# Black hole “no hair” theorem

The "no hair" theorem (or conjecture), suggests that black holes can be entirely described by their mass, angular momentum and charge. All other details of the BH formation are lost.

Is there a simple way of understanding why you would be able to tell the difference between a BH made from protons and one made from electrons, but not between one made from matter rather than anti-matter?

i.e. What is special about charge, and why aren't lepton number, baryon number, strangeness etc, also black hole properties?

• Electric charge isn't quite as special as you're thinking. The no-hair theorems you usually hear about are for electrovac assumptions. So the special role of electric fields as opposed to any other fields is put in as an assumption. There are known counterexamples if you allow other fields. See livingreviews.org/lrr-1998-6 . – Ben Crowell Oct 23 '14 at 0:13
• The no hair conjecture assumes that the hole lives in quiet, flat space. Since quiet, flat space does not exist, its not of much use in the real universe, even if its true. The constant addition of energy in stochastic non - symmetric modes leaves the heavy GR object without even much of a singularity, much less no hair. – Tom Andersen Aug 18 '15 at 22:13

The simplest answer lies in a combination of Gauss's law and Birchoff's theorem. These say, alternately, that the electric field and gravitational field of a spherically symmetric charge distrubtion only depends on the charge and mass-energy enclosed in the spherical shell${}^{1}$.

Therefore, if I have a spherically symmetric distribution of charged mass, and I'm out at some radius $R$ outside of it, I can't tell whether it collapsed to a black hole or not just by making local measurements of the fields. This means that the information about the charge and mass of the black hole must live outside of the horizon. There is no similar set of theorems that tells you about whether the distribution was made out of electrons or antiprotons, or lepton number, or the like. So, you lose the information about that stuff, but not the mass and charge.

${}^{1}$Note that the mass energy in a shell depends on the enclosed electromagnetic field, so it's natural that the Nordstrom metric differs from the Schwarzschild metric.

• I like the simplicity of this argument. Is it really that simple? – Rob Jeffries Oct 24 '14 at 8:26
• @RobJeffries: there's more to the no-hair theorem, certainly, because angular momentum muddies the whole thing. There is no strong theorem like this when you relax spherical symmetry to axisymmetry, which you need to do to consider spin. – Jerry Schirmer Oct 24 '14 at 12:55
• After all, a straight wire with an oscillating current is axisymmetric, but it will also radiate EM radiation. – Jerry Schirmer Oct 24 '14 at 12:57

A very brief, although somewhat incomplete, answer would be that charge is related to a local symmetry, therefore with a gauge field that acts in the whole spacetime, even if the charges are inside the event horizon you could use the electromagnetic field to probe it. Lepton and Baryon number, or other flavour related quantities are related to global symmetries, and therefore once inside the horizon there is nothing that can physically probe this numbers.

The incompleteness of the answer is related to the other gauge fields, namely the carriers of weak and strong forces. Note that by no-hair conjecture the black hole should not contain this charges either. As I think this is somewhat more complicated I'll just say that Einstein equations coupled with (classical) Yang-Mills fields allow for black hole solutions which violate no-hair conjecture, although this solutions are unstable and therefore if they are a valid counterexample is up to dispute. Scalar fields may also cause hairs to grow (sorry for the pun)

There is a Living Review in Relativity by Chrusciel, Costa and Heusler that addresses the black hole uniqueness theorem. Maybe the introduction will give you an idea of what part is theorem, what is false and what remains conjecture.

Also in the seventies there were some papers giving detailed calculations which suggest that lepton and baryon numbers are ill-defined for black holes and that they are not able to carry weak and strong charges. A typical example would be Bekenstein's "Transcendence of the Law of Baryon-Number Conservation in Black-Hole Physics" and references therein, specially by Hartle and Teitelboim.