On dark stars, Planck cores and the nature of dark matter
OOn dark stars, Planck coresand the nature of dark matter
Igor NikitinFraunhofer Institute for Algorithms and Scientific ComputingSchloss Birlinghoven, 53757 Sankt Augustin, [email protected]
Abstract
Dark stars are compact massive objects, described by Einsteingravitational field equations with matter. The type we consider pos-sesses no event horizon, instead, there is a deep gravitational wellwith a very strong redshift factor. Observationally, dark stars canbe identified with black holes. Inside dark stars, Planck density ofmatter is reached, Planck cores are formed, where the equations aremodified by quantum gravity. In the paper, several models of darkstars with Planck cores are considered, resulting in the following hy-pothesis on the composition of dark matter. The galaxies are floodedwith low-energetic radiation from the dark stars. The particle typecan be photons and gravitons from the Standard Model, can also be anew type of massless particles. The model estimations show that theextremely large redshift factor z ∼ and the emission wavelength λ ∼ m can be reached. The particles are not registered directlyin the existing dark matter experiments. They come in a density suf-ficient to explain the observable rotation curves. The emission has ageometric dependence of density on radius ρ ∼ r − , producing flat ro-tation curves. The distribution of sources also describes the deviationsfrom the flat shape. The model provides a good fit of experimentalrotation curves. Outbreaks caused by a fall of an external object ona dark star lead to emission wavelength shifted towards smaller val-ues. The model estimations give the outbreak wavelength λ ∼ mcompatible with fast radio bursts. The paper raises several principalquestions. White holes with Planck core appear to be stable. Galacticrotation curves in the considered setup do not depend on the mattertype. Inside the galaxy, dark matter can be of hot radial type. Atcosmological distances, it can behave like the cold uniform type. Keywords: Planck stars, RDM-stars, TOV-stars, dark matter1 a r X i v : . [ a s t r o - ph . GA ] F e b igure 1: On the left and in the center: an RDM-star – a black hole, coupledto radial flows of dark matter. On the right: experimental rotation curvesfor three galaxies. Image from [3], data from [4]. Dark stars, also known as quasi black holes, boson stars, gravastars, fuzzballs,are solutions of general theory of relativity, which first follow the Schwarz-schild profile and then are modified. Outside they are similar to black holes,inside they are constructed differently, depending on the model of matterused. An overview of these models can be found in the paper by Visseret al. “Small, dark and heavy: but is this a black hole?” [1]. A recentadvance has been reported by Holdom and Ren in their paper “Not quitea black hole” [2]. Our contribution to this family are RDM-stars [3], quasiblack holes coupled to Radial Dark Matter. A typical configuration of anRDM-star is shown on Fig.1 on the left and in the center. It is a station-ary solution, including T-symmetric superposition of ingoing and outgoingradially directed flows of dark matter.An RDM-star can be used as the simplest model of a spiral galaxy. In thelimit of weak gravitational fields, the dark matter flows radially convergingtowards the center of the galaxy produce a typical geometric dependence ofmass density on the radius ρ ∼ r − , which corresponds to constant orbitalvelocity v = Const , flat rotation curve. It is a qualitatively correct behaviorfor many experimental rotation curves at large distances, see Fig.1 on theright. We will show that a distribution of RDM-stars in the galaxy alsoallows to describe correctly the deviations of rotation curves from the flatshape. The model of RDM-stars fits very well the experimental rotationcurves by Sofue et al. [4–8] and Salucci et al. [9–13].In strong gravitational fields, RDM-stars behave interestingly. First ofall, the event horizon, typical for real black holes, is erased. Instead, adeep gravitational well is formed, where the values of the redshift becomeenormously large. As a result, for an external observer the star looks black,2ike a real black hole. Simultaneously, the mass density increases rapidly,reaching and exceeding the Planck value.This is where Planck stars come into play. This model is based on thecalculations in quantum loop gravity, performed for a scalar field cosmol-ogy by Ashtekar et al. [14–16]. According to these calculations, the massdensity has a quantum correction: ρ X = ρ (1 − ρ/ρ c ) , where the criticaldensity ρ c ∼ ρ P is of the order of Planck value, ρ is the nominal densitybefore the correction and ρ X is the effective density participating in Ein-stein field equations. As a result of this correction, ρ = ρ c corresponds to ρ X = 0 , at the critical density the gravity is effectively switched off, while ρ > ρ c corresponds to ρ X < , in excess of critical density the effective neg-ative mass appears (exotic matter), with gravitational repulsion (quantumbounce phenomenon). In the Planck star model by Rovelli, Vidotto [17],Barceló et al. [18], a collapse of a star leads to the quantum bounce, isreplaced by extension, as a result, the black hole turns white.In this paper we consider a stationary version of a Planck star, sta-bilized under the pressure of the external matter (Planck core). We willconsider two stationary spherically symmetric models with a Planck corein the center. The subject is related to the stability of white holes, earlierinvestigated in papers by Ori and Poisson [19], Eardley [20], Zel’dovich,Novikov and Starobinskij [21]. It is also related to the origin of fast radiobursts and gives an unusual viewpoint on the nature of dark matter.The paper is organized as follows. In Section 2 the model of RDM-starsis considered. The main computations have been performed in the author’spaper [22], here a short overview of the results is given. In Section 3 themodel of TOV-stars with Planck core is presented. In Section 4 the nature ofdark matter according to the considered models is discussed. A theoreticallyinteresting question on stability of white holes is considered in the Appendix. RDM-stars and rotation curves of galaxies.
RDM-star geometry canbe used as a simplest model of dark matter distribution in spiral galaxies.Let us consider dark matter flows radially converging towards the center ofa galaxy, displayed on Fig.1 center, in the limit of weak gravitational fields.The one-line calculation ρ dm ∼ r − , M dm ∼ r, v = GM dm /r = Const (1)evaluates mass density, enclosed mass function and orbital velocity of stars.Adding a concentrated mass in the center, v = GM /r + Const , the ro-tation curve described by the sum of Keplerian and constant terms can be3btained. The real rotation curves, displayed on Fig.1 right, possess a sim-ilar structure, with Keplerian behavior at small distances and flat shape atlarge distances. The red line shows a segment 2-20kpc where the rotationcurve for the Milky Way can be considered as approximately flat, with theSun position at 8kpc. These plots show that the real rotation curves devi-ate from a simple sum of Keplerian and constant terms, revealing additionalstructures, oscillations. On the other hand, the model with a single RDM-star in the center of the galaxy is also a simplification. Further we consider amodel of distributed RDM-stars, able to capture the additional structures.Then we perform a calculation in the limit of strong gravitational fields toanalyze the interior structure of an RDM-star.The detailed description of rotation curves in the RDM-model is basedon two assumptions: (1) all black holes are RDM-stars; (2) their density isproportional to the concentration of the luminous matter in the galaxy. Asa result, the dark matter mass density can be represented by the integral ρ dm ( x ) = (cid:90) d x (cid:48) b ( | x − x (cid:48) | ) ρ lm ( x (cid:48) ) , b ( r ) = 1 / (4 πL KT ) /r . (2)Here ρ lm is the density of luminous matter, the kernel b ( r ) represents a con-tribution of a single RDM-star and L KT is a parameter of length dimension,regulating a coupling between the dark and the luminous matter. This formof coupling has been proposed earlier in a context of a different model inworks by Kirillov and Turaev [23, 24]. The physical meaning of the L KT parameter is the radius at which the enclosed mass of dark matter equals tothe mass of the luminous matter, to which it is coupled: M dm ( L KT ) = M lm . The detailed rotation curve of Milky Way, known also as GrandRotation Curve (GRC), has been constructed on the basis of various ex-perimental data by Sofue et al. [4–8]. This curve is presented on Fig.2 bydata points with errors. Here one can see several structures, including Ke-plerian contribution of the central black hole (BH), inner and outer bulges(LM1,2), galactic disk (LM3), followed by dark matter contribution (DM)and background outer part (bgr). The red line with the marked Sun positionrepresents the same 2-20kpc approximately flat interval as on the previousfigure. This part appears to be relatively small due to a much larger rangeof distances involved in the analysis.In paper [8], the distribution of luminous matter in the bulges is de-scribed by exponential spheroid model , representing the mass density byan exponent ρ lm ∼ exp( − r/a ) . For the galactic disk Freeman’s model [25] is used, with the surface mass density described by similar exponent ρ lm ∼ δ ( z ) exp( − r/R D ) . Taking these distributions, the integral (2) andthe resulting rotation curve v ( r ) can be evaluated analytically. The lengthy4igure 2: Detailed rotation curve for Milky Way, fitted by RDM-model.Blue points with error bars – data from [8]. Green curve – fit by RDM-modelfrom [22] for three coupling scenarios (s1-3). Contributions of differentgalactic structures are also shown.explicit expressions per every structure are given in [22], also used there asbasis functions for the fitting procedure. For stability of the fit, the relativecoupling of dark matter to different structures of luminous matter has beenfixed as shown in Table 1. The λ -constants are used as multiplicative factorsto integrals (2). Since the different galactic structures may possess a differ-ent density and different population of black holes, we can select differentcoupling constants for them. This procedure is equivalent to a readjust-ment of the corresponding L KT -parameters, while we prefer to use a single L KT -parameter and adjust the individual couplings by relative λ -factors.Three scenarios have been considered in Table 1, the first one assigns alldark matter coupling to the galactic disk, the second one introduces equalcoupling among all structures, the third one describes a prevailing couplingfor the central structures.The result of the fit is shown by curves on Fig.2. The green line repre-sents the total rotation curve, a quadratic sum over all structures. It hasalmost the same shape for all three scenarios. Also, the separated contribu-tions of different structures are shown. They depend on the scenario, e.g.,the third scenario with prevailing dark matter coupling to central structuresalso shows a considerable contribution of dark matter in the center. Table 2presents the obtained fitting parameters – the total masses and geometricsizes of the structures. In the considered modeling, the dark matter halo issharply cut at the radius R cut , further providing Keplerian fall of the outerpart of the rotation curve, followed by its linear increase due to the uniformbackground density. Interestingly, the parameters, characterizing the outerpart of the rotation curve, the total mass of dark matter halo M dm ( R cut ) and the background density ρ bgr , appear to be approximately the same forthe considered three scenarios. 5able 1: GRC fit: coupling coefficients for 3 scenarios λ KT s1 s2 s3 λ smbh λ λ λ disk Table 2: GRC fit: the results* par s1 s2 s3 M smbh . × . × . × M . × . × . × a . . . M . × . × . × a .
13 0 .
13 0 . M disk . × . × . × R D . . . L KT . . . R cut
58 45 53 M dm ( R cut ) 2 . × . × . × ρ bgr
646 653 649 * masses in M (cid:12) , lengths in kpc , density in M (cid:12) /kpc mag are separated. Other galaxies can be modeled with a concept of a Universal RotationCurve (URC) introduced by Salucci et al. [9–13]. It represents averagedexperimental rotation curves of more than 1000 galaxies. Before averaging,the galaxies are subdivided to bins over the magnitude mag and the curves v ( r, mag ) are normalized to the values at optical radius: v/v opt , r/R opt .Here, v opt = v ( R opt ) and the optical radius of the galaxy R opt = 3 . R D isdefined as a distance, under which 83% of the luminous mass is located. Theaveraging smooths the individual features of the curves, their local minimaand maxima. The resulting experimental curves appear to be more smoothand are shown by points with errors on Fig.3.On these plots, the radius and velocity are presented in a linear scale,rather than the logarithmic one used in previous plots. As a result, theearlier described central structures are shrinked to a single unresolved cen-tral contribution. The modeling is accordingly simplified, preserving onlythe central and the disk contributions. The basis functions are explicitlywritten in [22], the result of the fit is presented by green curves on Fig.3.The presented plots show that the model of distributed RDM-stars,based on the Newtonian weak field limit and the proportionality assump-tions above, provides a good fit of the experimental rotation curves, forboth GRC and URC types. 7igure 4: RDM-star model in strong fields. A typical solution in differentcoordinates (see text). RDM-star model in strong fields.
The system to solve is combinedfrom Einstein gravitational field equations and geodesic equations: G µν = 8 πG/c · T µν , u ν ∇ ν u µ = 0 , ∇ µ ρu µ = 0 . (3)We consider the model with T-symmetric non-interacting superposition ofingoing and outgoing flows of dark matter. Therefore, geodesic equationscan be applied separately for every flow, described by velocity field u µ andintrinsic mass density ρ . A static spherically symmetric metric is chosen: ds = − Adt + Bdr + Dr ( dθ + sin θ dφ ) , (4)where the profile A ( r ) > describes the redshift and time delay effects, B ( r ) > measures geometric deformation in radial direction. D = 1 canbe put by convention, so that r is aerial radial coordinate, the area of r -sphere is πr . Time t is measured by the clock of a distant observer, where A → can be set. Energy-momentum tensor is taken in a form T µν = ρ ( u µ + u ν + + u µ − u ν − ) , u ± = ( ± u t , u r , , , (5)a sum of T-symmetric radial flows of non-interacting dust matter.The equations (3) have been solved in [3,22]. Geodesic equations possessanalytical solution πρ = c / (cid:16) r u r √ AB (cid:17) , (6) u t = c /A, u r = (cid:113) c + c A/ √ AB, in G = c = 1 normalization. The integration constants c − will be consid-ered in details later. The Einstein equations have a form rA (cid:48) = − A + AB + 4 c B (cid:113) c + c A, (7)8 B (cid:48) = B/A (cid:18) A − AB + 4 c c B/ (cid:113) c + c A (cid:19) , they can be solved numerically. The typical solution is shown on Fig.4left, in ( A, B ) -coordinates. Initially, near the point x , the curve has ahyperbolic form, typical for Schwarzschild solution. The difference startsnear the point x , where the Schwarzschild solution goes to infinity, theevent horizon is formed. In the considered solution, the dark matter actslike a barrier, preventing the formation of the horizon. The solution thengoes rapidly towards very small values of A and B , where it exhibits astrong redshift and possesses a small proper length. Then the solution goesto large values of A , a strong blueshift. The same solution is shown in thecentral part of this figure, in logarithmic coordinates, and on the right part,presenting a Misner-Sharp enclosed mass function: x = log r, a = log A, b = log
B, M = (1 − B − ) r/ . (8)In these coordinates the equations obtain the form more convenient for anumerical solution a (cid:48) x = − e b + c e b − a √ c e a , (9) b (cid:48) x = 1 − e b + c e b − a / √ c e a , (10) c = 4 c c , c = c /c . (11)The convenience follows from the resolution of singularities, typical for poly-nomial formulation, so that the resulting equations can be easily solved, e.g.,by Mathematica
NDSolve algorithm. Also each term in these equations hasa clearly defined range of domination, so that normally only one term in theequation is active. This simplifies the asymptotic analysis of the system.The behavior of the mass function on Fig.4 right shows that the solutionis bounced off horizon line M = r/ , then falls very rapidly. This fall isrelated to the phenomenon of mass inflation, described in the paper byHamilton and Pollack [26]. There is a positive feedback loop in black holesolutions with counterstreaming matter flows: (1) increasing energy of thecrossing flows leads to (2) increasing pressure, that leads to (3) increasinggravity, that leads again to (1). As a result, an accumulation of very largemass in the counterstreaming region happens. For the considered solutions,the function M ( r ) decreases with decreasing r . To explain this property, onecan imagine spherical shells of positive mass consequentially removed fromthe star. Finally, the mass arrives to a negative central value, a concentratednegative mass. It corresponds to the well known Schwarzschild singularityof naked type and explains the appearance of a blueshift region in thesolution. On the other hand, the singularity is coated in a massive shell9ppearing due to the mass inflation phenomenon, so that the total massof the system remains positive. Also, below we will introduce a quantumgravity cutoff in the model, which will remove the naked singularity withmost of surrounding structures.The integration constants c , > , while c = u µ u µ = − , , +1 cantake three discrete values, corresponding to the type of dark matter parti-cles: massive, null or (theoretically) tachyonic. Interestingly, the solutionin strong fields ( A (cid:28) ) depends on the matter type very weakly, since thecorresponding term c A in the equations becomes small. Solution in weakfields ( A ∼ ) depends, at first, on the parameter c that defines asymptoticradial velocity of the dark matter: for c < − , the massive radial flow hasa turning point, the matter cannot escape; c > − , possible for all mattertypes, the matter can escape to large distances, the case further considered: c = c √ c , c = c / √ c , (cid:15) = ( c + c ) / . (12)The parameter (cid:15) defines an asymptotic gravitating density ρ grav = ρ eff + p eff . The effective density and pressure, produced by counterstreamingdark matter flows, are defined as components of energy-momentum tensor T νµ = diag ( − ρ eff , p eff , , , where ρ eff = c / (8 πr A ) / (cid:112) c A, p eff = c / (8 πr A ) · (cid:112) c A. (13)In the weak field limit A ∼ we obtain ρ grav = (cid:15)/ (4 πr ) , M grav = (cid:15)r ,as in (1). This makes (cid:15) a directly measurable parameter, in physical units (cid:15) = ( v/c ) , where v is the orbital velocity of stars at large distances fromthe galaxy center, for Milky Way v ∼ km/s, (cid:15) ∼ · − . Quantum gravity cutoff.
Further, we omit index eff in the formulae, as-suming that the effective density and pressure are always considered. Also,for definiteness, we fix the dark matter to null type (NRDM). The resultingmodel is equivalent to a perfect fluid with the following equation of state(EOS): ρ = p r , p t = 0 , (14)there is a relativistic relation between mass density and radial pressure,while the transverse pressure is switched off. The formulae (14) become ρ = p r = (cid:15)/ (8 πr A ) . (15)Further, for illustration, we consider the solution for the Milky Way galaxywith a concentrated RDM-star in the center. Fig.5 left shows the corre-sponding metric profiles. The solution starts in point 1 far away from the10igure 5: On the left: quantum gravity cutoff in RDM-model for MilkyWay scenario. In the center: a mechanism for generating FRB in RDM-model. On the right: simultaneous analysis of rotation curves and FRBs inRDM-model. Images from [22].center, then in point 2 attempts to go to the Schwarzschild regime. We re-mind that the scale is logarithmic and the metric profiles jump many ordersof magnitude near the point 2. Then they fall into abyss due to the redsupershift phenomenon. Much earlier than the A -profile reaches minimumin point 3, the Planck density is achieved ρ ∼ ρ P . At this point we stop thesolution and place a Planck core below it. Since the B -profile is also verysmall at this point and according to the formula (8), the Planck core pos-sesses negative total mass, whose repulsive force supports the whole systemin equilibrium. In further computations, only the order of the magnitudeis important, on necessity, corrections can be applied via phenomenologicalfactors [22]. Taking into account that ρ P = l − P in the units used, where l P is Planck length, also that redshift factor falls rapidly at almost constant r ∼ r s , the value in the cutoff point becomes A QG = (cid:15) ( l P /r s ) / (8 π ) . (16)For the Milky Way, substituting the estimation of the (cid:15) -parameter aboveand the known gravitational radius r s of the central black hole from Ghezet al. [27], we obtain A / QG = 1 . · − . This value will be important forour further calculations. RDM-stars as sources of Fast Radio Bursts.
The common propertyfor all dark star solutions is the presence of high energetic phenomena andstrong redshift in their depths. Therefore, high energy photons created inthese phenomena on the way out can be shifted to a long wave diapason.11his makes dark stars natural candidates for sources of FRBs, the powerfulflashes of extragalactic origin, registered in radio band. The lowest FRBfrequency of 111 MHz has been reported by Fedorova and Rodin [28], thehighest of 8 GHz – by Gajjar et al. [29]. Detailed experimental characteris-tics of FRBs can be found in frbcat catalogue by Petroff et al. [30], there isalso a catalogue of existing FRB theories frbtheorycat by Platts et al. [31].At the time of this writing, 118 distinct FRB sources have been registeredand 59 FRB theories have been created.A particular scenario with an RDM-star generating an FRB has beenconsidered in [22]. An object of an asteroid mass falls onto the RDM-star. The gravitational field inside the star acts as an accelerator withsuper-strong ultrarelativistic factor γ = A − / QG ∼ . The nucleons N composing the asteroid enter in the inelastic collisions with particles X forming the Planck core, producing the excited states of a typical energy E ( X ∗ ) ∼ √ m X E N . The high-energy photons formed by the decay of X ∗ with energy E ( γ, in ) ∼ E ( X ∗ ) / are subjected to super-strong redshiftfactor γ − = A / QG ∼ − . The γ -factors do not compensate each other dueto the presence of the square root in the formula. Thus, the outgoing energy E ( γ, out ) ∼ (cid:112) m X m N / (2 γ ) , the wavelength λ out = √ λ X λ N γ . Taking λ X ∼ l P , we obtain a formula for FRB wavelength λ out = 2(2 π ) / (cid:112) λ N r s /(cid:15) / , (17)containing only Compton wavelength of nucleon λ N and ( r s , (cid:15) ) -parametersof the RDM-star. Interestingly, Planck values are canceled out of the for-mula. Further, taking λ N = 1 . · − m, r s = 1 . · m, (cid:15) = 4 · − ,the wavelength and frequency of FRB are λ out = 0 . m , ν out = 0 . GHz , (18)that falls in the observed range 0.111-8GHz of FRB frequencies.Further evaluations can be found in [22]. A snowball mechanism is in-troduced for generating a sequence of excited states, which produces theenergy spectrum of photons cut from above by the computed E out value.The spectrum is open towards low energy values, however, the increas-ing scatter broadening dilutes the signal there. A common mechanism ofstimulated emission (aka LASER) can generate a short pulse of coherentradiation, by the scheme displayed on Fig.5 center. Other parameters, suchas pulse width and pulse delay, spectral and beam efficiency, as well as po-larization, repetition and periodicity, observed for some FRBs, have beenalso discussed in [22]. Most of these parameters are insensitive to the na-ture of the FRB source, being imparted by local environment and/or in-terstellar/intergalactic medium on the way of signal propagation. These12arameters can be described by the known source independent astrophys-ical mechanisms, such as scatter broadening and signal dispersion, as wellas scenarios with an FRB source passing through a planetary system or anasteroid belt.The estimation above has been made for a simplified scenario with aconcentrated RDM-star in the center of the galaxy with Milky Way alikeparameters. Fig.5 right shows more possibilities. The coordinates ( r s , (cid:15) ) are the gravitational radius and the parameter defining the contributionof a particular black hole (=RDM-star) to the galactic dark matter halo, (cid:15) = GM λ/ ( c L KT N ) , where M, λ, L KT are parameters from the fit of thegalactic structures explained at the beginning of this section, N is the num-ber of black holes in the structure. The band shows detected FRB frequen-cies between the lines (a) and (b), according to (17). Two horizontal linesshow two classes of solutions, supermassive and stellar black holes, accordingto the scenarios considered above: (c) s2, (d) s3, (e) corresponds to the min-imal velocity value v ∼ km/s on the plots Fig.2, (f) (cid:15) = 4 · − dividedto N = 10 stellar black holes, (g) the same with N = 10 , the estimationsof the number of stellar black holes are from Wheeler and Johnson [32] andreferences therein.The main conclusion from the analysis of the plot on Fig.5 right is thatthe band and the horizontal lines have an intersection, therefore the modelof RDM-stars is able to describe simultaneously the rotation curves andFRBs. Moreover, two solution classes exist, stellar and supermassive blackholes. The plot is constructed on the basis of Milky Way data, extractedfrom its highly detailed rotation curve, and is valid for galaxies of similarstructure. It would be interesting to populate it with data from other galax-ies, that depends on the availability of rotation curves with a comparabledetalization. In this section we consider Tolman-Oppenheimer-Volkoff (TOV) stars. It iswell known system, described by EOS wρ = p r = p t , (19)differing from RDM-stars EOS (14) by the presence of two components oftransverse pressure T µν = diag ( − ρ, p r , p t , p t ) , equally distributed with theradial one (isotropic pressure). Parameter w regulates the composition andtemperature of the star. Small values w = kT / ( mc ) correspond to anideal gas of massive particles of a given temperature. In this section we willmainly consider an ultrarelativistic plasma or photon gas, corresponding to13he value w = 1 / . The Einstein equations have a form (see, e.g., Blau [33]): wρ (cid:48) r = − ( ρM/r )(1 + w )(1 + 4 πr wρ/M )(1 − M/r ) − , (20) M (cid:48) r = 4 πr ρ, h (cid:48) r = 4 πr (1 − M/r ) − ρ (1 + w ) , (21)where the metric coefficients are chosen as A = e h f, B = f − , f = 1 − M/r. (22)A consequence of this system is so the called hydrostatic equation r ( p + ρ ) A (cid:48) r + 2 Arp (cid:48) r = 0 , (23)possessing an analytical solution πwρ = k A k (24)with constants k = 4 πρ w, k = − (1 + 1 /w ) / , k = log k . (25)The system can be rewritten in logarithmic variables (8), to the form con-venient for a numerical solution: a (cid:48) x = − e b + 2 e x + k a + b + k , (26) b (cid:48) x = 1 − e b + (2 /w ) e x + k a + b + k . (27)They are complemented by initial data a = 0 , b = − log(1 − M /r ) ,where w = 1 / , k = − , k = log(4 πρ / , ρ and M at a large r aregiven. The typical solution is shown on Fig.6 left. The coordinates are x = log r and arcsinh M , the last one possesses asymptotics ± log | M | atlarge | M | , convenient to display all features of a solution in a single plot.Usually a regular solution is investigated, satisfying a condition M = 0 in the center. This solution is shown by a thick line on the figure. Weinvestigate what happens if this condition is relaxed. If a solution with thesame ρ is started with smaller M , below the regular line, it simply endsbelow this line in a negative central value. More interesting, if the solution isstarted above the regular line, it will not end in a positive central value andwill not cross a horizon. Instead, it bounces off the horizon, goes throughthe mass inflation and ends in an even more negative central value. Thesesolutions are clearly singular in the center, however, they are of interestto us, since the quantum gravity cutoff considered below can remove thesesingularities, replacing them with a regular Planck core.For completeness we also consider a case of Fig.6 center, when the so-lution is started above the horizon line, physically under the horizon. It14igure 6: TOV-star solutions. On the left: solutions with negative cen-tral mass, in the center: with positive central mass, with phenomenon ofZel’dovich-Novikov-Starobinskij explained, on the right: a typical behaviorof metric coefficients.similarly bounces off the horizon from inside and goes to the positive centralvalue. If one reverts the integration, the solution started under the horizonfrom a positive mass Schwarzschild singularity will stay inside the horizon.This phenomenon was discovered by Zel’dovich, Novikov, Starobinskij [21]investigating the formation of white holes under the influence of matterejected from the central singularity. The system is described by similarequations and the result is that the ejected matter never leaves the hori-zon and the white hole under the described circumstances cannot explode.This effect (internal ZNS instability) is one of instability types inherent towhite holes, the other one (external Eardley instability) will be consideredbelow in the Appendix. Mainly, in this paper, we consider a dual solution,possessing negative mass Schwarzschild singularity and evolving outside ofthe instability region.The plot on Fig.6 right shows the typical evolution of metric coefficients,appearing to be very similar to such plots for RDM-stars. An importantdifference is that the redshift fall and the mass inflation for a TOV-starappear to be much more moderate in comparison with an RDM-star ofsimilar parameters. Table 3 shows the scenario with a stellar mass compactobject in a cosmic microwave background, described by TOV equations.Although the variation of metric coefficients and enclosed mass in physicalunits is very large, it is still much smaller than the analogous variations forRDM-stars.Considering this scenario in more details, we see that r − r s ∼ − m,the object comes very close to the gravitational collapse. This is a distancewhere the matter terms, initially weak, representing cosmic microwave back-ground, are amplified and start to dominate in the equation. Although thisis the result of a purely classical model, quantum considerations can change15able 3: TOV-star, scenario with a stellar mass compact object in cosmicmicrowave background model parameters M = 10 M (cid:12) , w = 1 / , ρ = ρ cmb = 4 · − J/m starting point of r = 10 m, a = 0 , b = 0 . ,the integration M /M (cid:12) = 10 r = 29532 . m, a = − . , b = 53 . ,supershift begins M /M − − . · − , r − GM /c = 1 . · − m r = 20638 . m,supershift ends a = − . , b = − . , log ( − M /M (cid:12) ) = 46 . minimal radius r = 1 . · − m,(Planck length), a = − . , b = − . ,end of the integration M = 1 . M Table 4: micro TOV-star, the critical case model parameters w = 1 / , ρ = ρ cmb = 4 · − J/m , a QG = − . starting point of r = 1 . · − m, a = 0 , b = 0 . ,the integration M = 7 . · kg r = 1 . · − m,supershift begins a = − . , b = 73 . , r − GM /c = 2 . · − m r = 7 . · − m,supershift ends a = − . , b = − . , log ( − M /M (cid:12) ) = 54 . minimal radius r = 1 . · − m,(Planck length), a = − . , b = − . ,end of the integration M = 1 . M log ( − M /M (cid:12) ) ∼ , in comparison with the mass of theobservable universe: log ( M uni /M (cid:12) ) ∼ . Thus, the considered compactobject contains a core of negative mass, by absolute value much greaterthan the mass of the universe, compensated by the coat of TOV matterwith almost the same positive mass. A similar computation for RDM-modelgives an even larger number: log ( − M /M (cid:12) ) ∼ .These enormous numbers could be the result of model idealization.Their origin is the unrestrained phenomenon of mass inflation. It can bechanged if a (non-gravitational) interaction between the counterstreamingflows and corresponding corrections to EOS will be taken into account. Also,the considered solutions are stationary and can take enormous amount oftime to form. A qualitative interpretation of the obtained solutions is that apermission of negative mass (Planck core) leads to a polarization of the so-lution to the parts with highly positive and highly negative masses, almostcompensating each other in the result.The other origin of large numbers is Planck density: ρ P = 5 · kg/m .Straightforward estimation for the Planck density core of only R = 1 mmradius gives the mass M = (4 / πR ρ P = 2 · kg, gravitational radius R s = 2 GM/c = 3 · m, much larger than the mass and the radius of theobservable universe M uni = 10 kg, R uni = 4 · m. Such a core will im-mediately cover the universe by its gravitational radius, with a large margin.To place such objects in our universe, a mechanism for mass compensationis necessary. For instance, the one of this paper, effectively negative massescreated by quantum gravity and coated by positive mass shells until theequilibrium with a moderate mass value is reached.Enormous reserve of energy hiding inside TOV-stars can fuel extremelyhigh-energetic phenomena. Figuratively speaking, if such a bubble burstssomewhere, the consequences can be felt throughout the universe. Thus,it is natural to consider these objects as potential sources of FRBs and wewill do this, at first considering the quantum gravity cutoff and formationof Planck core in the center of TOV-star. Quantum gravity cutoff.
Setting w = 1 / in a solution of TOV hy-drostatic equation (24), obtain ρ ∼ A − . Therefore for the consideredscenario with cosmic microwave background: ρ P /ρ cmb = A − QG . Taking ρ P = 4 . · J/m and ρ cmb = 4 . · − J/m from Longair [34],in energetic units, have A QG = ( ρ cmb /ρ P ) / = 3 · − , a QG = − .The question now is whether such value can be reached. For RDM-starswith physically interesting parameters the redshift fall is enormous and thePlanck density can be always reached before achieving the minimum in17 -dependence. For a TOV-star, the redshift fall and associated density in-crease in solutions are moderate. The solutions shown in Fig.6 right andTable 3 pass the minimum a before reaching a QG and the Planck densityfor these solutions is not reached. The necessary condition for formation ofthe Planck core is a < a QG . To investigate a satisfaction of this condition,we have performed the following numerical experiment. Keeping the outerdensity fixed to ρ cmb , we changed the solution mass, or associated parame-ter x s = log r s , where r s is Schwarzschild radius in meters, in the range x s ∈ [ − , . After the integration of TOV-equations, we detected theminimum a and found that it is well approximated by linear dependence a = − .
089 + 4 . · x s . As a result, a < a QG condition is satisfiedat r s < r s,crit = 0 . mm (micro TOV-stars), M < . · kg, approxi-mately Moon’s mass. The critical case is shown in Table 4. The equality a = a QG and the resulting r ∼ r s confirms this computation. TOV-stars as sources of Fast Radio Bursts.
Let us consider a pho-ton of initially Planck energy, E in ∼ E P , λ in ∼ l P , on the surface ofthe Planck core. After applying the redshift, the outgoing wavelength λ out = l P A − / QG = 0 . mm. Experimentally it is λ exp = 37 . mm, for thehighest 8 GHz FRB detection of FRB121102 source [29]. The deviationfactor λ exp /λ out ∼ can still be considered as a good hit, taking intoaccount 127 orders of difference in the input density parameters ρ cmb /ρ P .Technically, it can be compensated by an attenuation factor E in = E P /N ,the initial photon is N ∼ times weaker than Planck energy. A part ofthis factor can be related with (1 + z ) cosmological redshift of the source, z ∼ . − . , the remaining factor to explain is N ∼ .The analytical formula for the wavelength is also interesting: λ out = l P ( ρ cmb /ρ P ) − / , or, in Planck units, simply λ out = ρ − / cmb , depending onlyon the cosmic microwave background density.Consideration of other FRB parameters proceeds similar to [22]. Mostof the parameters depend not on the source, but on its environment andpropagation medium of the signal. Here we consider one question: can thebursts repeat? For the critical case r s = r s,crit and isotropic estimationof the total burst energy from Cao et al. [35], there is an inner reserve ofenergy for . · kg · c / (10 − J ) ∼ − bursts. The energy can bealso refilled from the environment, e.g., a companion, an asteroid belt, etc.In this refilling, when the threshold r s > r s,crit is passed, the conditions forPlanck core existence disappear. This can trigger the FRB, that will returnthe system to r s < r s,crit state.In summary, TOV-stars can also be the sources of FRB, or may representa species of these signals. Differently to FRB from RDM-star, triggered by18he fall of an asteroid, TOV-star signals can be autogenerated, possessingalso a mechanism for autonomous oscillations around the critical state. Comparison of different models.
While both RDM- and TOV-starscan generate FRBs, the asymptotically flat rotation curves are generatedonly by RDM-stars. Only they possess the necessary ρ ∼ r − dependence,while the considered TOV solutions have ρ → Const > asymptotics.On the other hand, in Barranco et al. [36] a different solution of theTOV system has been investigated, possessing ρ ∼ r − dependence. It isa well known analytical self-similar solution, whose existence follows fromscale-invariance of the system, see, e.g., the work by Visser and Yunes [37].Due to the appropriate density profile, this solution can be used to describethe rotation curves, a configuration known as isothermal dark matter halo.In addition, the asymptotic velocity on this solution appears to be ( v/c ) =2 w/ (1 + w ) . The experimentally observed velocities are non-relativistic,achievable only for small w . From here a conclusion is drawn, that the darkmatter composing the galaxies “must be cold”.If one uses RDM model instead of TOV, a physically different systemwith the absence of transverse pressure is formed. Here all types of darkmatter produce the same density profile ρ ∼ r − and the same asymptoti-cally flat rotation curves. The value of the orbital velocity is defined by theparameter ( v/c ) = (cid:15) , while the type of the matter by the other parameter c . As a result, the consideration of rotation curves in the RDM model doesnot impose a restriction on the type of dark matter in the galaxies.Self-similar solutions of the TOV system form a very special class, dif-ferent from the ones considered in this section. The regular type solutionswe consider look like a ball of almost constant density, with a little bump ofdensity in the center, due to self-gravitation. The singular solutions we con-sider have the same outer asymptotics, just possess a concentrated negativemass or a regular Planck core in the center. Self-similar solutions possesssuch a strong self-gravitation, that the whole solution shape is changed, alsoat large distances. This is possible only at a very large mass of solution.Especially, for the photon gas we consider, the mass should be enormous tomake the light condense under its own gravitation. The computation shows ρ ∗ = (cid:15) ∗ / (4 πr ) , M ∗ = (cid:15) ∗ r , (cid:15) ∗ = 2 w/ (1 + 6 w + w ) , in geometrical units,for self-similar solution. With w = 1 / , at r = 3 . · m, the outer rangeof the Milky Way galaxy, it is ρ ∗ = 0 . J/m , M ∗ = 4 . · M (cid:12) , beingcompared with ρ cmb = 4 · − J/m , M cmb = (4 π/ ρ cmb r = 2 . · M (cid:12) .Thus, the mass characteristics of the system we consider are 13 orders ofmagnitude below the formation of self-similar solutions.The other question is an ability of Planck stars directly generate FRBs,19nvestigated by Barrau, Rovelli, Vidotto in [38]. The BRV model considersa collapse of primordial matter to a black hole going through the quantumbounce to the eruption of the white hole. The eruption appears at a delayedtime due to strong gravitation. The time of recollapse depends on the massof the star and is estimated to t = 0 . M , in Planck units. Equating it withHubble time, the mass and the size of Planck stars are estimated, createdat the Big Bang and exploding “today”: M = (5 t H ) / = 1 . · kg, r s = 2 M = (20 t H ) / = 0 . mm. This estimation comes close to the criticalsize of TOV-stars r s,crit = 0 . mm obtained in our model.The BRV model predicts an observable FRB signal at λ ∼ r s ∼ . mm.The cosmological redshift correction can be also applied. The result isnumerically similar to our model ( λ ∼ . mm), although obtained in acompletely different setup: recollapse of Planck stars vs redshift of photonsof initially Planck energy that arise in stationary TOV solutions with thePlanck core in thermal equilibrium with CMB. Our prediction λ ∼ ρ − / cmb and the BRV formula λ ∼ (20 t H ) / coincide up to a numerical factor ∼ . ,if one takes into account cosmological constraints Ω cmb = ρ cmb /ρ crit =4 . · − , ρ crit = 3 H / (8 π ) , t H = 1 /H .In the original BRV model of Planck stars only non-repeating FRBs arepossible. The work by Barceló et al. [18] proposes repeating recollapses andfinal stabilization of an object due to dissipative effects. Such a stationaryobject can be equivalent to the RDM- and TOV-stars discussed here. Af-ter its formation, it can produce both repeating and non-repeating FRBsdepending on the environment.The further paper by Barceló et al. [39] seems to “close” the topic ofPlanck stars, referring to Eardley instability of the white hole part. Dueto this instability, the white holes under the influence of external radiationwould turn into black holes, not having time to emit the FRB. Below, inAppendix, we will bring a contra-argument, showing that Eardley instabil-ity can be eliminated if the core of the white hole possesses negative mass.Physically, it can be the Planck core, formed as the result of quantum grav-ity corrections when the Planck density is reached. Therefore the modelsfrom the Planck star family, as well as the FRB estimates based on them,avoid the white hole instabilities in a self-consistent way. In this section we consider three hypotheses on the composition of darkmatter, based on the considered dark star models.20 ypothesis 1: the galactic dark matter can be cold, can be hot, produc-ing the same rotation curves.It follows from the solution properties of the RDM model, the orbitalvelocity depends only on intensity factor (cid:15) , not on matter constitution(cold/hot, M/N/T cases, controlled by the other constant c ). It can be anew type of particles, which can be sterile for interactions with the knownmatter sectors, i.e., enter only in gravitational interactions with them. Itcan be almost sterile, i.e., other interactions allowed at high energies inPlanck cores, while extremely weak at low energies outside.One more fascinating possibility is that the dark matter is composedof known particles, placed in an unusual condition. Let consider a photonof Planck energy, emitted from the surface of the Planck core of an RDM-star: E in ∼ E P , λ in ∼ l P . Applying the redshift A / QG ∼ − , have λ out = l P A − / QG ∼ m. It is an extremely large wavelength, about 4light days, 16 times the Sun-Pluto distance. Such longwave photons cannot be registered by usual means, e.g., via radio telescopes. Although theenergy of every such photon is extremely small, they come in numbersproviding the necessary mass density to explain the rotation curves of thegalaxies. The detailed consideration shows that at the Planck core onePlanck energy particle per Planck area per Planck time is emitted, thatcorresponds to Planck density and pressure on its surface. After that thefactor A / QG is applied twice, for redshift and gravitational time dilation, thenthe geometrical ( r s /r ) factor corresponds to the measured halo density ρ = p r = 1 /l P · A QG · ( r s /r ) = (cid:15)/ (8 πr ) , where for the redshift factor theformula (16) is used.In the considered scenario the particles should be massless. For massiveparticles the Compton length must be greater than λ out ∼ m, obviously,excluding lightest neutrino species and other massive particles. Those par-ticles do not overcome the gravitational barrier and remain bounded insidethe RDM-star. From the Standard Model, the only appropriate particlesfor this scenario can be photons and gravitons. Scenarios with massiveparticles should have a larger starting energy to overcome the barrier. Hypothesis 2: the emission of galactic dark matter from a Planck coreis T-symmetric, in future and in past directions.We remind that an RDM-star contains two T-symmetric flows, ingoingand outgoing ones. Fig.7 left shows the mass shells for momentum P orvelocity u vectors. There is a one-sheet tachyonic shell, containing bothingoing and outgoing directions, and two-sheet massive/null shells, wherethese directions are separated. In any case, we assume that all mass shells21igure 7: Illustration to hypothesis 2. On the left: the mass shells formassive, null and tachyonic particles. In the center: a difference betweenBig Bang and Planck core light cones structure. On the right: the worldlines of dark matter particles captured by a wormhole.become completely occupied at the Planck core. The reason can be an ex-tremely high temperature, in Planck range T ∼ T P , the one achievable atBig Bang. It is so hot there, that the vicinity of the Planck core becomesinsensitive to the external thermodynamical time arrow and develops anown, T-symmetric thermodynamics. An important difference in this con-text is that RDM singularity and Planck core are timelike, while Big Bangsingularity is spacelike. Different orientation of light cones can lead to theabsent time arrow (recovered T-symmetry) near the Planck core and itspresence near/after the Big Bang. This difference is shown in Fig.7 center,the Big Bang light cones have only the upper part, while the Planck corelight cones have both, T-symmetrically occupied parts.One technical remark about P < parts of the mass shells. Althoughthey formally correspond to negative energy and seem to be related withnegative mass exotic matter, really they just correspond to T-conjugatedflows of the same particles as P > counterparts. To verify this, considerT-reflection, that reverts P µ and u µ vectors, as well as orientation of theworld lines, while preserves the action A = m (cid:82) dτ | x (cid:48) µ x (cid:48) µ | / and the energy-momentum tensor T µν = ρu µ u ν , which are only physically important.One more exotic possibility for T-symmetric emission is that the worldlines of dark matter are captured by a geometry of wormhole, as shown onFig.7 right. The ingoing flows from one universe become outgoing in theother universe and vice versa. In a stationary scenario, their T-symmetricsuperposition can be chosen.Independently on the detailed properties of the considered models, theemission of T-symmetric type is necessary due to simple physical reasons. Ifonly outgoing flows would be present, the total mass of a dark matter halo22igure 8: Illustration to hypothesis 3. On the left: evolution of photon gas instandard cosmology. In the center: the same with RDM-stars. On the right:Swiss cheese model with galaxies filled by hot dark matter, surrounded bycold dark matter, in expanding universe.could not be greater than the mass of (quasi) black holes, from where itoriginates. Experimentally, the halo mass is much greater than the mass ofblack holes. In the considered setup, ingoing and outgoing flows compensateeach other and allow for arbitrary ratio between halo and black hole masses. Hypothesis 3: the cosmological dark matter mimics cold type.The common opinion is that dark matter both in the galaxies and inbetween them is cold, i.e., is composed of massive non-relativistic particles.The hot cosmological dark matter would lead to a different expansion rateof the universe. Let us consider an evolution of uniform photon gas instandard cosmology, as shown schematically on Fig.8 left. There is aninitial flash, then the energy and the number density of photons fall in theexpanding universe. For cold dark matter only the density falls. This makesa difference to the evolution of the energy-momentum tensor.On the other hand, the distribution of dark matter photons in the RDMmodel is different, see Fig.8 center. Their initial energy at Planck core isfixed: E ∼ E P , the exit energy is also fixed by the local A / QG factor. Ifthe resulting distribution will possess a constant temperature, then in thelong-range evolution it will behave like cold dark matter.The other possibility is that EOS of cosmological dark matter is notidentical to the galactic one. There is a class of Swiss cheese models, wherethe galaxies and their halos do not change their size and structure under cos-mological expansion and move as a whole. The cosmological expansion actsonly on the level where the matter distribution can be considered as uni-form. The galaxies coated in massive halos can behave like macro-particlesof cold dark matter, as shown on Fig.8 right. Internally they can be filledwith hot radiation, externally produce the same gravitational fields as coldmassive particles. 23s we have mentioned earlier, while fitting the Milky Way rotationcurve, the experimental data are compatible with the presence of a cut ofdark matter halo at R cut ∼ kpc. While this cut was taken in the modeljust phenomenologically, the physical mechanisms for it can be constructed.Two-phase distribution, with hot radial dark matter inside the galaxy joinedto cold uniform dark matter outside, can be used. It resembles a knownphenomenon of termination shock on the border of the Solar system, wherethe solar wind meets the interstellar medium. This can be modeled similarlyin galactic scales. A suitable mechanism for the termination can be anykind of interaction of dark matter particles in the ingoing/outgoing flowsand the outer medium. In particular, it can be an absorption or a scatterof longwave dark photons by the intergalactic medium. In this paper we have experimented with the insertion of Planck core inseveral earlier known astrophysical models. What becomes possible as aresult of such modification:(1) RDM solutions can be properly continued to the strong field mode.These are stationary solutions describing black holes, coupled to the radialflows of dark matter. In weak fields, such configuration of dark matter canbe used as a model of spiral galaxies possessing realistic rotation curves.In this model, the geometric dependence of the density on the distance ρ ∼ r − , typical for the RDM configuration in a single center approximation,gives flat rotation curves, while assuming the coupling of all black holes inthe galaxy to RDM, deviations of the rotation curves from the flat shapeare also described. In strong fields, a peculiar phenomenon of erasing theevent horizon occurs; instead, a spherical region of super-strong redshift isformed. This phenomenon is accompanied by the effect of mass inflation byHamilton-Pollack, in a thin layer near the gravitational radius a very largepositive mass is accumulated, approximately compensated by the negativemass of the Planck core. Outside, such an object, an RDM-star, is perceivedas a Schwarzschild black hole of limited mass.(2) when an external body, for example, an asteroid, falls on an RDM-star, a flash of high-energy photons occurs, then the super-strong redshift ofthe RDM-star moves the flash frequency to the radio band. This process canbe considered as a mechanism for generating fast radio bursts. The calcula-tions lead to the formula for the wavelength λ out = 2(2 π ) / ( λ N r s ) / /(cid:15) / ,where λ N is the Compton wavelength of the nucleons that make up theasteroid, r s is the gravitational radius of the RDM-star, (cid:15) = ( v/c ) is theparameter determining the orbital velocity of stars v in the galaxy. Eval-24ation with the parameters of the Milky Way galaxy gives the wavelength λ out = 0 . m and the frequency ν out = 0 . GHz, in the range 0.111 ... 8 GHzfor the observed FRB frequencies.(3) the Tolman-Oppenheimer-Volkoff system of equations describing theequilibrium of isotropic matter, for the Planck core located in the center,also has an interesting structure of solutions. For the matter, we considerphoton gas stitched with cosmic microwave background at infinity. As aresult of the calculation, a stationary solution with the parameters of amicro black hole r s < . mm is obtained. With a smooth change inthe external boundary condition (slow accretion of external matter), au-tonomous oscillations arise in the system, accompanied by self-generationof photonic flashes. In this model, a formula for the outgoing wavelength is λ out = l P ( ρ cmb /ρ P ) − / , numerically λ out = 0 . mm, ν out = 333 GHz, thatfall close to the observed FRB range.(4) white holes become stable. We have examined the dynamics ofwhite holes in the Ori-Poisson model and showed that the insertion of nega-tive mass core eliminates both Eardley and Zel’dovich-Novikov-Starobinskijtypes of instability.Based on the considered models, in frames of this work, we have pro-posed three hypotheses about the composition of astrophysical dark matter.(1) In galaxies, the dark matter can be cold or hot, massive, null or eventachyonic, producing the same rotation curves. Particularly, it can be com-posed of massless particles with initially Planck energy, finally redshifted tothe extremely large wavelength λ out ∼ m. More particularly, it can becomposed of low energy photons with such wavelength.(2) The emission of dark matter particles happens in a T-symmetricway, in future and in past directions. This allows to explain the abundanceof dark matter in comparison with the masses of its sources.(3) At cosmological distances, the dark matter behaves like it is cold.Several mechanisms for this behavior have been proposed. Acknowledgements
The author thanks the organizers and participants of the XXIII Bled Work-shop “What comes beyond the Standard models?” for fruitful discussions.The author also thanks Kira Konich for proofreading the paper.
Appendix: Stability of white holes
The model by Ori-Poisson [19], describing a white hole or, more precisely, its juncturewith a black hole, can be considered as a simplification of the stationary models presentedin this paper. Instead of continuous superposition of ingoing and outgoing shells of matter, there are precisely two shells, one ingoing, one outgoing. The model is notstationary, it has a white hole as the initial state and a black hole as the final state. Theadvantage of the model is the existence of an analytical solution. The system is usuallyconsidered unstable, but we will now show that it is not.In addition to the previously described internal ZNS instability, there is externalinstability found in the work by Eardley [20]. In Ori-Poisson formulation, the instabilitycan be illustrated by the Penrose diagram on Fig.9 left. The original white hole (1)explodes, completely releasing its mass into an outgoing null shell (2). There is aningoing null shell (3) of originally small energy. This shell cannot enter the white hole,in principle (like a shell that cannot exit from the interior of a black hole). Instead, itslows down at the horizon and after a long wait (one can substitute here Hubble time, forinstance) receives a super-strong blue shift. It forms a thin super-energetic “blue sheet”.Upon its collision with the outgoing shell, an analytically computable rearrangement ofenergy occurs. As a result, a negligible part of the initial energy comes out (4). Theingoing blue sheet disappears under the horizon of a newly formed black hole (5).In Barceló et al. [39] an even more asymmetric scenario is considered, depicted onFig.9 center. Here not only the ingoing, but also the outgoing shell disappears in theblack hole. Absolutely nothing comes out of this system towards an external observer.One can immediately ask, how can it be that the black holes are stable, but the whiteholes are not? What about T-symmetry? As a resolution of this paradox, a T-symmetricscenario can be constructed, according to the principle “for each incoming blue sheet,there is the same outgoing”. It is displayed on Fig.9 right. Warning: the negative massis required in the scenario. The original white hole (1) explodes, emitting more energythan its own mass in the form of an outgoing blue sheet (2). It leaves behind the coreof negative mass (2’). After the collision between equal blue sheets (2) and (3), a shellof the same energy as the ingoing one comes out (4). The negative mass of the core iscompensated by the incoming blue sheet, a black hole of the same mass as the originalwhite hole is formed (5).The details of the computation are the following. The solution consists of 4 Schwarz-schild’s patches, marked ABCD on the figures. Their masses are m A = M − E, m B = M − dm, m C = m , m D = M, (28)where M is the mass of the white hole with the ingoing shell, dm is the mass of theingoing shell, E is the mass/energy of the outgoing shell ( G = c = 1 ), m is a remainder.The computation is based on Dray-’t Hooft-Redmount (DTR) relation [40]: f A f B = f C f D , f i = 1 − m i /R, i = A, B, C, D, (29) here R is the radius at shells intersection. Relative parameters are introduced: ξ = R/ (2 M ) − , α = dm/M, β = m /M, η = E/M, (30)where ξ measures the relative distance to the horizon, η is the relative efficiency of whitehole eruption. DTR relation results in the expression for the efficiency: η = (1 − α − β ) ξ/ ( α + ξ ) . (31)As a side remark, in [19] a different global structure linking Schwarzschild’s patches witha cosmological model was used, however, this does not influence the obtained efficiencyformula. Now, let us consider the exponential evolution of the shell ξ = ξ exp( − τ / (4 M )) . (32)Here τ = 2 t is the total time for the ingoing shell to reach the point of collision and forthe outgoing shell to reach the distant observer, therefore a double factor in the formula.To bring in some values, for r s = 1 . · m it is an exponential process where thedistance is halved every minute and for τ = 13 . · years the distance factor takes anenormously small value ξ ∼ exp( − ) , practically insensitive to the starting ξ .Considering (31) for β = 0 , as in original Ori-Poisson paper, in the limit < ξ (cid:28) α (cid:28) , obtain η ∼ ξ/α ∼ exp( − ) , a vanishingly small efficiency of eruption.Calculation with β < reveals a different class of solutions: β ∼ − α/ξ results in η ∼ , 100% efficiency, while for η = α , E = dm , the T-symmetric case is reachedselecting β = − ( α − ξ + 2 αξ ) /ξ ∼ − α /ξ. (33)The result of this computation demonstrates that in Ori-Poisson model Eardleyinstability can be eliminated if the system has a core of negative mass. Being combinedwith the earlier obtained result, both types of instability, Eardley and ZNS, can beremoved from the white hole models by the introduction of a Planck core. A noticeablenumerical difference between Ori-Poisson energetic parameters in comparison with thosein the other considered models appears mainly due to the under-exponent Hubble timedelays for the processes, that run continuously in the RDM and TOV models. References [1] M. Visser, C. Barceló, S. Liberati, S. Sonego: Small, dark, and heavy:But is it a black hole?, Proc. of Science , Black Holes in GeneralRelativity and String Theory, 010 (2008); arXiv:0902.0346.[2] B. Holdom, J. Ren: Not quite a black hole, Phys. Rev.
D95 , 084034(2017); arXiv:1612.04889.[3] S. V. Klimenko, I. N. Nikitin, L. D. Nikitina: Numerical solutions ofEinstein field equations with radial dark matter, Int. J. Mod. Phys.
C28 , 1750096 (2017); arXiv:1701.01569.[4] Y. Sofue, V. C. Rubin, Rotation curves of spiral galaxies: Ann. Rev.Astron. Astrophys. , 137-174 (2001); arXiv:astro-ph/0010594.275] Y. Sofue, M. Honma, T. Omodaka, Unified Rotation Curve of theGalaxy – Decomposition into de Vaucouleurs Bulge, Disk, Dark Halo,and the 9-kpc Rotation Dip: Publications of the Astronomical Societyof Japan , 227-236 (2009); arXiv:0811.0859.[6] Y. Sofue, Pseudo Rotation Curve connecting the Galaxy, Dark Halo,and Local Group: Publications of the Astronomical Society of Japan , 153-161 (2009); arXiv:0811.0860.[7] Y. Sofue, A Grand Rotation Curve and Dark Matter Halo in the MilkyWay Galaxy: Publications of the Astronomical Society of Japan ,75 (2012); arXiv:1110.4431.[8] Y. Sofue, Rotation Curve and Mass Distribution in the Galactic Center– From Black Hole to Entire Galaxy: Publications of the AstronomicalSociety of Japan , 118 (2013); arXiv:1307.8241.[9] M. Persic, P. Salucci: Rotation Curves of 967 Spiral Galaxies, Astro-physical Journal Supplement , 501 (1995); arXiv:astro-ph/9502091.[10] M. Persic, P. Salucci, F. Stel: Rotation curves of 967 spiral galaxies:Implications for dark matter, Astrophysical Letters and Communica-tions , 205-211 (1996); arXiv:astro-ph/9503051.[11] M. Persic, P. Salucci, F. Stel: The Universal Rotation Curve of SpiralGalaxies: I. the Dark Matter Connection, Monthly Notices of the RoyalAstronomical Society , 27-47 (1996); arXiv:astro-ph/9506004.[12] P. Salucci et al.: The Universal Rotation Curve of Spiral Galaxies. IIThe Dark Matter Distribution out to the Virial Radius, Monthly No-tices of the Royal Astronomical Society , 41-47 (2007); arXiv:astro-ph/0703115.[13] E. V. Karukes, P. Salucci: The universal rotation curve of dwarf diskgalaxies, Monthly Notices of the Royal Astronomical Society ,4703-4722 (2017); arXiv:1609.06903.[14] A. Ashtekar, T. Pawlowski, P. Singh: Quantum Nature of theBig Bang, Physical Review Letters , 141301 (2006); arXiv:gr-qc/0602086.[15] A. Ashtekar, T. Pawlowski, P. Singh: Quantum Nature of the BigBang: An Analytical and Numerical Investigation, Phys. Rev. D73 ,124038 (2006); arXiv:gr-qc/0604013.2816] A. Ashtekar, T. Pawlowski, P. Singh: Quantum Nature of the BigBang: Improved dynamics, Phys. Rev.
D74 , 084003 (2006); arXiv:gr-qc/0607039.[17] C. Rovelli, F. Vidotto: Planck stars, Int. J. Mod. Phys.
D23 , 1442026(2014); arXiv:1401.6562.[18] C. Barceló, R. Carballo-Rubio, L. J. Garay, G. Jannes: The life-time problem of evaporating black holes: mutiny or resignation, Class.Quantum Grav. , 035012 (2015); arXiv:1409.1501.[19] A. Ori, E. Poisson: Death of cosmological white holes, Phys. Rev. D50 ,6150-6157 (1994).[20] D. M. Eardley: Death of White Holes in the Early Universe, Phys.Rev. Let., , 442 (1974).[21] Ya. B. Zel’dovich, I. D. Novikov, A. A. Starobinskij: Quantum effectsin white holes, Sov. Phys. JETP , 933-939 (1974).[22] I. Nikitin: On dark stars, galactic rotation curves and fast radio bursts,J. Phys.: Conf. Ser. 1730 (2021) 012073; arXiv:1812.11801, 1903.09972,1906.09074.[23] A. A. Kirillov, D. Turaev: On modification of the Newton’s law of grav-ity at very large distances, Phys. Lett. B532 , 185 (2002); arXiv:astro-ph/0202302.[24] A. A. Kirillov, D. Turaev: The Universal Rotation Curve of SpiralGalaxies, Monthly Notices of the Royal Astronomical Society , L31-L35 (2006); arXiv:astro-ph/0604496.[25] K. C. Freeman: On the Disks of Spiral and S0 Galaxies, The Astro-physical Journal , 811 (1970).[26] A. J. S. Hamilton, S. E. Pollack: Inside charged black holes: II.Baryons plus dark matter, Phys. Rev.
D71 , 084032 (2005), arXiv:gr-qc/0411062.[27] A. M. Ghez et al.: Measuring Distance and Properties of the MilkyWay’s Central Supermassive Black Hole with Stellar Orbits, The As-trophysical Journal , 1044-1062 (2008); arXiv:0808.2870.[28] V. Fedorova, A. Rodin: Detection of Fast Radio Bursts on the LargeScanning Antenna of the Lebedev Physical Institute, Astronomy Re-ports , 39-48 (2019); arXiv:1812.10716.2929] V. Gajjar et al., Highest Frequency Detection of FRB 121102 at 4-8GHz Using the Breakthrough Listen Digital Backend at the GreenBank Telescope, The Astrophysical Journal , no.1, 2 (2018);arXiv:1804.04101.[30] E. Petroff et al.: FRBCAT: The Fast Radio Burst Catalogue, Pub-lications of the Astronomical Society of Australia , 1-27 (2019); arXiv:1810.05836; frbtheorycat.org[32] J. C. Wheeler, V. Johnson: Stellar Mass Black Holes in Young Galaxies,The Astrophysical Journal , 163 (2011); arXiv:1107.3165.[33] M. Blau: Lecture Notes on General Relativity
Confrontation of Cosmological Theories with Observa-tional Data , Springer Science and Business Media, 1974.[35] X. F. Cao, M. Xiao, F. Xiao: Modeling the redshift and energy distri-butions of fast radio bursts, Res. Astron. Astrophys. , 14 (2017).[36] J. Barranco, A. Bernal, D. Nunez: Dark matter equation of state fromrotational curves of galaxies, Monthly Notices of the Royal Astronom-ical Society , 403 (2015), arXiv:1301.6785.[37] M. Visser, N. Yunes: Power laws, scale invariance, and generalizedFrobenius series: Applications to Newtonian and TOV stars near criti-cality, Int. J. Mod. Phys. A18 , 3433-3468 (2003), arXiv:gr-qc/0211001.[38] A. Barrau, C. Rovelli, F. Vidotto: Fast Radio Bursts and White HoleSignals, Phys. Rev.
D90 , 127503 (2014); arXiv:1409.4031.[39] C. Barceló, R. Carballo-Rubio, L. J. Garay: Black holes turn whitefast, otherwise stay black: no half measures, J. High Energ. Phys. , 157 (2016), arXiv:1511.00633.[40] T. Dray, G. ’t Hooft: The Effect of Spherical Shells of Matter on theSchwarzschild Black Hole, Commun. Math. Phys.99