Holographic principle Posted by: Tarun
Holographic principle
Posted by: Tarun
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The holographic
principle is a property of string
theories and a supposed property
of quantum gravity that
states that the description of a volume of space can be thought of as encoded on a boundary to
the region—preferably a light-like boundary like a gravitational
horizon. First proposed by Gerard
't Hooft, it was given a precise
string-theory interpretation by Leonard
Susskind[1] who combined his ideas with previous ones of 't Hooft
and Charles Thorn. As
pointed out by Raphael Bousso, Thorn
observed in 1978 that string theory admits a lower-dimensional description in
which gravity emerges from it in what would now be called a holographic way.
In
a larger sense, the theory suggests that the entire universe can be seen as a two-dimensional information structure "painted" on the cosmological horizon, such that the three
dimensions we observe are an effective
description only atmacroscopic
scales and at low energies.
Cosmological holography has not been made mathematically precise, partly
because the cosmological horizon has a finite area and grows with time.
The
holographic principle was inspired by black hole thermodynamics, which conjectures that the maximal entropy in any region scales with the radius squared,
and not cubed as might be expected. In the case of a black
hole, the insight was that the
informational content of all the objects that have fallen into the hole might
be entirely contained in surface fluctuations of the event horizon. The
holographic principle resolves the black hole
information paradox within the framework of string
theory. However, there exist classical solutions to the Einstein equations
that allow values of the entropy larger than those allowed by an area law hence
in principle larger than those of a black hole. These are the so-called
"Wheeler's bags of gold". The existence of such solutions is in
conflict with the holographic interpretation and their effects in a quantum
theory of gravity including the holographic principle are not yet fully
understood.
Black hole entropy
An
object with entropy is
microscopically random, like a hot gas. A known configuration of classical
fields has zero entropy: there is nothing random about electric and magnetic
fields, or gravitational waves.
Since black holes are exact solutions of Einstein's equations, they were thought not to have any entropy either.
But Jacob
Bekenstein noted that this leads to a
violation of the second law of thermodynamics. If one throws a hot gas with entropy into a black hole,
once it crosses the event
horizon, the entropy would disappear. The
random properties of the gas would no longer be seen once the black hole had
absorbed the gas and settled down. One way of salvaging the second law is if
black holes are in fact random objects, with an enormous entropy whose increase is greater than the entropy carried by
the gas.
Bekenstein
assumed that black holes are maximum entropy objects—that they have more
entropy than anything else in the same volume. In a sphere of radius R,
the entropy in a relativistic gas increases as the energy increases. The only
known limit is gravitational;
when there is too much energy the gas collapses into a black hole. Bekenstein
used this to put an upper
bound on the entropy in a region of
space, and the bound was proportional to the area of the region. He concluded
that the black hole entropy is directly proportional to the area of the event
horizon.
Stephen
Hawking had shown earlier that the
total horizon area of a collection of black holes always increases with time.
The horizon is a boundary defined by lightlike geodesics; it is those light rays that are just barely unable to
escape. If neighboring geodesics start moving toward each other they eventually
collide, at which point their extension is inside the black hole. So the
geodesics are always moving apart, and the number of geodesics which generate
the boundary, the area of the horizon, always increases. Hawking's result was
called the second law of black hole thermodynamics, by analogy with the law of entropy increase, but at first, he did not take the analogy too seriously.
Hawking
knew that if the horizon area were an actual entropy, black holes would have to
radiate. When heat is added to a thermal system, the change in entropy is the
increase inmass-energy divided
by temperature:
If black holes have a finite entropy,
they should also have a finite temperature. In particular, they would come to
equilibrium with a thermal gas of photons. This means that black holes would
not only absorb photons, but they would also have to emit them in the right
amount to maintain detailed
balance.
Time
independent solutions to field equations don't emit radiation, because a time
independent background conserves energy. Based on this principle, Hawking set
out to show that black holes do not radiate. But, to his surprise, a careful
analysis convinced him that they
do, and in just the right way to come
to equilibrium with a gas at a finite temperature. Hawking's calculation fixed
the constant of proportionality at 1/4; the entropy of a black hole is one
quarter its horizon area in Planck
units.
The
entropy is proportional to the logarithm of the number of microstates, the ways a system can be configured microscopically while
leaving the macroscopic description unchanged. Black hole entropy is deeply
puzzling — it says that the logarithm of the number of states of a black hole is
proportional to the area of the horizon, not the volume in the interior.
Black hole information paradox
Hawking's
calculation suggested that the radiation which black holes emit is not related
in any way to the matter that they absorb. The outgoing light rays start
exactly at the edge of the black hole and spend a long time near the horizon,
while the infalling matter only reaches the horizon much later. The infalling
and outgoing mass/energy only interact when they cross. It is implausible that
the outgoing state would be completely determined by some tiny residual
scattering.
Hawking
interpreted this to mean that when black holes absorb some photons in a pure
state described by a wave
function, they re-emit new photons in a thermal mixed state described by a density
matrix. This would mean that quantum
mechanics would have to be modified, because in quantum mechanics, states which
are superposition with probability amplitudes never become states which are
probabilistic mixtures of different possibilities.
Troubled
by this paradox, Gerard
't Hooft analyzed the emission of Hawking
radiation in more detail. He noted that
when Hawking radiation escapes, there is a way in which incoming particles can
modify the outgoing particles. Their gravitational field would
deform the horizon of the black hole, and the deformed horizon could produce
different outgoing particles than the undeformed horizon. When a particle falls
into a black hole, it is boosted relative to an outside observer, and its
gravitational field assumes a universal form. 't Hooft showed that this
field makes a logarithmic tent-pole shaped bump on the horizon of a black hole,
and like a shadow, the bump is an alternate description of the particle's
location and mass. For a four-dimensional spherical uncharged black hole, the
deformation of the horizon is similar to the type of deformation which
describes the emission and absorption of particles on a string-theory world sheet.
Since the deformations on the surface are the only imprint of the incoming
particle, and since these deformations would have to completely determine the
outgoing particles,‘t Hooft believed that the correct description of the
black hole would be by some form of string theory.
This
idea was made more precise by Leonard
Susskind, who had also been developing
holography, largely independently. Susskind argued that the oscillation of the
horizon of a black hole is a complete descriptionof both the infalling and
outgoing matter, because the world-sheet theory of string theory was just such
a holographic description. While short strings have zero entropy, he could
identify long highly excited string states with ordinary black holes. This was
a deep advance because it revealed that strings have a classical interpretation
in terms of black holes.
This
work showed that the black hole information paradox is resolved when quantum
gravity is described in an unusual string-theoretic way assuming the
string-theoretical description is complete, unambiguous and non-redundant.[12] The space-time in quantum gravity would emerge as an
effective description of the theory of oscillations of a lower-dimensional
black-hole horizon, and suggest that any black hole with appropriate
properties, not just strings, would serve as a basis for a description of
string theory.
In
1995, Susskind, along with collaborators Tom Banks, Willy
Fischler, and Stephen
Shenker, presented a formulation of the
new M-theory using a holographic description in terms of charged
point black holes, the D0 branes of type IIA string theory. The Matrix theory they proposed was first suggested as a
description of two branes in 11-dimensionalsupergravity by Bernard
de Wit, Jens Hoppe, and Hermann Nicolai. The later authors reinterpreted the same matrix models as
a description of the dynamics of point black holes in particular limits.
Holography allowed them to conclude that the dynamics of these black holes give
a complete non-perturbative formulation
of M-theory. In 1997, Juan
Maldacenagave the first holographic
descriptions of a higher-dimensional object, the 3+1-dimensional type IIB membrane,
which resolved a long-standing problem of finding a string description which
describes a gauge theory.
These developments simultaneously explained how string theory is related to
some forms of supersymmetric quantum field theories.
Limit on information density
Entropy,
if considered as information (see information entropy),
is measured in bits.
The total quantity of bits is related to the total degrees of freedom of matter/energy.
For
a given energy in a given volume, there is an upper limit to the density of
information (the Bekenstein bound) about the whereabouts of all
the particles which compose matter in that volume, suggesting that matter
itself cannot be subdivided infinitely many times and there must be an ultimate
level of fundamental particles. As the degrees of freedom of a particle are the product of all the degrees of
freedom of its sub-particles, were a particle to have infinite subdivisions
into lower-level particles, then the degrees of freedom of the original
particle must be infinite, violating the maximal limit of entropy density. The
holographic principle thus implies that the subdivisions must stop at some
level, and that the fundamental particle is a bit (1 or 0) of information.
The
most rigorous realization of the holographic principle is the AdS/CFT correspondence
by Juan Maldacena.
However, J.D. Brown and Marc
Henneaux had rigorously proved already
in 1986, that the asymptotic symmetry of 2+1 dimensional gravity gives rise to
a Virasoro algebra, whose corresponding quantum
theory is a 2-dimensional conformal field theory.
High-level summary
The
physical universe is widely seen to be composed of "matter" and
"energy". In his 2003 article published in Scientific American magazine, Jacob
Bekenstein summarized a current trend
started by John Archibald Wheeler, which suggests scientists may "regard the
physical world as made of information, with energy and matter as incidentals."Bekenstein asks "Could we, as William
Blake memorably penned, 'see a world
in a grain of sand,' or is that idea no more than 'poetic
license,'" referring to the
holographic principle.
Unexpected
connection
Bekenstein's
topical overview "A Tale of Two Entropies" describes potentially
profound implications of Wheeler's trend, in part by noting a previously
unexpected connection between the world of information theory and
classical physics. This connection was first described shortly after the
seminal 1948 papers of American applied mathematician Claude introduced today's most widely used measure of
information content, now known as Shannon
entropy. As an objective measure of the
quantity of information, Shannon entropy has been enormously useful, as the
design of all modern communications and data storage devices, from cellular
phones to modems to
hard disk drives and DVDs,
rely on Shannon entropy.
In thermodynamics (the branch of physics dealing with heat), entropy is
popularly described as a measure of the "disorder" in a physical system of matter and energy. In 1877
Austrian physicist Ludwig
Boltzmann described it more precisely in
terms of the number of distinct microscopic states that the
particles composing a macroscopic "chunk" of matter could be in while
still looking like the same macroscopic "chunk". As
an example, for the air in a room, its thermodynamic entropy would equal the
logarithm of the count of all the ways that the individual gas molecules could
be distributed in the room, and all the ways they could be moving.
Energy,
matter, and information equivalence
Shannon's
efforts to find a way to quantify the information contained in, for example, an
e-mail message, led him unexpectedly to a formula with the same form as Boltzmann's. In an article in the August 2003 issue of Scientific
American titled "Information in the Holographic Universe", Bekenstein
summarizes that "Thermodynamic entropy and Shannon entropy are
conceptually equivalent: the number of arrangements that are counted by
Boltzmann entropy reflects the amount of Shannon information one would need to
implement any particular arrangement..." of matter and energy.
The only salient difference between the thermodynamic entropy of physics and
the Shannon's entropy of information is in the units of measure; the former is
expressed in units of energy divided by temperature, the latter in essentially
dimensionless "bits" of information, and so the difference
is merely a matter of convention.
The
holographic principle states that the entropy of ordinary mass (not
just black holes) is also proportional to surface area and not volume; that
volume itself is illusory and the universe is really a hologram which is isomorphic to the information "inscribed" on the
surface of its boundary.
Recent
work
Nature presents two papers authored by Yoshifumi
Hyakutake that bring computational evidence that Maldacena’s conjecture is
true. One paper computes the internal energy of a black hole, the position of
its event horizon, its entropy and other properties based on the predictions
of string theory and
the effects of virtual
particles. The other paper calculates the
internal energy of the corresponding lower-dimensional cosmos with no gravity.
The two simulations match. These papers have received positive appreciation
from Maldacena himself and Leonard
Susskind, one of the founders of string
theory. The papers do not suggest that the universe we actually live in is a
hologram and are not an actual proof of Maldacena's conjecture for all cases
but a demonstration that the conjecture works for a particular theoretical
case. The situation they examine is a hypothetical universe, not a universe
necessarily like ours. The new work is a mathematical test that verifies the
AdS/CFT correspondence for a particular situation.