9. Final remarks

The purpose of this presentation was to illustrate selected features of strong gravitational fields. Black holes are the sources of the strongest gravitational field in the sense that an event horizon has developed. Let us briefly consider the point 'black but not dark'. The presence of black holes may be recognized primarily due to gravitational interactions: the dynamics of their environment. In this sense they may be regarded as a component of a dark matter sector. Accepting such an oversimplified or naïve point of view for a while, one may ask about the character of this component. Partly, the answer is obvious: this is baryonic matter, as massive stars collapsing into black holes are composed of baryonic matter. But due to instability, there are no extremely massive stars, so BHs of millions of solar masses have a different origin (eternal black holes), so they might not necessarily be composed of baryonic matter. In principle, as they evaporate, they emit radiation; also, they could be charged so they could therefore affect their environment not only gravitationally. Hence, although they are black they are hardly 'dark'. As mentioned at the beginning of this exposition, BHs constitute a small fraction of the density of baryonic matter, so they are interesting objects in the Universe rather for the local properties imposed by their gravitational field, than for other reasons (at least so it seems to us at the moment).

The outcome of the presence of the horizon of the BH is a dramatic difference in the symmetry properties of the exterior and interior of the BH. Energy conservation related to the time-like Killing vector is changed into a corresponding momentum component conservation as the Killing vector is converted into a space-like one. That is a consequence of the static spacetime outside the horizon being transformed into a homogeneous one, along the t-direction, but it also becomes a dynamically changing spacetime inside the horizon: expanding along the homogeneity direction and contracting perpendicularly to that direction. On the one hand, this leads to the information paradox. On the other hand, the presence of the BH's event horizon may be interpreted as an interchange of the roles of the time and radial coordinates. And this leads to unexpected scenarios, with some surprising processes and phenomena taking place outside the horizon yet with even more striking properties of the interior of the horizon. It should be underlined that the discussion presented here has dealt mostly with eternal BHs, which have not been created due to gravitational collapse but rather have existed forever (since the Big Bang). However different these may seem, they have a lot in common. They both decay due to Hawking radiation [2]; as suggested by various authors [16], the interior of gravitationally collapsing black holes is also of a cylindrical shape, and both eternal and collapsing BHs share one more common but bizarre property, their volume is infinite [16, 35]. Hence, though it is not guaranteed that the interiors are the same their properties might turn out to be quite similar. But there is a still a deeper problem of a much more fundamental character: could the interior of black holes be described by the approach presented here? Or more specifically, could a very strong gravitational field, inside the BH horizon, be described in terms of the theory of relativity? Or is a new physical approach necessary, as emphasized by G. t'Hooft [19] (see also [36]) involving quantum mechanical aspects also? As usual, the answer will come in time, but even if the answer is satisfactory, in this case, it will probably never be the final word.
