Black Holes are unique objects which allow for meaningful theoretical studies
of strong gravity and even quantum gravity effects. An infalling and a distant
observer would have very different views on the structure of the world.
However, a careful analysis has shown that it entails no genuine contradictions
for physics, and the paradigm of observer complementarity has been coined.
Recently this picture was put into doubt. In particular, it was argued that in
old Black Holes a firewall must form in order to protect the basic principles
of quantum mechanics. This AMPS paradox has already been discussed in a vast
number of papers with different attitudes and conclusions. Here we want to
argue that a possible source of confusion is neglection of quantum gravity
effects. Contrary to widespread perception, it does not necessarily mean that
effective field theory is inapplicable in rather smooth neighbourhoods of large
Black Hole horizons. The real offender might be an attempt to consistently use
it over the huge distances from the near-horizon zone of old Black Holes to the
early radiation. We give simple estimates to support this viewpoint and show
how the Page time and (somewhat more speculative) scrambling time do appear.Comment: 4 pages; minor changes, a few references adde

Golovnev, Alexey

English

arXiv

Alexey Golovnev

Springer - Publisher Connector

Smooth horizons and quantum ripples

Eur. Phys. J. C (2015) 75:185
DOI 10.1140/epjc/s10052-015-3395-8
Letter
Smooth horizons and quantum ripples
Alexey Golovneva
High Energy Physics Department, Saint Petersburg State University, Ulyanovskaya ul., d. 1, Petrodvoretz, 198504 Saint-Petersburg, Russia
Received: 19 August 2014 / Accepted: 8 April 2015 / Published online: 1 May 2015
© The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract Black holes are unique objects which allow for
meaningful theoretical studies of strong gravity and even
quantum gravity effects. An infalling and a distant observer
would have very different views on the structure of the world.
However, a careful analysis has shown that it entails no gen-
uine contradictions for physics, and the paradigm of observer
complementarity has been coined. Recently this picture was
put into doubt. In particular, it was argued that in old black
holes a firewall must form in order to protect the basic princi-
ples of quantum mechanics. This AMPS paradox has already
been discussed in a vast number of papers with different atti-
tudes and conclusions. Here we want to argue that a pos-
sible source of confusion is the neglect of quantum gravity
effects. Contrary to widespread perception, it does not nec-
essarily mean that effective field theory is inapplicable in
rather smooth neighbourhoods of large black hole horizons.
The real offender might be an attempt to consistently use it
over the huge distances from the near-horizon zone of old
black holes to the early radiation. We give simple estimates
to support this viewpoint and show how the Page time and
(somewhat more speculative) scrambling time do appear.
It is an amazing fact about black holes that they can emit
particles [1]. Assuming that this radiation is purely thermal
and that a black hole will eventually evaporate completely,
the information which has been swallowed by the black hole
during its lifetime is lost, in contradiction to the unitary nature
of quantum mechanics. It was then established that unitar-
ity can still be saved if an effective description of the outer
near-horizon region is allowed in terms of a Planck-width
stretched horizon which can absorb, thermalise and emit
information [2]. Of course, due to the equivalence principle,
an infalling observer should not experience anything special
while crossing the horizon. However, there would be no way
for him to share this knowledge with the distant colleague,
and therefore one can avoid running into a contradiction by
adopting the fancy viewpoint that there is no need for a uni-
ae-mails: agolovnev@yandex.ru; golovnev@hep.phys.spbu.ru
versal global effective field theory description of physics in
the whole spacetime. It marked the birth of the black hole
complementarity paradigm [2].
In order to better understand the relevant time scales of
evaporation, recall that a black hole of mass M has Hawking
temperature
kBT = h¯c
3
8πGM
with the usual notations for the Planck constant h¯ ≡ h2π ,
the speed of light c, the gravitational constant G, and the
Boltzmann constant kB. Under the Stefan–Boltzmann law,
the emissive power scales as T 4 ∝ 1
M4
while the horizon area
∝ M2 which implies M˙ ∝ 1
M2
for the rate of energy loss, and
therefore the characteristic time of evaporation grows with
mass as M3 [3], or more explicitly t ∼
(
M
mPl
)3
tPl where the
subscript Pl stands for the Planckian quantities.
The problem [4] presents itself when the black hole has
emitted one half of its total entropy,1 or at the Page time
tP ∼
(
M
mPl
)3
· tPl.
In this case the already emitted radiation contains practi-
cally all information which has gone into the black hole [8],
and therefore must be fully entangled with the late radiation
appearing in the near-horizon zone. On the other hand, from
the viewpoint of an infalling observer, both sides of the hori-
zon are just the two halves of very flat (almost Minkowski)
space. Under the spell of the equivalence principle, the freely
falling observer has all the legal rights to expect meeting
the vacuum state there. The vacuum state is a very regular
thing characterised by maximal entanglement of those two
halves. Of course, being simultaneously maximally entan-
gled with two different systems is impossible. The corre-
sponding general principle of quantum mechanics is known
under the beautiful name of monogamy of entanglement.
1 Note that, under the name of energy curtains, this paradox was earlier
reported in Ref. [5], see also Refs. [6,7].
123
185 Page 2 of 4 Eur. Phys. J. C (2015) 75 :185
There is apparently a strong tension between the equiva-
lence principle and quantum mechanics which becomes evi-
dent at the Page time. But it was also argued [4] that the same
should be true much earlier, at the scrambling time when the
information of constituent matter has already been scram-
bled (thermalised) by the black hole. This time scale is much
smaller. Actually, there are some reasonably good reasons to
believe that it must be of order
tscr ∼
(
M
mPl
log
M
mPl
)
· tPl,
which makes the black holes amazingly fast scramblers [9].
The status of this time scale is however unclear [10].
With one time scale or another, the way out proposed
in Ref. [4] was a firewall just behind the horizon. In other
words, the equivalence principle is sacrificed in such a way
that we cannot trust the usual effective field theory in the
zone. Adopting entanglement with the early radiation, we
can no longer afford entanglement between the two halves
of the near-horizon region which translates into presence of
energetic quanta around the horizon, or a firewall.
A natural attempt to fully identify the interior with the dis-
tant radiation is not only extremely non-local but also leads to
the frozen vacuum [11] or inability of the infalling observer to
excite the near-horizon state which violates the equivalence
principle no less than a firewall.
A possible alternative would be to resort to strong com-
plementarity by arguing that it is not a problem when two
observers see absolutely different physical content of the
zone if they cannot communicate their findings to each other.
It does not seem to work out in an ideal way because an
infalling observer might perform a precise measurement of
the early radiation before entering the zone, or because there
is some time for a freely falling observer to change his
mind and turn around inside the zone before crossing the
horizon [12]. However, the contradictory measurements and
inferences might turn to be computationally unfeasible [13],
extremely fine-grained [14], overwhelmingly affecting the
black hole state, or even be akin to observing quantum super-
positions of macroscopic worlds [15].
The problem remains controversial, and it is only clear
that new insights are needed and, hopefully, expected.
They would certainly deepen our understanding of quan-
tum mechanics and gravity; and the relevant issues are really
exciting. For example, one line of reasoning [16] asserts
that the very measurement made by the distant observer cre-
ates high energy quanta, or a firewall, which would kill the
infalling colleague. If this process is to be causally conceiv-
able, and if we do insist on causality of so violent behaviours,
then Einstein–Rosen bridges between the black hole and its
early radiation must be invoked [17].
Instead, we would like to offer a different approach to the
AMPS problem which might actually point at better integrity
of physical description. We argue that the paradoxes might
be resolved by taking quantum gravity effects into account
in the form of unavoidable entanglement with microscopic
geometrical configurations of spacetime.
Our main idea is that locally an effective field theory
description can be valid and very precise everywhere in the
low curvature regions, although taken all the way over huge
spacetime distances, the tiny errors might accumulate con-
siderably enough to entail the loss of purity of the early
Hawking radiation. Note, for an illustration, that the mean
calendar year length difference of Gregorian and Julian cal-
endars is <11 min and at first glance seems impractical, but
in 400 years it sums up to three full days. Random errors do
not grow as fast as a gradual change but still can eventually
matter.
Below we give some simple estimates of quantum gravity
effects on propagating radiation, and show how the Page and
scrambling times can naturally appear from such considera-
tions.
Let us first address the clear-cut problem at the Page time
tP ∝ M3. If we want to take the quantum gravity effects into
account, then probably it would be safest and fairly model-
independent to assume that the wavelength of a typical pho-
ton of Hawking radiation
λ ∼ M
mPl
· lPl
cannot be determined with precision better than the Planck
length lPl. We treat it as an intrinsic fluctuation of the wave-
length
δλ ∼ lPl.
If the photon has propagated over a huge number N of wave-
lengths, then the statistical uncertainty of the path length
L = Nλ amounts to
δL ∼ √N · lPl
from which we see that δL reaches λ when
L ∼ λ
3
l2Pl
∼
(
M
mPl
)3
· lPl ∼ ctP.
Therefore, to the Page time, the information about the rel-
ative phases of different photons is definitely lost. Of course,
the reason is that we treat the geometry as a mute background
arena for electrodynamics. But the actual quantum state of
photons gets dynamically entangled with quantum fluctua-
tions of geometry and, after a long enough time, tracing over
the states of geometry produces a very blurred image of the
emitted light. Since it is no longer pure, there are no obstruc-
tions for the late radiation to be entangled with the black hole
interiors. Quantum gravity has gone into full play despite the
incredible smallness of all its local effects.
123
Eur. Phys. J. C (2015) 75 :185 Page 3 of 4 185
The scrambling time tscr ∝ M log M is much trickier to
discuss in this context. But we argue that a more careful
treatment of the decoherence features allows one to naturally
come at this time scale, too. Let us assume that the dynamics
of photons in presence of small quantum gravitational cor-
rections can be described as an open quantum system with
the Lindblad equation [18,19],
ρ˙ = Lˆρ
for the density matrix ρ. It provides a natural framework
for discussing phenomenology of quantum gravity, see for
example the paper [20] and references therein for possible
effects in the oscillations of neutral kaons. The dynamical
semigroup generator Lˆ consists of two terms: the commu-
tator with the Hamiltonian which reproduces the standard
Schrödinger equation and the additional Lindblad operator,
the required properties of which we do not need to discuss
now.
The order of magnitude of the coefficients in the matrix of
the Lindblad operator depends on the adopted level of coarse
graining. We want to address situations in which an effective
field theory description is just marginally enough to come
to a contradiction. It is natural then to adopt a resolution at
the level of ultraviolet cutoff scale which is presumably the
Planck scale. According to our considerations above, we can
expect to start confusing the neighbouring states after the
time period τ ∼ λc , and therefore the matrix entries of the
Lindblad operator are expected of order c
λ
∼ mPlMtPl .
Note then that we are not interested in simply convert-
ing one state into another with off-diagonal elements of the
Lindblad operator because it does not automatically entail
decoherence, even though it may produce very interesting
effects such as CPT-violation [20]. We would rather like to
find an independent growth of probabilities for other states
bringing the system to a statistical mixture. For the ini-
tial Cauchy data, we can assume that a given photon has
been in a given pure state with the probability p1 practi-
cally equal to 1. However, due to quantum gravity effects,
the other states could not have been totally absent. Their ini-
tial probabilities p j (0) can be estimated as ∼mPlM , or some
mild power of it. We see that typically log p j starts grow-
ing with time as mPlMtPl · t . Extrapolating this trend far beyond
any reasonable limit, an undoubtedly mixed state is reached
when
log
M
mPl
∼ mPl
MtPl
· t,
or at the fast scrambling time. Admittedly, it sounds rather
speculative, but so is the issue of scrambling time itself in
the context of firewalls.
Of course, it was always clear that somehow relaxing the
assumption of entanglement between the early and the late
radiation would give a way to resolving the paradox [21].
Moreover, a concrete realisation was proposed in the con-
text of the many worlds interpretation of quantum mechan-
ics [22]. A black hole randomly emits really huge amounts
of low-energy quanta, and therefore, for an old specie, its
position must be very indefinite due to recoil effects [3].
Accordingly, we have to face macroscopic superpositions
in the system. (We note in passing, it might be interesting to
compare these superpositions with those which appear in the
Ref. [15].) It is certainly a logical possibility that after speci-
fying a certain branch for the macroscopic world the unitarity
is lost despite being safe in the full picture [23]. However,
it seems rather radical an idea which probably could make
macroscopic quantum superpositions (too easily) observable.
Our proposal is different. We talk about small fluctua-
tions of geometry irrespective to the foundational issues of
quantum mechanics. Note also that what we mean is not just
gravitons radiated from the black hole which are of little or
no interest for resolving the paradox, but it is really an effect
of a slightly random medium with the spacetime foam on the
way of the photons. Of course, it would be completely legit-
imate to wonder how the distant fluctuations and the nearby
emission effects contrive to avoid the potential tensions and
save the day.
Unfortunately, we are not so much aware of the details
of quantum gravity and even its real degree of non-locality.
However, our point is that these non-local quantum gravity
effects need not be locally observable with any deviations
from effective field theory or with other conceivable types of
anomalies such as the frozen states. What matters is only the
Planck scale physics.
Although we are very far from giving the final and defini-
tive solutions, the estimates look very interesting because
they produce the relevant time scales from a completely dif-
ferent side. Actually, it is not the first time when black holes
teach us non-trivial lessons about the Nature. Very remark-
ably, black holes obey the usual laws of thermodynamics
[24] which should not be expected of a simple and fairly
isolated system. Somehow, general relativity has given the
hints to a deeper parent theory which has to describe black
holes as statistical systems with many degrees of freedom.
Now we might learn some new lessons about quantum grav-
ity regimes from an unexpected direction. We ought to be
ready and open-minded for new insights. Black holes have
a good credit history, and it would be a nice idea to take
seriously what they say.
Note added in Proof After submission of this paper many
works on the Black Hole information paradox have been
published. Among those which might have some relation
to our attempt on resolving the unitarity crisis with quantum
gravity corrections we would like to mention F. Bezrukov, D.
Levkov, S. Sibiryakov. arXiv:1503.07181 and M. Requardt.
arXiv:1503.07312.
123
185 Page 4 of 4 Eur. Phys. J. C (2015) 75 :185
Acknowledgments The author is grateful to A. Andrianov for useful
discussions. The Author was supported by Russian Foundation for Basic
Research Grant No. 12-02-31214 in the initial stages of the project and is
supported by Dynasty Foundation now. Support by the Saint Petersburg
State University travel grant No. 11.42.1297.2014 is also acknowledged.
OpenAccess This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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Black holes are unique objects which allow for meaningful theoretical studies of strong gravity and even quantum gravity effects. An infalling and a distant observer would have very different views on the structure of the world. However, a careful analysis has shown that it entails no genuine contradictions for physics, and the paradigm of observer complementarity has been coined. Recently this picture was put into doubt. In particular, it was argued that in old black holes a firewall must form in order to protect the basic principles of quantum mechanics. This AMPS paradox has already been discussed in a vast number of papers with different attitudes and conclusions. Here we want to argue that a possible source of confusion is the neglect of quantum gravity effects. Contrary to widespread perception, it does not necessarily mean that effective field theory is inapplicable in rather smooth neighbourhoods of large black hole horizons. The real offender might be an attempt to consistently use it over the huge distances from the near-horizon zone of old black holes to the early radiation. We give simple estimates to support this viewpoint and show how the Page time and (somewhat more speculative) scrambling time do appear

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Smooth horizons and quantum ripples

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