Editing Holographic principle (section)

==Black hole entropy==
{{Main|Black hole thermodynamics}}
An object with relatively high entropy is microscopically random, like a hot gas. A known configuration of classical fields has zero entropy: there is nothing random about [[Electric field|electric]] and [[magnetic field]]s, or [[gravitational wave]]s. Since black holes are exact solutions of [[Einstein field equations|Einstein's equations]], they were thought not to have any entropy.

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 [[entropy]] that increases by an amount greater than the entropy of the consumed gas.

Given a fixed volume, a black hole whose event horizon encompasses that volume should be the object with the highest amount of entropy. Otherwise, imagine something with a larger entropy, then by throwing more mass into that something, we obtain a black hole with less entropy, violating the second law.<ref name="SusskindArXiv" />

[[File:Deriving Emergent gravity from microscopic theories.jpg|thumb|upright=1.2|The illustration demonstrates the way of thinking of the entropic gravity, holographic principle, entropy distribution, and derivation of Einstein General Relativity equations from these considerations. The Einstein equation takes the form of the first law of thermodynamics when Bekenstein and Hawking equations are applied.]]
In a sphere of radius ''R'', the entropy in a relativistic gas increases as the energy increases. The only known limit is [[gravitation]]al; when there is too much energy, the gas collapses into a black hole. Bekenstein used <!-- Redshift~golden number~greenshift  -->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.<ref>{{cite journal |first=Jacob D. |last=Bekenstein |title=Universal upper bound on the entropy-to-energy ratio for bounded systems |journal=Physical Review D |volume=23 |issue=215 |pages=287–298 |date=January 1981 |doi=10.1103/PhysRevD.23.287 |bibcode = 1981PhRvD..23..287B |s2cid=120643289 }}</ref> [[Gravitational time dilation]] causes time, from the perspective of a remote observer, to stop at the event horizon. Due to the natural limit on [[Speed of light|maximum speed of motion]], this prevents falling objects from crossing the event horizon no matter how close they get to it. Since any change in quantum state requires time to flow, all objects and their quantum information state stay imprinted on the event horizon. Bekenstein concluded that from the perspective of any remote observer, 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 light-like [[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 [[Second law of thermodynamics|law of entropy increase]].

At first, Hawking did not take the analogy too seriously. He argued that the black hole must have zero temperature, since black holes do not radiate and therefore cannot be in thermal equilibrium with any black body of positive temperature.<ref>{{Cite journal |last1=Bardeen |first1=J. M. |last2=Carter |first2=B. |last3=Hawking |first3=S. W. |date=1973-06-01 |title=The four laws of black hole mechanics |url=https://doi.org/10.1007/BF01645742 |journal=Communications in Mathematical Physics |language=en |volume=31 |issue=2 |pages=161–170 |doi=10.1007/BF01645742 |bibcode=1973CMaPh..31..161B |s2cid=54690354 |issn=1432-0916}}</ref> Then he discovered that black holes do radiate. When heat is added to a thermal system, the change in entropy is the increase in [[mass–energy equivalence|mass–energy]] divided by temperature:
::<math>
{\rm d}S = \frac{{\rm\delta }M \ c^2}{T}.
</math>
(Here the term ''δM&nbsp;c<sup>2</sup>'' is substituted for the thermal energy added to the system, generally by non-integrable random processes, in contrast to d''S'', which is a function of a few "state variables" only, i.e. in conventional thermodynamics only of the [[Kelvin]] temperature ''T'' and a few additional state variables, such as the pressure.)

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 do not 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 [[Hawking radiation|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]].<ref>{{cite journal | first = Parthasarathi | last = Majumdar | title = Black Hole Entropy and Quantum Gravity | arxiv = gr-qc/9807045 | journal =   Indian Journal of Physics B| date = 1998|bibcode = 1999InJPB..73..147M | volume = 73 | issue = 2 |page=147 }}</ref>

The entropy is proportional to the [[logarithm]] of the number of [[microstate (statistical mechanics)|microstates]], the enumerated 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.<ref name="sciam2003">{{cite magazine | first = Jacob D. | last = Bekenstein | author-link = Jacob Bekenstein | url = http://www.sciam.com/article.cfm?articleid=000AF072-4891-1F0A-97AE80A84189EEDF | title = Information in the Holographic Universe – Theoretical results about black holes suggest that the universe could be like a gigantic hologram |magazine= [[Scientific American]] |date=August 2003 |page=59 }}</ref>

Later, [[Raphael Bousso]] came up with a [[Bousso's holographic bound|covariant version of the bound]] based upon null sheets.<ref>{{cite journal
| last         = Bousso
| first        = Raphael
| year         = 1999
| title        = A Covariant Entropy Conjecture
| journal      = Journal of High Energy Physics
| issue        = 7
| doi          = 10.1088/1126-6708/1999/07/004
| arxiv        = hep-th/9905177
|bibcode = 1999JHEP...07..004B
| volume=1999
| page=004| s2cid = 9545752
}}</ref>