View source for No-hair theorem

{{Short description|Black holes are characterized only by mass, charge, and spin}}
{{distinguish|Hairy ball theorem}}
{{general relativity|expanded=theorems}}

The '''no-hair theorem''' states that all stationary [[black hole]] solutions of the [[Einstein field equations#Einstein–Maxwell equations|Einstein–Maxwell equations]] of [[gravitation]] and [[electromagnetism]] in [[general relativity]] can be completely characterized by only three independent ''externally'' observable [[Classical physics|classical]] parameters: [[mass]], [[angular momentum]], and [[electric charge]].{{Citation needed | date=November 2024}} Other characteristics (such as geometry and magnetic moment) are uniquely determined by these three parameters, and all other information (for which "hair" is a metaphor) 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 after the black hole "settles down" (by emitting [[gravitational wave|gravitational]] and [[electromagnetic waves]]). 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.<ref>Archived at [https://ghostarchive.org/varchive/youtube/20211211/BIHPWKXvGkE Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20121230050811/http://www.youtube.com/watch?v=BIHPWKXvGkE Wayback Machine]{{cbignore}}: {{cite web |title=Interview with John Wheeler 2/3 |date=May 2009 |via=[[YouTube]] |url=https://www.youtube.com/watch?v=BIHPWKXvGkE&t=6m }}{{cbignore}}</ref>

<blockquote>
[[Richard Feynman]] objected to the phrase that seemed to me to best symbolize the finding of one of the graduate students: graduate student Jacob Bekenstein had shown that a black hole reveals nothing outside it of what went in, in the way of spinning electric particles. It might show electric charge, yes; mass, yes; but no other features{{snd}} or as he put it, "A black hole has no hair". Richard Feynman thought that was an obscene phrase and he didn't want to use it. But that is a phrase now often used to state this feature of black holes, that they don't indicate any other properties other than a charge and angular momentum and mass.<ref name="web_of_stories">[https://www.webofstories.com/play/john.wheeler/84 Transcript: John Wheeler – Feynman and Jacob Bekenstein], Web of Stories. Listeners: Ken Ford, Duration: 1 minute, 19 seconds, Date story recorded: December 1996, Date story went live: 24 January 2008.</ref>
</blockquote>

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.<ref>{{cite journal |last=Israel |first=Werner |journal=Phys. Rev. |volume=164 |pages=1776–1779 |date=1967 |doi=10.1103/PhysRev.164.1776 |title=Event Horizons in Static Vacuum Space-Times |issue=5 |bibcode=1967PhRv..164.1776I }}</ref> The result was quickly generalized to the cases of charged or spinning black holes.<ref>{{cite journal |last=Israel |first=Werner |journal=Commun. Math. Phys. |volume=8 |pages=245–260 |date=1968 |doi=10.1007/BF01645859 |title=Event horizons in static electrovac space-times |issue=3 |bibcode=1968CMaPh...8..245I |s2cid=121476298 |url=http://projecteuclid.org/euclid.cmp/1103840605 }}</ref><ref>{{cite journal |last=Carter |first=Brandon |journal=Phys. Rev. Lett. |volume=26 |pages=331–333 |date=1971 |doi=10.1103/PhysRevLett.26.331 |title=Axisymmetric Black Hole Has Only Two Degrees of Freedom |issue=6 |bibcode=1971PhRvL..26..331C }}</ref> 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.&nbsp;Robinson, under the additional hypothesis of non-degenerate event horizons and the technical, restrictive and difficult-to-justify assumption of [[Real analytic function|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 (i.e., the global charges [[baryon]]ic number, [[lepton]]ic number, etc., all of which would be different for the originating masses of matter that created the black holes) are conserved in the black hole, or if they are conserved somehow then their values would be unobservable from the outside.{{citation needed|date=March 2015}}

==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]] <math>M</math>,
* [[electric charge]] <math>Q</math>,
* [[position (vector)|position]] <math>\textbf{X}</math> (three components),
* [[linear momentum]] <math>\textbf{P}</math> (three components),
* [[angular momentum]] <math>\textbf{J}</math> (three components).
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 field]]s and massive [[vector field]]s ([[Proca action|Proca]] fields, etc.).{{citation needed|date=July 2013}}

It has since been extended to include the case where the [[cosmological constant]] is positive (which recent observations are tending to support).<ref>{{cite journal |last1=Bhattacharya |first1=Sourav |last2=Lahiri |first2=Amitabha |arxiv=gr-qc/0702006 |title=No hair theorems for positive Λ |date=2007 |doi=10.1103/PhysRevLett.99.201101 |pmid=18233129 |volume=99 |issue=20 |pages=201101 |journal=Physical Review Letters |bibcode=2007PhRvL..99t1101B|s2cid=119496541 }}</ref>

[[Magnetic monopole|Magnetic charge]], if detected as predicted by some theories, would form the fourth parameter possessed by a classical black hole{{Citation needed | date=November 2024}}.

==Counterexamples==
Counterexamples in which the theorem fails are known in spacetime dimensions higher than four; in the presence of [[Non-abelian gauge theory|non-abelian]] [[Yang–Mills field]]s, non-abelian [[Proca action|Proca fields]], some [[minimal coupling|non-minimally coupled]] [[scalar fields]], or [[skyrmion]]s; 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".<ref>{{cite arXiv |last=Mavromatos |first=N. E. |author-link=Nikolas Mavromatos |eprint=gr-qc/9606008v1 |title=Eluding the No-Hair Conjecture for Black Holes |date=1996 }}</ref> It has been proposed that "hairy" black holes may be considered to be bound states of hairless black holes and [[soliton]]s.

In 2004, the exact analytical solution of a (3+1)-dimensional spherically symmetric black hole with minimally coupled self-interacting scalar field was derived.<ref>{{cite journal |last=Zloshchastiev |first=Konstantin G. |journal=Phys. Rev. Lett. |volume=94 |pages=121101 |date=2005 |doi=10.1103/PhysRevLett.94.121101 |title=Coexistence of Black Holes and a Long-Range Scalar Field in Cosmology |issue=12 |bibcode=2005PhRvL..94l1101Z |arxiv = hep-th/0408163 |pmid=15903901|s2cid=22636577 }}</ref> This showed that, apart from mass, electrical charge and angular momentum, black holes can carry a finite [[Scalar field theory|scalar charge]] which might be a result of interaction with [[Inflation (cosmology)|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 results from the [[first observation of gravitational waves]] in 2015 provide some experimental evidence consistent with the uniqueness of the no-hair theorem.<ref name="BBC_11Feb16">{{cite news |title=Gravitational waves from black holes detected |url=https://www.bbc.co.uk/news/science-environment-35524440 |work=BBC News |date=11 February 2016}}</ref><ref>{{Cite journal |url=https://physics.aps.org/articles/v9/52 |title=Viewpoint: Relativity Gets Thorough Vetting from LIGO |journal=Physics |volume=9 |date=2016-05-31 |last1=Pretorius |first1=Frans |page=52 |doi=10.1103/physics.9.52 |doi-access=free}}</ref> This observation is consistent with Stephen Hawking's theoretical work on black holes in the 1970s.<ref>[https://www.facebook.com/stephenhawking/posts/965377523549345 Stephen Hawking].</ref><ref>[https://www.bbc.com/news/science-environment-35551144 Stephen Hawking celebrates gravitational wave discovery].</ref>

== Soft hair ==
A study by [[Sasha Haco]], [[Stephen Hawking]], [[Malcolm Perry (physicist)|Malcolm Perry]] and [[Andrew Strominger]] postulates that black holes might contain "soft hair", giving the black hole more degrees of freedom than previously thought.<ref>{{Cite journal |last1=Hawking |first1=Stephen W. |last2=Perry |first2=Malcolm J. |last3=Strominger |first3=Andrew |date=2016-06-06 |title=Soft Hair on Black Holes |url=https://link.aps.org/doi/10.1103/PhysRevLett.116.231301 |journal=Physical Review Letters |volume=116 |issue=23 |pages=231301 |doi=10.1103/PhysRevLett.116.231301 |arxiv=1601.00921 |pmid=27341223 |bibcode=2016PhRvL.116w1301H |s2cid=16198886}}</ref> 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.<ref>{{Cite journal |last=Horowitz |first=Gary T. |date=2016-06-06 |title=Viewpoint: Black Holes Have Soft Quantum Hair |url=https://physics.aps.org/articles/v9/62 |journal=Physics |language=en |volume=9 |page=62 |doi=10.1103/physics.9.62 |doi-access=free}}</ref> This was the subject of Hawking's final paper which was published posthumously.<ref>{{cite journal|last1=Haco|first1=Sasha|last2=Hawking|first2=Stephen W.|last3=Perry|first3=Malcolm J.|last4=Strominger|first4=Andrew|title=Black Hole Entropy and Soft Hair|journal=Journal of High Energy Physics|year=2018|volume=2018|issue=12|page=98|arxiv=1810.01847|doi=10.1007/JHEP12(2018)098|bibcode=2018JHEP...12..098H|s2cid=119494931}}</ref><ref>{{Cite web|date=2018-10-10|title=Stephen Hawking's final scientific paper released|url=http://www.theguardian.com/science/2018/oct/10/stephen-hawkings-final-scientific-paper-released|access-date=2021-09-14|website=the Guardian|language=en}}</ref>

==See also==
* [[Black hole information paradox]]
* [[Event Horizon Telescope]]

==References==
{{reflist|30em}}

==External links==
* {{cite journal |last=Hawking |first=S. W. |author-link=Stephen Hawking |arxiv=hep-th/0507171 |title=Information Loss in Black Holes |date=2005 |doi=10.1103/PhysRevD.72.084013 |volume=72 |issue=8 |pages=084013 |journal=Physical Review D|bibcode=2005PhRvD..72h4013H |s2cid=118893360 }}, Stephen Hawking's purported solution to the black hole [[unitarity (physics)|unitarity]] paradox, first reported in July 2004.

{{black holes}}

{{DEFAULTSORT:No-Hair Theorem}}
[[Category:Black holes]]
[[Category:Theorems in general relativity]]