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 classical parameters: mass, electric charge, and angular momentum. All other information about the matter that formed a black hole or is falling into it "disappears" behind the black-hole event horizon and is therefore permanently inaccessible to external observers. Physicist John Archibald Wheeler expressed this idea with the phrase "black holes have no hair," which was the origin of the name. In a later interview, Wheeler said that Jacob Bekenstein coined this phrase. The first version of the no-hair theorem for the simplified case of the uniqueness of the Schwarzschild metric was shown by Werner Israel in 1967. The result was quickly generalized to the cases of charged or spinning black holes. There is still no rigorous mathematical proof of a general no-hair theorem, and mathematicians refer to it as the no-hair conjecture. Even in the case of gravity alone, the conjecture has only been partially resolved by results of Stephen Hawking, Brandon Carter, and David C. Robinson, under the additional hypothesis of non-degenerate event horizons and the technical, restrictive and difficult-to-justify assumption of real analyticity of the space-time continuum.
Example
Suppose two black holes have the same masses, electrical charges, and angular momenta, but the first black hole was made by collapsing ordinary matter whereas the second was made out of antimatter; nevertheless, then the conjecture states they will be completely indistinguishable to an observer outside the event horizon. None of the special particle physics pseudo-charges are conserved in the black hole, or if they are conserved somehow then their values would be unobservable from the outside.
Every isolated unstable black hole decays rapidly to a stable black hole; and stable black holes can be completely described at any moment in time by these eleven numbers:
These numbers represent the conserved attributes of an object which can be determined from a distance by examining its gravitational and electromagnetic fields. All other variations in the black hole will either escape to infinity or be swallowed up by the black hole. By changing the reference frame one can set the linear momentum and position to zero and orient the spin angular momentum along the positive z axis. This eliminates eight of the eleven numbers, leaving three which are independent of the reference frame: mass, angular momentum magnitude, and electric charge. Thus any black hole that has been isolated for a significant period of time can be described by the Kerr–Newman metric in an appropriately chosen reference frame.
Extensions
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 and massive vector fields. It has since been extended to include the case where the cosmological constant is positive. Magnetic charge, if detected as predicted by some theories, would form the fourth parameter possessed by a classical black hole.
Counterexamples
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". It has been proposed that "hairy" black holes may be considered to be bound states of hairless black holes and solitons. In 2004, the exact analytical solution of a -dimensional spherically symmetric black hole with minimally coupled self-interacting scalar field was derived. This showed that, apart from mass, electrical charge and angular momentum, black holes can carry a finite scalar charge which might be a result of interaction with cosmological scalar fields such as the inflaton. The solution is stable and does not possess any unphysical properties; however, the existence of a scalar field with the desired properties is only speculative.
Observational results
The LIGO results provide some experimental evidence consistent with the uniqueness of the no-hair theorem. This observation is consistent with Stephen Hawking's theoretical work on black holes in the 1970s.
Soft Hair
A study by Hawking and Malcolm Perry postulates that black holes might contain "soft hair," giving the black hole more degrees of freedom than previously thought. This hair permeates at a very low energy state, which is why it didn't come up in previous calculations that postulated the no hair theorem.
