No-hair theorem

The no-hair theorem postulates 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.^[1] All other information (for which "hair" is a metaphor) 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. Physicist John Archibald Wheeler expressed this idea with the phrase "black holes have no hair"^[1] which was the origin of the name. In a later interview, John Wheeler says that Jacob Bekenstein coined this phrase.^[2]

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.^[3] The result was quickly generalized to the cases of charged or spinning black holes.^[4][5] 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 (i.e., zero electric fields), 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 is made out of ordinary matter whereas the second is made out of antimatter; nevertheless, they will be completely indistinguishable to an observer outside the event horizon. None of the special particle physics pseudo-charges (i.e., the global charges baryonic number, leptonic number, etc.) are conserved in the black hole.

Changing the reference frame

Every isolated unstable black hole decays rapidly to a stable black hole; and (excepting quantum fluctuations) stable black holes can be completely described (in a Cartesian coordinate system) at any moment in time by these eleven numbers:

• mass-energy M,
• linear momentum P (three components),
• angular momentum J (three components),
• position X (three components),
• electric charge Q.

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 which 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 (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).^[6]

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".^[7] 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 (3+1)-dimensional spherically symmetric black hole with minimally coupled self-interacting scalar field was derived.^[8] 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 scalar field with desired properties is only speculative.

Observational results

The LIGO results provide some experimental evidence consistent with the uniqueness or no-hair theorem.^[9][10] This observation is consistent with Stephen Hawking's theoretical work on black holes in the 1970s.^[11][12]

See also

• Black hole information paradox
• Event Horizon Telescope

References

External links

• Hawking, S. W. (2005). "Information Loss in Black Holes". arXiv:hep-th/0507171v2. , Stephen Hawking’s purported solution to the black hole unitarity paradox, first reported in July 2004.

Black holes
Types	Schwarzschild Rotating Charged Virtual Binary
Size	Micro Extremal Electron Stellar Intermediate-mass Supermassive Quasar Active galactic nucleus Blazar Large quasar group
Formation	Stellar evolution Gravitational collapse Neutron star Related links Compact star Quark Exotic Tolman–Oppenheimer–Volkoff limit White dwarf Related links Supernova Related links Hypernova Gamma-ray burst
Properties	Thermodynamics Schwarzschild radius M–sigma relation Event horizon Quasi-periodic oscillation Photon sphere Ergosphere Hawking radiation Penrose process Blandford–Znajek process Bondi accretion Spaghettification Gravitational lens
Models	Gravitational singularity Penrose–Hawking singularity theorems Primordial black hole Gravastar Dark star Dark-energy star Black star Eternally collapsing object Magnetospheric eternally collapsing object Fuzzball White hole Naked singularity Ring singularity Immirzi parameter Membrane paradigm Kugelblitz Wormhole Quasi-star
Issues	No-hair theorem Information paradox Cosmic censorship Alternative models Holographic principle Black hole complementarity firewall (physics) ER=EPR Final parsec problem
Metrics	Schwarzschild Kerr Reissner–Nordström Kerr–Newman
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Related	Timeline of black hole physics Rossi X-ray Timing Explorer Hypercompact stellar system Black hole starship