Aspects of black holes

Here is just a short draft of a few remarks about black holes.

Black holes were one of my favorite topics of interest in my free time in high school : just after I found the equations for the expansion of the universe (Friedmann equations) in my own calculations, I also managed, by the same tools, to find equations for black holes: first the Schwarzchild metric (spherically symmetric, no charge) and then the Reissner–Nordström metric (spherically symmetric, nonzero electric charge). So while I do not figure out well the rotating black hole (which would be harder to calculate by lack of symmetries), I can speak with confidence of the spherically symmetrical ones, and I think that with a little bit of work, anyone who followed that simple reasoning which gave the Friedmann equations can manage apply the same reasoning to do these calculations of the black hole as well.

Of course, for the electrically charged case, one needs to know the energy-momentum tensor of an electric field, which can be shortly described as follows. This field splits the 4 dimensions of space-time into a pair of two orthogonal 2D planes, each of which keeps its rotational symmetry. One is formed of the space-like "direction of the field" (the vertical direction z, for a charged spherically symmetric black hole) together with time; along it, is negative pressure together with positive energy density, with the same amount in order for this to be invariant by "rotation" (boost). Along the other plane (x,y), that is the pure space-like plane orthogonal to the field, is positive pressure, which in any of its direction is the exact opposite to the negative pressure along the field. The value of this pressure in each direction is, essentially, half the square of the intensity of the field.

The horizon

To understand what happens near the horizon, a special picture of the (vertical space)-time plane is needed. Let us denote (z,t) the relevant local coordinates system near the horizon, with which Special Relativity is approximately valid. In this map we shall give the following 2 fields:

The altitude field r that is the radial coordinate, though it is not a radius in any meaningful sense. The definition of this quantity is that, following spherical symmetry, in each sphere of constant altitude r at some fixed time, the geodesics have length 2πr.
Absolute time T, defined as follows: an event of bouncing back a vertical light beam, is said to happen at time T if T is the middle time between emission and reception of this beam by a "steady" observer far away from the black hole and at rest with respect to it. Of course, this definition only works above the horizon.

Now (r, T) is obtained from (z,t) as the analogue, in space-time geometry, of a polar coordinate system: as long as r is close to the "altitude of the horizon" r_s the following formulas approximately hold

r−r_s is proportional to the squared radial coordinate z²−(ct)²
T is proportional to the angular coordinate artanh(ct/z) which can also be written (1/2).ln((z+ct)/(z−ct))

As measured by these coordinates, space-time around the horizon is divided into the following zones:

The zone (-z<ct<z) is the one "outside the black hole".
The zone (-ct < z < ct, thus r < r_s) is the one "inside the black hole".
The half-line (0<z=ct, thus T = + infinity) is the horizon.
The zone (z+ct<0) does not actually exist
Its border, (z+ct=0) concentrates the history of the formation of the black hole and whatever subsequently fell into it long ago.

So, the horizon is crossed without noticing, as a virtual wave plane coming at the speed of light.

Inside the charged black hole

In the Schwarzchild black hole, things end by a singularity, but in the Reissner–Nordström black hole the end is different : instead of a singularity, it is another horizon with a smaller value of r (which approaches 0 when the charge of the black hole approaches 0), and other values of the proportionality coefficients. It needs a change of coordinates that works the same as in the above case of horizon, but with a different list of zones. Only the zone (ct<z<-ct) actually exists (so from any event there, there is a limit of available time before the end). It ends by its 2 borders, that are the internal "horizons". While the external horizon is naturally empty and crossed without noticing, each of both internal horizons are ends of the story, as they are walls of infinite energy.

The horizon of the past, is an infinite physical concentration of a bit of the past history of the the black hole, from its creation, with all things that fell into it in between. This wall is coming "from the bottom" (the same way as the external horizon); it is normally the one smashing first for objects which naturally fell into the black hole, as they are "moving to the botton" (the "speed" is described by the decrease T which inside the black hole is no more a time coordinate but a vertical coordinate, that keeps endlessly decreasing from the +infinity of the external horizon to the -infinity of the internal horizon).
The horizon of the future is the one coming "from the top", and concentrates the history of all what is going to fall into the black hole "long after" the considered object. Only a strong boost upwards in between, or a sufficient electric charge with the same size as the black hole, can reverse the decrease of T so that the horizon of the future is met first.

Recently (beginning of 2022) I analyzed in more details how things are going at the internal horizon of black holes with possibly both a charge and an angular momentum, and elucidated aspects which seemed to have been missed by the existing literature: see section 8 of this article.

Orbits

The orbits of particles around Schwarzchild black holes are not exactly given formulas but, thanks to the use of conserved quantities (energy and angular momentum) and a change of variables (taking 1/r as new variable and taking the angular coordinate in guise of time if I remember well), reduce to a kind of well-known, rather simple differential equation: an equation of the form (dx/dt)² = (polynomial of degree 3 in x), the solutions of which have a known name in the mathematics literature, but I forgot which (on the way I noticed that the case of degree 4 polynomials is reducible to the case of degree 3 by an homographic transformation which sends one of the roots to infinity...).

Hawking radiation of charged or rotating black holes

With only a limited understanding of QFT, having never checked any mathematical study about the Hawking radiation, I happened to notice a ridiculous mistake seemingly committed by experts in the field. Here is an excerpt of a correspondence I had with one of them (occurring after I invited him to check the above linked section 8):

Me: ...I have been recently puzzled when looking through abstracts of a few articles on the topic, as if the authors had missed the following two basic clues (I am not sure they really missed these as I did not check the literature in details, but the abstracts I saw seemed that way):

A Kerr black hole does not have a single temperature, but a field of temperatures varying from Eastwards to Westwards radiating directions (that might be expressed by considering temperature not as a quantity, but as a vector in some multidimensional space I will not care to precisely describe). At least, the temperature depends on the direction of emission if and only if the entropy depends on the angular momentum. Now I see an article claiming that the entropy only depends on the mass and not on the angular momentum. I did not try to follow the details but I just cannot believe such a conclusion.
When a charged black hole which radiates away its mass approaches the lower limit of possible masses for its given charge, that does not stop it anyway from smoothly continuing evaporation to its final explosive end, because its charge is also radiated away by emitting charged particles (mainly electrons or positrons since these have the lowest masses)

Him: Right, I'd say that most people wouldn't agree with the results of that article - angular momentum is definitely a property of a black hole that carries information and therefore contributes to the entropy. I assume you're reading Volovik's works; what he does is consider not only the Hawking radiation from the event horizon reaching Earth, but also the Hawking radiation from the inner horizon that travels farther inward toward the black hole's center (or I suppose towards a new Universe in the fully extended Kerr metric). However, entropy is related to a measure of how much information is accessible to an observer, and since any radiation produced inside of the event horizon will never reach us, it therefore shouldn't be included in the calculation.
Regarding your first point, I would still say that a single temperature can be defined for a Kerr black hole. When Hawking and Bekenstein originally formulated the idea, the temperature of a black hole is simply proportional to the surface gravity κ at the event horizon (for Earth the surface gravity is g=9.8m/s^2, and for black holes there's an analogous definition). For Kerr, the surface gravity depends on M and a, but it otherwise is constant and doesn't depend on any coordinates. So regardless of where you are, this surface gravity will be the same. Thus the black hole is in equilibrium with its surroundings. (...) And with your second point, that's a great insight and I don't think anyone would disagree with you. Usually what gets said is that as a charged black hole evaporates, it radiates mostly photons at first (because they are massless), but eventually, it would reach the extremal limit where it can't lose any more mass for a given charge. Then it will still eventually evaporate (perhaps much slower because it's harder to radiate particles with mass like electrons), but it will remain extremal. This is mostly used as a justification for why people do calculations with extremal black holes, since most black holes will become extremal once they get small enough.

Me: To be precise about the temperature of a rotating black hole: I expect the back hole to be in thermal equilibrium with the black-body radiation naturally appearing inside an opaque ROTATING box containing the black hole at its center. The rotation of the box needs to conform somehow with the rotation of the black hole, otherwise there is no thermal equilibrium and the radiation forms a kind of viscosity link between the box and the black hole, which will cause some entropy creation on the black hole until the box and the black hole end up conforming their rotation to each other.

Him : Right, it is defined as the temperature of radiation received by an observer at infinity, which happens to depend only on the surface gravity of the event horizon. The trajectory-dependent radiation I mentioned was my attempt to explain what you were saying about how the temperature varies, since radiation will look different once you get close to the black hole. But for an observer at infinity, the temperature should always be the same, regardless of what azimuthal angle the observer is at. The Kerr spacetime is azimuthally symmetric, so by construction there can't be any difference at different positions. You could treat this as a rotating blackbody box, but at infinity, any rotation effects are negligible (in that regime, Kerr asymptotically approaches flat Minkowski spacetime).

Me: I disagree with the terms of this claim. Only an observer located on the polar axis will see a definite temperature of radiation. Others will see not a definite temperature but a mixture of temperatures, which they can resolve by a good telescope which will show different temperatures emitted between the Eastern and Western edges of the black hole's image. There is a big technical contradiction if you try to combine the concepts of a rotating blackbody box and views approaching infinity : given a fixed angular speed (more exactly a solid move, which is the concept needed to define the stable blackbody radiation), there is a distance from the center at which the velocity of the box exceeds the speed of light. There is no way to neglect the effect of this need for the needed observer to be going at superluminal speed then.

(He then finally admitted my point)

Now thinking further on the radiation of charged particles by charged black holes. While the normal Hawking radiation could be described as having a specific temperature, that could be put in equilibrium with some kind of thermal bath, black holes with a significant charge cannot be described in such a way: they are violently far from any conceivable thermal equilibrium. They run the risk of a sudden discharge by lightning. If a black hole's charge is x times (x<1) the limit for its mass (such as half of it), then the typical energy of emitted single electrons or positrons (same sign of charge as the black hole) is about x times the Planck energy (Ep) times the square root of the fine structure constant. No, it does not get colder for bigger black holes.

I have not the tools to compute the limit of electric disruption, but let us fancy it here to be the electric field intensity such that a single electron moving the size of an atom (10e-10 m) gets an amount of energy comparable to its mass energy (Em).

This means that for a black hole to approach extremality without disruption, requires its radius to be > (Ep/Em)*sqr(α)*10e-10 m, which is about 2e+11 m.

By comparison, our galactic black hole has radius 12 million km, that is 1.2e+10 m.

So, yes, it is harder to radiate more massive particles than lighter ones. However, electrons are very light already : much lighter than the Planck mass. For the evaporation to stall in an extremal configuration, the black hole needs to start evaporation when it was so massive that the charge it had, initially a very small fraction of its mass in natural conditions (so that it does not dissociate the atoms of infalling matter to reject the charges of the same size), ends up being dominant before the black hole's mass becomes "as small as" 20 times that of our Galactic black hole. And of course, it should be kept along the way from any presence of infalling matter, which would otherwise cause disruption.

In conclusion, it looks like something went wrong in the physicists usual approximations on the topic.

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