6. The meaning of a strong gravitational field

Let us underline the rather unexpected and counterintuitive observations that accompany the presence of the event horizon of a Schwarzschild BH. The strange intimate symmetry of the outer versus inner region: static observers outside the horizon and observers at rest inside the horizon measuring the Doppler shift of signals incoming from MS would record basically the same outcomes. The speed of a test particle falling towards the BH appears to be impeded after crossing the horizon. As described elsewhere, the speed of a test particle uniformly accelerated inside the horizon after reaching its maximal value starts to diminish. A null geodesic follows exactly half a circular orbit within the horizon. Signals exchanged within the horizon seem to mimic the cosmological model expanding along one specific direction and contracting perpendicularly to this direction. All of these are manifestations of the presence of such a strong gravitational field that the event horizon of the BH is developed.

Inside the horizon of a Schwarzschild BH, one comes across a unique phenomenon: an interchange of the roles of the r and t coordinates. Outside the horizon r > rS, the radial coordinate is an ordinary spatial coordinate, which may change from rS to ∞ in both directions, dr ¼ �j j dr and the time coordinate t is a temporal one, that is, such that, dt > 0. Inside the horizon r < rS, and coordinate r becomes a temporal one: r changes from rS to 0 and dr < 0; coordinate t then plays the role of a spatial coordinate: �∞ < t < ∞ and dt ¼ �j j dt .

Such an interchange results in a dramatic difference of the symmetry properties of the spacetime. As mentioned earlier, the Schwarzschild spacetime outside the horizon is static, independent of time and isotropic; this results in the conservation of energy and angular momentum, respectively. Inside the horizon, the spacetime is still independent of t but this is now a spatial coordinate in that spacetime, leading to t-component momentum conservation; it is no longer static but instead dynamically changing, being r-dependent. Inside the horizon of the Schwarzschild BH, spacetime is cylindrical-like, homogeneous along the t-axis and spherical-like, of radius r perpendicularly to this axis (see also [2, 16]).

All of this presents the above-seemingly unexpected or counterintuitive phenomena in a new perspective. The speed of the infalling test particle is measured as 'distance'/'time' so the interchange of the roles of 'distance' and 'time' leads to the inverse expressions to those ~ exterior to the horizon, V and interior, V; hence, the speed turns out to decrease inside the horizon. The cylindrical-shape BH interior is a dynamically changing spacetime: expanding along the t-axis and contracting perpendicularly to this axis. This results in both red- and blueshifts, respectively [12, 17]. Hence, it actually is a realization of a specific cosmology. The fact that light rays propagating perpendicularly to the cylinder axis occupy a semicircular photon sphere analogue is found to have a deeper significance [18]. The same value π is found for other kinds of black holes, and this appears to be a fundamental discovery; it may be a symmetry property linked to a 'new physics' of black holes [19]. Also, other observations may need deeper analysis but, whatever the interpretation, they are caused by the strong gravitational fields that form the BH horizon.

Let us emphasize that the common sense property of the BH, namely. 'nothing, not even light can leave their interior' takes on a new sense now: crossing the event horizon, a test object can never reach it again as this would mean travelling backwards in time.

There is a more formal interpretation of the interchange of the role of radial and temporal coordinates in the theory of relativity. The Killing vector representing time independence symmetry, being time-like outside the horizon becomes space-like inside the horizon—this actually means that the time-like component of the momentum four vector is converted into a space-like momentum component, respectively. This opens the door for radiation emitted by black holes—Hawking radiation.

## 7. Astrophysical black holes

Generations of thermonuclear reactions support stars against gravitational collapse [3, 20]. The first stage is a process of hydrogen burning to make helium. When a substantial amount of hydrogen is exhausted, gravitational contraction raises the temperature until helium burning, the so-called triple alpha process, can start. This evolution eventually leads, for massive stars, to the last stage where an element with the largest binding energy per nucleon, <sup>56</sup> <sup>26</sup>Fe, is produced. What happens then?

One can consider the state of a star of mass M and radius R, which exhausted its thermonuclear fuel, T ¼ 0. It is supported by a nonthermal pressure, due to the fermionic nature of electrons, protons and neutrons. There are two competing contributions to the energy of such an object. A negative one arises from a gravitational origin

$$E\_g \propto -\frac{M^2}{R} \tag{41}$$

and a positive one, the kinetic energy of the electronic gas:

$$E\_k \propto nR^3 \langle E \rangle \tag{42}$$

where n denotes the density of electrons and h i E is the electronic mean energy. Taking the following relation between the characteristic electron momentum, pF, and the corresponding wavelength, <sup>1</sup>=<sup>3</sup> λ∝ n� ,

$$p\_F \propto \lambda^{-1} \propto n^{1/3} \tag{43}$$

<sup>2</sup> one obtains for <sup>a</sup> nonrelativistic range of energies, h i <sup>E</sup> <sup>∝</sup> pF

$$E\_k \propto \frac{M^{5/3}}{R^2}.\tag{44}$$

It appears that the kinetic energy term dominates in the range of decreasing values of R, preventing further contraction. However, for more massive stars, higher energies are available and the electrons would be regarded as relativistic, E ∝pF h i and then,

$$E\_k \propto \frac{M^{4/3}}{R} \tag{45}$$

In such a case for a mass M larger than the Chandrasekhar limit, MWD ≈ 1:4M<sup>⊙</sup> (for white dwarfs), the pressure of the electron gas could not support a star against its gravitational contraction.

For even more massive stars, one comes across inverse beta decay leading to the formation of a neutron star core. In such a case, the Pauli exclusion principle, this time for neutrons, prevents gravitational collapse, up to some specific limit, Mcr ∝2 � 3 M⊙. For masses larger than this limiting case, nothing can stop the ongoing gravitational collapse eventually leading to a singular state of matter—a black hole.

#### 8. Entropy and Hawking radiation

In early 1970s, it was indicated by Bekenstein [21] and Hawking [22] that BH entropy is proportional to their surface area:

$$S = k\_B \frac{4\pi r\_S^2}{4l\_P^2} = k\_B \frac{4\pi M^2}{l\_P^2} \tag{46}$$

where lP denotes the Planck length and kB Boltzmann's constant. It was also recognized that BHs may be regarded as simple thermodynamic systems (the black hole 'no hair' theorem) characterized by three parameters, mass M, angular momentum J and charge Q. Accordingly, one can identify four different kinds of black hole: Schwarzschild (nonrotating and uncharged, characterized by their mass M), Reissner-Nordstrom (charged but nonrotating, characterized by M and Q), Kerr (rotating, characterized by M and J) and Kerr-Newman (rotating and charged, characterized by M, J and Q). In the case of Schwarzschild spacetime, one can apply a simple thermodynamic formula [21, 22].

$$d\mathcal{U}I = TdS\tag{47}$$

and identifying U ¼ M to determine the BH temperature TBH as being proportional to its inverse mass,

$$T\_{BH} = \frac{\hbar c^3}{8\pi M \mathbf{k}\_B \mathbf{G}}\tag{48}$$

where we use standard notation. It was Hawking's idea that black holes should lead to a new kind of uncertainty [23], other than the one having a quantum mechanical origin. When matter or radiation falling in towards a black hole crosses its horizon, the information it carries is inevitably lost. This led to two controversies. Firstly, information itself is lost. Secondly, one can consider black hole formation due to the gravitational collapse of matter (or radiation) as the unitary evolution of a pure quantum state. After the formation of the horizon, further evolution has to be regarded in terms of mixed states due to the loss of information. This means the breakdown of quantum mechanical predictability. Both elements of such an information problem, loss of information and breakdown of unitary quantum evolution, were objected to from the very beginning.

Hawking himself [24] formulated the idea of black hole decay. Due to the existence of an event horizon and the conversion of one of the Killing vectors from a temporal to a spatial one, a pair of entangled particles, one of positive and one of negative energy, would be created in the proximity of the horizon. Two scenarios are then possible when one (the one with negative energy) or both of the particles fall behind the horizon. The point is that the particle with negative energy could not 'survive' in our part of the Universe for fundamental reasons, but it could exist within the horizon. This is so because the energy, the t-component of the particle momentum vector within the horizon, takes on a spatial character, so it might then be either positive or negative. Hence, one of the particles, the one with positive energy, departs to infinity, being recorded as Hawking radiation and the other member of the pair, with negative energy, falls behind, 'tunnelling through' [25] the horizon and reducing the BH mass. This is the meaning of BH evaporation. Hawking evaporation is the radiation of a black body of temperature, TBH (8.2).

Therefore, BHs turn out to be evaporating nonequilibrium systems with a decay time

$$t\_{BH} \cong 10^{74} \left(\frac{M}{M\_{\odot}}\right)^3 \tag{49}$$

fifty seven orders of magnitude larger than the age of the Universe for moderate BH masses M. According to the generalized second law of thermodynamics, the entropy during evaporation is an increasing function of time. Indeed, during evaporation, the BH entropy decreases,

$$d\mathcal{S}\_{BH} = -\frac{d\mathcal{U}}{T\_{BH}}\tag{50}$$

yet the entropy of the respective BH radiation is larger by one-third [26].

$$d\mathcal{S}\_r = \frac{4}{3} \frac{d\mathcal{U}}{T\_{BH}}.\tag{51}$$

One may suspect that information lost due to the presence of the horizon may be retrieved due to evaporation, thus restoring this fundamental aspect of quantum mechanical unitary evolution [27–29]. A closer scrutiny shows that this is not so obvious: at the initial stages of the BH decay, both BH and radiation are close to their maximally mixed states, thus no information is retrieved. Although the process of releasing information might be of a non-perturbative character, the information problem (referred to as the information paradox) still remains unsolved. It was indicated that smooth quantum mechanical unitary evolution should lead to the breakdown of the smoothness of the proximity of the event horizon, leading to a 'firewall' [30]. This concept was objected to in more recent papers [31–33]; nevertheless, the information paradox is still far from being removed. It may currently be formulated in many different ways and one of those ways can be expressed as follows:

Hawking radiation consists of particles born as entangled pairs; those recorded far away are then entangled with a diminishing BH. Finally, the BH disappears. What, then, are those particles recorded at distant locations still entangled with [34]?
