On phase transitions during collisions near the horizon of black holes

During particle collisions in the vicinity of the horizon of black holes, it is possible to achieve energies and temperatures corresponding to phase transitions in particle physics. It is shown that the sizes of the regions of the new phase are of the order of the Compton length for the corresponding mass scale. The lifetime is also on the order of the Compton time. It is shown that the inverse influence of the energy density in the electro-weak phase transition in collisions on the space-time metric can be neglected.


Introduction
The works of A.A. Friedman [1,2], written 100 years ago, in which solutions were obtained for an expanding homogeneous isotropic universe [3], are the theoretical basis of the modern standard cosmological model.The discovery in 1965 of relic radiation [4,5] indicates that in the model of the expanding early Universe there were times when the temperature of matter was so high that phase transitions predicted by the theory of elementary particles could occur.There are three such phase transition in the standard model of particle physics [6,7,8]: 1) Between quark-gluon plasma and hadrons at energies E of the order of 200 MeV.The corresponding temperature T = E/k B ≈ 10 12 K, where k B ≈ 1.38 • 10 −23 J/K is the Boltzmann constant, may have taken place in the expanding Universe during the order of 10 −6 s after the Big Bang.
2) Electro-weak phase transition at energies of the order of E W ≈ 100 GeV.The corresponding temperature T W ≈ 10 15 K could take place during the order of 10 −12 s after the Big Bang.
3) The grand unification phase transition at energies E GUT ≈ 10 16 GeV.The temperature corresponding to the energy of the grand unification phase transition T GUT = E GUT /k B ≈ 10 29 K, may not have been achieved in the early universe, in models with an inflationary stage in which the heating temperature is significantly lower than T GUT .In models with a radiation-dominant stage in the early Universe, the temperature T GUT could be reached at times of the order of 10 −38 s.
The study of the properties of the matter at such temperatures and the phenomena at these phase transitions is of undoubted theoretical interest.Is it possible to achieve such temperatures in experiments on the Earth?The maximum high temperature for macroscopic parts of the substance is achieved at the time of nuclear explosion and can be on the order of 10 8 K.It is significantly less than the temperature of even the quark-gluon phase transition [9].
The highest temperature achieved in experiments on Earth refers to microscopic quantities of matter and is obtained when heavy element nuclei collide in particle accelerators.Temperature in 4 • 10 12 K was obtained from a collision of gold nuclei in Brookhaven National Laboratory (United States) in 2010 [10].In 2012, it was reported that a temperature of 5•10 12 K was reached when the lead nuclei collided at the Large Hadron Collider [11].At such temperatures, hadron matter transforms to the quark-gluon plasma state.However, such temperatures are more than two orders of magnitude less than the temperature of the electro-weak phase transition.
Thus, macroscopic amounts of matter in the state of phase transition of elementary particle physics in laboratories on Earth cannot be obtained, and microscopic amounts can be obtained only for the phase transition in the quark-gluon plasma state.
Is it possible to observe matter at the temperatures of phase transitions of the particle physics in astrophysical processes at present?Brightly luminous accretion discs formed when matter falls into black holes have a visible temperature of hundreds of millions of Kelvin degrees [12].As shown in our work [13], in the processes of collisions of particles near the horizon of black holes, it is possible to achieve energies in the system of the center of mass of colliding particles on the order of the energy scale of the electro-weak phase transition.A summary of these results is presented in Section 2.
Here we will consider questions about the size of the regions of the phase transition region obtained in a collision and the lifetime of such a region.To do this, in Section 3 we apply formulas for the energy density and radiation intensity of a gas of relativistic particles.The possibility of obtaining an electro-weak phase transition in a macroscopic volume during a collision in the vicinity of supermassive black holes is studied in Section 4. The influence of the matter energy-momentum tensor in the phase transition region on the space-time metric will be evaluated in Section 5.

The high-energy collisions near the horizon of black holes
The Kerr metric of a rotating black hole [14] in the Boyer-Lindquist coordinates [15] has the form G is the gravitational constant, c is the speed of light, M and aM are the mass and angular momentum of the black hole, respectively.We accept that 0 ≤ a ≤ GM/c.The event horizon of the Kerr black hole has the radial coordinate According to [16] the squared energy of collision of two particles with the mass m with the angular momenta L 1 and L 2 in the center-of-mass system, which are nonrelativistic at infinity and are freely incident on a black hole with the angular momentum aM, is given by the expression where x = rc 2 /GM, l = Lc/GmM, and A = ac/GM.The expression (5) has a singularity on the event horizon.In general case the limit value of the collision energy for two particle with masses m 1 , m 2 , energies E 1 , E 2 and angular momenta J 1 , J 2 is where J nH = 2E n r H /A. If the angular momentum of one of the particles tends to J nH , then the expression for the energy (6) diverges.This is the so-called Banados-Silk-West effect.Note that despite the unlimited increase in collision energy in the center of mass system, the energy that can be extracted at a large distance from a black hole cannot exceed E 1 + E 2 (assuming no Penrose effect [17]).This follows from the law of energy conservation.
A particle having a critical angular momentum value can get from infinity to the event horizon of a black hole only in the case of an extremely rotating black hole A = 1.In other cases, particles with large angular momentum values are prevented from falling onto the horizon by the potential barrier of the effective potential.As shown in [18,19], the superhigh center-of-mass energy can be achieved in multiple collisions near nonextreme black holes.To reach the horizon, particles incident from infinity should have an angular momentum low in absolute value.The angular momentum of one of the particles necessary for a high-energy collision can be acquired either in multiple collisions or in the interaction with the electromagnetic field of the accretion disk.The similar effect for electrically charged black holes was discovered in [20].Real astrophysical black holes are surrounded by matter (for example, they have an accretion disk).The possibility of particles colliding with unlimited energy near the horizon of such "dirty" black holes also takes place [21].
The value of the collision parameters corresponding to the temperature of of the elementary particles phase transitions may depend on the type of black holes.In the case of Kerr black holes the estimates for the distance from the horizon, where the collision energies required for phase transitions of elementary particles, can be achieved are given in our work [13].So, for elementary particles with mass m the value of temperature T is reached near the extreme rotating black hole at the distance For the proton mass the electro-weak temperature can be reached at a distance r − r H = 2 × 10 −4 r H .This amounts to tens of centimeters for stellar-mass black holes and hundreds of thousands of kilometers for supermassive black holes.In the mechanism of multiple collisions near the horizon of (not extreme) rotating black holes, such temperatures can be achieved at larger distances [13].Therefore, collisions in which phase transition temperatures are reached can, in principle, occur in the vicinity of stellar-mass black holes for elementary particles, and in the case of supermassive black holes for macroscopic bodies.
Next we estimate the size of the phase transition region and the lifetime of the state with the new phase.

Size and lifetime of the new phase
The energy density of the photon gas can be calculated by the known formula: (see (63,14) in [22]) where σ is the Stefan-Boltzmann constant, is the reduced Planck constant.At ultra-high temperatures, other elementary particles should also contribute to the energy density of matter.Their contribution is taken into account using the factor g eff describing the number of effective massless degrees of freedom of particles of the standard model of particle physics.
Under such definition, the photon's contribution to g eff is 2, according to the photon's two polarization states.In general case one has [6] Here it is assumed that the equilibrium temperature T i of particles of type i may differ from T .For example, in the Universe at present the temperature of the cosmic microwave background radiation is is equal to 2.7 K, and estimates for the temperature of the neutrino gas give 1.95 K. Photon gas after the moment of the last collisions of cosmological neutrinos with cosmological plasma at energies of 2-3 MeV was still heated up in the process annihilation of cosmological positrons with electrons.
The value of g eff in standard model of particle physics depends on temperature.For T in the interval 1 MeV < T < 100 MeV taking into account neutrinos, electrons and positrons leads to g eff = 10.75.At temperatures above 300 GeV, all standard model particles (photon, W ± , Z 0 bosons, 8 gluons, 3 generations of quarks and leptons, Higgs boson, must contribute to (10), which leads [6] to the value g eff = 106.75.The graph of g eff depending on temperature is presented in [6], page 65, Fig. 3.5.
Denoting k B T = mc 2 , where m is the characteristic mass scale, we get from (10) for the energy density of radiation of all types of particles where l C = /mc is the (reduced) Compton wavelength corresponding to the mass m.The pressure corresponds to a value three times less The size R 0 of the area in which after a collision can form the heated drop of a new phase of matter, we estimate from the relation It is assumed that the region of the new phase is a sphere of radius R 0 .Then one obtains Assuming that the collision energy is of the order of magnitude E c.m. ∼ g eff mc 2 , we find that the size of the region phase transition is of the order of the Compton wavelength l C for a particle of mass m.
Let us estimate the lifetime of a drop of a new phase formed as a result of a collision, generalizing the formula for the radiation intensity of black body to the case the presence of additional degrees of freedom described by the quantity g eff Let us write the energy balance equation for an infinitesimal time interval dt d(εV where V is the volume of new phase drop, S -its surface area.When obtaining estimates by order of magnitude, we will assume that the drop is spherical, and its radius may depend on time due to expansion into the surrounding space.We also assume that during the life of a drop of a new phase, thermodynamic equilibrium takes place in it and, therefore, we can talk about the temperature of the entire drop, the dependence of temperature on time and use formulas for the equilibrium state of the corresponding relativistic gas.Then from (17) we get R 3 dε = − (Jdt + εdR) .
Using ( 10) and ( 16), we obtain the equation By integrating this equation we get If the drop radius does not change R ≈ R 0 = const, then the solution is Thus, the temperature decreases exponentially and the lifetime of the new phase is of order τ ≈ R 0 /c.Since according to the equation ( 15), the size of the new phase region is assumed to be Compton, then the lifetime corresponds to Compton time τ C = /(mc 2 ) for a particle of mass m, corresponding to the phase transition energy.
Taking into account the possible expansion of the area of the new phase, the lifetime of the new phase can only decrease.Let's give formulas under the assumption of a constant expansion rate dR/dt = v ≈ const.Then and after the integration of ( 19) we obtain a dependence of the temperature of the region with a new phase on time In the limit v/c → 0, one obtains the expression (21).Thus, the lifetime of the new phase obtained in a collision of elementary particles has the order of Compton time /(mc 2 ) for a particle of characteristic mass scale m.For the quark-gluon phase transition this time is τ ≈ 3 • 10 −24 s, for the electro-weak phase transition τ ≈ 7 • 10 −27 s.

Phase transition in macroscopic volume
To perform a quark-gluon or electro-weak phase transition in a macroscopic volume, it is necessary to collide with ultra-relativistic energies of macroscopic amounts of matter.When ordinary macroscopic bodies collide with such energies, the regions of the new phase can make up a macroscopic volume only if the density of the bodies is comparable to the Compton density characteristic of the phase transition of mass m (see (12)).Only in this case, the lifetime of the new phase can significantly exceed the Compton time τ C .Such density of matter occurs only in neutron stars.Collisions of macroscopic objects with ultra-relativistic velocities are possible in the vicinity of the horizon of extremal rotating black holes [13].The collision of compact objects with star masses near supermassive black holes was considered in [23].
When falling towards the event horizon of a black hole, macroscopic bodies can be destroyed by tidal gravitational forces.Let's estimate the mass of black holes in which it is possible to fall to the event horizon of neutron stars without destruction by tidal forces.For evaluation, we assume that a star is destroyed if the tidal forces for the points of the center of mass and the surface exceed the force of attraction of the points of the surface to the center of the falling body.Let us assume that the falling object (neutron star) is a uniform ball of density ρ and radius R. Also let's consider only the nonrotating black hole and radial tidal forces.Then the condition for the falling to the horizon without destruction has the form 2GM or (after simple transformations) where M ⊙ is the Sun mass, ρ w = 10 3 kg/m 3 is the water density.Neutron stars have the density ρ ∼ 10 17 − 10 18 kg/m 3 .Therefore, neutron stars fall to the horizon of black holes with mass M > 20M ⊙ without destruction.Of course, a collision with ultra-relativistic velocities of neutron stars in the vicinity of a massive black hole should be considered a very unlikely event.Estimates in [13] show that in the collision near the vicinity of the horizon extreme rotating black hole with a mass of 10 9 M ⊙ in points with radial coordinate r H + 7 • 10 5 km the maximum collision energy in the center system mass can reach 100mc 2 .In nucleon-nucleon collisions, this is the electro-weak unification energy.The masses of neutron stars range from one to three solar masses, and their radii are about 10 -20 km.
The gravitational radius of a black hole with a mass of 100 solar masses is approximately 300 km.Therefore, with such collision energy of two neutron stars, a black hole should form and it will not be possible to obtain a substance in a state of an electro-weak phase transition outside the event horizon.Thus, it is impossible to obtain macroscopic quantities of a substance with electro-weak phase transition with a lifetime significantly exceeding Compton time for the electro-weak scale due to collisions in the vicinity of black holes.

The influence of spontaneous symmetry breaking on the space-time metric
Let us consider a real scalar field with self-action [24] V Here, µ = μc/ , μ is a mass parameter, λ is the dimensionless self-action constant.Stable equilibrium states of such field are located at two points The potential function (26) can be written as Both lower states have zero energy, and the unstable equilibrium with ϕ = 0 has an energy density Using the representation ϕ = ϕ 0 + χ one obtains Thus, the mass of the χ field is √ 2λϕ 0 = √ 2µ.In the case of Higgs boson m H = 125.3GeV and we have where l H C = /(m H c). For the electro-weak interaction, quantum corrections lead to the limitation [24] λ where e is the elementary electric charge, ε 0 is the electric constant.
To estimate the inverse influence of the scalar field on the curvature of space-time we use Einstein's equations where Λ is the cosmological constant, T ik is the energy-momentum tensor of background matter.The energy-momentum tensor for a constant scalar field with minimal coupling with curvature is [25] T and is similar to the contribution of an additional cosmological constant.On the appearance of a non-zero cosmological constant under spontaneous symmetry breaking, it was indicated in the work [26].Phase transition in electro-weak interactions was discussed in cosmology by Kirzhnits and Linde [27,28], Weinberg [29] and others.Estimates of changes in the value of the cosmological constant during phase transitions in the early Universe were made at work [30].
If there is only a constant scalar field and the energy-momentum tensor of the background matter is equal to zero T (0) ik = 0, then the solution to the Einstein equations (33) will be the de Sitter space-time.In de Sitter space one has and it follows from (33) where l Pl is the Planck length For the electro-weak case under ϕ = 0 from (31) we have For the radius of curvature we obtain This value is many orders of magnitude greater than the Compton wavelength of the particle and the size of the region in which the phase transition occurs.It should be expected that in order for special collisions with ultra-high energy to occur, in such volumes r 3 there must be a large number of particles falling onto the black hole.Their total mass will be much greater than the mass of the electro-weak scale.Thus, the inverse effect of energy density in the electro-weak phase transition in collisions on the space-time metric can be neglected.

Conclusion
An integral part of the standard model of particle physics is the mechanism of spontaneous symmetry breaking.The discovery at the Large Hadron Collider of the Higgs boson in 2012 makes us take seriously the possibility of a phase transition from one vacuum to another at high temperatures, as it is the case in quantum non-relativistic many-body theory, where the ground state plays the role of the vacuum.In our work [13] was shown that in the processes of collisions of particles near the horizon of black holes, it is possible to achieve energies in the system of the center of mass of the order of the energy scale of the electro-weak phase transition.
In this article we showed that the region of phase transition in such collisions is microscopic.In order of magnitude, the size of the region is equal to the Compton wavelength of the Higgs boson.Using formulas for black body radiation, we show that the lifetime of such region is of the order of the Compton time for the electro-weak phase transition scale.
During a phase transition in the case of spontaneous symmetry breaking, the energymomentum tensor corresponds to the emergence of an effective cosmological constant.It is shown that for phase transitions during particle collisions its influence on the space metric in the phase transition region can be neglected.
Note that despite the short time existence and microscopic volumes of a new phase of matter during an electro-weak phase transition in collisions in the vicinity of the black hole horizon, its very existence is of fundamental importance for the study of elementary particle physics in the ultra-high energy region, unattainable on Earth.
Author Contributions: A.A.G. and Y.V.P. have contributed equally to all parts of this work.All authors read and agreed to the published version of the manuscript.
Funding: This research received no external funding