# Why do some call the no-hair conjecture the no-hair theorem?

This excellent answer to Why would a black hole's magnetic hair being short-lived not violate the no-hair conjecture, but long-lived hair would? How long is “long-lived”? has got me thinking because it references no-hair as a theorem rather than a conjecture.

Wikipedia's No-hair theorem says:

There is still no rigorous mathematical proof of a general no-hair theorem, and mathematicians refer to it as the no-hair conjecture.

Wikipedia's Theorem says:

In mathematics and logic, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms or on the basis of previously established statements such as other theorems. A theorem is hence a logical consequence of the axioms, with a proof of the theorem being a logical argument which establishes its truth through the inference rules of a deductive system. As a result, the proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.

Astronomers, astrophysicists and cosmologists are celebrated for using a bit of artistic license when naming things and concepts; Big Bang, Hanny's Voorwerp, Dark Doodad, Gomez's Hamburger, The Great Annihilator1, European Extremely Large Telescope, Massive Monolithic Telescope, Kilodegree Extremely Little Telescope, etc.2

But I'm not sure that promoting a conjecture to a theorem sans proof falls into this category.

Question: Why do some call the no-hair conjecture the no-hair theorem?

1several of these are from Make Your Day Better With These 8 Cool Space Things That Have Totally Ridiculous Names

2several of these are from The silliest names scientists have given very serious telescopes

## 1 Answer

Why do some call the no-hair conjecture the no-hair theorem?

Perhaps they mean something specific, or perhaps they don't know any better... you'd have to examine it on a case-by-case basis. Perhaps a better question is: are there reasons why someone would be motivated to say theorem instead of conjecture?

Sometimes people call unproven mathematical statements "theorems" before there are publicly known proofs, typically because there are strong motivations to think it is true (often utilizing Occam's razor, leaning on empirical data, etc...). As @Peter Erwin pointed out, Fermat's theorem was called such long before it was proven rigorously mathematically. Laplace was famous for omitting proofs of his "theorems," which were found to most often be (mostly) correct. I think this kind of terminological inconsistency is not as bad as some other kinds, such as how string models are called "string theory" even though the fundamental assumption is not testable and the correspondence of the models to previously known theories is not well understood.

The no-hair "theorem" has not been rigorously proven mathematically in generality. From the wiki article:

"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."

The use of the word "all" above is important: it is very difficult to prove a statement about all solutions to the Einstein field equations. However, special solutions to the Einstein field equations are known for which theorems of uniqueness have been proven: e.g., Schwarzschild (spherical symmetry), Kerr-Newman (axisymmetric with charge). These can be thought of as specific examples of no-hair theorems, but they should not be confused with the general case which is an unproven theorem (also known as a conjecture).

The wiki article also states: "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."

Also, extensions of general relativity and classical field theory approaches offer a more complicated picture for black hole hair. For example, "black hole solutions with Yang-Mills and scalar hair... [where] ... high degree of symmetry displayed by vacuum and electro-vacuum black hole space-times ceases to exist in self-gravitating non-linear field theories."

To conclude, it is misleading to refer to the general "no-hair conjecture" as a theorem, but the use of "theorem" is justified in special cases.