Definitions

# Holographic principle

The holographic principle is a physical property of quantum gravity theories, proposed by Gerard 't Hooft and Leonard Susskind, which resolves the black hole information paradox. It has been given a precise mathematical formulation within string theory, largely as a result of Susskind's work.

The principle states that the description of a volume of space should be thought of as encoded on a boundary to the region, preferably a light-like boundary like a gravitational horizon. For a black hole, the principle states that the description of all the objects which will ever fall in is entirely contained in surface fluctuations of the event horizon.

In a larger and more speculative sense, the theory suggests that the entire universe can be seen as a two-dimensional information structure "painted" on the cosmological horizon, so that the three dimensions we observe are only an effective description at low energies. Cosmological holography has not yet been made mathematically precise, partly because the cosmological horizon has a finite area and grows with time.

## 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 you throw a hot gas with entropy into a black hole, once it crosses the 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. The second law can only be salvaged if black holes are in fact random objects, with an enormous entropy whose increase more than compensates for the entropy carried by the gas.

Bekenstein argued 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 limit is gravitational - when there is too much energy the gas collapses into a black hole. Beckenstein 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 divided by the Planck area.

Stephen Hawking had earlier shown 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 was an actual entropy, black holes would have to radiate. When heat is added to a thermal system, the change in entropy is the increase in mass-energy divided by temperature:


dS = {dMover T} 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, they would 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 the logarithm of the number of ways an object can be configured microscopically, while leaving the macroscopic description unchanged. Black hole entropy is deeply puzzling--- it says that the number of states of a black hole is proportional to the area of the horizon, not the volume in the interior.

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 at the beginning 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 matter 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 wavefunction, they reemit 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 superpositions with probability amplitudes never become states which are probabilistic mixtures of different possibilities.

Troubled by this paradox, '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 description of 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. The space-time in quantum gravity should emerge as an effective description of the theory of oscillations of a lower dimensional black-hole horizon. This suggested 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 found the first mathematically complete formulation of then new M-theory using a holographic description in terms of charged point black holes, the D0 branes of type IIA string theory. In 1997, Juan Maldacena gave the first holographic descriptions of a higher dimensional object, 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 quantum chromodynamics, and afterwards holography gained wide acceptance.

## 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.

In a given volume, there is an upper limit to the density of information 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 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, a collaborator of Albert Einstein, which suggests scientists may "regard the physical world as made of information, with energy and matter as incidentals." Bekenstein quotes William Blake and questions whether the Holographic principle implies that seeing "the world in a grain of salt," could be more than "poetic license".

### 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 E. Shannon 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, all 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 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. 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 .

## References

General

• Bousso, Raphael (2002). "The holographic principle". Reviews of Modern Physics 74 825–874. 0203101. | doi = 10.1103/RevModPhys.74.825

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