Black Hole Information Paradox

Article originally appears at Maths & Physics Club, IIT Bombay and aims to explain the recent paper published by Hawking, Malcolm & Strominger claiming a resolutions to the (in)famous black hole paradox. I aim to ply the reader with enough knowledge to conclude with the answer to the question – ‘Does the baffle still remain?’.

black-hole
Introduction

From a talk given on Aug ‘15 at a conference held at Stockholm[1]; Stephen Hawking, the well-celebrated british physicist, proposed a resolution of the infamous Information loss paradox, a child of his own theory of Hawking Radiation[2][3] that overturned the widely perpetuated belief that nothing escapes from a black hole’s event horizon. In the interim, this left most of his contemporaries bemused since his theory had no mathematical backbone. A rigorous paper[4] collaborated by Hawking, Malcolm & Strominger in the January of ‘16, humorously entitled ‘Soft Hair on Black Holes’, attempts to pave a mathematical map to resolve the paradox once and for all. But Hawking et al conclude[4] that they’ve just introduced fresh & concrete tools to address the paradox and a complete description still remains an open challenge.

What exactly IS the paradox?

The information loss paradox is yet another acclaimed scenario where Quantum Mechanics, describing the subatomic world with the highest precision and General Relativity, equally successful at the cosmic scale, act stubbornly incompatible[10].
General Relativity implies that once a system ‘falls’[5] in the black hole, any information regarding the system can not affect an outside observer, thus for all practical purposes the information is irretrievable. Notice, that we use irretrievable instead of destroyed, i.e. it doesn’t posses a problem if the black hole remains an ever absorbing body.


1. The Hawking Radiation

 


Around 1975, due to a remarkable connection made by Hawking[6] and Beckenstein[7] between Thermodynamics, Quantum Mechanics and Black Holes, Hawking Radiation came into picture which collapsed the possibility of everlasting black holes to null.

Hawking Radiation stated that black holes, in fact, aren’t as black as we expected them to be. Courtesy to locally small quantum effects[2], (which are ignored in the process of formation and evolution of black holes since the radius of curvature of spacetime outside the event horizon is very large compared to the Planck length[11], the length scale on which quantum fluctuations of the metric are expected to be of order unity) add up to produce a significant effect over the lifetime of the Universe. This simply implies that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole of M solar masses was similar to a blackbody, with temperature of about 6 × 10-8/M kelvins[8].
The fact that a black hole will emit photons when none escape is indigestible.
In layman language[9], Quantum Mechanics capacitates a constant potential for “virtual” Particle – Antiparticle pair to be created and annihilated, leaving a net zero state. Same phenomenon at the knife edge of the event horizon permits a possibility when one escapes, while the other plunges immediately into the black hole. Rather than instant annihilation, the “virtual” particle becomes real and escapes; making a petty energy drain on the black hole. The rate at which the energy dissipates from a black hole of M solar masses spares a massive 1071M3 seconds[8] as any black hole’s life period. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s[2]. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs.

2. Aftermaths of Hawking Radiation Theory


In order to understand why the information loss paradox is a problem, we need first to understand what it is.

Let’s design a thought experiment where a quantum system in pure state enters a black hole. The no-hair (or more aptly, three-hair) theorem[15] compels Hawking radiation to be completely independent of the state of the quantum system entering the black hole. Because part of the information about the state of the system is lost down the hole, the final situation is represented by a density matrix[11] rather than a pure quantum state (which it was, originally). In quantum physics, a scattering matrix or S-matrix is a unitary matrix[12] that maps the scattering channels (the quantum states of system before or after the collision). The scenario could have been picturised as the initial pure state quantum system interacting with a boundary condition and the Hawking radiation as the final scattering channel. But since the transformation from a pure quantum state to a density matrix is non-unitary, this means there is no Scattering-matrix[13] for the process of black-hole formation and evaporation. Or to say, the black hole has performed a non-unitary transformation on the state of system![14]  As you may recall, non-unitary evolution is not allowed to occur naturally in quantum theory because it fails to preserve probability & reversibility.
In layman terms what we start with is a pure state and a black hole of mass M.  What we end up with is a mixed state and a black hole of mass M.  In transforming between a mixed state and a pure state, one must throw away information.  For instance, in our example we took a state described by a set of eigenvalues and coefficients, a large set of numbers, and transformed it into a state described by temperature, one number.  All the other structure of the state was lost in the transformation.
In the face of such evolution, quantum mechanics falls apart, and we are faced with a dilemma.  Do black holes really defy the tenets of quantum theory, or have we missed something in our thought experiment.
Thus, therein lies a paradox!

The Way Ahead

1. Possible Solutions To The Fate of Black Holes

There have been many proposed solutions to the paradox, each equally bewildering as another. They are generally classified first into radical options, which require a quantum theory of gravity which has large deviations from semi-classical physics on macroscopic scales, such as non-locality or endowing horizons with special properties not seen in the semiclassical approximation, and conservative options, which do not need such help. Among the conservative options, it is argued that restoring unitary evolution relies on elimination of singularities.[16]

2. Hawking’s leads : Soft Hair on Black Holes

AdS-CFT correspondence theory[17], a theory launched in 1993, brought what is called as a Holographic Principle into center play in string theory. Holographic principle bestows same properties to universe as its root word indicates, a dimensional reduction. A hologram, which we encounter is a 2-D representation of a 3-D object. Holographic principle was first introduced by G. ‘t Hooft in 1933 as an attempt to a solution to the Information loss paradox and the widely proclaimed war he and L. Susskind had waged against hawking’s earlier claims that information is indeed lost. ‘T Hooft opined[18] that the requirement that physical phenomena associated with gravitational collapse should be duly reconciled with the postulates of quantum mechanics implies that at a Planckian scale our world is not 3+1 dimensional. Rather, the observable degrees of freedom can best be described as if they were Boolean variables defined on a two-dimensional lattice, evolving with time. This observation, deduced from not much more than unitarity, entropy and counting arguments, implies severe restrictions on possible models of quantum gravity. This dimensional reduction implies more constraints than the freedom we have in constructing models. This is the main reason why so-far no completely consistent mathematical models of quantum black holes have been found.
A year later, L. Susskind published a paper, ‘The World as A Hologram’[19], backing up ‘t Hooft’s theory. In July ‘04 Hawking conceded the bet since the AdS/CFT correspondence has shown there is no information loss. Then Hawking set working on the question : How does the information of the quantum state of the infalling particles re-emerge in the outgoing radiation?[1] He proposes that the information is stored, not in the interior of the black hole (as one might expect), but on its boundary, the event horizon. This is a form of holography. Classically, a black hole is independent of its past history. This is also true in the quantum domain. How then can a black hole emit the information about the particles that fell in? The answer, as explained above, is that the information is stored in a supertranslation associated with the shift of the horizon that the ingoing particles caused (Ingoing particles increase the mass of the black hole resulting into an outward shift in the horizon).

 

A supertranslation, in layman terms, are the result of a kind of symmetry. In Strominger’s terms[20], Individual light rays can’t talk to each other—if you’re riding on a light ray, causality prevents you from talking to somebody riding on an adjacent light ray. So these light rays are not tethered together. You can slide them up and down relative to one another. That sliding is called a super-translation. It turns out that adding a soft graviton at the event horizon has an alternate description as a super-translation in which you move some of these light rays back and forth relative to one another.

 

The supertranslations form a hologram of the ingoing particles at the horizon(For each particle that goes in, a soft-graviton is added). Thus, the jargon soft hair. The varying shifts along each generator of the horizon leave an imprint on the outgoing particles in a chaotic but deterministic manner. There is no loss of information and Information can be recovered in principle, but it is lost for all practical purposes.

Bibliography

[1] The Information Paradox for Black Holes
http://arxiv.org/pdf/1509.01147v1.pdf

[2] Black Hole Explosions?

http://www.nature.com/nature/journal/v248/n5443/abs/248030a0.html

[3] Particle Creation by Black Holes

http://inspirehep.net/record/101338?ln=en

[4] Soft Hair on Black Holes

http://arxiv.org/pdf/1601.00921v1.pdf

[5] Event Horizon

https://www.wikiwand.com/en/Event_horizon

[6] Breakdown of Predictability in Gravitational Collapse

http://journals.aps.org/prd/pdf/10.1103/PhysRevD.14.2460

[7] Black Holes & Entropy

http://journals.aps.org/prd/abstract/10.1103/PhysRevD.7.2333

[8] Hawking Radiation

http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/hawking.html

[9] WordPress Blog

https://starguyspeaks.wordpress.com/2016/02/21/on-steven-hawkings-new-theory-of-black-holes/

[10] The Final Contradiction

http://theory.caltech.edu/people/jhs/strings/str115.html

[11] Description of States in Quantum Mechanics by Density Matrix and Operator Techniques

http://journals.aps.org/rmp/pdf/10.1103/RevModPhys.29.74

[12] Unitary Operator

https://www.wikiwand.com/en/Unitary_operator

[13] S-Matrix

https://www.wikiwand.com/en/S-matrix

[14] Black Hole Information Loss Problem

http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/info_loss.html

[15] No-Hair Theorem

https://www.wikiwand.com/en/No-hair_theorem

[16] Conservative solutions to the black hole Information loss problem

http://arxiv.org/pdf/0901.3156v1.pdf

[17] The AdS/CFT Correspondence

https://arxiv.org/pdf/1501.00007.pdf

[18] Dimensional Reduction in Quantum Gravity

http://xxx.lanl.gov/abs/gr-qc/9310026

[19] The World as Hologram

http://xxx.lanl.gov/abs/hep-th/9409089

[20] Interview by co-author Andrew Strominger

http://blogs.scientificamerican.com/dark-star-diaries/stephen-hawking-s-new-black-hole-paper-translated-an-interview-with-co-author-andrew-strominger/

 

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