Smoothed one-core and core--multi-shell regular black holes

We discuss the generic properties of a general, smoothly varying, spherically symmetric mass distribution $\mathcal{D}(r,\theta)$, with no cosmological term ($\theta$ is a length scale parameter). Observing these constraints, we show that (a) the de Sitter behavior of spacetime at the origin is generic and depends only on $\mathcal{D}(0,\theta)$, (b) the geometry may posses up to $2(k+1)$ horizons depending solely on the total mass $M$ if the cumulative distribution of $\mathcal{D}(r,\theta)$ has $2k+1$ inflection points, and (c) no scalar invariant nor a thermodynamic entity diverges. We define new two-parameter mathematical distributions mimicking Gaussian and step-like functions and reduce to the Dirac distribution in the limit of vanishing parameter $\theta$. We use these distributions to derive in closed forms asymptotically flat, spherically symmetric, solutions that describe and model a variety of physical and geometric entities ranging from noncommutative black holes, quantum-corrected black holes to stars and dark matter halos for various scaling values of $\theta$. We show that the mass-to-radius ratio $\pi c^2/G$ is an upper limit for regular-black-hole formation. Core--multi-shell and multi-shell regular black holes are also derived.

One-and multi-parameter-dependent mathematical distributions smoothing the Dirac's δ distribution are needed in areas of science where the notion of locality is being abandoned. For instance, in quantum gravity the noncommutativity of coordinates is phenomenologically explained by the nonlocality of matter distributions [1]. The singularities arising in classical physics are due to the hypothetical point-like matter distributions. Such a point-like or Dirac distribution is mathematically useful in getting closed-form simple expressions for the physical and geometric entities one is concerned with. In a sense, the Schwarzschild, Reissner-Nordström, Kerr and other classical solutions of general relativity are extremely simplified models of nature and should exist in a real world only asymptotically. In some other instances of science, as is the case with regular black holes sourced by nonlinear electrodynamics, such distributions were not needed. That remains true, however, as far as one is concerned with macroscopic scales; for scales of the order of the Compton wavelength or the Planck length, the contribution of the vacuum, namely its radial pressure sustaining matter from collapsing, renders mass distributions extended.
In a first tentative one may think to replace the Dirac distribution for mass by a central-decreasing as one moves away from the source-extended distribution. For spherically symmetric solutions, a Gaussian mass distribution with width θ, G(r, θ) = e −r 2 /(2θ 2 ) (2π) 3/2 θ 3 , where r is a radial coordinate, satisfies the abovementioned requirement. However, the resulting metric and fields are not in closed-forms and are not easily handled numerically-not to mention analyticallyvia computer algebra systems [1]. Do extended distributions for charge and spin (if the solution is rotating) follow the same mass-distribution model? In Ref. [2] it was argued that, if masses follow Gaussian distributions, charges, rather, should follow extended Weibull distributions to ensure a de Sitter behavior of the solution in the vicinity of the origin. Whether the gravitational quantum effects are well understood or not, introducing them phenomenologically via mass, charge, and spin distribution functions seems to be a fruitful way as this cures singularities, skips the matching problems, and preserves the asymptotical behavior. There remains to understand how the vacuum responds to the mass, charge, and spin extended distributions to generate negative pressures sustaining matter from collapsing. The only known process to advance an explanation for that is vacuum fluctuations but so far no concrete formulation seems to exist.
For the Gaussian distribution the substitution rule Dirac-to-Gaussian reads where the numerical coefficients in (1) and (2) have been determined on observing the normalization conditions Let D(r, θ) ≥ 0 be some spherically symmetric, not necessarily central, distribution with the normalization condition ∞ 0 D(r, θ) 4πr 2 dr = 1. ( If D is a mass distribution one may think of it to be central, however, the vacuum negative radial pressure, too assumed to be spherically symmetric, may push more matter from the center rendering the distribution noncentral. This is the case with charge distributions [2], which are of Weibull character. Let m(r, θ) denote the mass inside a sphere of radius r. This is given by (4) where M is the total mass of the solution. To simplify the notation we have set which is the cumulative distribution. The above substitution rule (2) is replaced by It is understood that the distribution D(r, θ), smoothing the Dirac's one, is assumed to be finite everywhere. We further require that D ′ (r, θ) has a finite value at r = 0 (here the prime denotes differentiation with respect to r). The convergence of the integral in (3) implies that D must go to 0 faster than 1/r 3 in the limit r → ∞. These requirements are expressed mathematically as In Sec. II we discuss the generic properties of any mass distribution obeying (7)-(9) and of its resulting metric solution. In Sec. III we define new twoparameter, (n, θ), mathematical distributions mimicking the Gaussian distribution and reduce to the Dirac distribution in the limit of vanishing parameter θ (for all n) and discuss their specific properties and the properties of their resulting metric solutions. In Sec. IV we discuss some limiting cases. In Sec. V we provide instances of applications ranging from noncommutative black holes, quantum-corrected black holes to stars and dark matter halos for various scaling values of θ. An Appendix section has been added to complete the discussion of, and to derive some equations pertaining to, Sec. II. We conclude in Sec. VI.

II. GENERIC PROPERTIES OF THE METRIC
We seek a static spherically symmetric solution of the form where m(r, θ) is given by (4). In the following we discuss the generic properties of (10) for a matter distribution obeying the minimum set of constraints (7), (8), and (9).
a. Behavior near the origin.-Since D(r, θ) is assumed to be finite everywhere, for r ≪ 1, we may replace D(r ′ , θ) in (4) by D(0, θ) to obtain Thus, any distribution with nonvanishing value at the origin [D(0, θ) = 0] yields a metric having a de Sitter behavior there with an effective "cosmological constant" linearly proportional to the total mass M as far as the width θ does not dependent on the mass. The distribution need not be central to yield such a behavior for f : All that we need is to have D(0, θ) = 0.
b. The scalar invariants.-With the further assumption that D ′ (r, θ) has a finite value at r = 0 (8), it straightforward to show that the curvature and Kretschmann scalars are finite at the origin: Since m(r, θ) behaves as r 3 (11) near the origin, we see that both expressions of R and R αβµν R αβµν have finite limits as r → 0. Thus, the singularity at the origin has been removed.
c. Horizons.-The horizons, all denoted by r h , are solutions to the equation f (r h ) = 0, which reduces to where we have used (4) and (5). In the r h y plane, the horizons are the intersection points of the straight line y = c 2 r h /(2MG) and the curve y = D(r h , θ) among which we find the point r h = 0, which we exclude. By (11) the graph of y = D(r h , θ) is flat at the origin (in the limit r h → 0) and by (3) it is also flat asymptotically (in the limit r h → ∞). Since D(r, θ) is the cumulative distribution it is an increasing function of r [D ′ = 4πr 2 D > 0 (7)], so its shape looks like a flat S, as depicted in Fig. 1. This generic graph of D(r, θ), as is the case with any cumulative distribution, does not depend on θ and on whether the latter depends on M or whether θ depends on M or not. Let x ext ≡ r ext /θ and generally We show in the Appendix that x ext is a solution to Thus, if D does not depend on the mass M, this will be the case for x ext too. For large, massive black holes one of the two nonzero horizons (inner horizon r h− ) shrinks to 0 while the other one (outer horizon r h+ ) goes to infinity. For the latter horizon we let r h → ∞, so that the r.h.s of (16) is 1 by (3) implying where r S denotes the Schwarzschild radius. It is clear from (3) and (4) that the metric (10) whereh and k B are the reduced Planck and Boltzmann constants. We obtain D ′ upon differentiating (5) with respect to r and we use (16) to express (D/r)| r h+ in terms of c 2 /(2MG). Finally, we arrive at where T is seen as a function of r h+ . Using (9) and (17), we see that for large massive black holes (MG/c 2 )r 2 h+ D(r h+ ) → 0 and which is the well-known expression for the Schwarzschild black hole temperature. Note that r ext is the minimum value of r h+ . In the Appendix we show that T(r ext ) = 0. Now, since T vanishes at r ext and it is positive (23) for large values of r h+ , it must reach some maximum value for somer h+ > r ext where then, by (23), goes to zero as r h+ approaches infinity.
Here the prime denotes differentiation with respect to r. We see that an evaporation process which starts at some value of r h+ >r h+ leads, after some loss of matter, to a configuration where the temperature becomes initially larger than the temperature of the starting point, it attains its maximum value atr h+ , then it drops to zero as r h+ reaches the value r ext , which marks the end of the evaporation process for there will be no black hole. The remaining mass is a cold, at T = 0, regular non-black-hole solution (17).
e. The stress-energy tensor.-In this approach the effects of noncommutativity, corrections due to quantum effects, or presence of dark matter halos regularizing the geometry of spacetime, are taken into consideration upon extending the mass distribution without modifying the field equations. So, we assume that the classical field equations still hold and remain valid to all orders. In this regard, the stress-energy tensor sourcing the metric (10), which is defined by G µν = (8πG/c 4 )T µν , is split into two non-interacting perfect fluids: A pressureless dust with energy density ǫ d = c 2 ρ m and a quantum vacuum having ǫ v = 0, a negative radial pressure p r = −c 2 ρ m < 0, and a transverse pressure where In the vicinity of the origin we certainly have p t < 0 but its sign may change as r increases. The negative radial pressure of vacuum is necessary for preventing matter from collapsing and forming a singularity. By the method of superposition of fields and its generalization [6]- [10], using the same reference frame one can add the components of the two fluids to have, for instance,

III. NEW DISTRIBUTIONS D D D
In this section we define new mathematical distributions D that mimic to a large extent the Dirac's one, then we discuss the special properties of the metric (10).

A. Definition
Let A(n, z) be the function defined by If z is real and n > 0 (as we shall see later, we will require n > 3 to ensure convergence of the integral), then is the incomplete beta function 1 . One brings (27) to (28) upon setting t = −u n . Using the new variable x = r/θ (18), we define the distribution D n (r, θ) to be the function related to A(n, x) by where n is a real number and to ensure the convergence of the integral in (3) we have required n > 3 (9). It is obvious from the definition that the distribution (30) reduces to the Dirac δ in the limit θ → 0. The cumulative distribution takes the form Both D n (r, θ) and D n (r, θ) take the simplified expressions: n D n (r, θ) D n (r, θ) c n x ext 0.561 1.679 There does not seem to be a special name given to the distributions of the form (30). A distribution of the form is called the standard Cauchy 2 distribution [11]. It has been used to model dark haloes in spiral galaxies in the center and in the outer spatial regions [13]; the model is widely accepted. An advantage in using the distribution (32) is that it depends on two parameters (θ, n). The distributions (32) and (1) have their denominators proportional to θ 3 , thus holding θ constant and varying n one can generate a distribution (32) mimicking to a large extent the Dirac's one, as shown in Fig. 2, which is not possible with the Gaussian distribution (1). Another advantage is that the cumulative distribution (33) can be brought to a closed-form (for all n > 3) in terms of arctan and ln elementary functions. Table I provides some distributions D n (r, θ) with their cumulative functions D n (r, θ). 2 Its generalization [12], know as the generalized Cauchy distribution f (z), is proportional to where θ is the location parameter, σ is the scale parameter, and p is the tail constant.

B. Special properties of the metric (10) and the physical scales
In the previous section we discussed the general properties of the metric (10) that are independent of the special form of the distribution D(r, θ). In this section we focus on other, rather specific, properties of (10) that result from the application of the distribution (32): These are the properties of (34).

large M
Introducing the parameters x h = r h /θ and x S = r S /θ we bring (16) to For x large, the cumulative distribution is easily brought to the form Using this in (35) we solve it by iteration and obtain the outer horizon The area of the outer horizon A = πr 2 h+ expands for large r, that is for large M, as The area spectrum follows the Bekenstein [14] law A = bN with b ≡ 4ℓ 2 P ln 2 and N ∈ N + . (39) Here N is a positive integer and ℓ P = √h G/c 3 ≃ 1.616 × 10 −35 m is the Planck length. If the black hole emits a quanta, that is, if N changes by 1 (dN = −1), this yields a change in the mass parameter M given by the first order approximation which is independent on θ. Since b ≪ 1 and M is supposed large, this implies that the change in M or the mass loss is almost continuous.

Extremal horizon
The value of x ext is solution to (19), which takes the form The solution of which yields an x ext independent on the mass M.

Temperature and its maximum value
With D given by (32) the expression of T (24) reduces to For this type of distributions (32) it is easy to solve (24) and determine the value of the outer horizonr h+ that yields a maximum temperature. We find yielding (44) Note that for a Gaussian distribution (1)r h+ = √ 2 θ, which is larger than the value given in (43) for all n.
The inner horizon r h− is obtained upon solving the algebraic equation: For instance, for n = 5 we obtain where ν 5 = 5 sin(3π/5)/(2π) ≈ 582/769 ≃ 0.756827. The metric (45) is a quintessence-like metric. Knowing that a Gaussian distribution (1) does not accurately describe [15] the galaxies rotation curves [16] as does, for instance, the pseudoisothermal model [13]. This shows another advantage in using the distributions (32), for they can model dark matter distributions better than a Gaussian distribution and provide best fits compared to the standard models [13,17,18]. From this point of view θ is of the order of the stellar or core radius.
With these identifications the outer (37) and inner (47) horizons coincide with the solutions given in Eqs. (41) and (42) of Ref. [20]. The parameters ν 5 and γ > 0 [20] being of the order of unity, we see that from this point of view θ is of the order of the Planck length. Considering regular particle-sized black holes with a Gaussian mass distribution (1), it was argued in Ref. [1] that a qualitative realization of the UV self-completeness of quantum gravity could be achieved taking θ of the order of the Compton wavelength of a particle of mass M: θ ∼ 1/M. This scheme can be easily realized using our model for mass distributions given by (32).
Thus, the parameter θ provides three length scales of application: 1. A cosmological scale where θ is of the order of the stellar or core radius; 2. A subatomic scale where θ is of the order of the Compton wavelength of a particle of mass M;

A Planckian scale for describing quantumcorrected black holes.
For black hole or particle-like solutions there are, however, other means by which one may constrain the values of θ, as we shall discuss in the remaining sections.

Radius of fuzzy matter distributions
It is straightforward to show that the transverse pressure (25) is up to a constant factor given by for all r. This vanishes in the limit r → ∞. It is negative for 0 ≤ r < r 0 , null for r = r 0 , and positive for r > r 0 where r 0 ≡ 2 n − 2 1/n θ. (50) One may call the value r 0 the distributional radius of the black hole or that of the galaxy. It is a measure of the distance beyond which the effects of vacuum due to the fuzzy distribution of matter tend to be neglected. For a Gaussian distribution (1), r 0 = ∞, that is, the tangential pressure is negative for the whole range of the radial coordinate. One sees that the distributions (32) provide more realistic models for describing fuzzy matter distributions or galactic dark matter halos. Note that r 0 is justr h+ (43). This is a mere "specific" coincidence and it depends on the nature of the mass distribution D; for a Gaussian distribution (1) we havẽ r h+ and r 0 = ∞. The fact that r 0 =r h+ means that black holes with an outer horizon given by (43) have maximum temperature and their transverse pressure is positive outside the horizon.
For black hole solutions one may constrain the values of θ upon requiring that all the fuzzy matter distribution be confined within the inner horizon. This allows one to describe classically the geometry outside the event horizon.

IV. LIMITING VALUES
Using (32) we bring (13) to If we assume that the smallest value of Λ is the cosmological constant Λ csm , this yields the maximum value for θ where we replaced n sin(3π/n) by its upper bound 3π.
An upper bound for Λ could be set requiring the width θ to be of the order of the Compton wavelength h/(Mc) for the mass M, which expresses the inability to localize a single particle in a region of sizeh/(Mc). We obtain where we replaced n sin(3π/n) by its upper bound 3π. For describing quantum-corrected black holes θ could be of the order of the Planck length. For these holes, an upper bound for Λ is rather   [26], [27], and [28].

V. NEW METRIC SOLUTIONS
Selecting the simplest solution given in Table I we are led to the following regular metric This is a substitute to the singular Schwarzschild metric resulting from the substitution rule (2) where we have replaced the Gaussian distribution by (32) taking n = 6. The corresponding continuous mass density ρ m (r) and mass m(r) within a sphere of radius r are given by Plots of (55) are shown in Fig. 3 for different values of MG/(πc 2 θ). This solution models a regular noncommutative black hole where the effects of noncommutativity of coordinates are phenomenologically played by a smeared, extended, mass distribution. For larger values of n, the mass distribution (32), being almost a step function (see Fig. 2), is more confined in a region around the black hole and the solution represents a classical black hole.
For smaller values of n the distribution (32) is more extended, like a Gaussian distribution, and the solution represents a semi-classical black hole.
One may ask: What is the upper limit of the ratio M/θ, where θ is a measure of the extent of matter, that prevents the occurrence of horizons? The answer is as follows.
For modeling dark matter halos one may apply the mass distribution (57) to halos with stellar radius a and mass M such that M < c 6 πc 2 a G and c 6 = 0.284, so as to avoid the formation of black-hole dark matter halos. The mass within a sphere of radius r is given by (57) on replacing θ by a.
Such an upper limit on M is not absolute, that is, larger dark matter halos are modeled by the distribution (32) taking n < 6. This will set another upper limit for the mass for such halos similar to (58) with a new coefficient c n larger than, but remains of the same order of 0.284 for the values of n considered in Table I: This upper limit is at least satisfied by the dwarf galaxies with stellar radii 10-30 kpc as can be seen from the dark matter profiles [18] derived from the data of rotation curves of the DDO 154, DDO 105, NGC 3109, and DDO 170 spiral galaxies reported in Refs. [22], [23], [24], and [25], respectively. The scaling empirical Eq. (3) of Ref. [18] correlates the dark matter mass M inside a sphere of radius a with a where the ratio M/a remains of the order of 10 20 kg/m. For modeling stars one may apply the mass distribution (57) to stars with radius a and mass M such that (58) is satisfied so as to avoid the formation of a black hole. This is justified since the graph of D 6 , shown in Fig. 2, is almost similar to that of a step function; the mass distribution vanishes almost identically for r > θ, and vanishes faster than a Gaussian distribution in the vicinity of r θ. For lighter stars we may take n > 6 in (32) so that the mass remains confined inside the sphere of radius θ (the radius of the star). We reach the same conclusion as before, in that, the ratio M/a remains bounded from above by the constant µ defined in (60). For the stars of Table II, the data of which has been reported in Refs. [26], [27], and [28], the ratio M/a ∼ 10 23 kg/m < µ.
For modeling elementary particles we take θ to be of the order of the (reduced) Compton wavelength, a = θ =h/(Mc), and n ≥ 6 so that the shape of the mass distribution be that of a step function. In this case D n depends on the mass of the particle via θ. The condition (59) ensuring the absence of horizons reduces to where we have dropped c n . This is the well known property stating that the masses of elementary particles are much smaller than the Planck mass m P . We draw the general conclusion that any mass distribution of extent θ and mass M is exempt of, or freed from, horizons if The largeness of the constant µ (62) is behind the difficulty in manufacturing laboratory black holes by compressing solids. To achieve that one should reduce the size of the solid, with given mass M, to below M/µ. This may apply to the whole universe itself: If the ratio (mass of the universe/extent of the universe) is bigger than µ, we may be living inside a two-horizon black hole, most likely inside the inner horizon. Otherwise, the space around us is freed from horizons.
Can the metric (55), which corresponds to n = 6, describe a quantum-corrected Schwarzschild black hole? In Sec. III B 4 we have seen that such a black hole can be described by a mass distribution (32) provided we take n = 5. The metric expansion (45) with n = 5, which describes a quantum-corrected Schwarzschild black hole, has been first derived in [19] upon evaluating the self-energy insertion tensor (SEIT) [29] due to the inclusion of a single-closed loop, which is a quantum correction. The finite piece of the SEIT contains some arbitrary parameters while the infinite piece is supposed to be canceled by appropriate counter-terms in the Lagrangian. However, it is all possible that these canceling counter-terms may alter the values of the parameters in the finite piece of the SEIT causing the final expression of the metric (63) to include, say, a term proportional to 1/r 4 or other powers of 1/r instead of a term proportional to 1/r 3 .

VI. CONCLUSION
A way to describe phenomenologically dark matter halos, stars, effects of noncommutativity and quantum corrections to stellar objects is to model them by extended mass distributions.
A variety of such mass distributions as well as a charge distribution have been put forward for the sole purpose mentioned above. To the best of our knowledge only a Gaussian mass distribution has received a twofold application: Constructing noncommutative black holes and describing dark matter halos.
We have discussed the generic properties of these mass distributions. Their resulting metric solutions all have a de Sitter behavior near the origin, finite scalar invariants, and finite temperature if they describe black holes. In the latter case, the evaporation processes are marked by the finiteness of the temperature which first increases to a maximum value then decreases to absolute zero at the end of the process, contrary to the Schwarzschild case where the temperature unceasingly increases to infinity during the process of evaporation.
Then we have specialized to a new class of mass distributions. We have defined and used step-like mass distributions. Being dependent on two independent parameters, these distributions are multi-fold and they apply to a variety of physical configurations ranging from noncommutative black holes, quantum-corrected black holes to stars and dark matter halos depending on different scaling values of one of the two parameters.
The resulting regular metric solution is always given in closed form in terms of the arctan and ln elementary functions. For linear mass densities exceeding πc 2 /G, the geometry is that of a two-horizon regular black hole; otherwise the geometry is freed from horizons and describes a regular non-black-hole configuration that could be a quantum particle, a star, a dark matter halo, the whole universe, or a compressed quantum solid.

Appendix: Equation yielding the extremal horizon for generic mass distribution
Using the new variable x = r/θ we bring (5) and (16)  Excluding the point x = 0, the line y = x/x S is tangent to the curve y = D(x) at the only point of intersection x ext . Thus, x ext is solution to the system where the prime denotes differentiation with respect to Note that (A.6) may be arranged as 1 − 4πx S θ 3 D(x ext )x ext 2 = 1 − 4πr S D(r ext )r ext 2 = 0, which implies that the temperature (22) vanishes at r ext .