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Before getting to the essential point of whether black hole destroys information or not, it is necessary to clarify the meaning of information (Figure 09xaa). According to Leonard Susskind, who played a principle role in the debate, entropy is hidden information. This view considers the basic units (or bits) of information to be the microscopic particles, which can be atoms, elementary particles, or down to the smallest unit the size of Planck length (~ 10^{-33}cm). They are hidden because its existence is not known to us. Once we are aware of its presence (such as an electron in a collision experiment), it becomes information, and generates events that we can keep track of. The information is said to be conserved because we can always recover the original event by time reversal operations (at least in theory, which asserts that the laws of physics are the same in both forward and backward time directions, the "Second Law of Thermodynamics" notwithstanding because it is actually applicable to one or at most few particles).
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## Figure 09xaa Black Hole Information Paradox |

Such explanation in layman's language is rather confusing, it is impossible to follow the arguments for the paradox if it is taken literally. Here's the clarification involving mathematical definitions of :

- Information - This terminology became much in vogue ever since the creation of "Maxwell's Demon". However, it is a rather vague concept - carrying different meanings such as : (1) Negative of Entropy; (2) A message like a sentence (see Figure 09xa); (3) Alphanumeric sequence such as the catalog number of a book in library; (4) An unique sequence like the genetic code in DNA; (5) The qubits as "Quantum Information" in quantum computing (also see "Different Kinds of Information). In the context of the "Information Paradox", it is none-of-the-above. What these scientists are/were talking about is related to the superposition of states governed by quantum theory. Specifically by the Schrodinger Equation, for which the wave function composed by superpositioning of many eigen-states (stationary states) is also a solution of the equation, that is (by taking the example of the infinite square well, see text in there for details) :
#### Figure 09xa Typical Information

[view large image]#### Figure 09xb Eigen-states

**The total probability is P = 1 (absolute certainty) again regradless of which state it collapses into (Figure 09xb). Such process is called decoherence.** - Conservation of Information - It means that the probability amplitudes c
_{n}'s do not change from the initialization of the superposition to the measurement. The system usually exists in ground state (at lowest energy level) with n = 0. It requires certain perturbation to excite it into various mixed states. The mathematical details (taking the infinite square well as example again) :

**Figure 09xc shows the time varying probability in spite of the invariance form of the total probability P. Figure 09xd illustrates the collapse of the superposition via a measurement using the mixed spin states as example.**#### Figure 09xc Unitary State as Info

#### Figure 09xd Quantum Measurement

[view large image]BTW, a deterministic process such as the Schrodinger Equation is time-reversible, i.e., it is invariant under a change of time t - t.

In quantum field theory, the wave function is replaced by the interacting particles (see Figure 09xd1) as depicted by the Feynman diagram. It obeys the same kind of "Information Conservation" with the S-matrix element (also called probability amplitude) S_{fi}in place of the c_{n}, i.e.,_{f}S*_{fi}S_{fi}= 1, which guarantees that probability is conserved in the process. Such relationship indicates that the processes have an inverse,

A good example comes from the discovery of the Higgs particle, the detection of which depends on its decay into various modes with different % value of S*which in turn implies that it is possible to return to the initial state from the final state at least in principle although the probability is almost zero in practice so that the second law of thermodynamics is " *almost*" never violated. This is another example of the conservation of information. Here's a Feynman diagram for e^{-}+ e^{+}#### Figure 09xd1 S-Matrix

[view large image]scattering with probability amplitude S _{11}(see Figure 09xd1) . Detection of the process is equivalent to decoherence of quantum states._{fi}S_{fi}for each process in it. See "Higgs Boson Decay Modes" and "Different Perspectives of Quantum Mechanics and Quantum Field Theory (or lst and 2nd Quantizations)".

Now the explanation has run a full circle. The original presentation does make some sense if the mysterious "hidden information" is identified to the S-matrix element S*_{fi}S_{fi}.

The Paradox - Since a black hole is characterized only by its mass, angular momentum (for a rotating one), and electric charge (if it contains any), the "No Hair theorem" (Figure 09xd2) dictates that all other information (called "hair") "disappears" behind the event horizon and is permanently inaccessible to external observers. It is argued that the event horizon acts like a measuring device to collapse the superposition in the decoherent

process. If that's the case, there would be no problem with the loss of information, because the superposition is terminated normally. The paradox is about the existence of the "measuring device" - there is none at the event horizon according to general relativity. Or else the superposition is somehow trapped inside the black hole, for which plenty of opinions are proposed about its fate - whether it is lost forever or can be recovered eventually. | |

## Figure 09xd2 No-hair Theorem [view large image] |

- Here's some of the purported resolutions :
- Hawking Radiation (Figure 09xe) - In spite of his own discovery in 1974 of the radiation by his namesake, which can return matter-energy back to the outside world (Figure 09xf), Hawking himself insists that information would be destroyed because such radiation has the characteristics of blackbody radiation. It is generated by a randomized process that would not convey any information. He illustrates such argument with the $-matrix in place of the regular S-matrix to indicate the blurring of the various processes to the point of no more "Conservation of Information". It seems
- Black Hole Complementarity - According to the space-time diagram of the black hole, local observer would cross the event horizon uneventfully without even noticing it. However, a distant observer would perceive a slowing down of all processes, and the wavelength of the emitting light would be red shifted. Eventually, everything comes to a standstill and the wavelength becomes infinite at the event horizon.

In 1993, 't Hooft and Susskind proposed the "Principle of Black Hole Complementarity" via the prescription of "black hole space-time diagram". As shown in Figure 09xga, for the outside observer, matter (the elephant) would be gradually reduced to thermal radiation at the event horizon to return as scrambled information, and it takes an infinite time for the elephant to cross such boundary according to observer A.While for observer B inside the black hole, the elephant crosses the event horizon into the black hole, nothing untoward happens until the tidal force takes over ... information is carried to the singularity (see "Black Hole Space-time"). According to the quantum theorists the information is not lost from both points of view. #### Figure 09xga Black Hole Complementarity

[view large image]This was when the "war" reached a stalemate. - AdS/CFT (Anti-de Sitter/Conformal Field Theory) Correspondence - Following the idea that microscopic quantum ripples at the event horizon can encode information inside the black hole (see "Black Hole Entropy"), it is subsequently shown in 1998 that black hole in (4+1) dimension corresponding to hot radiation in (3+1) dimension (see "Holographic Space-time" ). This discovery is promptly taken over by the quantum theorists to argue that there is no mysterious information loss as the black hole evaporates. In spite of the facts that (1) The discovery is based on the unproven superstring theory, (2) The Anti-de Sitter space is very different from the standard cosmological model, (3) Such black hole exists in a (4+1) space (4 spatial + 1 temporal), not the (3+1) world we are living in, and (4) The radiation is hot (randomized)
#### Figure 09xgb Holographic Space-time [view large image]

(see Figure 09xgb). The quantum team claimed victory anyway despite such shaky ground. It took a while for Hawking to concede defeat. He called a press conference in 2004 and announced to the world that he had changed his mind. Black holes did not, after all, irreversibly annihilate information. The bet with Preskill (his gambling partner) was duly settled in the form of a baseball encyclopedia. Meanwhile he lost another wager (on conservation of infromation) with Don Page (one of his students) and had it settled in 2007. Here's a photocopy of the original contract with his signature and fingerprint [view large image]. **Stephen died on March 14, 2018.** - Fire Wall -
The verdict of "information wager" was over turned for re-trial in 2013. On closer examination of the involement of entanglement, it is discovered that the emitting particle is forbidden to entangle with both its former partner inside the black hole and the Hawking radiation
at the same time. The paradox can be resolved by severing the link with the twin (inside the black hole), but it is a violent process. The energy holding up the entanglement would incinerate the event horizon with a wall of fire (Figure 09xh). General relativity maintains that crossing over the event horizon should be uneventful but the firewall erects a signpost in violation of this doctrine. Figure 09xi illustrates the strange episodes surrounding the paradox. #### Figure 09xh Black Hole and Information

[view large image]#### Figure 09xi BH Firewall

_{}**N.B.**If the emitting particle becomes entangled with the environment (the Hawking radiation), then it is not maximally entangled with its former partner any more, and polygamy is permitted (see "monogamy").

Anyway, according to such argument the original information paradox is incarnated as either to abandon general relativity (due to the signpost) or to give up quantum theory (due to the quantum entanglement creating the signpost). For the case of firewall, the observer plunging into the black hole will be incinerated instead of crushed to death by tidal force (see insert in Figure 09xi and Figure 09d). A meeting was convened at CERN on March 2013 to grapple with the issue, but no resolution is in sight so far.

Then a few months later a new idea claims that entanglement of the photon in the Hawking radiation and its twin in the black hole can be replaced by a wormhole (Figure 09xj). Such connection may not be a problem for quantum monogamy in the way that normal entanglement would cause. The firewall could thus be dispensed with this scheme. Accordingly, there could be wormholes permeating over all the universe; what would happen when they encounter something ? would those things be sucked into the worm holes, going to another universe ? Or the wormholes exist only in the bulk dimensions, then we have to accept the reality of extra-dimensions. It seems that the scenarios are getting more and more bizarre and very difficult to comprehend. #### Figure 09xj Wormhole Entanglement

- Apparent Horizon - Hawking posted a paper in arXiv on 22 January 2014 claiming that the firewall violates the CPT invariance of quantum gravity. He suggested that gravitational collapse produces apparent horizon but no event horizon behind which information is lost. This proposal is alleged to be supported by ADS-CFT and is the only resolution of the information paradox compatible with CPT. His verdict now is "no information loss".

Hawking etal. deposited yet another preprint with the title "Soft Hair on Black Holes" to arXiv on Janaury 05, 2016 (see a slightly more comprehensive version in "Stephen Hawking's New Black-Hole Paper, Translated: An Interview with Co-Author Andrew Strominger" by the editor of Scientific American).

This idea is strikingly similar to the "Principle of Black Hole Complementarity" in which the in-falling particle (inside the black hole) appears to an outside observer as a footprint emitting radiation with very long wavelength (due to gravitational red shift) and the process takes an infinite long time to complete (due to gravitational time dilation, see Figure 09xk).Essentially, it suggests that in-falling mass (to the black hole) would leave behind soft graviton (very long wavelength gravitational wave) at the event horizon, while the in-falling electric charge would leave behind soft photon (very long wavelength electromagnetic wave) there. It is not clear if the surface (at the event horizon) would be able to contain an infinite amount of such soft hair, but it is a step forward to pinpoint the substance that preserves all the information falling into the black hole. #### Figure 09xk Black Hole, Soft Hair

_{}

## Figure 09xe Hawking Radiation |
## Figure 09xf Black Hole Evaporation |
that the opinions are divided into two camps. There is the General Relativity (GR) proponents (the relativists) who favor information loss. They blame the controversy on the incompleteness of quantum theory, in particular the discontinuous jump from superposition to a definite state. |

Here's the lineups of the two sides :

GR Team : Stephen Hawking, Kip Thorne, and Roger Penrose.

Quantum Team : Leonard Susskind, Gerard 't Hooft, and Juan Maldacena.

See "Black Hole Information Paradox" for further detail; and the history plus various bizarre proposals in the image below from "New Scientist".

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