The final stages of stellar evolution. Gravitational collapse. The formation of a black hole.

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4.2. The final stages of stellar evolution. Gravitational collapse. Formation of a black hole.
Thus, most of the active life of stars is a quasi-static phase (10¹⁰ years long for ordinary stars, but may be less than 10⁶ years for giant stars), during which fusion nuclear reactions take place and heat and radiation pressures balance the gravitational force. In the previous chapter, "Thermonuclear reactions inside stars", we discussed the whole sequence of thermonuclear reactions - from hydrogen fusion to helium, fusion of helium to carbon, subsequent synthesis of oxygen, magnesium, silicon, ...., to the last iron in sufficiently massive stars. However, each star contains only a finite amount of "nuclear fuel", so time must come, when all the energy-releasing nuclear reactions stop, and the star's active life ends.
After the end of thermonuclear reactions, the stars are definitely seized by gravity, which squeezes the star "as far as possible" - to high densities; the greater the mass of the star, the higher the densities. These stage star evoution and subsequent phenomena are referred to as the final stages of stellar evolution. The ultimate fate of a star is determined by its remaining mass *) at the end of evolution (ie, the initial mass, minus the mass of all matter, particles, and radiation ejected and radiated by the star during its evolution) after depletion of thermonuclear reactions.
*) During their evolution, stars lose a large amount of their mass - they eject it during instabilities in eruptions and in a stable stellar wind, they emit a lot of energy by radiation. The more massive a star is, the more weight it loses over the course of its life. It is estimated that stars born with a mass of 6-8 M_¤ at the end of their evolution will have a residual mass of only about 1-1.5 M_¤, so they will probably end up as a white dwarf. Most stars formed with an initial mass of about 10-15 M_¤ will have a final residual mass of about 1.5-2 M_¤, will explode as supernovae, and their evolution will end at the neutron star stage. Only highly massive stars with an initial mass greater than about 20 M_¤ will have a residual mass >2 M_¤ and will succumb to complete gravitational collapse with the formation of a black hole.
For simplicity, we will first consider a spherical star, around which it will be according to Schwarzschild-Birkhoff's theorem 3.3 the Schwarzschild geometry of outer spacetime (3.13); inside the star, it will be seamlessly followed by the metric of Schwarschild's inner solution. First, we outline the final stages of stellar evolution globally according to Fig.4.2.

White Dwarf
After consuming all the nuclear fuel and extinguishing all the fusion nuclear reactions releasing energy, the star reaches its lowest energy state (if we do not consider gravitational energy). Due to gravitational forces the star is compressed from the original few hundred thousand kilometers in diameter to several thousand kilometers, and the density of the order of thousand kilograms per cm³. The substance of the star is fully ionized and the gravitational forces are balanced mainly by the Fermi pressure of the degenerate electron gas.
Fermi pressure of degenerate gas
Particles with spin 1/2, such as electrons, protons and neutrons, are classified as fermions - their sets are governed by the so-called Fermi-Dirac statistics. The basis here is Pauli's exclusion principle, according to which only one fermion can occupy a single energy state (or at most two particles with oppositely oriented spin). At high densities of matter, all electron energy levels are occupied up to a certain maximum energy, which corresponds to a certain maximum momentum; this condition is called degeneration *), it is a degenerate electron gas. Every other electron in a given volume must occupy a new higher energy level and thus have a higher momentum. The pressure here therefore increases significantly faster than corresponds to the equation of state of an ideal gas. The pressure of the degenerate electron gas is applied in white dwarfs, at even higher densities the degenerate neutron gas is applied in neutron stars. The behavior of a substance at high pressures and densities is discussed in more detail below in the section "Behavior of a substance at high pressures; neutronization".
*) Lat. degeneratus = different from its species, perverted, declining, with loss of diversity.
The way in which the spin of particles determines the statistical behavior of sets of particles is shown in the passage "Indistinguishability of particles" - "Spin, symmetry of the wave function and statistical behavior of particles" in the monograph "Nuclear physics and physics of ionizing radiation".
In the end, therefore, all the matter inside the star is gravitationally compressed into a compact structure with a diameter of only a few thousand kilometers with a very high density and temperature. A star in this state is called a white dwarf. It has a mass roughly like the Sun, but a size similar to the Earth's; the density here is higher than about 10⁴ -10⁶ g/cm³. A large supply of thermal energy is accumulated inside it, coming from earlier thermonuclear reactions and from gravitational contraction, which radiates only very slowly due to the small surface. Therefore, the white dwarf can shine even without the ongoing nuclear reactions, by the remaining heat, for hundreds of billions of years. It is only after this very long time that it gradually cools down; then, after radiating thermal energy, it becomes an infrared and eventually a black dwarf *). The best-known white dwarf is binary star guide of the Sirius, a Sirius B.
*) The Fermi pressure of the degenerate electron gas has a non-thermal origin and acts even after the white dwarf has cooled down - it then maintains the gravitational balance of the black dwarf. However, due to the small surface area and the insulating plasma layer, the white dwarf has a relatively low luminosity (one hundred to a thousand times smaller than the Sun), so its cooling time is in the order of billions of years.
As the white dwarf cools inside, under high pressures, carbon atoms may gradually coalesce into a crystalline form known as diamond. After cooling down, a black dwarf remains, inside which are single diamond crystals, which in certain circumstances may reach perhaps even planetary dimensions (!).
If the white dwarf is part of a close binary with a giant star, the substance may overflow from the giant to the white dwarf. The gradual accumulation of matter on the white dwarf then leads to instabilities and cataclysmic processes. A thicker layer of hydrogen can form on the surface of the white dwarf, in which, due to high temperature and pressure, a thermonuclear reaction of an explosive nature can be ignited, accompanied by a sudden release of energy and a flash of radiation - a nova explosion (discussed in the previous §4.1). This process can be repeated several times. The accumulation of mass on a white dwarf can eventually lead to exceeding the Chandrasekhar stability limit (1.4 M_¤ - see below) and the collapse of the star, which results in a supernova explosion (type Ia - thermonuclear explosion, or collapse into a neutron star - type II and other subtypes ). These events are discussed in more detail below in the section "Supernova explosion. Neutron star. Pulsars." and in the passage "Types of Supernovae and Their Astronomical Classification".

Stability of the white dwarf. Chandrasekhar limit.
As S.Chandrasekhar showed already in 1930, the white dwarf is stable only when its weight is not too great. The limit of stability for a spherical body of mass M and radius R, containing the total number N of fermions of mass m_f, can be roughly determined by the following model consideration: Since the concentration of fermions is r_f~ N /R³, in the context of Pauli's principle the volume occupying per one fermion is ~ 1/r_f = R³/N. According to the quantum uncertainty relation, the momentum of a fermion is ~ h.r_f^1/3. The relativistic energy of fermions is then E_f ~ h.r_f^1/3/c ~ h.c.N^1/3/R. The gravitational energy per fermion is E_g ~ -G.M.m_f/R. The total energy is
                          E = E _f + E _g ~ h.c.N^1/3 / R - G.M.m _f / R   .
Equilibrium configuration is achieved at the minimum value of total energy E. The analysis of the equation for total energy shows that at low mass M the energy E is positive, with increasing radius R decreases to negative values, reaches minimum and at R ® ¥ approaches zero - at a certain final value of R there is a configuration of stable equilibrium between the gravitational force and the Fermi pressure of the degenerate particles. For high masses, the total energy E is negative and as R decreases, the value of E decreases indefinitely - the equilibrium state does not exist and gravitational collapse occurs.
Thus, the maximum mass at which an equilibrium can still occur is given by the condition E = 0 in relation to the total energy, ie h.c.N^1/3 = G.M.m_f .
We can now distinguish two boundary cases of the composition of the star's matter :
¨ 1. If the mass of the star is formed only by those N fermions, which at the same time create Fermi pressure, then the total mass of the star is M = N.m_f . In practice, this situation may occur in degenerate neutron gas, so m_f = m_n, where m_n is the mass of the nucleon (it doesn't matter if we use the mass of the proton or neutron). The solution of the equation E = 0 then for the maximum number of nucleons N_max and for the maximum mass M_max of the degenerate star gives the relation :
                   N_max ~ [h.c/G.m_n²]^3/2 » 2.10⁵⁷ ,   M_max= N_max.m_n ~ [h.c/G ]^3/2.1/m_n² » 1,5M_¤   .
In this basic approximation, not considering numerical corrections depending on the chemical composition of the substance, the maximum mass of a degenerate star is given only by the basic physical constants.
¨ 2. The Fermi pressure is caused by electrons, so m_f = m_e, while the gravitational mass of a star is made up mainly of nucleons (protons and neutrons in the nuclei of the star's mass); so it is with the white dwarf . The total mass of a star is M = N_n .m_n, where N_n is the total number of nucleons, related to the number of electrons N by the relation N_n = N.Z/A, where Z is the proton number and A is the mass (nucleon) number of atoms of a stellar substance. The solution of the equation E = 0 then for the maximum number of nucleons and the maximum mass of the white dwarf gives :
                 N_max ~ [h.c/G.(Z/A).m_n.m_e]^3/2 ,   M_max s M_Ch ~ [h.c/G ]^3/2.(A/Z)^3/2.(1/m_e)^3/2.(1/m_n)^1/2  .
This maximum possible mass of the white dwarf M_Ch is called the Chandrasekhar limit. In addition to the basic physical constants, it also depends on the chemical composition substances of the white dwarf, on the ratio of the number of protons and neutrons.
The above calculations are only model and have the character of rather dimensional estimates. More accurate values of the limit masses of compact stars can be obtained by solving the Oppenheimer-Volkov-Landau equation (4.3) using an appropriate equation of state, eg the Harrison-Wheeler equation of state (see below).
The Chandrasekhar limit for a hypothetical star from hydrogen itself (proton star), ie Z/A = 1, is 2.74 M_¤, for the realistic case Z/A = 0.5 (helium, carbon, .. calcium, ... iron ) is M_Ch = 1.44 M_¤ .
White dwarfs, which occur very abundantly in space, are therefore the final stages in the evolution of lighter starssimilar to our Sun, in whose interior only lighter elements such as carbon and oxygen were created by thermonuclear fusion.

Supernova explosion. Neutron stars. Pulsars.
Exceeding the Chandrasekhar mass limit for the stability of a white dwarf basically occurs in two situations :
1.A white dwarf is part of a close binary star with a standard ordinary star, from whose surface layers it "sucks" gases, thereby increasing its mass, until exceeding the Chandrasekhar limit.
2.The massive star, after the fuel is exhausted and all thermonuclear reactions have ended, has a residual mass exceeding that of Chandrasekhar, so the white dwarf does not even stabilize - the contraction and gravitational collapse of the star's core continues, during the supernova explosion.
So if the mass of the white dwarf greater than the afore mentioned Chandrasekhar limit (which is about 1.4 solar mass M_¤ *), the presure of degenerate electron gass is no longer able to balance such large gravitational forces. There is another contraction - the collapse of the star, during which two diametrically different phenomena can occur, depending on the composition of the white dwarf :
-->Thermonuclear explosion
If the white dwarf is composed of light elements (hydrogen, helium, carbon, oxygen), the increase in pressure and temperature during contraction will trigger a rapid thermonuclear reaction (such as the fusion of carbon and oxygen into nickel) throughout the white dwarf's volume. The large released energy leads to a thermonuclear explosion and the scattering of the white dwarf, which manifests itself as a supernova explosion, according to the astronomical classification of type Ia (the spectral lines of hydrogen are not represented in the radiation spectrum: at the end of its evolution, the star has already consumed hydrogen in its core and blown off the outer hydroen layers in the red giant stage). In the place where the star once was, after the Ia supernova, only an expanding cloud of gases will remain, no compact object will be formed.
-->A neutron star
If the white dwarf does not contain sufficiently large concentrations of light elements, a massive thermonuclear reaction will not ignite and the gravitational contraction will initially continue undisturbed. Soon, the density and temperature increase so much, that high-energy electrons are "pushed" into the nuclei and absorbed by them; they combine there with protons to form neutrons and flying neutrinos: e^- + p⁺ ® n^o + n '_e - so-called inverse beta-decay (see §1.2, section "Radioactivity beta" , passage "Inverse beta decay") - lower part Fig.4.2. As aresult, the electron content in the star decreases and the Fermi pressure therefore decreases. Substance stars is becoming more easily compressible, there is therefore a further contraction, thereby electrons become even faster and easier are absorbed by nuclei. This is considerably unstable situation and the process will continue with avalanche increasing speed. Due to gravity, so there is a sharp shrinkage (a "implode") of a star, in which almost all the protons and electrons combine to neutrons; at this stage, equilibrium can be occur again. This created a neutron star, which has a diameter of only a few tens of kilometers and its density is of the order of density in atomic nuclei ~10¹⁴ g/cm³ (a teaspoon of such a mass would weigh billions of tons!). The gravitational forces are balanced by the Fermi pressure of the degenerate neutron "gas". A neutron star is a kind of gigantic "nucleus" composed mostly of neutrons **) and held together by its own gravity of total matter (the structure of a neutron star is discussed below "Internal structure of neutron stars").
*) Chandrasekhar's limit of 1.4 M_¤ applies to non-rotating (or slowly rotating) white dwarfs. With rapid rotation, this limit can be up to ~3 M_¤ [73].
**)On the part of nuclear physics, the question may arise about the stability of neutrons forming a neutron star. Free neutrons, without strong interaction with protons, are unstable and have a half-life of less than 15 minutes decays by b^- decay into protons, electrons and (anti) neutrinos. This is also common in atomic nuclei with an excess of neutrons (radioactivity b^-), see §1.2, passage "Radioactivity beta" of the book "Nuclear Physics and Physics of Ionizing Radiation".
The force that prevents the massive neutron decay in a neutron star is gravity. Not directly, but indirectly, by inducing such a density and pressure that the Fermi energy of the electrons is higher than the maximum energy of the beta-electron during the decay of the neutron (which is 780keV). In such a case, the electrons created by the decay of neutrons acquire such an energy as their number increases (Pauli's exclusion principle) that they are pushed back into the protons to form neutrons. In the neutron - proton - electron plasma, the formation of a neutron star creates an equilibrium between b^-neutron decay and electron capture of protons, ie between direct and inverse beta decay. Then the electrons contained in the plasma are occupied by all energies (including high energies), so event. electrons from neutron decay "have nowhere to emit" energetically (in terms of phase space) and therefore do not fly out - no further b- decay of neutrons occurs.
Thus, in a neutron star, there is a certain amount of high-energy electrons in the mixture with neutrons, sufficient to prevent the decay of neutrons (and, of course, the same number of protons to maintain electrical neutrality). In the simplest approximation, it can be shown that this ratio will be 1: 8 (see below).

Fig.4.2. A general simplified scheme of the final stages of stellar evolution and gravitational collapse: a white dwarf, a neutron star and a black hole in a cross-sectional space-time diagram (on the horizontal axis, the radial dimension is spatial, on the vertical axis is time). In this way, ie through all three stages, however, the collapse could only take place in very special cases. At masses lower than the respective limit, the collapse actually stops in the stage of a white dwarf or neutron star, at high masses these stages do not stabilize and a black hole is formed.

During the implosion leading to the formation of a neutron star, a large amount of energy is released rapidly - both gravitational energy during collapse and energy during specific nuclear reactions inside. This energy radiates in the form of electromagnetic (in the non-spherical case also gravitational) waves, and is carried away by neutrinos (the largest part!) and the upper layers of the star, which expand rapidly into space to form a glowing nebula: the neutron star is accompanied by a massive explosion supernovae (middle part of Fig.4.2) - according to the astronomical classification of type II. Such a supernova glows with the intensity of hundreds of millions of Suns for several days to weeks.
Note:This scenario outlined above is just one of four of the possible mechanisms of a supernova explosion (these other mechanisms are briefly discussed below "Supernova Types and Their Astronomical Classification") . Although, according to astromic observations, it may be a minority, from our point of view of gravitational physics and the formation of compact gravitationally collapsed objects, we will consider it basic here.
Supernova radiation. Light curve.
In addition to the primary energy of corpuscular particles and photons during the explosion itself, other downstream processes can also contribute to the observed bright radiation of a supernova :
-Shock wave, which is formed when the very rapidly expanding gases from the inner regions of the star "catch up" with the more distant and slower layers and the "stellar wind" that the star emitted before the explosion. The kinetic energy of this collision strongly heats the expanding cloud.
-Radioactivity of elements that were synthesized in the star (and especially formed during the supernova explosion) mostly in the form of radioactive isotopes *). These then gradually decay into other (more stable) isotopes, emitting high-energy radiation (especially electrons b ^- and photons g and X). As a result of the energy released in this way, the expanding cloud glows for some time thermal and fluorescent radiation. In addition to short-term radionuclides, there is undoubtedly a large number of longer- lived isotopes (T1/2 > 10² years), thanks to whose radioactivity the expanding supernova cloud still glows intensely in the X and gamma range for hundreds or thousands of years (but it is still difficult for current detectors to measure and view...). The radioactivity of some long-lived radioisotopes, as is iodine ¹²⁹I, aluminum ²⁶Al and iron ⁶⁰Fe, which may have been used in the formation of the protoplanetary disk, planets and asteroids at the beginning of the solar system. And extremely long - term radionuclides (T1/2 > 10⁹years), especially ⁴⁰K potassium , thorium ²³²Th and uranium ^238,235U, persist for billions of years; moreover with us on earth is still preserved (§1.4 "Radionuclides" passage "Natural radionuclides" in the monograph "Nuclear physics and ionizing radiation") ...
*) When massive nuclei absorb electrons in a supernova explosion, releases huge amounts of neutrons from part of which is absorbed by the cores of light and medium heavy elements. These nuclear reactions produce a large number of radioactive isotopes (eg Al-26, Ni-56, Fe-60, I-129, ...) - see §1.3 "Nuclear reactions", passage "Neutron-induced reactions", monograph "Nuclear physics and ionizing radiation physics".
An important characteristic of a supernova is its light curve - time course of radiation intensity (magnitude) of the supernova. The light curves of supernovae depend on explosion mechanisms and properties of ejected material (its transparency and content). After a very sharp initial increase in brightness, a maximum is reached within a few days, after which the radiation intensity begins to decrease gradually, over tens to hundreds of days. The fastest decrease is observed in type Ia supernovae: after reaching the maximum, the decrease first begins with a half-life of about 6 days, which is attributed to the radioactive decay of nickel⁵⁶Ni beta^- radioactivity with a half-life of T1/2 = 6 days for cobalt ⁵⁶Co. It is also radioactive, beta⁺ -radioactivity and by electron capture is converted by a half-life of 77 days to stable iron ⁵⁶Fe; roughly with this half-life, the slower phase of the supernova brightness decrease continues. A further decrease in the brightness of the supernova is already very slow, in addition to adiabatic expansion, it is involved in the radioactive decay of long-lived radioisotopes (some of which have been mentioned above) .
During the absorption of electrons and the neutronization of matter inside the supernova, huge amounts of neutrinos are emitted, which carry away energy and effectively cool the collapsing central portion of the star's burnt matter. Due to their weak interaction, neutrinos practically do not create pressure and fly into the surrounding universe without resistance. Effective cooling by neutrino radiation aids in the rapid gravitational collapse of the central part of the supernova, which can stop even the Fermi pressure of the degenerate neutron "gas" (see below "Behavior of the substance at high pressures; neutronization") .
After radiating enormous energy for several months, its core remains in place of its original star, collapsed into a compact structure with only a few kilometers in diameter and an unimaginable density of the order of 10¹⁴ g/cm³ (nuclear density), composed mainly of neutrons - neutron star. Further shrinkage is then prevented by the Fermi pressure of the degenerate neutron gas, caused by Pauli 's fermion exclusion principle.
Note:However, a very massive star can overcome Pauli's exclusion principle by its gravitational shrinkage. The neutrons approach each other at such a small distance that the asymptotic freedom of strong interaction between quarks is exerted. The neutron substance "melts" into a mixture of free quarks and gluons - the quark-gluon plasma - see "Internal structure of neutron stars" below. And with an even heavier weight, no more force can stop the gravitational shrinkage, complete gravitational collapse occurs (described in more detail below in the section "Complete gravitational collapse. Black hole.").
The site of the supernova's explosion is surrounded by a rapidly expanding nebula from the ejected outer parts of the star. Very well known is the Crab Nebula, which is the remnant of a supernova explosion observed in 1054 by Chinese astronomers :


Supernova explosion observed in 1054 in China. Today, a Crab Nebula containing a pulsar inside - a rapidly rotating neutron star - is observed at that place.

Destruction of the planetary system
If the original star had a planetary system, it will probably be destroyed in a supernova explosion. The nearby inner planets usually evaporate with a huge flow of energy, and their gas is blown into interstellar space by the pressure of radiation. The orbits of more distant planets will become unstable due to the loss of much of the star's mass during a supernova explosion. However, protoplanets can condense again from the gas disk around the neutron star, and later new (exo)planets perhaps may also form..?..
Supernovae - space killers and also creators of new life-giving worlds !
A supernova explosion is the biggest disaster what can hit heavy stars. Not only is its own star destroyed, but it is scattered and evaporated possible orbiting planets, even nearby partner stars (in a binary or multiple system) can be ejected. From the point of view of nuclear physics, it can be said that the supernova explosion is, among other things, the biggest radiation accident in the universe! If one of the surrounding stars (tens of light-years away) exploded like a supernova, our Earth would be hit by such intense radiation that it would exceed the lethal radiation dose to humans more than 100 times (see §5.6, passage "Astrophysics and cosmology: - human hopelessness?")..!.. A supernova explosion "sterilizes" the vast surroundings of many light-years in space, where life will not be possible for a long time.
On the other hand, these destructive and murderous supernovae "positively" contribute to the further evolution of matter in the universe. The supernova-ejected cloud contains a large number of heavy elements - including biogenic ones - which will enrich the surrounding interstellar matter. The explosion also creates a massive shock wave that can compress gas clouds in the surrounding space. This can initiate the gravitational contraction of these gas-dust clouds, which can result in the formation of new young stars, enriched with heavier elements. In planetary systems around such stars, life can then develop in principle (discussed in more detail in the section "Origin and Evolution of Life") of the work "Anthropic Principle or Cosmic God"). Without a cataclysmic supernova explosions, we probably wouldn't be here..!.. General importance of supernovae for the universe is discussed below in the passage of "Astrophysical importance of supernovae".
Types of supernovae and their astronomical classification
In astronomy, supernovae are extremely bright stellar objects of an explosive nature that suddenly appear in the sky and whose brightness then decreases again by many orders of magnitude over the course of weeks to months. The name is derived from Latin. words nova, ie new, because visually it seemed that a "new star" was born (stella nova). But now we know that this is really the opposite - a manifestation of the demise of the old star, which has reached the final stages of its life and irreversibly transforms into an object substantially different from ordinary stars. In the previous §4.1, in the passage on the basic evolution of stars, the instabilities, pulsations, and "smaller" explosions of a star were mentioned, which lead to a sudden brightening of a weaker star; this phenomenon is astronomically observed as a nova - the original star is usually not visible in smaller telescopes, it looks like the birth of a "new star". The name super nova expresses that this is a much more grandiose cosmic phenomenon.
The oldest surviving records of supernova sightings come from 1006 in Egypt and Mesopotamia, the most famous is the above-mentioned supernova from 1054 observed in China (gave rise to the Crab Nebula), then from 1181 in China and Japan. Significant was the supernova in 1522, which Tycho Brahe observed and called it the "nova stella", and the supernova from 1604, which was observed by J.Keppler and which Galilei mentioned as an argument against the then dogma of the immutability of heaven, traditionally from the Aristotle period *). The current observation of the SN1987A supernova in the Large Magellanic Cloud, is very significant, including the capture of neutrinos in the SuperKamiokaNDE device (see the "Neutrinos" passage in §1.2 "Radioactivity" of the monograph "Nuclear Physics and Ionizing Radiation Physics"), allowing to be tested current theories of formation and dynamics of supernova explosions.
*) The supernova explosion is an astronomically very short, fast and transient event that is not easy to observe and analyze in detail. The above-mentioned historical observations of supernovae took place visually either directly or on small simple telescopes, without the possibility of quantification and spectral analysis. The observation data are therefore only very rough and incomplete. Even current observations are often not enough to capture the initial period of an explosion, before reaching maximum brightness.
However, astronomy offers a certain possibility of "retrospective" observation of supernova explosion. The explosion, accompanied by extremely intense radiation, takes place not only towards us, but also in the opposite direction, to more distant areas. In these more distant regions, the radiation may strike a cloud of interstellar dust, which will then emit secondary (much weaker) radiation of the same time course and spectrum as the primary radiation of the supernova. In principle, this radiation can then be analyzed additionally and later, according to the distance of the respective cloud. Another promising possibility is the observation of images of a supernova split by a gravitational lens (§4.3, passage "Gravitational lenses in space"), coming with different time shifts.
Astronomical naming of supernovae consists of the abbreviation "SN", the year of discovery and finally possibly letters of the alphabet denoting the order of multiple supernovae discovered in the same year - eg the afore mentioned SN1054, or the current SN1987A. Now, with the help of large telescopes, several supernovae in more distant galaxies are observed every year.
How often do supernovae explode? - continuous fireworks lasting billions of years !
Until recently, supernovae were considered a rare event, observed in the deep stillness of the night sky about once a century. However, it is only an optical illusion caused by three circumstances :
× The enormous vastness of the universe, in which the stars are very sparsely distributed; the vast majority of stars are extremely far from us.
× Long lifespan of stars - millions and billions of years.
× Very short duration of a supernova explosion - hours, days, months.
The vast majority of supernova explosions take place very far from us (billions of light years). So even though, at the peak of their activity, supernovae can shine brighter than a billion of our Suns, we can't either see them at all, or they appear to us as tiny light powders that we can easily overlook in the vast starry sky. Only those supernovae that explode in our Galaxy can be observed directly through the eyes (that is, at intervals of hundreds of years); in the surrounding galaxies by astronomical telescopes (several supernovae per year). Extrapolation of astronomical observations *), as well as astrophysical analysis of the life of massive stars - in relation to their number in the universe - leads to an estimate that roughly every few seconds (and perhaps an average of once per second) somewhere in our observable universe explodes some supernova. It is a bit of an exaggeration to say, that this amount of supernovae is a continuous, billion years running "cosmic firework show" - but extremely diluted in the vast space of universe...
*) Special telescopes equipped with robotic systems for searching large bands of the sky are being prepared any new resources that appear. This will allow the registration of many distant supernovae that would escape the attention of large telescopes with a narrow field of view.
Four different mechanisms of a supernova explosion
Contemporary nuclear astrophysics presents four possible, diametrically opposed, supernova explosion scenarios :
1.The classical Chandrasekhar scenario explained in more detail above: after crossing the limit of instability, electrons are absorbed by atomic nuclei to form neutrons during the rapid shrinkage of the star's interior. The decreasing Fermi pressure of the disappearing electrons leads to the implosion of the star, creating a neutron star.
2.Thermonuclear explosion of the star: by an enormous increase in pressure and temperature during shrinkage, a rapid thermonuclear reaction of carbon and oxygen in the whole volume of the shrunken star occurs, while the released energy leads to "scattering" of the star. This manifests itself as a supernova explosion. No neutron star is present inside the rapidly expanding cloud formed by this mechanism.
3.Thermonuclear explosion of a star due to e^- e⁺-pair instabilities (the process is discussed in more detail in §4.1, passage "Formation of electron-positron pairs").
4.Gravitational collapse of the central part of the star into a black hole .
The various mechanisms of supernova formation will be briefly discussed below in connection with the astronomical distribution of supernovae :
The astronomical classification of supernovae
originated at a time when the events taking place there were not yet known, so it has no clear and logical connection with the mechanisms of supernova formation. Supernovae are astronomically classified according to the presence of spectral lines of various elements in the spectrum of their radiation and according to the shape of the light curve (time curves of supernova magnitude, especially the dynamics of brightness decrease). If the spectrum of a supernova does not contain hydrogen lines, it is classified as type I; if it contains Balmer lines of hydrogen, it is referred to as a type II supernova. Each of these two categories is further subdivided into subgroups according to the presence of other spectral lines or the shape of the light curve. Type Ia supernovae do not contain helium lines in their spectrum either, but a silicon absorption line ( Si II at 615 nm) is present, especially in the region of the brightness peak. Type Ib supernovae contain a helium line (He I at 587.6 nm). Type II supernovae are divided into type II-P with flat light curve and type II-L with linear decrease of the light curve.
From an astrophysical point of view, this classification is essentially irrelevant. A more accurate division is into two types: "thermonuclear supernovae" for type Ia and "core collapse supernovae" for types Ib, Ic and II, which distinguishes the internal mechanism, not what they look like from a distant observation... It was distinguished right in the opening passage of this of "Supernova explosion. Neutron star. Pulsars." .

Supernova formation processes
In terms of the dynamics of formation and mechanism, we can divide supernovae into three (or 4) basic types :

Astrophysical importance of supernovae
Supernova explosion, which belongs to the most massive and the most dramatic phenomena which universe we observe has several important astrophysical implications :
× Chemical evolution of the universe
Above all, supernovae contribute to the chemical evolution of the universe - thrown substance enriches the surrounding outer space of the heavier elements that were synthesized inside the star in thermonuclear reactions (as discussed in more detail in §4.1, section "Thermonuclear reactions inside the star"). These ejected heavier elements in space may become part of future generations of stars and planets. Figuratively speaking, in cosmic nucleogenesis, stellar generations "pass the baton" of elements created during the demise of the old generation to the formation of other heavier elements by the stars of the new generation. Thanks to this nucleosynthesis, there is also living nature and we humans - without carbon from inside the stars and the explosion of supernovae, we could not exist as a carbon form of life! We are all made of stellar ash ...
Type Ia supernovae, which arise from white dwarfs with a mass of 1.4 M_¤, enrich the surrounding space mainly with carbon, oxygen and other lighter elements (in less massive stars, the end product of thermonuclear fusion is mainly carbon and oxygen). Type II supernovae, formed from massive stars with heavier elements synthesized down to iron, enrich the surrounding universe with these heavy elements - and even more heavily: this is because a supernova explosion releases a large amount of neutrons, which are effectively trapped in the nuclei of the expanding layers within a few seconds, where heavy neutron-rich nuclei are formed. Their repeated b^- transformations create heavy and very heavy nuclei (including uranium and transuranium) in the expanding envelope. This rapid capture of neutrons, called the r-process, has created about half of all elements heavier than iron in the universe (see syllabus "Cosmic alchemy" or passage "Cosmic alchemy - we are the descendants of the stars!" in monograph "Nuclear Physics, ionizing radiation"). The other half of them is formed with s- process - slow neutron fusion inside heavy stars in the late stages of this thermonuclear development (as shown in §4.1, 'The thermonuclear reaction in stars" passage "Neutron capture and generate heavy elements").
Particularly efficient nucleogenesis can be expected in the explosion of rapidly rotating supernovae with a strong magnetic field, where a large amount of neutron-rich plasma is ejected and intense neutron r-capture occurs. This significantly enriches the ejected substance with heavy nuclei up to uranium. Due to magneto-hydrodynamic processes, this enriched substance is ejected into the surrounding space at high speed in the narrow cones of the poles of the axis of rotation.
× Cosmic radiation
Huge number of energetic particles and radiation emitted in the supernova take off into the space and is probably important source of cosmic rays propagating through space - see the discussion "Cosmic radiation" in the above monograph; in §4.8 "Astrophysical significance of black holes" we will see that another possible source of cosmic radiation can be massive jets from the interior of rotating accretion disks around massive black holes.
× Stimulation of star formation
The shock wave created by a supernova explosion can stimulate the gravitational contraction of gas dust clouds in the surrounding interstellar mass and thus the formation of other stars.
Neutron stars - permanent destruction and trapping of heavy elements ?
As discussed above, the supernova explosion "liberates" the heavier elements thermonuclear synthesized by the star and ejects them into the surrounding universe. If a star's core, which contains most of the nuclei of heavy elements from stellar nucleosynthesis, collapses into a neutron star during a supernova explosion, these nuclei are destroyed and their neutronized "remains" are forever imprisoned by massive gravity inside the neutron star - so for cosmic nucleogenesis are "lost". However, this only applies to solitary neutron stars. If neutron stars are part of a binary (or multiple) system, then during their mutual circulation, gravitational waves are generated, which gradually carry away the kinetic energy of the orbital motion, whereby the orbiting bodies gradually approach each other until they finally merge. During this fusion of neutron stars, a large amount of neutron matter can be ejected, which immediately explodes (rapid decompression from nuclear density) and "nucleonizes" to form the nuclei of heavy elements.
In §4.8, the passage "Collisions and fusions of neutron stars", is discussed this possibility of the formation of heavy elements in collisions and fusions of "already finished" neutron stars (after many millions of years) in the binary systems of two neutron stars (or neutron star + black hole) by the mechanism of "nucleonization" of the ejected neutron substance. This is an "additional" formation of heavier elements from matter that would otherwise be lost to the chemical evolution of the universe..!..
When can we expect an observable supernova explosion ?
In the above discussion, "How Often Do Supernovae Explode? - A Continuous Billion-Year Space Fireworks!" we put a relatively high astronomical observation frequency of supernova explosions in the observable universe. The vast majority of these events, however, occur in billions of light years distant universe. Relatively early supernova explosion, the time horizon of a million years, expected for a very massive stars observed in the phase of the red giant *). Such "old" stars at the end of their lives already in his heart burn all hydrogen, the envelope strongly "inflated" and cooled, and in the shrinking cores occurs thermonuclear "burning" of helium and other heavier elements (as was explained in more detail in the previous §4.1, section "Thermonuclear reactions inside stars"). This less energy-intensive "fuel" is only enough for a few million years. Once everything burns, the star will rapidly collapse into a neutron star or black hole, with a huge explosion of a type II supernova.
*)One such "endangered" relatively close massive star is Betelgeuse in the constellation Orion, about 600 light-years from Earth. It is a red giant the size of about 1000 solar radii, the luminosity is about 100,000 times greater than our Sun, the mass is about 15-20 M_¤. The spectral class of radiation M1-2 IAB shows that the star is already in a very advanced stage of its development. This period of the red giant is highly unstable, it is a precursor to extinction by a supernova explosion; this is also indicated by the observed variability of the star (with a semi-regular period of about 6 years). The explosion of Betelgeuse as a type II-P supernova can be expected in about 1 million years! When this happens, this supernova will be the brightest object in the night sky (perhaps brighter than the Moon) and for 2-3 months it will be seen as a bright shining point even in the daytime sky. Fortunately, the Betelgeuse's axis of rotation is not rotated toward Earth, so we are unlikely to be threatened by an intense flash of ionizing radiation (cf. the following paragraph "Danger from supernovae") .
Furthermore, a supernova explosion - type Ia - can occur in one of the many binary systems of a white dwarf and a normal (or giant) star by mass overflow, the mechanism described above (in the section "Types of supernovae and their classification").
Danger from supernovae
A supernova explosion releases such a huge amount of radiant energy, that if one of the nearby or "neighboring" stars in our Galaxy explodes like a supernova, intense ionizing radiation could seriously endanger the existence of life here on Earth! - issues of threat to life by cosmic radiation are discussed at the end of the already mentioned treatise "Cosmic radiation", passage "Biological significance of cosmic radiation".

Strong magnetic field of neutron stars
Thanks to the law of conservation of the star's rotational angular momentum during shrinkage, white dwarfs, and especially neutron stars, will rotate very quickly - one to several hundred revolutions per second (possible explanation of such high speeds see below "Pulsars"). Neutron stars can also have a very strong magnetic field. As already mentioned in the previous §4.1, passage "Compact objects", due to the shrinkage of the star in the final stages, the magnetic field lines of the original field are compressed and the intensity (induction) of the magnetic field near this object increases sharply. Even a relatively weak magnetic field of a normal star, which is of the order of B » 10^-4T, so due to the "compaction of the lines of force" (assuming that 4p.R² B » const.) during contraction, it will increase at the surface to a huge value of B » 10⁸ Tesla and higher.
Magnetar
If a neutron star rotates at high speed (frequency), the intensity (induction) of the magnetic field of the neutron star can reach extreme values in some cases with the magnetohydrodynamic effect » 10¹⁰ - 10¹² Tesla. A neutron star with such an extremely strong magnetic field is called a magnetar. The rotating strongly magnetized neutron star acts as a massive "alternator" that converts some of the mechanical rotational energy into the energy of a variable electromagnetic field. This weakens the magnetic field and the magnetar gradually becomes a normal neutron star. Mechanical changes or defects in the crust of such a neutron star (similar to an earthquake - "star quake") can lead to a sudden rearrangement of magnetic field lines ("magneto shake"), which induces massive magnetohydrodynamic currents in the surrounding plasma leading to strong heating and energy release - this is accompanied by short , but with a very intense flash of electromagnetic radiation, radio waves including X and g- radiation .

P u l s a r s - fast-rotating neutron stars
In 1968, at the Radio Astronomical Observatory in Cambridge, experts led by A.Hewish (pulses were measured mainly by J.Bell) when examining the radio signals from space, registered very regular pulses, coming with nanosecond accuracy. Their source objects were called pulsars (short for "pulsating radio source", or "source emitting radio pulses"). It seemed so strange that some astronomers even initially thought it might be the signals of other civilizations that were often discussed at the time. Eventually, however, Hevish and other astronomers tended to believe that they were very fast-rotating compact stars -neutron stars. The periods of most pulsars range from 0.03 sec. *) to 4 seconds. No normal star is able to rotate so fast without being torn apart by centrifugal forces. It must be a highly compact object with a high mass, the inertia of which ensures such a high stability of the rotational frequency, resistant to environmental influences; no other mechanism (perhaps pulsation) could do this. And only a neutron star, thanks to strong gravity, "can withstand" very fast rotation (up to about 600 rpm), without tearing by centrifugal forces.
*) The formation of such a rapid rotation is difficult to explain by the rotational angular momentum of the original star. A possible mechanism for the "additional spin" of a neutron star could be accretiongas flowing, for example, from a co-rotating companion in a binary system. This gas would form an accretion disk around the neutron star and, when absorbed by the neutron star, would impart additional angular momentum (cf. the analogous mechanism of "spinning" a black hole by an accretion disk discussed in §4.8, section "Accretion disks around black holes"). Lone neutron stars rotate more slowly, with longer periods of the order of tenths to units of seconds.

Fig.4.3. Pulsar like a fast rotating neutron star.
a) Global beacon model of a neutron star as an inclined magnetic rotator.
b) Formation of synchrotron radiation during the motion of a relativistic electron in a magnetic field. The electron acts as a glowing "reflector" orbiting in a spiral path.
c) A somewhat more detailed model of the pulsar shows that the radiation does not arise at the surface of a neutron star, but in its magnetosphere at the interface between a stationary plasma and a rotating plasma entrained by a neutron star.

As fast-rotating neutron stars, pulsars therefore, they are now being considered. The mechanism of why we observe very regular rapid flashes of radiation in pulsars is not yet known in all details. The so-called the beacon model (Fig.4.3) explains the pulsar as a neutron star with a strong "frozen" magnetic field, that rotates around an axis making a small angle with the axis of the magnetic field. The interaction of a rapidly rotating magnetic field with electrically charged particles in the plasma surrounding a neutron star accelerates electrons to relativistic velocities. These accelerated electrons moving in a strong magnetic field are then a source of strong synchrotron radiation *) emitted anisotropically in a narrow cone in the direction of the magnetic axis. The electromagnetic radiation emitted in this way then "hits" the distant observer at regular intervals (equal to the period of rotation of the neutron star), similar to the cone of light of a rotating beacon reflector. However, the acceleration of charged particles is at the expense of the rotational energy of the neutron star, which contributes to the period of the pulsar slowly lengthening (see below the passage "Smooth and sudden changes in the rotational speed of pulsars").
*) The radiation arising from the movement of a relativistic charged particle along a curved path in a magnetic field is called synchrotron because it was first observed in 1947 at a 70 MeV synchrotron.The mechanism of synchrotron radiation is outlined in §1.6, passage "Cyclotron and synchrotron radiation" of the book "Nuclear physics and physics of ionizing radiation".
Neutron stars are also likely to be very intense sources of long-wave electromagnetic radiationmagneto-dipole character, generated at a frequency given by rotation. From an electrical point of view, a rotating magnetized neutron star acts as a massive "alternator" that converts some of the mechanical rotational energy into energy of a variable electromagnetic field carried away by long-wave electromagnetic waves. Here on Earth, however, this radiation cannot be detected due to the opacity of the interstellar plasma; although this plasma is very sparse, due to the large distance of the source, long-wave electromagnetic radiation is practically completely absorbed.
Due to the very weak light emission of neutron stars, they are not optically observable, we can only register their radiation as radio pulsars. "Lone" neutron stars can only be observed in this way if they are relatively young. The intensity of this pulsar radiation gradually decreases (as the ionized plasma disappears around the neutron star), so older neutron stars are difficult to observe. However, if neutron stars are part of tight binary stars with gas overflow, then when this substance falls on a neutron star, a large amount of gravitational energy is released, which is converted into thermal motion of particles - the gas is heated to temperatures of millions of degrees, so it shines in the X-ray spectrum.

Internal structure of neutron stars
It is in principle impossible for any observer composed of known forms of matter to directly examine (visually or experimentally) the interior of neutron star. No electromagnetic signal gets out from inside the neutron star, no neutron sample can be taken (neutrinos and gravitational waves that could theoretically get out of the neutron star, due to the absence of nuclear reactions and axially symmetric rotation in the neutron star do not arise). Therefore, we can try to reconstruct the internal structure of neutron stars only theoretically, on the basis of an analysis of the properties of the substance of which neutron stars are composed - its equations of state, gravitational, nuclear, mechanical and hydrodynamic behavior. Based on current knowledge in these areas (see also below "The behavior of materials under high pressures; neutronization") was created by the current model of the internal structure of neutron stars :
Around the neutron star is probably only a very thin gaseous atmosphere of dense hydrogen and helium, the thickness of only a few meters . The surface of a neutron star is formed by a rigid outer crust (several hundred meters thick) composed of a crystal lattice of iron nuclei and heavier nuclei, along with electron gas. The density here ranges from »10⁶g/cm³ in the upper layers to »10¹¹g/cm³ in the lower layers, while in the depths there are nuclei with an increasing proportion of neutrons (it is more energetically advantageous to combine electrons with protons in the nuclei to form neutrons).
When, in depth, in the inner crust, the density exceeds about 10¹¹g/cm³, neutrons are released from the nuclei and form a neutron liquid penetrating the nuclear crystal lattice (Area 4 in the section "Behavior of a substance at high pressures; neutronisation"). With the increasing density in the depth increases the proportion of free neutrons, when it would exceed the density of » 2.10¹¹g/ cm³ nuclei completely dissolve and the substance is in the form of neutron liquid in admixture with about 10% of protons and electrons (region 5 in "Behavior of the substance at high pressures; neutronization"). This area is called the outer core of neutron stars. Free neutrons inside a neutron star are due to extremely high pressure in a highly degenerate state, where nuclear forces can cause neutron pairings of opposite spins, analogous to Cooper's electron pairs in superconductivity; these paired neutrons can form a so-called Bose-Einstein condensate (§1.5 "Elementary particles and accelerators", passage "Fermions as bosons; Superconductivity" in the book "Nuclear Physics and Physics of Ionizing Radiation"), causing a partial superfluidity of the neutron fluid. This effect can also cause better compressibility than the Fermi pressure of degenerate neutron gas.
The innermost region, the inner nucleus, of mass neutron stars can reach a density 2 to 10 times higher than the nuclear density. Here, specific properties of interactions of elementary particles at high energies with the participation of quarks can come into play :
Quark matter, quark-gluon plasma ?
It is possible that a hyperon or quark mass could form inside a massive neutron star. Fast-rotating neutron stars lose some of their energy and rotational angular momentum from the emission of radio waves, electrons and other charged particles from their magnetosphere. This shrinks the star and increases the pressure inside it, which can lead to the merging of nucleons into hyperons, or even the destruction and decay of baryons into quark mass - quark-gluon plasma (see §1.5 "Elementary particles", section "Quark structure hadrons" in the book "Nuclear physics, ionizing radiation"). Under normal circumstances, quark-gluon plasma is highly unstable, for a brief moment, about 10^-22 sec. experiencing its hadronisation, conversion into baryons and mesons. However, extremely high pressures act in the neutron star, so that the hadrons are pressed so close to each other that they "intertwine" with each other's quark structure, lose their "identity" and "dissolve" into a mixture of almost free quarks and gluons. Hadronisation no longer occur, there is "no place" for hadrons. Extreme gravitational pressure can thus stabilize the quark-gluon plasma inside the neutron star. Here, too, quarks, pushed very close to each other by massive gravity, could form Cooper's pairs, behaving like bosons, and form superfluid condensate..?.. It could also lead to better compressibility than the Fermi pressure allows; massive neutron stars could thus shrink to a smaller size than would be expected for a composition of neutrons alone.
These possible processes in neutron stars can therefore be summarized in a remarkable statement from the point of view of (sub)nuclear astrophysics :

The neutron star is the only object in the universe, that can produce and "tame" an otherwise highly unstable quark-gluon plasma in large quantities, stabilize it inside with powerful gravitational forces and hold it for billions of years !

Strange quark mass ?
A hypothesized was made (E.Witten, 1984) that if a sufficient number of "strange" s -quarks (in addition to the usual quarks u and d forming nucleons) are present in a quark-gluon plasma , it can prevent hadronization and such "strange quark mass" may be stable, even under normal conditions; strong interaction holds it together. In a situation where the quarks are "pushed" close to each other and all the lower fermion quantum states are occupied, the quarks s are practically unable to transform into u quarks, because there is no more free space for the new u quarks thus formed .
Opposite transformations can occur, so that an equilibrium configuration of the quarks u, d, s in the fermion gas is established, which is more energetically advantageous than hadronization. The resulting formation could then be stable, held together by a strong interaction. The strange quark mass is able to absorb neutrons, decompose them into quarks and form another strange quark mass.
A suitable quark mass has very unusual properties. These unusual properties would also have hypothetical "strange stars" composed of strange quark matter. In particular, the stability of such a star would not be determined by the above gravitational criteria. Unlike a neutron star, a strange star has no minimum mass, it is stable for any small mass: not held together by gravity, but by strong interaction. The maximum mass here is » 2M_¤, at higher mass it would collapse into a black hole similar to a neutron star. The radial course of the density of a strange star is quite different from that of a neutron star: the density of the strange quark mass changes very little from center to edge (from the outside, the density on the surface changes almost abruptly from zero to ~ 10¹⁴g/cm³; however, a quark star may be surrounded by a thin "crust" of normal material such as electrons).
The question is, how could such strange quark stars form? Conventional baryon mass does not contain any strange s-quarks, randomly formed strange particles (K-mesons, hyperons) containing s-quarks are highly unstable and decay rapidly. It has been hypothesized that small macroscopic islands of strange quark matter could form during the high-energy processes of a supernova explosion, or could persist in space from the hadron era just after the Big Bang. Ordinary proton-containing matter hardly interacts with this strange quark mass due to its repulsive electrical force. However, the neutron can be absorbed by this quark mass and decomposed into quarks. So if a macroscopic "piece" of strange quark matter enters a neutron star, it will absorb neutrons rapidly, thereby it growing and absorbing neutrons even more efficiently. Thus, such a strange quark mass can initiate an avalanche-like process of transforming a neutron star into a strange quark star , in which a huge amount of energy is suddenly released. This energy does not destroy a strongly bound quark star, but it probably radiates in the form of a massive flash g .
There is no experimental evidence for such an exotic state of "strange quark matter", as well as observational indications for "strange quark stars".
So the strange quark mass probably isn't ...

Pulsars - precision "clockworks" in space
The neutron star, as a very massive compact structure, rotates at a constant speed for a long time, so that the pulsar emits very regular pulses of electromagnetic radiation. As if "accurate watches were ticking" there. The registration of these regular electromagnetic pulses of pulsars and their small changes can be used to analyze some subtle astrophysical phenomena in outer space :
- Theaccretion of matter to a neutron star changes its rotational angular momentum, which also changes the frequency of pulses. Accretion disks around neutron stars are mostly co-rotating, so accretion increases the frequency (for neutron stars without accretion disks, the rotation slows down due to the entrainment of the angular momentum of the "stellar wind" particle emissions and the acceleration of the charged pulsar particles) .
-The circulation dynamics of compact binary systems can be accurately analyzed on the basis of periodic changes in the pulse frequency of the pulsar, caused by the Doppler effect during its orbital motion. A typical example is the binary pulsar PSR 1913 + 16, for which the decrease of orbital energy by gravitational wave radiation has been measured (it is described in more detail in §2.7, section "Detection of gravitational waves", passage "Binary pulsar) .
- Detection of gravitational waves using pulsars.When regular electromagnetic pulses from pulsars pass through space containing low-frequency gravitational waves, there is a certain (albeit very weak) effect on their propagation - there may be a long-period modulation of short-period electromagnetic signals from pulsars due to gravitational waves (see again § 2.7., part "Detection of gravitational waves", pasage "Time modulaion of the period of signals from pulsars"). Microsecond pulsars are particularly suitable for this purpose, in which the effects of "star-shaking" and accretion, which may affect the period of the pulsars, are less pronounced. This phenomenon will hopefully be used in principle in the future to detect long-period gravitational waves in space - systems of a larger number of radio telescopes are being built for this detection using the Pulsar timing array method.

Smooth and sudden changes in the rotational speed of pulsars Although the neutron star rotates at a highly constant frequency due to the law of conservation of angular momentum, there are very small changes in rotational speed, which are twofold :
¨ Smooth changes in rotational frequency - very slow long-term deceleration due to three phenomena :
a) Emissions of "stellar wind" particles from the surface of a neutron star slowly carry away the rotational angular momentum.
b) Electromagnetic dipole radiation of a neutron star (mentioned in the previous passage).
c) The acceleration of charged particles in the pulsar magnetosphere is at the expense of the rotational energy of a neutron star.
¨Sudden step changes of rotational frequency - small short-term irregularities (disturbances - sudden shortening) in the period of pulses with relative amplitude dT/T » 10^-10 - 10^-5. Even though the cause is unknown with complete certainty, assume the two possible mechanisms related to the effects caused by the internal structure of neutron stars (as outlined above in the passage "Internal structure of neutron stars") :
1. Disorders and fractures the crystalline crust neutron stars (a "starquake") during the gradual reduction of its flattening during decelerating rotation. The neutron star has a high rotational speed after its formation and is flattened due to centrifugal force. During the gradual deceleration of the rotational speed this leads to a gradual increase in stress in the crust, after exceeding the "limit of strength" the crust bursts ("star shake") and takes on a less eccentric shape. This suddenly reduces the moment of inertia of the neutron star and, according to the law of conservation of angular momentum, suddenly increases the rotational frequency, or shortens the period T.
2. Changes in neutron fluid flow - turbulence and vortices in neutron "fluid" flow (which is possibly partiallysuperfluid) with a gradual slowing down of the rotation of a neutron star. The fluid inside a neutron star probably rotates somewhat faster than the crust, which is inhibited by electromagnetic radiation and particle emissions ("stellar wind"). When larger differences in the rotational speed of the crust and the neutron fluid are achieved, turbulences in the flow and vortices can occur at their contact, which can transfer part of the higher rotational energy from the inside to the crust and accelerate its rotation. Furthermore, the shear stress at a larger difference between the rotational speed of the inner fluid and the outer crust can lead to deformation and cracking of the crust, similar to point 1.
These step changes are accompanied by strong electromagnetic emission - a flash of radio waves, for magnetars it can also be X and gamma radiation. After a sudden abrupt acceleration of the rotational speed again results in a slow frequency "relaxation" to the original rotational speed; the relaxation time is several tens or hundreds of days. This is followed by the usual long-term slowing down of the neutron star's rotation. During the long life of a neutron star, accompanied by a gradual deceleration of rotation, many sudden abrupt changes in rotational frequency are likely to occur, which will be repeated with increasing intervals of occurrence.

Neutron star stability. Oppenheimer-Landau limit.
In the section on white dwarfs, it was shown above that the Fermi pressure of a degenerate electron gas has a limited ability to balance (self) gravitational forces - there is a Chandrasekhar limit for the mass of a white dwarf. Even the Fermi pressure of the degenerate neutron "gas" is not unlimited. Analogous considerations as outlined above for white dwarfs can be applied to neutron stars, with that this is the case 1. of the passage "Stability of the white dwarf. Chandrasekhar's limit.". The corresponding maximum possible mass of the degenerate neutron configuration, allowing further stability, is called the Oppenheimer-Landau limit. The above dimensional estimates resulted in a value of »1.5 M_¤, more accurate calculations based on the solution of the Oppenheimer-Volkov-Landau equation (4.3) using the "Harrison-Wheeler equation of state" (see below, Fig.4.5), give higher values, around 2-3 M_¤.

Complete gravitational collapse. Black hole.
Thus, like the white dwarf, even the neutron star has a limited mass from above. At too large masses, greater than about two masses of our Sun *) - Oppenheimer-Landau limit, the gravitational forces are already so great that they overcome both Fermi and nuclear forces between neutrons (nuclear forces have a short range - state of saturation); the star's substance no longer has any sources or mechanisms of sufficiently large internal repulsive forces to be able to balance such strong gravity (it is discussed in more detail below in the section "Behavior of matter at high pressures"). In this situation, the catastrophic gravitational collapse continues, Fig.4.2 above (we will not consider here the possible stages of the hyperon or even quark stars mentioned above) until the star falls below its gravitational (Schwarzschild's) radius r_g = 2G.M/c² (see §3.4), crosses the horizon and a black hole, also called a collapsar, is formed. The properties of black holes will be discussed in more detail in the remaining paragraphs of this chapter (§4.3 - 4.9). Here we will only outline some characteristics of the gravitational collapse and the formation of a black hole.
*) Here, too, rotation matters and, moreover, uncertainties in the theory of nuclear substance. The mass of a neutron star should probably be limited by [227] M_n» (5M_¤).(r_nuc/r')^1/2 , where r_nuc » 2 . 10¹⁴ g/cm³ is the ordinary nuclear density and r' » (0.5 - 5).r_nuc is the density at which there are significant deviations from the current theory of nuclear matter on a larger scale. The maximum mass of neutron stars is most often estimated in the range of 1.5 - 2.5 M_¤.
The direct formation of a black hole without a supernova explosion ?
When collapsing very massive stars, there is a theoretical possibility that after depletion of nuclear fuel, the interior star will reach the gravitational radius (horizon) before a supernova explodes. The formation of a black hole would then be "silent and inconspicuous" - the star simply "disappears", without the accompaniment of a more pronounced light phenomenon (it was mentioned above in the passage "Processes of supernova formation") .

Gravitational radius, event horizon
Gravitational forces are by far the weakest of all known types of interactions. However, with a sufficiently large accumulation of matter, these weakest gravitational forces, due to their universatility, can become dominant and can even be so powerful that nothing can withstand them, not even light.
Let us have some (non-rotating) spherical star or planet with total mass M and radius r . In order to have a body from surface of such planet or star to completely overcome its gravitation attraction, and freely to move away from it into space infinitelly, it must be given a radial velocity at least equal to (according to Newton's theory)

such velocity v₂ is called escape velocity or also 2. cosmic velocity (it was derived in §1.2, passage "Gravitational bodies" and "Movement of bodies in the gravitational field"). The escape velocity does not depend on the mass or the composition of the escaping body (universality of gravity), it depends only on the mass M of the gravitational body and the radius r from which the escaping body starts. For a body starting from the Earth's surface, the escape velocity is about 11.2 km/s - the second cosmic velocity. With increasing mass M or with decreasing radius r the escape velocity increases from the surface of the body *). Already in 1783 J.Mitchell and independently in 1795 P.Laplace, based, of course, on Newton's non-relativistic law of gravity and the corpuscular theory of light, pointed out that very massive and dense stars may not be visible at all because the escape velocity from their surface may be greater than the speed of light - they would be "dark stars". An emitting particle of light (the concept of a photon or electromagnetic waves was not known at the time) like an upwardly thrown stone, slowed down by the star's strong gravitational pull, stops, and then falls back onto the star. Thus, although "black hole physics", as a subdiscipline of astrophysics and the general theory of relativity, is one of the youngest disciplines, its ideological roots go far back.
*) If we take the for illustration as a basis the mass of the Sun M_¤ = 1.989 . 10³⁰ kg, the radius of which is R_¤ » 696 000 km, then according to the above formula the escape velocity v₂ from the gravitational field of a body of mass M with radius R can be expressed as v₂ = 617.7 . ( M / R )^1/2 [km/s], where the mass M º M/M_¤ and the radius R º R/R_¤ are expressed in "solar units". The coefficient of 617.7 km/s is equal to the escape velocity from the surface of the Sun. If we demand that the escape velocity v₂ be equal to the speed of light c = 299,792 km/s, we obtain for the body mass M critical radius r_g = 2.95 . M/M_¤, ie about 3 kilometers/s. for each "Solar mass".
The radius r_g, at which the escape velocity is just equal to the speed of light (for a spherically symetric body M), is called the gravitational or Schwarzschild radius :

This formula, which can be easily obtained within Newtonian theory by placing escapes velocity v₂ equal to the speed of light c , coincidentally applies exactly also in GTR; here, however, this Schwarzschild sphere has a profound significance of the event horizon causally separating the area inside and outside, as shown in §3.4 "Schwarzschild geometry" and as we will see in the following.
The first relativistic analysis of the gravitational collapse (for the simplest case of a spherical homogeneous cloud of free-falling dust particles) was performed in 1939 by Oppenheimer and Snyder [195], who came to the conclusion that in the final stages of collapse, a horizon of events emerges, ie according to today's terminology "black hole". However, the intensive development of black holes physics begin until about the 1960s.
The research of the English physicists S.Hawking and R.Penrose has the greatest credit for it; also made significant contribution to it B.Carter, J.A.Wheeler (who is the author of the name "black hole"), R.Kerr, D.Christodolau, R.Ruffini, W.Israel, J.Bekenstein, J.Zeldovic, I.Novikov, K.Thorne, J.Bardeen and many others.

Fig.4.4. Gradual closing of the exit cone of light rays from a point source located on the surface of the star during its collapse.
a) For bodies with a very large diameter in comparison with r_g = 2M, the gravitational field is relatively weak and the light rays from a point source located on the surface propagate practically in a straight line.
b) With the continuing collapse the gravitational field increase, the rays are curved, but if r > 3M, the exit cone still remains 180° .
c, d) In the late stages of collapse, the output light cone begins to narrow: more and more of the light emitted by the source is pulled back to the surface of the body by gravity; only rays in a narrow cone vertically upwards can be radiated into the space.
e) After exceeding the gravitational radius, no more emitted photon can get into the surrounding space, all the light is drawn by gravity towards the center - a black hole is formed.

Fig.4.4 shows one of the most interesting phenomena accompanying the gravitational collapse: the gradual narrowing and closing of the exit cone of light rays. An output cone (not to be confused with a space-time light cone !) means a space cone with a vertex at a given point, that only rays emitted in the direction inside this cone from a given point can enter the outer space, while rays in directions outside the output cone are absorbed by the gravitating body. If the body (planet, star) has a weak gravitational field, the rays propagate from each point on the surface practically in a straight line, so that the output "cone" is the entire half-space above the surface of the body (angle 180°) - Fig.4.4a. During the collapse, the gravitational field strengthens and the rays are curved (Fig.4.4b). In the late stages of collapse (after crossing the photon sphere - see §3.4) the gravitational field intensifies to such an extent that the rays emitted too "obliquely" are bent by gravity so that they strike the surface; only rays radiated in a narrow cone almost vertically upwards, can escape - Fig.4.4c,d. After exceeding the gravitational radius, the output cone is completely closed - all light is pulled back by gravity (Fig.4.4e) *), a black hole is formed. The spacetime around a black hole curves so much (an extremely strong gravitational field) that it "closes in on itself " and is interrupted in terms of the causality the connection with the outside world.
*) The difference between classical Newtonian and general-relativistic behavior of a black hole
The described scenario of the formation of a black hole qualitatively resembles the above-mentioned situation of the "dark star", which was speculated at the end of the 18th century Laplace and Mitchell. However, there are two significant differences between the Newtonian and relativistic versions of the "dark star" compressed below the gravitational radius :
1.From the usual Newtonian point of view (but it stops working here!) photons emitted from the star's surface will first fly out towards larger radii (they may even rise a little outside the critical gravitational radius) and then be turned by gravity and pulled back inwards - rays or photons the lights would fall back onto the star, much like the stones thrown upward hit the Earth.
However, according to the general theory of relativity, every photon emitted in any direction within the critical sphere (below the gravitational radius) will always move only inwards, to smaller and smaller radii. Not even for a moment the photon can flying up !
2.According to the laws of classical Newtonian mechanics, a star compressed below the gravitational radius can remain permanently in a static non-collapsing state, if the gravitational compression is balanced by its internal pressure. The light won't get out, but a brave astronaut in a powerful enough rocket could land on the surface, take a sample, and then take off and fly out into outer space.
According to the general theory of relativity, any star that compresses below the sphere of the gravitational radius will have such a strong gravitational compression that no internal counterpressure can equal it, and the star must inevitably collapse. No observer, even if equipped with a extremely powerful rocket, can reverse the direction of its motion and return to outer space after entering space below the gravitational radius; will inevitably fall inwards.

Where and how can black holes form ?
The basic rectilinear mode of formation of black hole with star mass was given above "Complete gravitational collapse. Black hole." :
A sufficiently massive star (M> »10-20 M_¤) after consuming all the thermonuclear fuel will have a residual mass higher than the Oppenheimer-Landau limit. The substance of the star no longer has any sources or mechanisms of sufficiently large internal repulsive forces that would be able to balance such strong gravity. In this situation, a complete gravitational collapse occurs, the star shrinks below its gravitational radius (r_g = 2GM/c² for a non-rotating star), crosses the horizon, and a black hole or collapsar is formed.
In a close binary system, a black hole can be formed by an indirect mechanism, so that from one component (which is a giant star) it flows a considerable amount of matter through the inner Lagrange point (see §1.2, Fig.1.1d, passage "Binary system") to the other component, which is white dwarf or neutron star. After a certain time, the Oppenheimer-Landau limit is reached by accretion, a complete collapse occurs and a black hole is formed. The flowing mass is then still absorbed by this black hole, around which an accretion disk is formed - §4.8, part "Accretion disks around black holes", Fig.4.26.
In large regions with a dense accumulation of gases and tight star formations, especially in the center of galaxies, giant - supermassive black holes can also form (it is discussed in §4.8, passage "How did central supermassive black holes form?").
It is possible to see a black hole ?
Thus, since no radiation emanates from the black hole (we do not consider the astronomically irrelevant Hawking radiation of quantum evaporation), it cannot be "seen" directly in the sense of the usual astronomical observation. At a sufficiently close or large (supermassive) black hole, we could at most observe its shadow image - "silhouette" - against the background of a diffuse light source - a galaxy, nebula (cf. discussion in §4.3, section "Gravitational lenses. Optics of black holes."); however, our telescopes are not enough for that yet (they were only succeeded by connecting several telescopes *))... In §4.8 "Astrophysical significance of black holes" it will be discussed how the black holes can be indirectly "observed" on the basis of radiation generated during the absorption of the surrounding mater. During this accretion, an otherwise non-radiant black hole becomes a brightly shining object ! More precisely, the glowing object is the absorbed gas in its immediate vicinity - §4.8, part "Accretion disks around black holes". And also huge jets of gases and radiation from the accretion disks of quasars and radio galaxies - §4.8, passage "Mechanism of quasars and active nuclei of galaxies".
*)Note:By connecting and precisely synchronizing .... large telescopes on several continents in the Event Horizon project, however, in 20..- 20 .. we managed to take a picture of the silhouette of a large black hole ..... in the center of the galaxy .. .... distant ..... light years ........
Planets around black holes ?
At greater distances from the black hole's gravitational field is exactly the same as around a normal star, so there after Keplerian circular or elliptical orbits in principle, orbiting the planet. If we were in a hypothetical (astrophysically unreal) the scenarios presented that the central star immediately collapsed into a black hole without the above-mentioned accompanying explosive phenomena, orbiting planets would not feel it at all (except for the light going out in a few minutes or hours, depending on the orbital distance). The gravitational field would not change at all, and the planets would continue to orbit in their original orbits.
However, previous dramatic phenomena in the final stages of stellar evolution, especially the supernova explosion, can destroy and destabilize the planetary system (discussed above) , so less orbiting planets can be expected around black holes..?..
Black holes - extremely exotic objects !
When astrophysicists learned in the 1940s of the unusual and incomprehensible phenomena that gravitational collapse could lead to, they sought "a law of physics that would prevent stars from making such nonsense" (as the prominent British astrophysicist A.Eddington put it). It turned out that such a law did not seem to exist, and now the consequences of the gravitational collapse are almost universally accepted. If the Sun collapsed into a black hole (but this cannot happen), its gravitational radius would be about 3 km; the gravitational radius of the Earth would be only 0.9 cm - you can already see how exotic objects black hole are! In general, as mentioned above, the gravitational radius of a black hole (non-rotating)we can simply determine by dividing its mass by the mass of the Sun and multiplying the result by a factor of 2.95 km - ie about 3 kilometers for each "Solar mass".
Lower mass - stronger gravity ..? ..
The following comparison may seem paradoxical: The star already has all that huge gravitational mass in it during its "active" life - and initially even greater (during its evolution, the star loses much of its initial mass stellar "wind" emissions and gas eruptions during instabilities), and yet light escapes uninterruptedly. And after the gravitational collapse of the remaining object, with less mass, does gravity no longer let light out ? The simplified explanation is as follows :
During the equilibrium evolution in the main sequence, the star has a diameter of hundreds of thousands to millions of kilometers, so the surface gravity in the photosphere, from which light is emitted, is relatively faint and light can escape almost undisturbed (only with a slight gravitational redshift) into space. And inward gravity even decreases (it's more complicated, depending on the density distribution of the substance).
During the gravitational collapse, the diameter of a star decreases sharply and the gravity on its surface increases dramatically. Finally, to the level where the escape velocity reaches the speed of light - an optical horizon is created , which according to the general theory of relativity is the horizon of events. The light will no longer come out. At greater distances, the intensity of gravity remains as low as before the collapse (or rather smaller - in proportion to how the total mass of the star decreased in the final stages of its evolution). Following the usual Kepler orbits around the star - now black holes - planets can continue to orbit here as before.
So globally - at greater distances - gravity decreases , while locally (near the center) gravity increases enormously !
A similar pattern applies to gravity in the immediate vicinity of the horizons "large" and "small" black holes. The gravitational field of giant black holes is, of course, more powerful, but it is more "spread" into space - there are only small gradients of gravitational forces around their horizon (we could temporarily survive here). But with smaller black holes of stellar masses, the tidal forces near the horizon are so great, that they ruptures each macroscopic body into a stream of atoms and particles..!.

"Pit" or "hole" in space-time
The name "black hole" (first used by J.A.Wheeler) describes very well the basic properties of a collapsar, creating a deep defect in space-time that does not emit any light. However, it is a very strange "hole", some of the properties of which are completely different from the usual "pit in the ground". We can measure its width (diameter), but we cannot measure its depth; it is "infinite" or indefinite - the question arises here "what is depth?". Every ordinary hole, pit, shaft, well, can be backfilled or filled when we no longer need it (or it would be dangerous) - if we try to do so, the black hole will increase even more, each mass we throw into it will increase the radius of its horizon (see also §4.6 "Laws of dynamics of black holes"); the bottomless pit will remain. Even with the "blackness" of a black hole, it's more complicated. It is black in the sense that it does not emit any radiation *), from an optical point of view it is an absolutely black absorbing body. Against a light background, it appears as a dark disk, but it does not overshadow anything, it has the properties of a gravitational lens, around which interesting light effects arise (for a more detailed discussion, see §4.3, section "Gravitational lenses. Optics of black holes.").
*) This is so from a classic point of view. In §4.7 "Quantum radiation and the thermodynamics of black holes" however, will show the possibility of radiation emission from a black hole under the influence of quantum effects. However, this possibility is probably only theoretical and does not occur with astronomical black holes .

Two different views of gravitational collapse - outer and inner
The collapse in the late stages, when approaching the gravitational radius, is already completely relativistic and appears completely different for an observer on a star than for a distant outer observer.
From a physical point of view, this is due to the principle of equivalence, according to which acceleration can compensate for the effect of the gravitational field. For observers falling near the Schwarzschild sphere with the acceleration of free fall, the strongly curved spacetime here seems to be "leveling out" locally and the horizon disappears (for this observer, on the contrary, due to the acceleration, a kinematic - Rindler - type horizon arises in the outer distant region, as if there were a strong gravitational field). The different external and internal manifestations of gravitational collapse are caused by the effect of gravitational dilation of time in the general theory of relativity (we derived it in §2.4 "Physical laws in curved spacetime", passage "Space and time in the gravitational field").
In §3.4 "Schwarzschild geometry", the passage "Radial motion of particles", we found that particle falling radially towards the center exceed the gravitational radius r=2M for a finite proper time interval (and will continue to move to singularity). While from the point of view of the external observer, due to the gravitational dilation of time near the radius r=2M, the particle's motion begins to slow down and stops completely at the gravitational radius - the particle would need an infinitely long coordinate time from the view-point of external observer to cross the horizon. And in the same way will behave the surface of the star, which gravitationally collapsing into a black hole :
¨ Exterior view
For the outside observer the collapse from the time t₀ when it gets into the relativistic area, starts gradually slowing down due to deceleration the passage of time by the gravitational field and never in the final time reaches gravitational radius - in horizon time stops, collapse "freezes". However, the decrease in the brightness of the star L and the increase in the gravitational redshift is exponential [285] :

(L_o is the luminosity and l_o wavelength of the star's light at time t_o) with a half-life roughly equal to the time of light passing through the distance r_g, so that the star practically "goes out" in a fraction of a second from the onset of relativistic influences. If we take into account the quantum nature of light, then in a finite (and very short) time, the surface of the collapsing star really leaves the last photon and the collapsar becomes "absolutely black".
As light laboriously "climbs" out of the pit of the collapsing star's strong gravitational field, losing energy and lengthening its wavelength, shifting to red. The flow of time on the surface of a collapsing star is gradually infinitely prolonged (it is "frozen"). Each light (regardless of its original color, wavelength or photon energy during its emission) will move very far beyond the red border, beyond the infrared and then the radio field as it moves away from the stellar surface. When light (and all electromagnetic radiation) is overcome by gravity, all of its energy is removed, making it cease to exist. The collapsing star, with its gravity, "visually cuts off" from the surrounding universe. What's more, it also cuts off in terms of causality ...
¨ Interior viewFor observers on a collapsing star (if he could stay alive) the horizon is no obstacle and can, after a finite (and very short!) interval of his own time, in principle overcome it without difficulty - there is no real space-time singularity on the horizon (§3.4 "Schwarzschild geometry"). However, below the gravitational radius, this observer can no longer send any information out; gravity "will not let out" nor light. No phenomena taking place below the horizon can in any way affect the outside world and cannot be observed from it in any way. The horizon is a kind of "membrane" permeable only in the inward direction. Once an object crosses the event horizon, it loses all hope of escaping or returning. If the falling body emits light or other radiation, it is also absorbed, so that the outside observer will never see it again. Whatever happens below the horizon (in a black hole), will stay there as well.
The deep connections between space, time and gravity in the general theory of relativity show (see §3.4 and §4.3), that after reaching the horizon, all bodies will move towards the center r = 0 with the same fatality with which time runs from the past to the future (spacetime light cones are completely turned inwards). Even if the observer was in the rocket, for example, even the greatest force of the engines could not prevent him from falling to the center. Once the gravitational radius is reached, no known (and perhaps none at all!) force can stop the gravitational collapse, because no force can turn back time. The collapse continues and after the final own time the star collapses to the point r = 0, to the so-called singularity *) with zero volume, infinite density and curvature of spacetime, with infinite pressures and gradients of gravity forces (this is at least according to the classical GTR).
*) Spacetime singularity (lat. singularis = unique, exceptional, extraordinary ) in the general theory of relativity is a terrible place where the curvature of spacetime becomes infinitely large and spacetime analytically ceases to exist (from a geometrical-topological point of view, singularities are analyzed in §3.7 "Spacetime singularities"). Infinite curvature of spacetime means infinite gradients of gravitational forces - infinitely large tidal forces acting on any object of non-zero size. Everything is destroyed here!
After the creation of the horizon of the black hole, the mass of the spherical star from the inner point of view will continue to collapse unstoppably and compress to zero volume and infinite densities - it will create a space-time singularity, into which it he plunges and "disappear" in it (limit prediction according to classical GTR; some alternative possibilities will be discussed in §4.7 "Quantum radiation and thermodynamics of black holes").
For a black hole, we can paraphrase a familiar proverb about hell : "What a black hole catches, it will never return!". Nothing, not even light, can overcome the gigantic gravitational force of this ghostly object...
Gravitational oscillation during collapse inside a black hole
During spontaneous shrinkage of mechanical systems (elastic bodies, tension springs, gas cloud), reflection and temporary opposite movement occur when the smallest volume is reached, replaced again by shrinkage, etc. - oscillations occur, which are gradually attenuated by dissipative processes. An analysis of the course of gravitational collapse, carried out in 1970 by V.Bìlinský, I.Chalatnikov and E.Lifšic, as well as independently by Ch.Misner, shows that a similar phenomenon occurs during gravitational collapse inside a black hole, although by a slightly different mechanism (collapsing the mass cannot "bounce" off the center with increased pressure and move upwards, because the space-time light cones are uncompromisingly turned inwards within the horizon). As matter falls toward singularity, tidal forces deform it into an ellipsoid - stretching in the direction of the fall and compressing in both perpendicular directions. This very fast dynamic shear curvature excites another dynamic gravitational field due to self -gravity (this nonlinear property of gravity in GTR was mentioned in §2.5 "Einstein's gravitational field equation"). This causes compression in the vertical direction of the fall materials and expansion in the horizontal direction. And the resulting self-gravity field again has the opposite effect, etc.... The result is an alternating stretching and compressing the collapsing matter - chaotic oscillations tidal forces as they approach singularity, leading to dynamic "kneading" and mixing of collapsing matter (Ch.Misner called it " mixmaster dynamics"). The oscillations of tidal forces around the singularity are very strong during gravitational collapse and the formation of a black hole. Then gradually dampen and weaken exponentially due to the intense radiation of gravitational waves (absorbed by the singularity), until they finally disappear. Again, however temporarily excites during accretion and absorb some mass by black hole. In the case of stellar mass black holes, tidal oscillations disappear in fractions of a second, but in giant black holes they can persist for several months.
An observer falling into a black hole
Imagine that as an observer we fall into a black hole and from the distance we are watched by another, external observer. For the outside observer, the time of the falling observer seems to stop near the horizon. For the inner observer, everything happens in his frame of reference as if there had not been a fall into a black hole (we assume here that it is a supermassive black hole, on the horizon of which only weak tidal forces act). However, when looking at the outer universe, he will observe that events flow faster and faster there, he will see a number of supernova explosions, as he will observe the slow and long evolution of stars very accelerated in time. At the moment of falling over the horizon of events, the whole life of the universe is projected to him very quickly.
  An outside observer records a completely different course of our fall into a black hole. He will see how the fall into the black hole is constantly slowing down near the horizon, and in the immediate vicinity of the horizon the movement seems to freeze and the fall into the black hole will never occur.
  So far, we were only interested in the issues of movement and passage of time for observers. At the end of the next paragraph (in the passage "A Man Falling into a Black Hole"), we will look at what the observer would physically feel and what would happen to him if he fell into a black hole.
What happens to matter, its atoms and molecules when it is absorbed by a black hole ?
Above all, this question can only be solved theoretically, but experimentally it is in principle untestable! Even if we had a black hole somewhere nearby. In §4.8 we will see that the substance creates a glowing accretion disk around the black hole, along the axis of which jets are formed, through which up to 25% of the absorbed mass can escape - in the form of ionized substance and radiation. And never in a finite time we could see a mass fall below the event horizon, or what is going on inside. If we jumped into a black hole behind a falling mass (see below "Man falling into a black hole"), we would go below the horizon of events, but thus we would seal the fate of our demise; nor could we send our information to our colleagues, the event horizon will not let out any signal. The course of this "suicide" case would depend on how big the black hole is. With "small" black holes of stellar masses, with a horizon diameter of several kilometers, we would not get alive into a black hole - huge gravitational field gradients (tidal forces - §1.2, passage "Gravitational gradients - tidal forces") would tear us far ahead of the horizon , we would fall into a black hole as wellas a "string of atoms". In the case of a large black hole, such as those in the cores of galaxies, we would not even notice that we have penetrated below the horizon, tidal forces would not be large yet. However, they would soon grow strongly and we would eventually be torn and crushed at the same time... The final fate of matter inside a black hole can be assessed from two points of view :
1. According to the "classical" general theory of relativity, all matter collapses into a point singularity with zero volume and infinite density. The paradox of such a state suggests that here the theory reaches the limits of its possibilities...
2. According to the quantum approach, this matter "dissolves" in the "topological space-time foam" quantum fluctuations geometry of space in the scales Planck order of 10^-33cm (cf. §B.4 "Quantum geometrodynamics") .
In both cases it is the irreversible termination of the substance, including particles of which it is composed. According to the theorem "Black Hole has no hair" (§4.5 "The "black hole has no hair theorem") no individual characteristics of absorbed matter are preserved for the outer universe, except mass, charge and angular momentum. Hypotheses about the possibility of penetrating other universes through black holes are not confirmed (shown in §4.4 "Rotating and electrically charged Kerr-Newman black holes"). In §4.7 "Quantum radiation and the thermodynamics of black holes" is critically assessed the "return" of the substance from the black hole by effect Hawking quantum evaporation.
People falling into the black hole
Imagine in an imaginary experiment astronaut - a suicide man, who jumps out of the rocket at a certain greater distance and falls feet forwards in direction towards the spherical black hole. We will briefly describe his feelings and destiny. As mentioned above, where a living person can enter or around a black hole depends on the size (weight) of the black hole. The larger the black hole, the closer to or deeper below the horizon an astronaut can penetrate and the longer it can survive. In the case of black holes in stellar masses, the tidal forces of an astronaut would immediately kill ("spaghetti") at a distance of tens of gravitational radii.
   In order to give the falling astronaut at least a little longer life, we choose a resting *) large black hole with a mass of the order of 10⁹M_¤; such giant black holes are found in the centers of galaxies and are often observed as quasars (§4.8. "Astrophysical significance of black holes"). In this case, the falling astronaut can, alive and well, cross the horizon and enter a black hole. At this moment (of his own time) he practically does not feel anything extraordinary **), he is still moving freely in a weightless state. However, his fate is inevitably sealed, he has only a few tens of hours left to live. As his fall gradually accelerates and gets closer to singularity, tidal gravitational forces begin to apply: he begins to feel a pull on his legs toward the black hole and on his neck and head in the opposite direction (and at the same time pressure from side). In the spherical case of tidal forces, all bodies expand radially and compress transversely. These forces will grow rapidly, so that his muscles and bones will no longer be able to withstand them, the body will rupture and the astronaut will perish (he will be "spaghetti"). And near the singularity, the remnants of the body are immensely deformed, tidal gravitational gradients ripping and crushing cells, then individual atoms, even electrons, protons, and quarks inside them. All these fragments eventually nest into the singularity and become part of it, they disappear in it ...
*)If an astronaut jumped into a black hole during a collapse, or just after the collapse or after a larger mass was absorbed by a black hole, he would still be far from the singularity torn and "kneaded" by the chaotic oscillations of tidal forces (mentioned above in the passage "Gravitational oscillations during a collapse inside a black hole"). Here we consider the situation after the complete suppression of gravitational oscillations inside a black hole (which can take several months for a giant black hole).
**)The hypothesis has emerged, that just below the horizon of a black hole there could be a kind of "fire wall" made of high-energy quanta and radiation (A.Almheiri, D.Marolf, J.Polchinski, J.Sully, 2012), which would burn a falling astronaut. It could allegedly arise from Hawking's quantum radiation, whose pairs of quanta below and above the horizon could remain "quantum entangled" even after emission. Such quantum effects could perhaps occur with black mini-holes, not with black holes in stellar masses (or even larger ones).
  The presence of a "fire wall" contradicts Einstein's principle of equivalence, on which the general theory of relativity is based. Therefore, the "fire wall" hypothesis is erroneous probably, so we will not consider it in the further analysis of the properties of black holes (except for a brief mention in §4.7, the passage "Paradox of information loss ?").

The paradox of the external and internal view of the gravitational collapse
The collapse of a star stops - it "freezes" forever when it reaches the horizon when it is observed in an external static frame of reference. But it does not freeze and, conversely, continues rapidly beyond the "freezing point" when viewed from the point of view of an object collapsing with the star's surface. Who is right? External or internal observer? In terms of the theory of relativity, both ! The collapsing star - as such - actually shrinks below the critical gravitational radius. The fact that this seems like a "freeze" when viewed from a distance can be considered a gravitational-optical-time "illusion". Or can it be reflected "pragmatically" in the spirit of the following passage "Are there 'complete' black holes in the universe? " :
Are there 'complete' black holes in the universe ?
As mentioned above, at first the rapidly accelerating gravitational collapse gradually slows down as it approaches the causal horizon due to relativistic phenomena, and immediately above the horizon it comes to a complete stop (all movement completely "freezes"). The horizon is reached only in infinite (external) time - so never! From the point of view of the outer universe, the horizon of events can never be crossed and a real black hole cannot be created. For the outside observer, the worldlines of all particles end on the horizon of events (Schwarzschild's sphere in the spherical case); whatever happens below the gravitational radius with the falling observer (in terms of his own time), as if it did not exist for the outside observer. From the point of view of events in the universe, only the outer part of the "black hole", which behaves like a compact gravitationally collapsed object, actually applies and manifests itself - see §4.8. "Astrophysical significance of black holes". From this "pragmatic" point of view, we could therefore say that "complete" black holes (including the "interior" below the horizon) do not effectively exist in the universe.
However, physics, as a universal and objective natural science, is obliged to deal with natural phenomena from all possible points of view. That is, also from the point of view of collapsing matter - an observer who "falls" in the gravitational field of a collapsed body together with matter. From this point of view, the collapsed object - the black hole - continues to live an intense "inner life" (invisible by distant observers), in which a relentless collapse dominates and some very unusual phenomena may occur there. We also analyze these phenomena from a theoretical point of view in a number of places in this chapter, as well as in the previous chapter 3 "Geometry and topology of spacetime", although they are probably not directly relevant for "practical" events in outer universe. However, they certainly have their gnoseological interest ...

The absolute nature of the black hole horizon raises an interesting paradoxical question: "How can the gravitational force get out from under the black hole horizon and act on external bodies when the horizon does not let anything out?". This issue will be discussed in §4.5 "Black hole has no hair", in the section "Preservation of interaction with matter absorbed by a black hole".

Behavior of the substance at high pressures; neutronization
To better understand why a star in which all nuclear fuel has burned, with a mass higher than a certain limit, can no longer withstand its own gravity, it is useful to study the lowest energy state of a system of a given number of atoms containing N nucleons [115].
In the "cold" substance *), in which neither the pressure caused by the thermal motion of the particles nor the radiation pressure is substantially applied, the main role is played by the Fermi pressure related to the Pauli principle. If we have a set of N fermions of mass m concentrated in a unit volume, it will be according to Pauli's principle, each fermion occupies an effective volume of 1/N, and thus according to Heisenberg's uncertainty relation, its momentum will be of the order of ~ h.N^1/3. The velocity of the fermion will be on average ~ h.N^1/3/m in the non-relativistic case (ie when h.N^1/3<<m), and of course practically equal to one (units c=1) in the relativistic case (for h.N^1/3> m). The pressure, which is the product of the momentum, velocity and density of particles, is then of the order of P ~ h²N^5/3/m in the non-relativistic case and P ~ h.N^4/3 for the relativistic Fermi gas.
*) A cold substance is considered to be a situation where the temperature is so low that it does not have a significant effect on the physical properties of the substance. This can be achieved even at temperatures of tens of thousands of degrees (eg with white dwarfs).

If the number of nucleons N is not too high (less than about ~10⁵², i.e. total mass less than ~10²⁵kg - so that the total gravitational force does not deform the atoms), the lowest energy state of such a system will be the crystal lattice of iron atoms Fe⁵⁶. The strongest here are nuclear forces, the minimum of which corresponds to the Fe⁵⁶ nuclei with the highest binding energy per nucleon *). In second place are the electromagnetic forces that determine the dimensions and shape of the crystal lattice. It can be said that the Fermi pressure is compensated by electric attractive forces between particles in the crystal lattice. The forces of self-gravity are practically negligible here and cannot prevail over the valent forces in the crystal lattice, much less over the nuclear forces.
*) See Fig.1.3.3 in §1.3 "Nuclear reactions and nuclear energy" in the book "Nuclear physics and physics of ionizing radiation".
For large weights (greater than about ~10²⁶kg), self- gravity already becomes substantial, the gravitational forces inside the system exceed the valent forces, and the crystal lattice disintegrates. The electrons then behave as free particles forming a degenerate electron gas. If the density is not very high, and these electrons are non-relativistic, the Fermi pressure is capable of balancing the gravitational force (in the nonrelativistic case the bulk of the pressure caused by the Fermi electrons because the value of 1/m for them is much higher than for nucleons and nuclei). The relationship between pressure and density in specific situations is described by the equation of state of matter. The relatively complex and realistic multicompartment equation of state of "cold matter" was constructed by Harrison and Wheeler [115] (Fig.4.5). In this equation of state, several significant areas differ according to which physical processes dominate here and how the pressure is balanced. We would gradually find all these regions on the way from the surface to the interior of the neutron star :

	Fig.4.5. Relationship between pressure and density for high pressure values occurring in stars. Top: Diagram of course Harrison-Wheeler equation of state (depending of the pressure p on the density r) of "cold material".
	Bottom: Diagram shows the dependence of exponent g = [ (p + r ) / p ] . dp / d r - a compressibility factor - on the density of the mass-energy equation of state for the transcribed shaped adiabat (polytropey): P = C . r ^g .

Area 1: r <10 ⁴ g / cm ³
This first area of low densities is sometimes divided into two sub-areas :
a) The lowest with r < ~ 50 g/cm³, where the usual laws of solid state physics apply and the properties of individual substances strongly depend on their chemical composition according to Mendeleev's periodic table. The pressure here is caused by electrons in the outer (valence) orbitals.
b) ~ 50 g/cm³ < r <~ 10⁴ g/cm³, where the elastic properties already depend only on the average Z (continuously), but not on a specific chemical composition. The electron orbitals are strongly compressed and the electrons in the lower orbits also contribute to the pressure.
Area 1 is not interesting for the analysis of the end stages of stellar evolution, but it plays an important role, for example, for the structure of planets .

Area 2: ~10 ⁴ g / cm ³ < r < ~10 ⁷ g / cm ³
At densities above ~10⁴g/cm³ the Fermi energy of electrons already exceeds their binding energy in the atom, these electrons are released and the substance takes the form "gas" mixtures of nuclei and electrons. The pressure here is caused practically exclusively by degenerate electron gas. As the density increases to a value of about ~10⁷g/cm³, these electrons become relativistic.

Area 3: ~ 10 ⁷ g / cm ³ < r <~ 10 ¹¹ g / cm ³If the density exceeds a value of about r » 1.5.10⁷g/cm³, electrons begin to enter the nucleus and combine there with protons *) to form neutrons and flying-out neutrinos. In this situation, the iron core with A=56 is no longer the core with the greatest stability. With increasing electron pressure, the range of mass numbers of the most stable nuclei, in b- equilibrium with such an electron gas, shifts to higher values.
*) This nuclear process is sometimes called inverse b-decay - see §1.2 "Radioactivity", part "Radioactivity b⁺ " in the monograph "Nuclear physics and physics of ionizing radiation".

Area 4: ~ 10 ¹¹ g / cm ³ < r <~ 10 ¹⁴ g / cm ³As the density increases further, around the value of r ~ 10¹¹g/cm³the nuclei become so heavy and enriched in neutrons, that they become unstable with respect to emission neutrons. As the density increases, more and more neutrons leave the nucleus, so that the substance consists of a mixture of neutrons, heavy nuclei and electrons. At densities close to 10¹⁴g/cm³, the individual nuclei disappear and the substance consists of a mixture of neutrons (mostly), protons and electrons. We say that the substance is neutronized.

Area 5: r ³ ~ 10 ¹⁴ g / cm ³
At these densities, the Fermi momentum of the baryons reach relativistic values. The neutron, electron and proton "gas" is in equilibrium with respect to the direct and inverse b- decay, so that the total energies of protons Ep , electrons Ee and neutrons En are related by the relation Ep + Ee = En and the Fermi momentum of the neutron is twice that of an electron or proton. The relative representation of individual types of particles will then be given by the ratios ne= np, nn= 8.np= 8.ne, ie nn : np : ne = 8 : 1 : 1. In this last area, however, there is great uncertainty in the equation of state, because in addition to the Fermi pressure of relativistic baryons, nuclear interactions between them and the formation of other particles. The precise nature of nucleon-nucleon interactions under such extreme conditions is not exactly known. Also, nothing definite is known about the nature of particle formation (eg whether to consider particles also from grandunification theories?). In the Harrison-Wheeler equation of state, at densities close to nuclear and densities higher, no nucleon-nucleon interactions or effects of new particle formation are considered, the substance is considered to be a mixture of non-interacting neutrons, protons and electrons, forming Fermi gas. It is assumed here that with increasing densification, the momentum of nucleons increases and thus the influence of nuclear forces on their motion decreases.
In fact, at very high densities of 2-10 . 10¹⁴ g/cm³, which could be in the center of massive neutron stars, however, extremely high pressures thicken baryons so close to each other that their quark structure connects to each other and "dissolves" into a mixture of almost free quarks and gluons - the so called quark-gluon plasma is formed, which can be stabilized by massive gravity. Quarks, pushed by strong gravity very close to each other, could form Cooper pairs, acting like bosons, and form superfluid condensate. These effects could lead to better compressibility than the Fermi pressure allows; massive neutron stars could thus shrink to a smaller size than would be expected for a composition of neutrons alone. It was discussed above in the passage "Internal structure of neutron stars".

In general, the equation of state of a "cold" substance is relatively reliably known for densities much lower than nuclear ones, while at high densities there are considerable uncertainties stemming from ignorance of the exact nature of elementary particle interactions at very high energies. Unfortunately, we know very little about the behavior of matter under very extreme conditions, rather we make certain more or less substantiated assumptions about how matter could behave under such conditions...
In connection with this, there is often an objection to complete gravitational collapse: "What if, after reaching a certain (albeit very high) density, the substance of the star is no longer compressible ?". The answer is: "If the accumulation of matter is such that the gravitational radius is reached, then in principle no matter can be incompressible !". In §3.4 "Schwarzschild geometry" we saw, that the Schwarzschild spacetime geometry that places such configuration will (if spherically symmetric), dictates each object below the horizon, to move towards the center r = 0. A each object must "listen" - the universal laws of space are superior to all other laws, because all physical phenomena are ultimately governed by the laws between quantities in space and time. Even the laws of special relativity prohibits absolute rigidity and incompressibility for bodies of non-zero dimensions, because "sound" (mechanical vibration) would have to propagate in them at infinite speed. In reality, however, the speed of sound v_acus.~ Ö(dP/dr) must in any case be less than the speed of light, ie dP/dr < c². This fundamental limit is however still unrealistically high, because for an isotropic medium (alone it may be well to define the pressure in the normal sense) satisfying the strong energy conditions (see §2.6, the relation (2.60) ), according to which a trace of the tensor of energy-momentum it must be positive definite, we get three times the lower limit for the ratio of pressure and mass density: dP/dr £ c²/3.
Even if we accept an unrealistic case of incompressibility, there will be a certain limit mass above which there can no longer be any equilibrium configuration. This is because pressure also appears in the energy-momentum tensor of a given substance and thus contributes to the excitation of the gravitational field; it also appears in the numerator of the relation (4.3). At high pressures can occur somewhat paradoxical situation, where the pressure does not prevent, but rather helps to further gravitational collapse ...
In practice, i.e. in gravitational kolaps with a sufficiently massive stars, no uncertainty in the equation of state super-dense substance at r l 10¹⁴g/cm³ have no influence, because the interactions responsible for them can come into play only below the event horizon; therefore, they cannot prevent a black hole from forming.

What's inside the black holes ?
The inquisitive question "What's inside something?" is completely legitimate, and one we commonly ask it for all macroscopic and even most microscopic objects. Exploring what is inside cells ("Cells - the basic units of living organisms") has fundamentally shifted our understanding of the nature of life and made biology and medicine a powerful and exact science. Examination of what is inside atoms has revealed the physical and chemical structure of matter and the nature of radiation ("Atom Structure"), has yielded a vast number of applications. In each area, the answer to the question "what is inside?" of some object, has substantially improved the level of our knowledge of the truth.
Black holes are somewhat resist this trend. And not just because there are no black holes within the distances available to us, and there is no hope of exploring them directly in the foreseeable future. There are mainly fundamental obstacles. How can we observe "what's inside" by observation, when no signal can ever get out of a black hole and give us an answer?! Whatever is inside the black hole, due to the event horizon, it cannot manifest itself outside the hole to affect the surrounding world. Even if a brave researcher set out to investigate the interior of a black hole, he would never be able to return and tell us his discoveries; nor could he send any message about them before his inevitable death. The only way to answer the inquisitive question is "what's inside the black holes?", is a theoretical study of what the laws of physics predict - their analysis and extrapolation. In the above passage "What happens to matter, its atoms and molecules when it is absorbed by a black hole?" we have already partially outlined what is going on there according to the general theory of relativity.
Basically, three possibilities are discussed, what could be hidden inside a black hole :
¨ Singularity with infinite gravitational forces. This is predicted by the "classical " general theory of relativity and concretizes it in more details the Penrose's and Hawking's theorems about singulatities (§3.7 "Spatio-temporal singularities ", §3.8 " Hawking's and Penrose's theorems about singularities "). However, singularity is not a very acceptable solution...
¨ Quantum foam in which the geometric properties of space and the causal properties of time end. The singularity "dissolves" in the quantum foam. This is, yet indeterminate, a view of quantum gravity and quantum geometrodynamics (§B.4 "Quantum Geodynamics") .
¨ A tunnel to another universe or other parts of space. This is suggested by the analysis of the complex geometric and topological structure of spacetime of rotating or electrically charged black holes (§4.4, part "Black holes - bridges to other universes?"). However, the real existence of such "tunnels" or "wormholes" is physically questionable (it is critically discussed in the mentioned §4.4) .
We do not yet know the definitive and irreversible answer. Perhaps the expected advances in unitary field theories will shed new light on these questions (§B.6 "Unification of fundamental interactions. Supergravity. Superstrings.") ...

The reality of the existence of black holes
If we recapitulate the results of the outlined analysis of the properties of the final stages of stellar evolution and gravitational collapse and compare it with the situation observed in space, we can draw the following conclusions :

This gives us a real basis and sufficient motivation to study the properties of black holes in the remaining paragraphs of this chapter. Here we can expect that in the analysis of black holes (which will be perhaps the most interesting part of the book for many readers) we can look forward to very unusual and fascinating phenomena !

Gravity, black holes and space-time physics :
Gravity in physics	General theory of relativity	Geometry and topology
Black holes	Relativistic cosmology	Unitary field theory
Anthropic principle or cosmic God
Nuclear physics and physics of ionizing radiation
AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy


4.1. Gravity and evolution of stars		4.3. Schwarzschild static black holes

Final stages of evolution - supernova, neutron star, black hole .