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 can reverse the decrease of T so that the horizon of the future is met first.

Orbits

I also calculated the orbits of particles around Schwarzchild black holes, but I forgot the details. They 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), reduced 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...).

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