1 From Black Holes to Neutrino Stars

Abstract Due to limitation of the binding energy of a self-gravitating matter, the radius of a body is at least twice larger than the Schwarzschild radius. The total energy is adsorbed at the body surface, giving rise of a size-dependent surface tension. Since the Hawking temperature appears to be the critical one, the black holes possess zero surface tension. Microscopic neutrino stars are also introduced.

A black hole is so condensed that even light cannot escape due to the gravitational attraction.Leaving apart any resistance due to particle repulsion and exclusion the black hole formation seems also energetically impossible.Imagine a self-gravitating star, emitting radiation in all.The kinetic energy of the matter particles is continuously decreasing and the star is shrinking towards a black hole.However, due to the mass-energy equivalence, the star is losing its mass as well, thus reducing the inner strength of the self-gravitational attraction.As the Schwarzschild radius decreases also with decreasing mass, this Zeno effect leads to the conclusion that only an empty black hole could form with zero size, when the mass of the star becomes zero.
To make the picture consistent, let us start from the Newton gravity.If the mass density  is radial-symmetrically distributed, the corresponding gravitational potential  obeys the Poisson equation The second integral form involves the mass 2 Since no other interactions are considered and the excess energy is freely released as radiation in vacuum, the body contracts permanently due to the gravitational attraction.At the end, a mass point is formed with 0 r mM  everywhere.According to Eq. ( 1), it corresponds to the classical Newton potential 0 /

GM r   
. As it is well known, U diverges for a mass point, which indicates a generic problem of the Newton gravity.
According to the Einstein special relativity theory, the total energy at rest is positively defined.The lowest value 0 E  limits the binding energy to the maximal 100 % mass defect.The integral and differential forms of The integration Eq. ( 3) leads straightforward to the radial mass distribution which is not constant anymore due to the binging energy limitation.Substituting Eq. (4) in Eq. (1) yields after integration the restricted gravitational potential Far from the center it tends to the Newton potential, while near the center the logarithmic po- . That is why the potential energy (2) is finite.
One can derive the mass density distribution by differentiating directly r m from Eq. ( 4) If the mass 0 M is very small, Eq. ( 4) approaches a mass point as expected.In the opposite case of large masses, Eqs. ( 4) and ( 6 It is zero outside the body but remarkably 2 pc  resembles inside an equation of state.As is seen, the limitation of the binding energy resolves the mass point singularity.However, since 0 R is one fourth of the Schwarzschild radius, a black hole singularity still holds.It points out that probably the maximal mass will prevent such a peculiarity as well. It is interesting what the effect would be of the binding energy limitation on black holes.To answer this question, we are going to repeat the analysis above, using the Einstein general relativity theory.In the frames of the latter, the problem is described by the Tolman-Oppenheimer-Volkoff (TOV) equations 1 from which Eqs. ( 1) and ( 7) follow, respectively, in the non-relativistic limit.Note that the relativistic mass M is smaller than 0 M and the difference is the binding energy mass defect.For a mass point, Eq. ( 8) reduces to and 0 p  .Integrating this equation results straightforward in the well-known relativistic potential, where Far from the center,  tends to the Newton potential / GM r  . The singularity at s r marks the event horizon, 3 where the surface of the black hole takes place.
Let us apply now the energy limitation to the TOV equations (8).In general, the pressure pX  is proportional to the energy density 2 c    and we got 1 X  in the semi-relativistic analysis above.Introducing 2 p X c  in Eq. ( 8) yields Searching for a black hole, we are looking for a compact body of self-gravitating matter.For radii larger than the body radius R the standard expressions r mM  and 0  hold.It follows immediately from Eq. ( 10) that Eq. ( 9) is the potential outside the body.It is well known that inside the body ( / )  are solutions of Eq. ( 10), 4 which is also supported by our semi-relativistic analysis.Introducing them in Eq. ( 10) leads to To determine the important value of the factor X , one can employ a general formula, 4 relating the relativistic mass M with the mass 0 M at the origin, The positive ratios / s Rr from Eq. ( 11) and the real ratios 0 / MM from Eq. ( 12) are plotted in Fig. 1.As is seen, there is no overlap between them for 0 X  .Moreover, the negative X is always related to a negative mass defect, which indicates lack of bounded body for dark matter.Looking for a positive binding energy, the necessary inequality MM  imposes that 0 X  .If the mat- ter is super-relativistic with 1/ 3 X  , the corresponding mass defect is 24.4 %.If the particles are moving much slower than the speed of light, the doubled non-relativistic factor 2 / 3 X  results in 28.6 % mass defect.The maximal mass defect about 29.3 % at 1 X  should correspond to the lowest body temperature in order to prohibit any further energy loss by quantum reasons.It is less than 30 % and the matter particles are probably at rest.Logically, 1 X  corresponds to the minimal body radius Its plot in Fig. 2 shows lack of singularity in contrast to the Schwarzschild potential (9).The potential (13) possesses a kink at R , which reflects the effect of a pressure jump at the body surface.Hence, the body possesses a capillary pressure, AR  is the area of the surface.It is a particular example of a size-dependent surface tension, described via the Tolman formula. 5It vanishes at a flat surface and the bending elasticity  coincides with the universal Einstein expression.Remarkably, the full energy 2

Mc
A  of the body is at the surface, which supports our expectation that the body is energetically empty and that is why it cannot shrink anymore.
The problem now is what causes the pressure 2 pc  if the particles are not moving.This equation looks very similar to the ideal gas equation of state / B p Nk T V  at constant temperature and we are going to explore such an identity.The characteristic potential of the body is the Helmholtz free energy ( , , ) F T V N as a function of the natural parameters.Substituting the ideal gas pressure in the thermodynamic relation , () where Considering the self-gravitating body with pressure 2 pc  , at constant temperature the gradients of the chemical and gravitational potentials cancel exactly in agreement with Eq. ( 10

M
, m  contains the total mass 0 of the self-gravitating matter.Using Eq.(1), one can calculate the overall potential energy

Fig. 1 .
Fig. 1.Plot of / s Rr (blue) and 0 / MM (red) as a function of the factor X from Eq. (11).According to Fig.(1), the radius of a selfgravitating body is at least twice larger than the Schwarzschild radius.

Fig. 2 . 2 ln
Fig. 2. The dimensionless potential is related to the surface tension  via the Laplace law 2/ RR pR      .The latter can be integrated directly to obtain one.Therefore, a black hole possesses zero surface tension 0  , which means a zero mass M .The mass of a gas particle k T c  is extremely small at moderate temperatures.Because any movement in the body is frozen, temperature causes solely some energy flucmass and energy are synonyms, m should be considered as the typical mass fluctuation as well.The temperature dependence of the free energy (15) follows directly from the following exponential Boltzmann distribution ), s Rr  to the Hawking temperature,