At World's End: Where Complementarity and Irreversibility meet in the Black Hole

0.12 MB·English

by  Giovanni Arcioni

At World’s End:
Where Complementarity and Irreversibility meet in the Black Hole
Giovanni Arcioni1 and Antoine Suarez2
1Statistical and Mathematical Modeling, Noustat S.r.l, Via Pirelli 30, 20124 Milano, Italy
2Center for Quantum Philosophy, P.O. Box 304, CH-8044 Zurich, Switzerland
[email protected], [email protected], www.quantumphil.org
(Dated: January 15, 2009)
It is argued that a slight modification of the complementarity principle may help to overcome
paradoxes about the observerwho falls through theevent horizon.
Key words: Infallingobserver, informationloss, quantum cloning, complementarity, irreversibility,death
boundary,world’send.
9
0
I. INTRODUCTION According to it the information has first to reach the
0
singularitybeforebecoming“reflected”andsubsequently
2
By assuming that quantum information falling into a encodedintheoutgoingradiationthroughaprocesssimi-
n
lartoteleportation. WhenBobcollectsAlice’sq-bitfrom
a black hole comes out encoded in the Hawking radiation,
the radiation, the original state doesn’t exists any more
J oneavoidstheinformationlossparadox,thatis,thecon-
within the black hole.[4]
6 clusionthattheobserveroutsidetheblackholewatchesa
However, complementarity leads to other quandaries
1 non-unitary evolution of a quantum state, in sharp con-
flict with the rules of quantum physics. However, if a in case of the infalling observer.
] Assuming that observers beyond the horizon are able
c quantumstatecrossesthe eventhorizonandatthe same
toperformquantumexperimentsisonlyaparticularcase
q timeitescapestheblackholeencodedintheoutgoingra-
- diation,itmeansthatthestatehasbeencloned,andthis ofthemoregeneralframeworkthatobserversbeyondthe
r horizon are able to generate new information, as for in-
g is again at odds with the principles of quantum physics.
stance sort of master work of literature or some relevant
[ To avoidthe cloning paradoxone can invoke the prin-
scientific discovery. In this context the question arises:
1 ciple of complementarity [1], which basically states that will this new information become destroyed during the
v cloningis possible provideditcannotbe observed. Ifthe
processofblackholeevaporation,orwillitalsoattheend
3 external observer Bob can receive quantum information
becomeencodedintheradiation? Accordingtothecom-
2 from the radiation but cannot get hold of the original
4 plementarityprinciplewhathappensbeyondthe horizon
state after its crossing the event horizon, no contradic-
2 does not belong to the physical reality for the observer
tion results. In other words duplication of information
. outside. As a consequence new information created by
1 behind the horizon and in the Hawking radiation and
the observer inside does not become encoded in the out-
0 similarcontradictionsneveroccursince,accordingtothe
9 going radiation. This is baffling because the observer
complementarity principle, the black hole interior is not
0 outside knows very well that, if he jumps into the very
inthe causalpastofanyobserverwhoisabletomeasure
: massive black hole, he will observe de facto many things
v the information content of the outgoing radiation.
i going on.
X Nevertheless,fora largeenoughblackholeonecanas-
Moreover, if information about Alice’s death due to
sume, in accordancewith the laws of General Relativity,
r tidal forces longtime after she crossed the horizon goes
a thatAliceremainsaliveforalongtimeaftercrossingthe
lost forever then Hawking radiation will reveal the full
event horizon and therefore she is able to perform quan-
quantum information defining Alice alive, i.e. the exter-
tum experiments. Suppose then that Bob recovers from
nal observer compels her resuscitation.
the radiation a copy of the quantum information Alice
This article proposes a new way for solving the mind
carries when she crosses the event horizon. Thereafter
boggling paradoxes of the infalling observer, and in par-
healsocrossesthe horizon. Alice couldthussendtoBob
ticularquantumcloning,by usinga slightmodified com-
her quantum information. One concludes that he would
plementarity principle. In Section II we first present the
beowingtwocopiesofthesamequantumstate: quantum
thought-experiment about quantum cloning. In Section
information would have been cloned.
III we define the concept of death boundary, and discuss
A possible way out of this new conundrum consists in different outcomes according to the possible locations
postulating a black hole’s information retention time. If of the death boundary with respect to the event hori-
ittakesacertaintimetoBobtocollectAlice’sq-bitfrom zon. In Section IV we introduce the concept of world’s
the radiation, then his jump into the black hole can be end and argue that a slight modification of the comple-
delayed enough so that he meets the singularity before mentarity principle may help to solve oddities regarding
receivinganymessagefromAlice. ThenBobisprevented the infalling observer. In Section V we show that the
from observing quantum cloning. [2, 3] new complementarity relates to irreversibility. Section
A different way is the hypothesis of a ”final state”. VI summarizes the conclusions.
2
singularity V = er/2R r −1 12 et/2R (2)
R
(cid:16) (cid:17)
U V
messagheorizornadiation Bob The curves forUcoVnsta=nterr/Rare1g−ivern by: (3)
R
(cid:16) (cid:17)
The singularity (r = 0) is given by UV = 1, and the
Alice event horizon (r =R) by U =0.
The proper time Alice measures between crossing the
horizon(U =0)atV =V andreachingcoordinateU is
A
given by
τ =kRV U (4)
A A
FIG.1: TheHayden-Preskillthought-experimentonquantum where k is a constant that depends on Alice’s initial
cloning: Alice crosses the event horizon carrying a quantum- data. When Alice falls freely from infinite k = 1/e. We
bitofinformation. ThesubsequentHawkingphotonsarecol- assumeAlicefreelyfallsfromnearhorizonsothatinthis
lected by Bob, who reconstructs Alices q-bit and thereafter case k is O(1).
jumps into the black hole too. Then Alice sends her q-bit
Notice that the total proper time for an observer to
to Bob by means of a photon message. If Bob has time to
fall from rest r =R to r =0 is finite and given by:
receive this message he gets to observe quantum cloning of
Alices original q-bit.
τ ∼R (5)
II. QUANTUM CLONING IN A BLACK HOLE
Consider now that Bob requires a time ∆t to collect
relevantinformationfromtheradiation,andthencrosses
ConsiderthesituationsketchedinFigure1. Alice falls
the horizon at V . From (2) one gets then
B
intoaSchwarzschildblackholeandcrossestheeventhori-
zon carrying a quantum-bit of information. The subse-
quentHawkingphotonsarecollectedbyBob,whoissoar- VB =e∆t/2R (6)
ing outside the black hole at a certain distance from the V
A
horizon. He can thus in principle reconstruct Alices bit.
To observe quantum cloning Bob has to receive Alice
Once Bob has collected this information he jumps into
message sending her q-bit before he hits the singularity
the black hole as well. Assuming a very massive black
atUV =1,i.e. atU = 1 . Substituting this value into
hole,AliceandBobcancontinuetoperformexperiments B VB
after crossingthe horizon. Suppose Alice sends her q-bit (4) and taking account of (6) one is led to:
to Bob by means of a photon message. By receiving it
Bob gets to observe quantum cloning of Alices original
V
q-bit. This would contradict the principles of quantum τA ∼R A =Re−∆t/2R (7)
V
mechanics [2, 5]. B
To analyze the experiment [2] it is convenient to use IfE istheenergyAlicerequirestosendinghermessage
the Kruskal coordinates U and V. If R denotes the to Bob, on the one hand the Uncertainty principle im-
Schwarzschild radius (i.e. R = 2MG/c2), then U, V poses: τ E =1. Ontheotherhand,theenergyavailable
A
are related to the Schwarzschild coordinates r,t by the to Alice is much less than the mass M of the black hole.
following equations: Thatis(inPlanckunits: G=c=1)E <M =R/2,and
- Inside the horizon, i.e. for 0≤r ≤R: therefore 1 <R/2. Substituting into (7) gives:
τA
U = er/2R 1− r 12 e−t/2R
R 1 R R2
V = er/2R(cid:16)1− r(cid:17)12 et/2R (1) Re∆t/2R < 2 ⇒e∆t/2R < 2 (8)
R
(cid:16) (cid:17) IfonedefinestheRindlerretentiontimeω =∆t/2R,
ret
then (8) imposes the following condition to forbid that
Bob observes quantum cloning:
- Outside the horizon, i.e. for R≤r ≤∞:
U = −er/2R r −1 21 e−t/2R ω >lnR (9)
R ret
(cid:16) (cid:17)
3
Assuming that it is entanglement what causes infor-
mation to become encoded in the Hawking radiation [6],
one can assume that information deposited in the black singularity
hole prior to the “half-way”point (before half of the en-
ttterhhvoaeepnpeyoverahmapateiosorrgnbaetespieornqnoucrioeacfekddlityashtebefodybrlaeaamwckmaityoht)imonrlgeeenmrrtae,aadiancianhstdeicsootnthnhecwireseiatalphefootdeiunruttn,tihtniiet-l U horizon = deatrh abdioautinodnary VBob
formation. In Planck units the time necessaryto radiate
half the entropy of the black hole is of order M3, that Alice
is, a Rindler time of order R2 which exceeds the bound
(9) by a too largeamount. Complementarity would look
more compelling if it just barely escapes inconsistency
[5].
However, if Alice jumps into the black hole when the
evaporationhasalreadyproceededbeyondthe “half-way
point”, and Bob has watched the system during a long
timebefore,thenAlice’squantuminformationmaybere-
vealed in the Hawking radiation very rapidly. Assuming
FIG.2: Theeventhorizonoverlapswiththedeathboundary,
that black holes scramble (thermalize) information very i.e. theboundaryatwhichhumanobserversarekilledbythe
fast, the estimate of a black holes information retention tidalforces. Even iftheradiation starts tocarry information
time is just barely compatible with the bound (9), what immediatelywhenobserverscrossthehorizon,thesewillnever
is considered a more gratifying situation [2, 5]. be able of observing cloning. In this case the event horizon
We wouldliketostressthataccordingtothis explana- marks a world’s end, i.e. the end of the physical reality for
any human observer.
tion the information is supposed to remain “stored” at
the “half-way” hyper-surface.
The “final state” explanation [4] meets even more se-
malreferenceframe(i.e. Alice’s geodesicequation). One
vere difficulties. The model is based on the assumption
can show that if the body falling towardsthe singularity
that reaching the singularity the information becomes
hasmassµ, heightLandwidthw, the radialcomponent
somehow “teleported” into the outgoing radiation, so
of the tidal force will produce [8] a pressure with value:
that the original state disappears before Bob can collect
information from the radiation. To this aim a measure-
ment just at the singularity is required. One has then µMGL
T = (10)
the problemofdefining howsucha measurementoccurs, ρρ 4w2r3
since there are no observers around to do any measure-
Similar expression holds for the angular components,
ments. Additionally any true measurementat the singu-
up to a different numerical factor.
larity would unavoidably lead to information loss. Thus
What is important to note in our context, however, is
the model works without true measurements. But then
the fact that there is a criticalvalue of the pressurep
it neglects the interaction between the collapsing matter crit
at which an irreparable damage occurs in any human
andtheinfallingHawkingradiationinsidetheeventhori-
brain. In other words, this critical value defines death,
zon [7] - a necessary condition to achieve the supposed
i.e. a damage that humans are not able to repair. In
“teleportation”.
some sense p can be interpreted as a new constant of
crit
Nature whose real value remains inaccurately measured
for the time being. We take the value of 10 atm just as
III. THE DEATH BOUNDARY
a first rough estimate.
Wecalldeath boundary thehyper-surfaceatwhichT
ρρ
Consider again the infalling observer Alice: she only reaches the critical value while R+ is the corresponding
experiences tidal forces in her reference frame. If the
value of the Schwarzschild coordinate r. From (10) it
blackholeisnotmassiveenough,tidalforcescanbecome
follows:
so large outside the horizon that Alice will be destroyed
even before crossing it. By contrast, if the black hole is
massive enough tidal forces at the horizon are negligible 1 µMGL 1/3
and Alice experiences nothing special when crossing it. R+ =(cid:18)p 4w2 (cid:19) (11)
crit
But as far as Alice keeps on falling inside the black
hole tidal forces strongly increase and she is going to be Referring to the previous section this implies that the
destroyed before getting to the singularity. reasoningleadingtoaboundforretentiontimeshouldbe
Tidal forces are measured by the components of the donenotwiththesingularityatr =0,butwiththecurve
RiemanncurvaturetensorwithrespecttoAliceorthonor- characterizingthedeath boundary atr =R+,whereBob
4
singularity singularity
U V U V
death boundary
messagheorizon Bob horizornadiation Bob
death boundary
Alice Alice
FIG.3: Thedeathboundarylieswithintheeventhorizon. If FIG.4: Thedeathboundaryliesoutsidetheeventhorizon. If
theradiationstartstocarryinformationwhentheevaporation the radiation starts to carry information when the observers
proceeds beyond the death boundary, then quantum cloning crossthehorizon,thesewillneverbeableofobservingcloning.
remainsforbidden. Thedeathboundarymarksaworld’send. The eventhorizon marksa world’s end.
IV. A SLIGHT MODIFICATION OF THE
gets killed. Though this does not give a significant dif-
COMPLEMENTARITY PRINCIPLE
ferenceforcalculatingaretentiontime,itopensthedoor
to another way of solving the paradoxes.
The conclusion that quantum information stored in-
One has different possible locations for the death
sideablackholeobeysthe rulesofunitaryquantumme-
boundary: inside, at or outside the event horizon:
chanics and thus cannot be destroyed is based on the
Consider firstthe casesketchedin Figure 2, where the assumption that any quantum measurement performed
deathboundaryoverlapswiththehorizonR=R+. Even beyond the horizon does not exist for the external ob-
if the radiation starts to carry information immediately server. Thus, according to the complementarity prin-
whenobserverscrossthehorizon,thesewillneverbeable ciple, the event horizon marks the end of the physical
of observing cloning. We assume for the human brain reality for the external observer at infinity.
stem approximative values of: pcrit= 10 atm , µ=20gr, As said, this sounds particularly awkwardbecause ac-
L = 5cm, w = 2cm. Imposing R = R+ formula (11) cording to general relativity the observer at infinity has
gives a value M = 6.74339×1035 (i.e. about 339 Solar to accept that he can cross the horizon without feeling
masses) and R+ =1000km. anytrouble,andaccordingquantumphysicshewillthen
Suppose now the case sketched in Figure 3, where observe things going on inside there.
the death boundary lies between the singularity and the Thisodditycanbeovercomebyaslightmodifiedcom-
event horizon, i.e. 0 < R+ < R. When the evaporation plementarity principle:
processgoeson,theeventhorizonwillgrowlowerasfast We denote world’s end any boundary to a region a
as M. So there is a moment where one reaches again human observer never will get information from (unless
the situation R = R+. If one assumes that information possibly after evaporation). If the death boundary lies
remains concealed till the evaporation proceeds beyond inside the event horizon (Figure 3), then this boundary
this point, then observerswill neverbe able of observing marks a world’s end. If the death boundary matches or
quantum cloning either. liesoutsidethe eventhorizon(Figures2and4), thenthe
Consider finally the case sketched in Figure 4, where horizon marks the world’s end.
the death boundary lies outside the event horizon, i.e. Instead of assuming that the physical reality ends at
R < R+ < ∞. Like in the case R = R+, even if the the event horizon for observers at infinity, we assume
radiation starts to carry information immediately when that it ends at the world’s end for any observer. Quan-
observers cross the horizon, they will never be able of tum states can be reduced by measurements performed
observing cloning. Notice however that in this case the by observers within the horizon, and in this case their
radiation could not start encoding quantum information evolutionisnotunitaryforexternalobserverseither. By
when this crosses the death boundary, since results from contrast, no state reduction happens beyond a world’s
eventualexperimentsconveniently arrangedbetweenthe end for any observer.
death boundary and the event horizon could directly Takingaccountoftheanalysisoftheprecedingsection,
reach observers at distances r > R+, who then would new complementarity implies that information falling
be able to observe cloning. into a black hole remains concealed until the evapora-
5
tion of the black hole reaches a world’s end, and it then information for a long time. Suppose a stone reaching
immediately emerges. Radiation emitted before or after the world’s end of a huge black hole. The information
the horizon becomes a world’s end does not contain any defining the stone will remain concealed until the evap-
information (in agreement with the original Hawking’s oration brings the horizon to match with the world’s
assumption). end, at which moment it gets encoded in the radiation.
At the end of the day it looks as if the stone would
disappear and reappear after a long delay. Actually any
V. NEW COMPLEMENTARITY AND modelassumingdiscretetimedisplaysasimilarbehavior
IRREVERSIBILITY thoughfor muchshortertime intervals. Inthis sense the
black hole acts like a magnifier of the fundamental unit
The proposed new complementarity involves the no- of time.
tionofdeathboundaryandtherebyrelatestotheconcept
of irreversibility. Indeed, this concept explicitly appears
in the clinical definition of death, which basically states: VI. CONCLUSIONS AND OPEN PROBLEMS
death occurs when the neural functions responsible for
certainspontaneousmovementsirreversibly breakdown.
In summary we have shown:
In establishing death this way, we are assuming as ob-
The world’s end hypothesis goes around quantum
vious that our capacity of restoring neuronal dynamics
cloning in a natural way.
(our repairing capability) is limited in principle, even if
Since beyond the world’s end it does not even make
we don’t yet know where this limitation comes from.
sense to talk about physical reality for any observer,the
The same concept of irreversibly can be applied to
very concept of singularity disappears.
measurement [9]. The amplification in a detector (e.g.
The region between the horizon and the world’s end
a photomultiplier) becomes irreversible in principle at a
belongs to the physical reality, and any new information
certain level, if beyond this level an operation exceeding
arising beyond the horizon becomes encoded in the out-
the human capabilities would be required to restore the
going radiation, and does not go lost for the external
photon’s quantum state. When such a level is reached
observer.
the detector clicks. Such a view combines the subjective
Since information characterizing living human brains
and the objective interpretation of measurement: on the
goes irreversibly lost, evaporation does not produce re-
onehandnohumanobserver(consciousornot)hastobe
suscitation.
actually present in order that a registration takes place;
ontheotherhandonedefinesthe’collapse’or’reduction’ New complementarity depends on the concept of irre-
with relation to the capabilities of the human observer. versibility. Although it seems that this concept refers to
In our model the informationcharacterizingliving hu- alimitationofthehumancapabilities,forthetimebeing
man brains goes irreversibly lost at a world’s end in the we do not know when precisely irreversibility happens,
same way in which information goes irreversibly lost in neither in death nor in measurement.
the detectionprocess. Any wellfunctioning humanbrain Newcomplementaritydoesnotaddresstheproblemof
stem breaksdownataworld’send. Sincethestemisthe howthe informationremainsconcealedwithintheblack-
mostprimitivepartofthebraininevolution,presumably hole and becomes encoded in the outgoing radiation [2,
any brain breaks down at a world’s end. 5]. Nevertheless it suggests that this problem may be
Inthissenseaworld’sendcanbeconsideredadetector relatedtothequestionofhowinformationlastsinmodels
reducing quantum neural states. working with discrete spacetime.
But it can also be viewed as an “information mirror” Reducing the problem of the infalling observer to
[2] reflecting other types of states. Although as a other known unsolved problems is somewhat already a
mirror working in a peculiar way, since it also conceals progress.
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