No hair theorem
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In astrophysics, the no-hair theorem states that all black hole solutions of the Einstein-Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three externally observable parameters: mass, electrical charge, and angular momentum. All other information about the matter which formed a black hole or is falling into it, “disappears” behind the black-hole event horizon and is therefore permanently inaccessible to external observers (see also the black hole information paradox. Thus the statement by John Wheeler: “Black holes have no hair.” That is, there are no features that distinguish one black hole from another, other than mass, charge, and angular momentum.
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 matter whereas the second is made out of anti-matter, they will be completely indistinguishable. None of the special particle physics pseudo-charges (baryonic, leptonic, etc.) are conserved in the black hole.
The no-hair theorem was originally formulated for black holes within the context of a four-dimensional spacetime, obeying the Einstein field equation of general relativity with zero cosmological constant, in the presence of electromagnetic fields (or optionally other fields such as scalar fields, massive vector fields (Proca fields), spinor fields, etc.). Counterexamples in which the theorem fails are known in spacetime dimensions higher than four; when the cosmological constant is nonzero; in the presence of nonabelian Yang-Mills fields, nonabelian 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. It has been proposed that “hairy” black holes may be considered to be bound states of hairless black holes and solitons.
[edit] Black holes in quantum gravity
The no-hair theorem is formulated in the classical space-time of Einstein’s general relativity, assumed to be divisible infinitely 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 gravity do away with this picture. Rather than having a potentially an infinite information capacity, it is suggested that the entropy of 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 radiation with a black body spectrum characteristic of that temperature. At a statistical level, this can be understood as a consequence of detailed balance following 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.
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 horizon of 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. 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.
In this way, the quantum black hole can still manage to look pretty much like the black hole of classical general relativity, not just at the event horizon, but for a substantial way inside it, despite actually possessing only a finite entropy.
So: does a quantum black hole have hair? It only has a 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, if we try to enforce a purely classical description, this 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 a non-zero entropy, calculations so far have been unable to relate this to any fluctuations from perfect isotropy.
The jury is therefore very much still out on whether quantum black holes may be shown to have any identifiable hair.
[edit] See also
[edit] References
- Hawking, S. W. (2005). Information Loss in Black Holes, arxiv:hep-th/0507171. Stephen Hawking’s purported solution to the black hole unitarity paradox, first reported in July 2004.
