A Reason Behind the "No-Hair Conjecture"?

Question

The famous "No-Hair Conjecture" states that a blackhole can have only 3 hairs: Mass, Angular Momentum, and Electric Charge. It occurred to me that the basic underlying reason behind this might be that the interior of a blackhole can't causally influence the exterior. For the interior to not affect the exterior, it appears to be a necessary condition that the observations made in the exterior should only depend on the conserved quantities of the interior. Otherwise, a cosmohiker who has passed the horizon may change a non-conserved quantity (say, the number of particles) and (through the dependence of exterior observations on this quantity) send a message to the exterior. But no matter what he does, a conserved quantity is not going to change and thus, the exterior observations can possibly depend on only such conserved quantities.

I realize that it just puts an upper limit on how many hairs the blackhole can have and there certainly is a greater number of conserved quantities than the three included in the "No-Hair Conjecture" and the rest of them are not considered as hairs of the blackhole, e.g., the baryon number. So, this reasoning can not explain the whole picture but can a reasoning based on the causal arguments explain the rest of the picture? More importantly, is this reasoning appropriate at least for what it seems to explain (that a hair of a blckhole must be a conserved quantity)? Also, if the blackhole hairs are related to conservation laws then one should be able to relate them to symmetries. Is there any such interesting link that is known?

Answer 1

Your line of argument cannot be fleshed out to make a proof of the no-hair theorems because it omits two necessary assumptions, stationarity and electrovac. If you omit stationarity, you have counterexamples such as the one in AGML's answer. If you omit electrovac, then there are counterexamples where the black hole is coupled to other fields besides the electromagnetic field.

More importantly, is this reasoning appropriate at least for what it seems to explain (that a hair of a blckhole must be a conserved quantity)

No-hair theorems (unlike Birkhoff's theorem) take stationarity as an assumption. So it's trivially true that if you can write down a definition of some property of a stationary black hole, then that property is conserved. For example, the maximum value of the Kretschmann scalar outside the horizon is guaranteed to be conserved. There are infinitely many such (automatically) conserved quantities. A no-hair theorem has to do something very different: it has to prove that out of all these conserved quantities, only three are independent parameters, and all the others can be determined from them.

Answer 2

Well, in the case of an isolated black hole, you certainly have some symmetries. Considering a horizon as a null 3-surface, with null tangent vector $\ell^{a}$ and (degenerate) 3-metric $q_{ab}$, the isolated horizon condition can be $£_{\ell}q_{ab} = 0$, which automatically makes $\ell$ a killing vector to the surface, and induces a conserved quantity, which we call the mass.

Furthermore, you have, normal to $\ell$, a spacelike 2-surface at each point on the horizon. Gauss's law works perfectly well here, which gives you the charge conservation law. And, if this surface has a rotational killing vector $\phi^{a}$, then you have an induced angular momentum.

Answer 3

Restricting to vacuum, the conserved mass and angular momentum are respectively conjugate to the timelike and rotational symmetries of the black hole. No-hair may roughly be equivalently stated as "isolated black holes do not radiate and are axisymmetric".

This is obviously not true of all black holes. For example the members of a black hole binary tidally deform one another and emit gravitational radiation, so you will need more than these parameters to describe them. Similarly, if you throw something really big into a black hole, it will wobble for a bit, which will break the symmetries. No-hair says rather these transient behaviours will be exponentially damped (i.e. a black hole has no stable ringing modes).

No-hair can be sort of thought of as an amped-up version of Birkhoff's theorem. The latter says that the Schwarzschild black hole is the unique spherically symmetric vacuum solution to general relativity. This doesn't quite put limitations on what "cosmohikers" could or could not do beneath the event horizon, because the theorem gives no reason to expect black holes to in fact be spherically symmetric. But it does imply that if you ever found a spherically symmetric black hole, it would have a conserved mass: spherically symmetric gravitational radiation is impossible, because that would constitute a spherically symmetric solution different from the Schwarzschild black hole.