Do black holes have magnetic fields?

[Ibix]

Box 32.2 in MTW discusses Price's theorem. It says:
"Price's theorem states that, as the nearly spherical star collapses to form a black hole, all things that can be radiated get radiated completely away - in part "off to infinity"; in part "down the hole"."

Yes, Black Holes and Time Warps also says that the EM radiation may go inwards into the hole. I think possibly just your wording "the hair doesn't have to be completely radiated away" (post #29) may be confusing, as it can be read as meaning that the black hole can still have hair at the end of the process. The hair all converts into radiation (or at least, something that can flow) and goes away, but some of the energy in the hair may go inwards into the hole and end up contributing to one of the non-hairy parameters.

 

[PeterDonis]

Box 32.2 in MTW

subsections A.6-A.9 of box 32.2 in MTW

None of these contradict what I said, or support your claim that "hair" such as a magnetic field can be radiated down the hole and become a property of a stationary hole. As I said, that would contradict the no hair theorem.

 

[JimWhoKnew]

your claim that ... and become a property of a stationary hole.

I claimed that?

 

[Ibix]

I claimed that?

I think you can read #29 that way, although I don't think you intended it so.

 

[PeterDonis]

I claimed that?

the hair doesn't have to be completely radiated away. Some of it can be back-scattered into the hole

The "hair" being back-scattered into the hole would mean it becomes a property of the hole. Perhaps what you intended is what @Ibix said in post #36, but if so, that was not at all clear.

 

[pines-demon]

The problem with stationary magnetic fields is that these are usually due to quantum mechanics (unless you have a dynamical system). So it is very possible that this question is not entirely possible to resolve without a valid theory for quantum gravity.

 

[JimWhoKnew]

I think you can read #29 that way, although I don't think you intended it so.

Yet, my understanding from the literature is that although hair can't fall into the hole to become a permanent property, it is not ruled out as a transient property.

Suppose an object falls radially (zero charge, zero angular momentum) into a perfect Schwarzschild BH. Between the crossing of the horizon and the crash at the singularity, would you consider it as hair?

 

[PeterDonis]

although hair can't fall into the hole to become a permanent property, it is not ruled out as a transient property.

Yes. If you read my previous posts where I talk about the no hair theorem, you will note that I say it is true for a stationary black hole. During the transient condition, the hole is not stationary.

Suppose an object falls radially (zero charge, zero angular momentum) into a perfect Schwarzschild BH.

It can't be a perfect Schwarzschild BH if an object is falling in, since the object affects the spacetime geometry.

Between the crossing of the horizon and the crash at the singularity, would you consider it as hair?

First, mass is already a property that a stationary black hole can have, so the "hair" question is irrelevant for it. (So are charge and angular momentum, so the same would apply if you postulated a charged object with nonzero angular momentum falling into a Kerr-Newman black hole.)

Leaving that aside, your question takes the wrong viewpoint. The right viewpoint is, what is the spacetime geometry? That is a 4-dimensional thing that already contains all the information about what fell in and what happened to it, and answers any "hair" questions one might have.

For the case of an object of non-negligible mass falling into a black hole, there is no known analytical solution for the spacetime geometry, so we either have to simulate it numerically or use approximations. The former has been done in the literature, but I don't have any references handy, so I'll stick to the latter. The obvious approximation is that we have a Schwarschild BH of mass $M$ in the far past, a transient region where an object of mass $m$ (or more precisely energy at infinity $m$) falls in, and a Schwarzschild BH of mass $M + m$ in the far future.

In this approximation, the hole has no "hair" in the stationary regions in the far past and far future, so any "hair" (note that, as above, mass itself is not considered "hair" since it is a property of a stationary black hole) carried by the object would have to go away in the transient region. As we've seen in the discussion in this thread, it would go away by being converted to radiation and either radiating away to infinity or going down the hole. Any radiation that goes down the hole adds further to the hole's mass, so as a hypothetical example, if the infalling object were carrying "hair" that converted to radiation with energy $E$, and of that, energy $R$ was radiated away to infinity, then the final mass of the hole in the far future stationary region would be $M + m + E - R$ instead of just $m$. In the transient region while the conversion of "hair" to radiation was still going on, the "hair" would be detectable, yes.