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 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 2π*r*. - 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.

*r*−*r*is proportional to the squared radial coordinate_{s}*z*^{2}−(*ct*)^{2}*T*is proportional to the angular coordinate artanh(*ct*/*z*) which can also be written (1/2).ln((*z*+*ct*)/(*z*−*ct*))

- The zone (-
*z*<*ct*<*z*) is the one "outside the black hole". - The zone (-
*ct*<*z*<*ct*, thus*r*<*r*) is the one "inside the black hole"._{s} - 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.

- 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.

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