### “No Hair” theorems

#### by Willie Wong

Hum, now I am a bit confused. I doubt any “prominent mathematician” will read this blog and comment, so I guess I’ll ask them in person next time I go to a conference.

The question is about the term “no hair theorem”.

A “theorem” (in quotes because it is only really proved in special cases and not in the most general situation) in black hole physics is the statement that “the Kerr-Newman family of black hole solutions exhausts all stationary solutions to the Einstein-Maxwell system if we assume non-degenerate event horizons.” A slighted dated (at the current time) review of developments for this statement can be found in this article of Piotr Chrusciel.

(Some more recent developments include:

- Rigidity results for stationary black holes without analyticity assumptions.
- Rigidity of domain of outer communication follows from rigidity of regular event horizon in the 1 black hole case. Vacuum — Alexandru Ionescu and Sergiu Klainerman. Electro-vacuum — Yours truly.
- Rigidity of axial symmetry under small perturbations. Vacuum — Spyridon Alexakis, Ionescu, and Klainerman. Electro-vacuum — Pin Yu and W, in preparation.

- Ruling out multi-black-hole configurations
- No two-black hole solutions under strong non-degeneracy condition — Jorg Hennig and Gernot Neugebauer
- Regularity results for electro-vac system with multiple holes, generalizing work of Li-Tian and Weinstein — Luc Nguyen

and there are probably a few I am not personally aware of…)

Now, that statement I mentioned earlier seems to be often referred to as the “No Hair Theorem”. Indeed, the article of Chrusciel that I cited calls it such, and as far as I know Alan Rendell also calls it that way. I believe the name goes back to (at least) Brandon Carter in his Les Houches lectures (collected in *Les Astres Occlus*), while folklore attributes the phrase to John Wheeler. The idea is that we expect the Kerr-Newman family to not only be the unique “regular” stationary solution, but that it represents an attracting fixed point: that all other black hole solutions will converge eventually toward a Kerr-Newman solution (an open problem now commonly known as the “stability of Kerr”). Under this strong assumption, the idea is that since Kerr-Newman black holes only carry information about mass, angular momentum, and charge, all other information (the “finer points”) are lost in the evolution. In other words, the black hole “sheds its hair” as it evolves.

But why do we call those information “hair”? It is because the solutions are uniquely characterized by three (or four, if you allow gauge rotation to magnetic charge) scalar quantities which represent the lowest order terms in the multipole expansion at infinity. Any additional information represents higher order terms in the expansion, which decay more quickly as we approach infinity. So from the point of view of an observer very far away from the black hole, those information cannot be “resolved”, similar to how when you look at a far-away individual you cannot see the individual strands of hair. Only when the observer gets closer to the black hole (or you get closer to the individual) can the “hair” be seen.

This at least was my interpretation of the phrase “no hair theorem”.

Recently I went to the AMS Southeastern Meeting at Boca Raton, Florida. There Gilbert Weinstein tried to convince first me, and then Lev Kapitansky, of an alternate interpretation. Gilbert is convinced that the hair refers to bifurcations: if one considers the family of Kerr-Newman solutions as a subset of all smooth metrics solving the Einstein-Maxwell equations, it is a precisely three parameter family. Locally this gives that the family is three dimensional at all its points. And in particular, it does not bifurcate and “grow” additional stationary solutions. Or, in slightly more mathematical language, the family of stationary solutions has no hair because locally it is a continuous embedding of a three dimensional manifold into the space of smooth solutions.

(Come to think of it, Gilbert’s interpretation may also fit what Carter wrote in his lectures. It has been a while since I last looked at those lectures to be sure.)

So the question is: which interpretation is more fitting?

Also see Alan’s response to my question on his blog.

The formulation given in Misner, Thorne and Wheeler’s

Gravitation1973, Box 33.1 A Black Hole has No “Hair” is:‘the external gravitational and electromagnetic fields of a stationary black hole are determined uniquely by the hole’s mass M, charge Q, and intrinsic angular momentum S’. The associated discussion notes that the theorems at that date (December 1972) involve ‘several highly technical assumptions, not stated here, about the global properties of spacetime’.Not sure that this could be described as ‘folklore’.

The (1998) review by Heusler (http://relativity.livingreviews.org/Articles/lrr-1998-6/index.html) is also useful (and it’s slightly more accurate in its citations).

The problem is one of primacy. Carter’s Les Houches lectures were also published in 73, and the lectures themselves were delivered in Summer of 1972. So it could be that Wheeler’s terminology was inspired by Carter. It is generally accepted that Wheeler has an active imagination and is the progenitor of many colourful names in general relativity. So I wouldn’t put it pass him to actually coin the term/conjecture: it is just that I personally do not know of any published references before Carter’s lectures that referred to them as such.

In response to your observation, I searched for the phrase “no hair” + theorem and came up with an article by Ruffini and Wheeler, ‘Introducing the Black Hole’, Physics Today, January 1971, which contains the phrase, suggesting that your idea about Wheeler coining the terminology is correct.

Ah, good old Physics Today. This is probably the end of the hunt. Knowing now the co-author and where it appeared, I also found references by Kip Thorne and Jacob Bekenstein attesting to this fact.

Thank you, sir, for your sleuthing.

Also, the highly technical assumptions is basically the following: analyticity of the space-time which guarantees that a stationary black hole is at least axially symmetric, and asymptotic fall-off properties that guarantees good structure at spatial infinity. Einstein’s equations then can be reduced to be a system of equations, which essentially has three unknown coefficients given by M Q and S which are specified at infinity.

The theorem MTW referred to is the one in Carter’s lectures. Basically Carter took the argument above one step further and showed that while for any set (M,Q,S) there may admit multiple solutions (in this sense, MTW’s statement is actually wrong strictly speaking for the knowledge they have at the time), those multiple solutions must be discrete in some topology of solutions of Einstein’s equations. Furthermore, each solution can be continuously deformed to another solution of different (M,Q,S) parameter. So the picture is something like this: take the space of all smooth Lorentzian metrics with asymptotic properties. Then the stationary electro-vac black holes may form a subset in it that is everywhere 3-dimensional with disconnected leaves that do not intersect. The values of (M,Q,S) are unique on each leaf, but may repeat between different leaves.

The second statement of Carter’s is that only one of the leaves admits solutions with S = 0. In other words, only one of the leaves contains the Schwarzschild solution. (So if you amend this condition to MTW, then their statement becomes more correct.)

It took the next decade or so to give a reasonably believable argument that there exists precisely one leaf of solutions. And there are still on-going work to strengthen the results (by weakening the myriad technical assumptions).