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No-hair theorem

## Summary

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.[1] 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. 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, Wheeler said that Jacob Bekenstein coined this phrase.[2]

"Richard Feynman, objected to the phrase that seemed to me to best symbolize the finding of one the graduate students: graduate student Jacob Bekenstein had shown that a black hole reveals by nothing outside it what went in, in the way of spinning electric particles. It might show electric charge; yes, mass; yes, but no other features - or as he put it, "A black hole has no hair" - and 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." [3]

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.[4] The result was quickly generalized to the cases of charged or spinning black holes.[5][6] 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 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 baryonic number, leptonic 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]

## 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 ${\displaystyle M}$ ,
• linear momentum ${\displaystyle {\textbf {P}}}$  (three components),
• angular momentum ${\displaystyle {\textbf {J}}}$  (three components),
• position ${\displaystyle {\textbf {X}}}$  (three components),
• electric charge ${\displaystyle Q}$ .

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 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 (Proca fields, etc.).[citation needed]

It has since been extended to include the case where the cosmological constant is positive (which recent observations are tending to support).[7]

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".[8] 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.[9] 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.[10][11] This observation is consistent with Stephen Hawking's theoretical work on black holes in the 1970s.[12][13]

## Soft hair

A study by Stephen Hawking, Malcolm Perry and Andrew Strominger postulates that black holes might contain "soft hair", giving the black hole more degrees of freedom than previously thought.[14] 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.[15] This was the subject of Hawking's final paper which was published posthumously.[16][17]

## References

1. ^ a b Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. pp. 875–876. ISBN 978-0716703341. Archived from the original on 23 May 2016. Retrieved 24 January 2013.
2. ^ Archived at Ghostarchive and the Wayback Machine: "Interview with John Wheeler 2/3" – via YouTube.
3. ^ 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
4. ^ 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.
5. ^ 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. S2CID 121476298.
6. ^ 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.
7. ^ Bhattacharya, Sourav; Lahiri, Amitabha (2007). "No hair theorems for positive Λ". Physical Review Letters. 99 (20): 201101. arXiv:gr-qc/0702006. Bibcode:2007PhRvL..99t1101B. doi:10.1103/PhysRevLett.99.201101. PMID 18233129. S2CID 119496541.
8. ^ Mavromatos, N. E. (1996). "Eluding the No-Hair Conjecture for Black Holes". arXiv:gr-qc/9606008v1.
9. ^ 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. PMID 15903901. S2CID 22636577.
10. ^ "Gravitational waves from black holes detected". BBC News. 11 February 2016.
11. ^ Pretorius, Frans (2016-05-31). "Viewpoint: Relativity Gets Thorough Vetting from LIGO". Physics. 9. doi:10.1103/physics.9.52.
12. ^ Stephen Hawking.
13. ^ Stephen Hawking celebrates gravitational wave discovery.
14. ^ Hawking, Stephen W.; Perry, Malcolm J.; Strominger, Andrew (2016-06-06). "Soft Hair on Black Holes". Physical Review Letters. 116 (23): 231301. arXiv:1601.00921. Bibcode:2016PhRvL.116w1301H. doi:10.1103/PhysRevLett.116.231301. PMID 27341223. S2CID 16198886.
15. ^ Horowitz, Gary T. (2016-06-06). "Viewpoint: Black Holes Have Soft Quantum Hair". Physics. 9. doi:10.1103/physics.9.62.
16. ^ Haco, Sasha; Hawking, Stephen W.; Perry, Malcolm J.; Strominger, Andrew (2018). "Black Hole Entropy and Soft Hair". Journal of High Energy Physics. 2018 (12): 98. arXiv:1810.01847. Bibcode:2018JHEP...12..098H. doi:10.1007/JHEP12(2018)098. S2CID 119494931.
17. ^ "Stephen Hawking's final scientific paper released". the Guardian. 2018-10-10. Retrieved 2021-09-14.