Physics near a black hole

Factual Questions
1

I asked a friend who owns a PhD in physics this question yesterday and got a totally unsatisfactory answer. My friend appears to know more math than physics and has spent his entire career in a computer science dept.

I have read in many places that a black hole has only three visible properties: mass, charge, and spin. My question is how do these appear? Charge, for example. Two electrons repel because a photon is emitted by one and carries momentum to the other one. An electron attracts a positron because, similarly a photon carries the opposite momentum in the same way (but how does it know which direction momentum to carry–subsidiary question). But a black hole cannot exchange a photon with an electron, so how is the charge mediated. (My friend’s answer: because there is an electric field–whatever that is.) Similarly, gravitational attraction is said to mediated by exchange of gravitons. Same question. And how is spin detected? I thought maybe the event horizon might be an ellipsoid, but no, my friend said it is always spherical.

I would like a qualitative explanation. Not because I fear equations, since I don’t, but I doubt that equations would explain anything.

2

However, the Ergosphere of a spinning black hole is an ellipsoid.

3

The electrostatic force between two charged objects, including the case where one or both objects is a black hole, can be mediated by virtual photons. These are temporary photon-like modes of the electromagnetic field, not subject to all of the constraints on real photons. For instance, virtual photons can have negative mass, and could therefore be emitted by a black hole to interact with a nearby electron or positron.

4

I’m not sure the ‘virtual photons’ explanation is the best possible one, in particular because it mixes general relativity and quantum field theory, which are notoriously on bad terms; there ought to be a classically (‘classically’ here meaning non-quantum, i.e. GR) valid explanation, as well.

Luckily, there is, if you consider how black holes form. First, the electromagnetic field at some point (or rather event, i.e. 4-dimensional spacetime point) x is determined by the past lightcone of x. If x thus is a point outside a black hole event horizon, since no point beyond the horizon can lie to the causal past of x, i.e. within x’s past lightcone, it seems that what’s behind the horizon has no effect on the electromagnetic field at x. Nevertheless, I claim that if the black hole is charged, there will be an appropriate electromagnetic field felt by any probe at x.

Why’s that so? Well, the black hole hasn’t existed forever – it has formed at some point in the past. Thus, all of the charge was at some point outside of the black hole’s horizon – and thus, within the causal past of x. So since in order to determine what gives rise to the electromagnetic field at x we need to look at the whole past lightcone of x, we find the cause of the field in these charges pre-collapse.

For a perhaps somewhat more intuitive picture, consider some static, uncharged black hole, in which you drop a charged sphere. The picture is probably familiar: the sphere gets closer and closer to the event horizon, without ever quite crossing (of course, for the sphere, the situation appears rather different). So, if there’s no point at which it vanishes as seen from the outside, the same goes for the charge it carries, and the electromagnetic field it produces. Effectively, the black hole is now charged.

5

Your explanation seems correct to me, HMHW. I suspect that if you write down the virtual photon modes they will be mathematically equivalent to the EM field components “left over” by pre-collapse charges.

6

These explanations are fascinating. Although the one about the spin makes no sense. The size of ergosphere depends in no way on the spin rate and it must disappear instantaneously when the spin drops to 0.

Physics is hard; math is easy.

7

Well, of course both pictures must ultimately yield the same physics – it’s effectively just a change of gauge. If we work in the Coulomb gauge, the fields will be transmitted instantaneously, and we’ll essentially get the description according to which they are mediated by virtual photons far off the mass shell. I just prefer a treatment that shows one does not have to appeal to quantum mechanics in order to solve this problem; after all, (classical) electromagnetism and general relativity play nice together, so there should be a resolution within this framework, as well.

8

The best way, perhaps, is just to see the charge, spin and mass of a black hole as parameters of the black hole that also have neat analogues in the Newtonian limit.

If we approach the idea from a geometric angle instead and classify black holes in terms of the geometric properties of the spacetimes that contain them. We’d start off propbably by just considering all isolated black holes in otherwise empty space (and by empty, I mean very empty, i.e. no cosmological constant, graviational waves, etc).

Firstly we’d have to decide what properties a 4-D Lorentz manifold (i.e. a spacetime) would require in order for it to describe such a black hole (of any type) and this turns out to be doable. Secondly we might then see if it was possible to parameterize all spacetimes and how many parameters we would need. This again we can do and it turns out that we need just 3 parameters and further that these 3 parameters are just analogous to mass, charge and spin (or functions of the 3).
How do these properties manifest themselves physically? Well general relatvity only really talks about, worldines, fields, metrics, etc. So exactly how they manifest themselves physically is just in the usual way that worldines, fields, metrics, etc, manifest themselves physically. That may seem a cop-out, but the actual detail doesn’t matter so much.

Let’s say we now introduce additional parameters which allow for a wider class of spacetimes, these parameters cannot be seen to be properties of the black hole themselves (that’s a very simplified argument btw).

9

The event horizon of a spinning black hole is ellipsoidal, but it’s a lot closer to spherical than the ergoregion is. And the ergoregion doesn’t disappear abruptly when the spin goes to zero; rather, it shrinks down and becomes more spherical as the spin decreases, until at zero spin it merges with the event horizon (thus, a black hole that’s spinning but only very slightly will have only a very narrow ergoregion).

10

The wiki article quoted above claims that the long diameter of the ergoregion is equal to that of the black hole. I found that statement bizarre.

On a related matter, if someone wants to address it, is the argument over the destruction of information when matter falls into the hole equivalent to whether there is a fourth property of the black hole, namely entropy?

11

Well, that’s a bit of a hairy subject, if you’ll excuse the pun. First of all, the ‘no hair’ theorem really talks about black holes in classical physics (GR+Maxwell equations), which black hole thermodynamics goes a bit beyond. The basic conundrum is simple: assume you have some hot, i.e. high entropy, system. Then, drop that system into a black hole of your choosing. Now, by the no hair theorem, the black hole will be describable with a very small number of parameters, and thus, have very little entropy – but the second law states that entropy in a closed system can never decrease!

But, as Hawking showed, there’s another quantity that never decreases – the area of a black hole. So, the idea was that if you assume the entropy of a black hole to be proportional to the area, everything works out in the end – and indeed, as Hawking could show later, black holes emit radiation that is consistent with them having a temperature corresponding to precisely that entropy.

But this leaves the puzzle open – where does all that entropy (in fact, the maximum possible amount for a given spacetime volume) come from? Ultimately, entropy measures how many microstates there are to a given macrostate – i.e. how many configurations of a system’s fundamental constituents give rise to the same macroscopic description. However, the classical black hole, described by just a couple of parameters, doesn’t have any microstates – its state is completely described by its charge(s), mass, and spin!

In order to answer this question, we’ll need a theory of the black hole’s fundamental constituents – which, the theory in question being quantum gravity, we don’t have, at least not in any complete and sound sense. However, both leading contenders for such a theory, string theory and loop quantum gravity, have put forward proposals to account for the black hole’s missing microstates, at least in certain cases.

12

One can in principle detect the spin of a black hole by examining objects that orbit around it. The best-known of these effects, known as Lense-Thirring precession, states that a spinning object rotating around a spinning black hole (or any spinning object) will precess, i.e., the axis of rotation of the satellite will change. This effect was recently verified for the Earth’s rotation (after many many years of frustration) by Gravity Probe B.

Even if you don’t have a spinning body, though, you can still infer the spin of a black hole by looking at the periods of orbits that go in different directions. A satellite orbiting a spinning object in the direction of the object’s rotation (they’re both going around in a clockwise direction, say), will take slightly less time to complete an orbit than a satellite orbiting opposite the central object’s rotation. This effect is pretty minuscule (if the Earth went around the Sun the other way, for example, it would extend the length of the year by less than a millisecond), but in principle you could measure it.

13

Is baryon number not conserved? Does a black hole have a baryon number even theoretically? If I increased the mass of a black hole by dumping in a neutron star would it be the same as if I dumped in an equal mass of pairs of electrons and positrons?

14

That should say short diameter. The ergosphere and horizon are tangent to each other at the poles.

And black holes do not conserve baryon or lepton number. Dropping a neutron star into a black hole will have the same effect as dropping in electrons and positrons, or an anti-neutron-star.

15

So if the conservation of baryon number is not absolute, does that imply a proton can decay?

16

Well, that’s hard to say. Certainly protons can “decay” by falling into black holes and eventually having their energy radiated back out as Hawking radiation, but that doesn’t really address the question of whether they can decay without falling into black holes. If gravity is unified with the other forces, then there presumably would be some sort of extremely low amplitude interaction which would do the trick, but neither of those is guaranteed.

There are, however, solid indications that the strong and electroweak forces can be unified. And most (so far as I know, all) models which do so, also end up predicting proton decay, even without involving gravity at all. Unfortunately no experiment has yet detected this, and the models all have adjustable parameters which can be set to make the lifespan pretty much arbitrarily long.

17

The ‘no-hairs’ theorum is infact more of a conjecture, I believe, even in a classical context (e.g. Einstein-Maxwell theory). As I said it’s been proved that an isolated black hole can only have a limited number of parameters but I think it’s only suspected that inserting a black hole in to a more complicated spacetime will only introduce a limited number of new parameters. Obviously there’s nothing stopping a new QG/TOE theory allowing even more parameters for a black hole, on the other hand the cost is that there must be more going on in spacetime than simply 4-D Lorentzian geometry.

18

General Relativity has perfectly acceptable, non quantum answers to how a black hole can have mass and angular momentum.

Mass shapes the geometry of spacetime. Likewise, spin shapes the geometry of spacetime (leading to the Lense-Thirring precession). Charge is sticker, because electrodynamics hasn’t been unified with gravity, and when it is, it will be a quantum theory. HMHW has it right, though. Electromagnetic information transmitted from his hypothetical sphere reaches an outside observer ever more slowly, because it redshifts climbing out of the gravitational well. Since static charge already has a frequency of zero, it can’t be redshifted, and an outside observer will always be able to observe the charge.

19

Electromagnetism goes very well with relativity including the general theory. Perhaps not suprisingly as relativity was first formulated precisely as a background for Maxwell’s equation. One point to note is unlike the modelling of, say, quantum fields in semiclassical gravity (where quantum fields have no effect on the gravitational field), the feedback between gravitational fields and classical electromagentic fields can be modelled ‘perfectly’ (for some values of ‘perfect’) in the context of general relativity.

Of course this isn’t the same as unifying gravity and electromagnetism, however Kaluzua-Klein theory is precisely a unifcation of gravity and classical electromagnetism, the cost of this though is an additional dimension, i.e. Kalzua-Klein has 5 dimensions of spacetime as opposed to 4. Kalzua-Klein theory is a slightly unfashionable area of mathematical physics precisely because it ignores quantum effects that are known to play a role in electromagentism and strongly suspected to play a role in gravity. However it is at the same time hugely influential, providing some of the basic framework for string theory.

20

But, but what about the observer suspended just outside the hole? It would seem that this observer, using the same reasoning as above, would see no EM field at all. In fact, in the limit, wouldn’t he extrapolate the sphere crossing the horizon at c?

Or are you saying your analogy only applies to a faraway observer?