- No hair theorem
The no-hair theorem in
astrophysicspostulates that all black holesolutions of the Einstein-Maxwell equations of gravitationand electromagnetismin general relativitycan be completely characterized by only three "externally" observable classical parameters: mass, electric charge, and angular momentum. All other information about the matterwhich formed a black hole or is falling into it, "disappears" behind the black-hole event horizonand is therefore permanently inaccessible to external observers (see also the black hole information paradox).
Origin of name
This theory gets its name from a comment by the famous astrophysicist John Wheeler, who stated that “Black holes have no hair.” [ cite journal | author=Ruffini, R., Wheeler, J. A. | date=Jan. 1971 |title=Introducing the Black Hole | journal=Physics Today
volume=24 | pages=30–41 ] Magnetic charge, if detected as predicted by some theories, would form the fourth parameter possessed by a classical black hole.
For example, if two black holes are "constructed" so that they have the same masses, electrical charges, and angular momenta, but the first black hole is made out of ordinary
matterwhereas the second is made out of antimatter, they will be completely indistinguishable "to an observer outside the event horizon". None of the special particle physicspseudo-charges ( baryonic, leptonic, etc.) are conserved in the black hole.
Independent from the reference frame
Like most ideas based on the
general theory of relativity, the "no hair" theorem is concerned only with properties which are independent of the frame of reference(point of view of the observer). The theorem therefore says nothing about a black hole's position or velocity.
The no-hair theorem was originally formulated for black holes within the context of a four-dimensional
spacetime, obeying the Einstein field equationof general relativitywith zero cosmological constant, in the presence of electromagnetic fields, or optionally other fields such as scalar fields and massive vector fields (Proca fields, spinor fields, etc.).
It has since been extended to include the case where the cosmological constant is positive (which recent observations are tending to support) [http://arxiv.org/abs/gr-qc/0702006No hair theorems for positive Lambda] .
Counterexamples in which the theorem fails are known in spacetime dimensions higher than four; in the presence of
non-abelian Yang-Mills fields, non-abelian Proca fields, some non-minimally coupled scalar fields, or skyrmions; or in some theories of gravity other than Einstein’s general relativity. However, these exceptions are often unstable solutions and/or do not lead to conserved quantum numbers so that "The 'spirit' of the no-hair conjecture, however, seems to be maintained" [http://arxiv.org/abs/gr-qc/9606008Eluding the No-Hair Conjecture for Black Hole states] . It has been proposed that "hairy" black holes may be considered to be bound states of hairless black holes and solitons.
Black holes in quantum gravity
The no-hair theorem is formulated in the classical spacetime of Einstein's general relativity, assumed to be infinitely divisible with no limiting short-range structure or short-range correlations. In such a model, each possible macroscopically-defined classical black hole corresponds to an infinite density of microstates, each of which can be chosen as similar as desired to any of the others (hence the loss of information).
Proposals towards a theory of
quantum gravitydo away with this picture. Rather than having a potentially infinite information capacity, it is suggested that the entropyof a quantum black hole should be a strictly finite "A"/4, where "A" is the area of the black hole in Planck units.
Along with a finite (non-infinite) entropy, quantum black holes acquire a finite (non-zero) temperature, and with it the emission of
Hawking radiationwith a black body spectrumcharacteristic of that temperature. At a statistical level, this can be understood as a consequence of detailed balancefollowing from the presumed micro-reversibility (unitarity) of the interaction between the quantum states of the radiation field and the quantum states of the black hole. This implies that if black holes can absorb radiation, they should therefore also emit radiation, with a black body spectrum characteristic of the temperature of the relevant part of the system.
Near the event horizon
From a different perspective, if it is correct that the properties of a quantum black hole should correspond at a broad level more or less to a classical general-relativistic black hole, then it is believed that the appearance and effects of the Hawking radiation can be interpreted as quantum "corrections" to the classical picture, as Planck's constant is "tuned up" away from zero up to "h". Outside the
Event horizonof an astronomical-sized black hole these corrections are tiny. The classical infinite information density is actually quite a good approximation to the finite but large black hole entropy, the black hole temperature is very nearly zero, and there are very few Hawking particles to disrupt the classical trajectories.
Within the event horizon
Very little changes for a test particle as the event horizon is crossed; classical general relativity is still a very good approximation to the quantum gravity outcome. But the further the particle falls down the gravity well, the more the Hawking temperature increases, the more Hawking particles there are buffeting the test particle, and the greater become its deviations from a classical path as the increasingly limited density of quantum states starts to pinch. Ultimately, much further in, the density of the quantum "corrections" becomes so pronounced that the classical variables cease to be good quantum numbers to describe the system. This deep into the black hole it becomes the quantum gravitational forces, above all else, that dominate the environmental interactions which determine the appropriate decohered states for sensibly talking about the system. Further in than this, the core of the system needs to be treated in its own, specifically quantum, terms.
A quantum black hole compared to a classical black hole
In this way, the quantum black hole can still manage to look like the black hole of classical general relativity, not just at the event horizon but also for a substantial way inside it, despite actually possessing only finite entropy.
A quantum black hole only has finite entropy and therefore presumably exists in one of a limited effective number of corresponding states. With reference to a careful description of the available states, this granularity may be revealed. However, trying to enforce a purely classical description represents a projection into a much bigger space, made possible presumably by probabilities supplied by environmental decoherence. Any structure implicit in the finite entropy against a quantum description could then be totally washed out by the huge injection of uncertainty this projection represents. This may explain why even though Hawking radiation has non-zero entropy, calculations so far have been unable to relate this to any fluctuations from perfect isotropy.
* Hawking, S. W. (2005). Information Loss in Black Holes, [http://arxiv.org/abs/hep-th/0507171 arxiv:hep-th/0507171] . Stephen Hawking’s purported solution to the black hole
unitarityparadox, first reported in July 2004.
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