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.

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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.

^{[citation needed]}
**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:

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. 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.

##
Four-dimensional space-time

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.).

^{[citation needed]}
##
Extensions

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 the first experimental observation of the uniqueness or no-hair theorem.

^{[9]}^{[10]} This observations are consistent with Stephen Hawking theoretical work on black holes in the 1970s.

^{[11]}^{[12]}
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See also

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References

Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). *Gravitation*. San Francisco: W. H. Freeman. pp. 875–876. ISBN 0716703343. Retrieved 24 January 2013.
https://www.youtube.com/watch?v=BIHPWKXvGkE&feature=youtu.be&t=6m
Israel, Werner (1967). "Event Horizons in Static Vacuum Space-Times". *Phys. Rev.* **164** (5): 1776–1779. Bibcode:1967PhRv..164.1776I. doi:10.1103/PhysRev.164.1776.
Israel, Werner (1968). "Event horizons in static electrovac space-times". *Commun. Math. Phys.* **8** (3): 245–260. Bibcode:1968CMaPh...8..245I. doi:10.1007/BF01645859.
Carter, Brandon (1971). "Axisymmetric Black Hole Has Only Two Degrees of Freedom". *Phys. Rev. Lett.* **26** (6): 331–333. Bibcode:1971PhRvL..26..331C. doi:10.1103/PhysRevLett.26.331.
Bhattacharya, Sourav; Lahiri, Amitabha (2007). "No hair theorems for positive Λ". arXiv:gr-qc/0702006v2.
Mavromatos, N. E. (1996). "Eluding the No-Hair Conjecture for Black Holes". arXiv:gr-qc/9606008v1.
Zloshchastiev, Konstantin G. (2005). "Coexistence of Black Holes and a Long-Range Scalar Field in Cosmology". *Phys. Rev. Lett.* **94** (12): 121101. arXiv:hep-th/0408163. Bibcode:2005PhRvL..94l1101Z. doi:10.1103/PhysRevLett.94.121101.
"Gravitational waves from black holes detected". *BBC News*. 11 February 2016.
"Gravitational waves detected 100 years after Einstein's prediction" (PDF). LIGO. February 11, 2016. Retrieved 11 February 2016.
https://www.facebook.com/stephenhawking/posts/965377523549345 Stephen Hawking
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External links