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# No-hair theorem

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 Title: No-hair theorem Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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]

There is still no rigorous mathematical proof of the 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 been only 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.

## Contents

• Example 1
• Changing the reference frame 2
• Four-dimensional space-time 3
• Extensions 4
• Counterexamples 5
• References 7

## 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:

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.).

## Extensions

It has since been extended to include the case where the cosmological constant is positive (which recent observations are tending to support).[3]

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".[4] 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 scalar field was derived.[5] 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.