falling objects will hit the singularity at
the horizon) r =0infiniteKruskal-Szekerestime. Onecanargue[19]
thatwhenasinglechargehitsthesingularity,thespheri-
ds2 ≈−dT2+dX2+dy2+dz2 . (58) calsymmetrywillberestoredandthetotalchargewillbe
8
√
uniformly distributed over the stretched horizon. In the and (cid:15) = δ/H. Hence, we will see Hall scrambling on
scrambling time, t an order one perturbation will decay de Sitter horizon with the electrodynamics θ-term. In
s
to size ∼ M /m, and all trace of it will be lost [10, 12]. particular, in the late time limit t>>t , we obtain:
P 1
Hence,thisshouldbethetimescaleforanyclassicalfields √
on the horizon to become spherically symmetric. δQe−2Ht
ρ (θ¯,φ,t)=− , (71)
H πH[δ/H2+r2]3/2
⊥
C. Cosmological horizon: Hall scrambling and Jθ¯ (θ¯,φ,t)=− θ¯Qe−2Ht , (72)
de-scrambling M πH[δ/H2+r2]3/2
⊥
Jφ (θ¯,φ,t)=θJθ¯ (θ¯,φ,t) , (73)
We can easily extend the discussion of this section for M M
cosmological horizons. Before, we use the results of the
where Hr = 2θ¯e−Ht. The corresponding scrambling
Rindler calculations, we should modify equation (4) for ⊥
time is given by,
cosmological horizons. We can simply follow the discus-
sions of section (II) and conclude that (cid:18) (cid:19)
1 2M
t ≈ ln P . (74)
(cid:20) ∂L (cid:21) s H H
JI =− n (67)
M µ∂(∇ φ )
µ I M The fact that the de Sitter horizon is a fast scrambler
where, n is the outward pointing normal vector to the indicates that a dual description, if exists, must be non-
µ
stretched horizon M. Without the θ-angle, equation (8) local in nature [11] and it should also be able to provide
now becomes a microscopic description of the Hall scrambling.
Discussion of this section can easily be generalized for
Jµ =−n Fµν| . (68)
M ν M arbitrarycosmologicalhorizonsandonecanshowthatin
thepresenceoftheelectrodynamicsθ-anglepointcharges
WewillmainlyfocusondeSitterspace. Itwaspointed
willHallscrambleontheapparenthorizonofaco-moving
out in [11] that charges also fast scramble across the de
observer if the expansion of the universe is accelerating.
Sitter horizon. It is not very surprising since the near
On the other hand, when the expansion of the universe
horizonregionofdeSitterspaceisRindler-like. However,
is decelerating, the observer sees the charges “Hall de-
it strongly suggests that the holographic description of a
scramble” as they re-enter the horizon. However, there
causal patch of de Sitter space must involves non-local
is a crucial difference: for arbitrary non-de Sitter cos-
degrees of freedom [11]. A comoving observer O in de
mological expansion both scrambling and de-scrambling
Sitter space only sees a static patch of de Sitter space
occur at a power law rate [28]. This perhaps indicates
with metric,
that it may be possible to describe the process of Hall
dr2 (de)-scrambling on arbitrary non-de Sitter cosmological
ds2 =−(1−H2r2)dt2+
1−H2r2 horizonsintermsoflocallyinteractingdegreesoffreedom
+r2(cid:0)dθ¯2+sin2θ¯dφ2(cid:1) , (69) [10, 11].
where, cosmological constant is given by Λ=3H2. Rest
isstraightforward: wewillreplacethemathematicalhori- V. AdS/CFT
zon by the stretched horizon at rH =1−δ. Let us now
imagineachargeQthattheobserverOcanseefort<t
1 Thegauge-gravitydualityortheAdS/CFTcorrespon-
and from (68) it is obvious that the total induced charge
dence [7, 29–31] has been successful at providing us with
on the horizon Q = −Q. At time t = t , the charge Q
H 1 theoreticalcontroloveralargeclassofgaugetheories. It
hits the stretched horizon at a point θ¯= 0. For t > t ,
1 is a remarkable achievement to compute observables of
unlike the Schwarzschild case, the Gauss’s law tells us:
strongly coupled large-N gauge theories in d-dimensions
total charge on the de Sitter horizon Q = 0. However,
H by performing some classical gravity computations in
we will not be able to see that from the Rindler approx-
(d+1)-dimensions. At finite temperature, gravity du-
imation which is valid only for small angle limit. Note
als of these field theories have black holes with horizons
that, following [28]it ispossible toperform anexact cal-
and at very long length scales the most dominant con-
culation in de Sitter space but we will not attempt it in
tributions come from the near horizon region [32]. So, it
this paper.
is somewhat expected that there is some connection be-
Inthenearhorizonlimit,atsmallangleapproximation
tween the low energy hydrodynamic description of these
the metric (69) becomes Rindler-like with
strongly coupled theories and the membrane paradigm
ρ=cos−1(Hr) , ω =Ht (70) fluid on the horizon [16, 17]. This connection was made
precise in [18], where the authors have shown that the
y =H−1θ¯cosφ , z =H−1θ¯sinφ
low frequency limit of linear response of the fluid of the
9
boundary theory is given by the that of the membrane VI. CONCLUSIONS
paradigm fluid on the black hole horizon.
In particular, let us consider U(1) gauge field in AdS- We have shown that in principle one can find out the
Schwarzschild in (3+1)−dimensions with the action valueoftheelectrodynamicsθ-anglebydroppingcharged
particles into a black hole and observing how the per-
√
(cid:90) (cid:20) g θ (cid:21) turbation scrambles on the horizon. This strongly sug-
S = d4x − F Fµν + (cid:15)αβµνF F . (75)
4g2 µν 8 αβ µν gests that any sensible quantum theory of the black hole
3+1
needs to be able to provide a microscopic description of
The U(1) gauge field in the bulk is dual to a conserved Hall-scrambling in the presence of the θ-angle. It will be
currentji intheboundarytheoryandDCconductivities extremely interesting to see if this effect has any astro-
are given by physical consequence.
Experiments set a rather strong limit on QCD θ-angle
Gij(k) |θ | << 10−9, however, at low energies, the theta
σij =− lim R , (76) QCD
ω,k→0 iω angles for QED and QCD appear as independent pa-
rameters and hence there is no reason to expect QED-
where,Gij(k)istheretardedGreenfunctionofboundary θ ∼θQCD. In grand unified theories such as for example
R
current ji. As shown in [18], the DC conductivity σij of SU(5),theθ-anglesforQEDandQCDarerelatedbythe
renormalization group. However, even in the context of
the strongly coupled (2 + 1)−dimensional dual theory
grandunification,inthepresenceofamechanisminvolv-
is given by that of the membrane paradigm fluid on the
ing axions to solve the strong CP-problem, it remains
horizonoftheSchwarzschildblackholeinAdS.Sincethe
a challenge to relate the θ-angles for QED and QCD at
nearhorizonregionofAdS-SchwarzschildisRindler-like,
energy scales far below the GUT scale.
we have
Our conclusions depend only on the Rindler-like char-
1 acter of the horizon and hence they are valid for any
σ11 =σ22 = , σ21 =−σ12 =θ . (77)
g2 horizon which is locally Rindler-like. However, it is in-
3+1
deed puzzling that an in-falling observer will not see any
Let us end by briefly discussing scrambling in the con- effects of the electrodynamics θ-angle whatsoever.
text of the AdS/CFT correspondence. Consider the pro-
totypecaseofN =4super-Yang-Millswithgaugegroup
SU(N) on a unit 3-sphere at finite temperature. The ACKNOWLEDGMENTS
AdS/CFTcorrespondencerelatesthistheorytotypeIIB
stringtheoryonasymptoticallyAdS5×S5 spacetime. In We would like to thank Tom Banks and Daniel Har-
the limit N (cid:29)1,λ=gY2MN (cid:29)1, the theory can be well low for the helpful conversations. SK would like to ac-
approximated by the classical supergravity with a large knowledge Marco Farina, Tom Hartman, Bithika Jain
AdS black hole. Since the boundary field theory is local, and Sachin Jain for the useful discussions. This material
itisobviousthattheboundarytheoryisnotafastscram- is based upon work supported by the National Science
bler. However, asarguedin[10], onlengthscalessmaller Foundation under Grant Numbers PHY-1316033 and by
than the AdS radius RAdS, AdS space is a fast scram- the Texas Cosmology Center, which is supported by the
bler. Anylocalizedperturbationonlengthscalessmaller College of Natural Sciences and the Department of As-
than RAdS in the gravity side is equivalent to a local- tronomy at the University of Texas at Austin and the
izedperturbationofaverysmallsubsetofN2 degreesof McDonald Observatory. The work of SK was also sup-
freedom of the dual CFT. So, any perturbation of a very ported by NSF grant PHY-1316222.
smallsubsetofN2 degreesoffreedomatsomepointfirst
will fast scramble among N2 degrees of freedom at that
point. Then it will scramble (not so fast) over the entire
sphere and hence the total scrambling time is [10]
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