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• An old story
• Step by step, paradox by paradox
• Soft hair for black holes
One paradox for four decades!
The famous “information paradox of black holes” has been last for more that 40 years.
Some of the most brilliant physicists has been thinking about the paradox and arguing
among themselves, but the subject is such controversial that there has been no answer
convinced the Physics society entirely. Yet they try to suggest a solution to the question
and don’t give up!
Seemingly, Physics left with such a paradox because physicists don’t want to abandon
any of basic principles such as causality, the uncertainty principle and the equivalence
principle. But Several decades ago, Hawking showed that assuming these principles,
information would destroy in a black hole! An unpleasable conclusion.
During these decades however, people has not found neither any flaw in Hawking’s
calculations nor a convincible reason for abandoning one of these principles.
An old question
Equivalence principle; building
block of GR
An observer falling in a gravitational field —
even the powerful one inside a black hole —
will see exactly the same phenomena as an
observer floating in empty space.
what would happen to
an astronaut who
dived into a black
hole. Obviously, he
would die. But how?
According to equivalence principle in general relativity, he wouldn’t feel any
thing while he is free falling toward the black hole . He even have no idea about
the existence of any kind of boundary like the event horizon. After a long time,
when he pass through the event horizon and enter the black hole, he realize that
the gravity force acts on his feet much more stronger than his head. This
difference becomes bigger and bigger and finally tears his body apart!
NATURE | VOL 496 | 4 APRIL 2013
New scenarios challenging old principles!
In 2012, Polchinski tried to do calculations from a new path which ultimately leaded to
a different end for this story! He showed that because of quantum effects, the event
horizon is a very hot soup of particles called “firewall” which will burn every thing near
it immediately.
The first sacrifice of emerging a firewall will be the
equivalence principle! Because the free falling observer
will observe the event horizon of the black hole.
Trying to Keep this principle still valid, Polchinski had to
propose a plan B in which such a firewall doesn’t form,
this time in the price of trampling quantum mechanics
principles!
A real crisis in
foundation of
physics may need
a revolution to
resolve!
QM
GR
Historical roots for firewall
The idea of firewall first stablished by Hawking in 1974. Using semi-classical calculations,
he showed that any isolated black hole has thermal radiations contain photons and other
particles. Via this procedure, black holes lose mass until they evaporate entirely. Although
GR will guarantee that free falling observer doesn’t notice the radiation, Hawking’s results
was breathtaking because by GR, black holes only can shallow and grow, not evaporate!
Where does the radiation come from?
The empty space is not empty. Because of quantum
fluctuations, vacuum space is full of particle-antiparticles
which are created and rapidly annihilate by recombining.
This may happen also near the black hole horizon where one of the created pairs may fall
into the black hole and the other one has to escape toward infinity because of momentum
conservation law. The escaped particles look like thermal radiation and carry positive
energy, so their infalling pairs must carry negative energy and that’s how the black hole
loses mass gradually.
Information paradox
Quantum mechanics claims that information cannot be missed or
destroyed because of unitarity. So in principle, one may figure out
every details about whatever has fallen in the hole by studying the
its radiations. However, Hawking radiation is thermal, so it
doesn’t contain any quantum information, means that one would
get the same radiation by tossing in a kilogram of rocks or a
bunch of computer chips full of data! Hence there is no way to
realize how a black hole has formed, even until it dies.
The information paradox split up physicists into two camps. One
group including Hawking believed that data will be disappeared
after falling into the black holes, and if this contradicts QM, then
we need to find new principles to stablish a new theory. Whereas
there were physicists stuck by QM and believed that there can not
exist any quantum like theory without conservation law for
information!
Holography, another new principle!
In 1997 Juan Maldacena came up with a new proposal states that any 3-D region of our
world can be described in full detail by information encoded on its 2-D boundary. In
other words, the whole universe is a projection of information on the boundary which is
not visible for the universe residents. More strictly, he investigated a 3-D universe
containing strings and black holes governed only by gravity. On the 2-D surface
however, fields and particles obey QM without any gravity. Any event happens in 3-D
universe can be understood by studying the boundary and vice versa. So holography
provides a mathematical dictionary between languages of these worlds.
Applying holography principle on a 3-D black hole, one can deduce that information
come out of the black hole is achievable on 2-D boundary where QM principles implies
information conservation. Hence, there should be a same conservation law also in 3-D
world and there should be a way for information to escape the black hole.
Paradox after paradox!
In the mid-1990’s Susskind realized that the information could be encoded in states of
escaping particles if they were entangled with their infalling pairs. But if it’s true, then
the particle have to be also entangled with the whole Hawking radiation emitted before.
Being entangled with two distinct systems at the same time is impossible by QM. So one
of these entanglements should be snipped. Susskind and others decided to keep the
entanglement between the emitted particle and the Hawking radiation. But breaking the
entanglement between pairs would release energy. Doing so for lots of pairs will cause a
noticeable amount of energy which make the event horizon as a fire ring and so violates
the equivalence principle! So the situation was clarified as the following statement which
was published by Susskind in arXiv:
“Either accept that firewalls exist and that general relativity
breaks down, or accept that information is lost in black holes
and quantum mechanics is wrong”!
After more than 40 papers on the same topic, no one could find any flaw in their logic.
So it seems really there is something inconsistent in our thinking
about black holes.
Impossible in real-world!
Later Daniel Harlow and his collaborator propose that it may be just a hypothetical
paradox, not a real one. They explain that to detect the real paradox, one has to first
decode a significant portion of the Hawking radiation and then dive into the black hole
to examine the infalling pairs. They showed that the decoding proses last for such a long
time that black hole would evaporate entirely before the observer was ready to jump! So
their claim is “There is no fundamental principle prevents observer from detecting
the paradox, however doing so is practically impossible”.
Giddings, however, argues that the firewall paradox requires a radical solution. He has
calculated that if the Hawking radiation stay entangled with the infalling pairs until the
escaping particles has travelled a short distance away from the event horizon, then the
energy released by braking their bounds will be much less. In this way, the equivalence
principle survive while some quantum lows need modifications.
Why physicists don’t like to think about Hawking’s argument again? It seems that all of
them have a deep respect to the Maldacena’s dictionary. As Polchinski mentioned “This
is the deepest ever insight into gravity because it links it to quantum fields”.
What’s new?
Recently, physicists start to study the infrared structure of quantum gravity in
asymptotically flat space. In 1962 Bondi, van der Burg, Metzner and Sachs (BMS)
realized that in addition to the known Poincare transformation, there is an infinite set of
diffeomorphisms called supertranslations under which physical data in past or future null
infinity transforms non-trivially. Moreover, it was shown that a certain antipodal
combination of past and future supertranslations is an exact symmetry of gravitational
scattering, means that there are infinite number of “supertranslation charges” and
conservation laws.
In the quantum theory, matrix elements of the conservation laws give an infinite number
of exact relations between scattering amplitudes. These relations have been studied
previously by Weinberg in 1965 and are known as the soft graviton theorem.
One may run the same argument backwards:
starting from the soft graviton theorem one may derive both the infinity of conservation
laws and supertranslation symmetry of gravitational scattering.
arXiv:1601.00921v1 [hep-th] 5 Jan 2016
Why are supertranslations important?
Supertranslations transform the Minkowski vacuum to a physically inequivalent zeroenergy vacuum. Since the vacuum is not invariant, supertranslation symmetry is
spontaneously broken. The soft (zero-energy) gravitons are the associated Goldstone
bosons. There are infinite number of inequivalent vacua differ from one another by the
creation or annihilation of soft gravitons. They all have zero energy but different
angular momenta. So there are two principle to doubt:
1) The vacuum in quantum gravity is not unique.
The information loss argument assumes that after the evaporation process is completed,
the quantum state settles down to a unique vacuum. In fact, the process of black hole
formation/evaporation will generically induce a transition among the infinitely
degenerate vacua. In principle, the final vacuum state could be correlated with the
thermal Hawking radiation in such a way as to maintain quantum purity.
2) Black holes have `soft hair'.
The information loss argument assumes that static black holes are characterized only by
their mass M, charge Q and angular momentum J (no-hair theorem) up to
diffeomorphisms. However BMS transformations are diffeomorphisms which change
the physical state. A Lorentz boost for example maps a stationary black hole to an
obviously physically inequivalent black hole with different energy and non-zero
momentum. Supertranslations similarly map a stationary black hole to a physically
inequivalent one.
In the process of Hawking evaporation, supertranslation charge will be radiated through
null infinity. Since this charge is conserved, the sum of the black hole and radiated
supertranslation charge is fixed at all times. This requires that black holes carry what
people call ‘soft hair’ arising from supertranslations. Moreover, when the black hole has
fully evaporated, the net supertranslation charge in the outgoing radiation must be
conserved. This will force correlations between the early and late time Hawking
radiation, similar to the correlations enforced by overall energy-momentum
conservation. Such correlations are not seen in the usual semiclassical computation.
The process of black hole formation/evaporation, viewed as a scattering amplitude
from
to
must be constrained by the soft graviton theorem.
Weinberg also proved the ‘soft photon’ theorem. This theorem implies an infinite
number of previously unrecognized conserved quantities in all abelian gauge theories electromagnetic analogs of the supertranslation charges. By a direct analog of the
preceding argument, black holes must carry a corresponding ‘soft electric hair’. The
structure in the electromagnetic case is very similar, but technically simpler, than the
gravitational one.
String-theoric black holes
The problem of black hole information has been discussed at the same time of
developments in string theory. In particular it was shown that certain string-theoretic
black holes store complete information about their quantum state in a holographic plate
constructed of quantum pixels and lives at the horizon. Moreover the storage capacity
was found to be precisely the amount predicted by the
Hawking-Bekenstein area-entropy law.
Whether or not string theory in some form is a correct theory of nature, the holographic
method it has presented results in a better understanding of storing information on the
black hole horizon, which might be employed by real-world black holes independently
of the ultimate status of string theory.
Indeed in this paper authors showed that soft hair has a natural description as quantum
pixels in a holographic plate. The plate lives on the two sphere at the future boundary
of the horizon. Exciting a pixel corresponds to creating a spatially localized soft
graviton or photon on the horizon, and may be implemented by a horizon
supertranslation or large gauge transformation. In a physical setting, the quantum state
of the pixel is transformed whenever a particle crosses the horizon.
The combination of the uncertainty principle and cosmic censorship requires all
physical particles to be larger than the Planck length, effectively setting a minimum
spatial size for excitable pixels. This gives an effective number of soft hairs
proportional to the area of the horizon in Planck units and hints at a connection to the
area-entropy law.
The End