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.
[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]
See also
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
External links
