4.1 – Constructing a Macro Holeum from the inside out

4.1 - Constructing a Macro Holeum from the inside out

Let us consider a Holeum that is a bound state of two microscopic black holes of masses m_{1} and m_{2} revolving around their center of mass under the action of gravity. From [1] and section 1.2 – A stable Holeum, the energy values of the Holeum are given by

E_{2}=-\frac{m_{1}m_{2}c^{2}\alpha_{2}^{2}}{2\left(  m_{1}+m_{2}\right)n_{2}^{2}} (eq. 4.1)


where \alpha_{2} is the gravitational coupling constant for the interaction between masses m_{1} and m_{2}, given by

\alpha_{2}=\frac{m_{1}m_{2}}{m_{p}^{2}} (eq. 4.2)


where

m_{p}=\left(  \frac{\hbar c}{G}\right)  ^{\frac{1}{2}} (eq. 4.3)


is the Planck mass, \hbar is the Planck’s constant divided by 2\pi, c is the speed of light in vacuum, and G is Newton’s universal constant of gravity. n_{2} is the principal quantum number of the bound state. From [1], the most probable radius of the Holeum is given by

r_{2}=\frac{\left(  R_{1}+R_{2}\right)  \pi^{2}n_{2}^{2}}{16\alpha_{2}^{2}} (eq. 4.4)


where

R_{i}=\frac{2m_{i}G}{c^{2}} (eq. 4.5)


is the Schwarzschild radius of the microscopic black hole of mass m_{i}. The mass of the bound state is given by

M_{2}=m_{1}+m_{2}+\frac{E_{2}}{c^{2}} (eq. 4.6)


We shall call this bound state of two microscopic black holes a di-Holeum. Now consider a bound state of the di-Holeum and a microscopic black hole of mass m_{3}. We call it a tri-Holeum. Its energy eigenvalues are given by

E_{3}=-\frac{m_{3}M_{2}c^{2}\alpha_{3}^{2}}{2\left(  m_{3}+M_{2}\right)n_{3}^{2}} (eq. 4.7)


where

\alpha_{3}=\frac{m_{3}M_{2}}{m_{p}^{2}} (eq. 4.8)


The bound state radius of the tri-Holeum is given by

r_{3}=\frac{\left(  m_{3}+M_{2}\right)  G\pi^{2}n_{3}^{2}}{8c^{2}\alpha_{3}^{2}} (eq. 4.9)


With the help of equation (4.6) this may be written as

r_{3}=\frac{\left(  R_{1}+R_{2}+R_{3}+\frac{E_{2}}{c^{2}}\right)  \pi^{2}n_{3}^{2}}{16\alpha_{3}^{2}} (eq. 4.10)


The mass of the tri-Holeum is given by

M_{3}=m_{3}+M_{2}+\frac{E_{3}}{c^{2}} (eq. 4.11)


Continuing in this manner we arrive at a k-Holeum, also called a Macro Holeum, consisting of k microscopic black holes. Its energy values, bound state radius, and bound state mass are given, respectively, by

E_{k}=-\frac{m_{k}^{3}c^{2}\left[  m_{1}+m_{2}+\ldots+m_{k-1}+\frac{\left(E_{2}+E_{3}+\ldots+E_{k-1}\right)  }{c^{2}}\right]  ^{3}}{2m_{p}^{4}n_{k}^{2}\left[  m_{1}+m_{2}+\ldots+m_{k}+\frac{\left(  E_{2}+E_{3}+\ldots+E_{k-1}\right)  }{c^{2}}\right]  } (eq. 4.12)


r_{k}=\frac{\left(  R_{1}+R_{2}+\ldots+R_{k}\right)  \left[  1+\frac{E_{2}+E_{3}+\ldots+E_{k-1}}{\left(  m_{1}+m_{2}+\ldots+m_{k}\right)  c^{2}}\right]\pi^{2}n_{k}^{2}}{\left(  \frac{m_{k}}{m_{p}}\right)  ^{2}\left[  \frac{m_{1}+m_{2}+\ldots+m_{k-1}}{m_{p}}\right]  ^{2}\left[  1+\frac{E_{2}  +E_{3}+\ldots+E_{k-1}}{\left(  m_{1}+m_{2}+\ldots+m_{k-1}\right)  c^{2}}\right]  ^{2}} (eq. 4.13)


M_{k}=\left(  m_{1}+m_{2}+\ldots+m_{k}+\frac{E_{2}+E_{3}+\ldots+E_{k}}{c^{2}}\right) (eq. 4.14)


The gravitational coupling constant \alpha_{k} for the interaction between a Macro Holeum containing k-1 microscopic black holes and the microscopic black hole of mass m_{k} is given by

\alpha_{k}=\frac{m_{k}M_{k-1}}{m_{p}^{2}} (eq. 4.15)


Using equation (4.14), this may be rewritten as

\alpha_{k}=\frac{m_{k}\left(  m_{1}+m_{2}+\ldots+m_{k-1}+\frac{\left(E_{2}+E_{3}+\ldots+E_{k-1}\right)  }{c^{2}}\right)  }{m_{p}^{2}} (eq. 4.16)


For the sake of simplicity we now consider an equal-mass case: m_{1}=m_{2}=\ldots=m_{k}=m, say. Then equations (4.12) – (4.16) reduce, respectively, to

E_{k}=-\frac{\left(  k-1\right)  ^{3}mc^{2}\alpha_{g}^{2}}{2kn_{k}^{2}}\left[\frac{g_{k-1}\left(  \alpha_{g},n_{2},n_{3},\ldots,n_{k-1}\right)  ^{3}}{f_{k-1}\left(  \alpha_{g},n_{2},n_{3},\ldots,n_{k-1}\right)  }\right] (eq. 4.17)


r_{k}=\frac{kR\pi^{2}n_{k}^{2}}{16\alpha_{g}^{2}\left(  k-1\right)  ^{2}}\left[  \frac{f_{k-1}\left(  \alpha_{g},n_{2},n_{3},\ldots,n_{k-1}\right)}{g_{k-1}\left(  \alpha_{g},n_{2},n_{3},\ldots,n_{k-1}\right)  ^{2}}\right] (eq. 4.18)


M_{k}=kmg_{k}\left(  \alpha_{g},n_{2},n_{3},\ldots,n_{k}\right) (eq. 4.19)


\alpha_{k}=\left(  k-1\right)  \alpha_{g}g_{k-1}\left(  \alpha_{g},n_{2},n_{3},\ldots,n_{k-1}\right) (eq. 4.20)


where

\alpha_{g}=\left(  \frac{m}{m_{p}}\right)  ^{2} (eq. 4.21)


Here n_{j} is the principal quantum number of the microscopic black hole of mass m_{j}; j=2,3,\ldots k. From equations (4.12) and (4.13) as well as from equations (4.17) and (4.18) we see that whereas E_{k} and r_{k} depend strongly upon n_{k}, the principal quantum number of the outermost microscopic black hole, their dependence on the rest of the principal quantum numbers is quite symmetric. None of the latter is singled out the way n_{k} is.

Hence for the sake of convenience we shall describe the k-Holeum in terms of a “valence” microscopic black hole, namely the outermost one and and the “core” microscopic black holes, namely, the rest of them, j=2,3,\ldots k-1.

f_{k-1} and g_{k-1} are given by

f_{k-1}\left(  \alpha_{g},n_{2},n_{3},\ldots,n_{k-1}\right)  =1+\frac{E_{2}+E_{3}+\ldots+E_{k-1}}{kmc^{2}} (eq. 4.22)


g_{k-1}\left(  \alpha_{g},n_{2},n_{3},\ldots,n_{k-1}\right)  =1+\frac{E_{2}+E_{3}+\ldots+E_{k-1}}{\left(  k-1\right)  mc^{2}} (eq. 4.23)


Note that we must have at least k=2 for a bound state formation. Thus in equations (4.22) and (4.23), we must define f_{1}=g_{1}=1. Note that the Schwarzschild radius, R_{k}, of the k-Holeum containing k microscopic black holes is given by

R_{k}=\frac{2M_{k}G}{c^{2}}=kRg_{k}\left(  \alpha_{g},n_{2},n_{3},\ldots,n_{k}\right) (eq. 4.24)


where M_{k} is given by equation (4.19). From equations (4.18) and (4.24) we get the ratio of the Schwarzschild radius R_{k} of the k-Holeum to its bound state radius r_{k} as follows:

\frac{R_{k}}{r_{k}}=\frac{16\alpha_{g}^{2}\left(  k-1\right)  ^{2}}{\pi^{2}n_{k}^{2}}\frac{g_{k}\left(  \alpha_{g},n_{2},n_{3},\ldots,n_{k}\right)  g_{k-1}\left(  \alpha_{g},n_{2},n_{3},\ldots,n_{k-1}\right)  ^{2}}{f_{k-1}\left(\alpha_{g},n_{2},n_{3},\ldots,n_{k-1}\right)  } (eq. 4.25)


Substituting for E_{j} from equation (4.17) into equations (4.22) and (4.23) we obtain

f_{k-1}=1-\frac{\alpha_{g}^{2}}{2k}\sum_{j=2}^{k-1}\frac{\left(  j-1\right)^{3}}{jn_{j}^{2}}\frac{g_{j-1}^{3}}{f_{j-1}} (eq. 4.26)


g_{k-1}=1-\frac{\alpha_{g}^{2}}{2\left(  k-1\right)  }\sum_{j=2}^{k-1}\frac{\left(  j-1\right)  ^{3}}{jn_{j}^{2}}\frac{g_{j-1}^{3}}{f_{j-1}} (eq. 4.27)


Since r_{k}>0 and M_{k}>0, for all k, we conclude from equations (4.18) and (4.19) that

f_{k-1}>0\text{, }g_{k-1}>0 (eq. 4.28)


From equations (4.26) – (4.28) we have

f_{k-1}\leq1\text{, }g_{k-1}\leq1 (eq. 4.29)


for all k and \alpha_{g}. In this equation and subsequent ones the equality holds if either \alpha_{g}=0 or n_{2}=n_{3}=\ldots=n_{k-1}=\infty. From equations (4.26) and (4.27) it trivially follows that

f_{k-1}\geq g_{k-1} (eq. 4.30)


From equation (4.29) it follows that

g_{k-1}\geq g_{k-1}^{2}\geq g_{k-1}^{3} (eq. 4.31)


From equations (4.30) – (4.31) we have

f_{k-1}\geq g_{k-1}\geq g_{k-1}^{2}\geq g_{k-1}^{3} (eq. 4.32)


From the latter we conclude that

\frac{g_{j-1}^{3}}{f_{j-1}}\leq1 (eq. 4.33)


From equations (4.33) and (4.17) we have

E_{k}\geq-\frac{\left(  k-1\right)  ^{3}}{2k}\frac{mc^{2}\alpha_{g}^{2}}{n_{k}^{2}} (eq. 4.34)


Letting p=k\alpha_{g} and k\gg2 in equation (4.34) we may rewrite it as

E_{k}\geq-\frac{p^{2}mc^{2}}{2n_{k}^{2}} (eq. 4.35)


If p\rightarrow\infty as k\rightarrow\infty in equation (4.35), then the energy of the system has no lower bound. Therefore the system will go on losing energy during interactions until it becomes a part of the infinite energy of the vacuum state of the universe. In this case, the system cannot have an independent existence. This is averted if p<M, where M is a positive constant. In this case the system can have a stable existence. Therefore it can be shown that p<M is the necessary and sufficient condition for the stability of the Macro Holeum. In the following we will assume p to be a positive constant. Note that the right hand side of equation (4.35) is identical in form to the equation for the energy eigenvalues of a hydrogen atom with m replacing the mass of the electron and p replacing the fine structure constant. Therefore it is clear that p may be interpreted as an effective coupling constant for the Macro Holeum.

Now we consider the problem of reducing this many-body problem to a two-body one. Here we are considering a truly gigantic system having a total mass comparable to or even much greater than the solar mass. Here k would be in the astronomical range, say, k=10^{50} to k=10^{100}. The huge mass of this system gives it a huge inertia. It will be very difficult to perturb such a system. Thus, we may assume that most of the system will remain unperturbed and only the outermost one or two microscopic black holes will be affected by the interactions of the system with its environment. In order to reduce this many-body problem to a two-body one, we will make the extreme simplification that all the microscopic black holes in the core are in the same quantum state described by n_{2}=n_{3}=\ldots=n_{k-1}=n, say, and that the outermost microscopic black hole is in an arbitrary quantum state described by the principal quantum number n_{k}. The first assumption embodies the great inertia of the system.

We recall here that in the nuclear many-body problem, the nucleus is assumed to be an infinite medium in which every nucleon moves in the same average potential produced by the rest of them. This reduces the many-body problem to a one-body problem. This, plus a heuristic spin-orbit potential with an arbitrarily chosen sign, fetches us good quantitative agreement with nuclear data in the nuclear shell model. Great complexity often yields to extreme simplification; provided the simplification embodies the physics of the system correctly. In our problem, if there is a greater excitation, we may describe it in terms of n,n_{k-1} and n_{k}; wherein the core now consists of k-2 microscopic black holes all in the same state of excitation n, and the two outer-most microscopic black holes are in arbitrary states of excitation described by n_{k-1} and n_{k}. But in the first approximation, in the following, we describe the system in terms of n and n_{k} only, apart from the other parameters.

Now we make a Taylor series expansion of f_{k-1}, equation (4.26), in powers of x=\alpha_{g}^{2}. Then we get

f_{k-1}(x)=1+xf_{k-1}^{\prime}(0)+\frac{x^{2}}{2!}f_{k-1}^{\prime\prime}(0)+\ldots (eq. 4.36)


From equation (4.26) we have

f_{k-1}^{\prime}(x)=-\frac{1}{2k}\sum_{j=2}^{k-1}\frac{\left(  j-1\right)^{3}}{jn_{j}^{2}}\frac{g_{j-1}^{3}}{f_{j-1}}-\frac{x}{2k}\sum_{j=2}^{k-1}  \frac{\left(  j-1\right)  ^{3}}{jn_{j}^{2}}\left[  \frac{3g_{j-1}^{2}g_{j-1}^{\prime}}{f_{j-1}}-\frac{g_{j-1}^{3}f_{j-1}^{\prime}}{f_{j-1}^{2}}\right] (eq. 4.37)


From equation (4.37) it follows that

f_{k-1}^{\prime}(0)=-\frac{S_{k-1}}{2kn^{2}} (eq. 4.38)


In these equations we have taken n_{2}=n_{3}=\ldots=n_{k-1}=n. Similarly from equation (4.27) it follows that

g_{k-1}^{\prime}(0)=-\frac{S_{k-1}}{2(k-1)n^{2}} (eq. 4.39)


where

S_{k-1}  =\sum_{j=2}^{k-1}\frac{\left(  j-1\right)  ^{3}}{j} =\frac{\left(  k-1\right)  k\left(  2k-1\right)  }{6}-\frac{3k\left(k-1\right)  }{2}+3\left(  k-1\right)  -C-\ln\left(  k-1\right) -\frac{1}{2\left(  k-1\right)  }+\sum_{j=2}^{\infty}\frac{A_{j}}{\left(  k-1\right)  k\left(  k+1\right)  \ldots \left(  k+j-2\right)  } (eq. 4.40)


where A_{2}=A_{3}=\frac{1}{12}, A_{4}=\frac{19}{80}, etc. and C=0.577216 is the Euler constant. Now substituting for S_{k-1} from equation (4.40) into equation (4.38) and keeping only the two leading terms we get

f_{k-1}^{\prime}(0)=-\frac{1}{2kn^{2}}\left[  \frac{k^{3}}{3}-\frac{3k^{2}}{2}+O(k)\right]  \ldots (eq. 4.41)


xf^{\prime}(0) =-\frac{\alpha_{g}^{2}k^{2}}{6n^{2}}+\frac{3\alpha_{g}^{2}k}{4n^{2}}+O(\alpha_{g}^{2}k^{0})=-\frac{p^{2}}{6n^{2}}+\frac{3p^{2}}{4kn^{2}}+O(\frac{p^{2}}{k^{2}})=-\frac{p^{2}}{6n^{2}}+O(k^{-1}) (eq. 4.42)


where p=k\alpha_{g} is taken as a constant. Differentiating equation (4.37) again with respect to x and letting x=0 we obtain

f_{k-1}^{\prime\prime}(0)=-\frac{1}{kn^{2}}\sum_{j=2}^{k-1}\frac{\left(j-1\right)  ^{3}}{j}\left[  3g_{k-1}^{\prime}(0)-f_{k-1}^{\prime}(0)\right] (eq. 4.43)


Substituting from equations (4.38) and (4.39) into equation (4.43), we obtain

f_{k-1}^{\prime\prime}(0)=-\frac{1}{2kn^{4}}\sum_{j=2}^{k-1}\frac{(2j+1)\left(  j-1\right)  ^{2}S_{j-1}}{j^{2}} (eq. 4.44)


Since we need only the leading order terms for k\gg2, we substitute only the first term of \ S_{j-1} from equation (4.40) into equation (4.44) to obtain

f_{k-1}^{\prime\prime}(0) =-\frac{1}{2kn^{4}}\sum_{j=2}^{k-1}\frac {(4j^{2}-1)\left(  j-1\right)  ^{3}}{6j}+ \text{ l.o.t.} =-\frac{1}{2kn^{4}}\sum_{j=2}^{k-1}\left[  \frac{2}{3}j^{4}-2j^{3}+\frac{11}{6}j^{2}-\frac{1}{6}j-\frac{1}{2}+\frac{1}{6j}\right]  +\text{ l.o.t.} (eq. 4.45)


where l.o.t. stands for lower order terms. Now we use the identity

\sum_{i=1}^{m}i^{4}=\frac{m^{5}}{5}+\frac{m^{4}}{2}+\frac{m^{3}}{3}-\frac{m}{30} (eq. 4.46)


where m=k-1\gg1. Therefore substituting only the leading term from equation (4.46) into the leading term in equation (4.45) we get

f_{k-1}^{\prime\prime}(0)=-\frac{k^{4}}{15n^{4}}+O(k^{3}) (eq. 4.47)


\frac{x^{2}}{2!}f_{k-1}^{\prime\prime}(0) =-\frac{k^{4}\alpha_{g}^{4}}{30n^{4}}+O(\frac{p^{4}}{k})=-\frac{p^{4}}{30n^{4}}+O(k^{-1}) (eq. 4.48)


Substituting equations (4.42) and (4.48) into equation (4.36), we obtain

f_{k-1}(x)=1-\frac{p^{2}}{6n^{2}}-\frac{p^{4}}{30n^{4}}+O(k^{-1})\ldots (eq. 4.49)


Now we further assume n\gg1. This allows us to keep only the first two terms in equation (4.49). Therefore we get

f_{k-1}(x)\simeq1-\frac{p^{2}}{6n^{2}}+O(k^{-1})\ldots (eq. 4.50)


Now for k\gg2, f_{k-1} \simeq g_{k-1}. Henceforth, for simplicity, we will drop the notation O\left(k^{-1}\right) and we will replace the symbol \simeq by = in the following. Therefore, for k\gg2 and n\gg1 we have

f_{k-1}=g_{k-1}=1-\frac{p^{2}}{6n^{2}} (eq. 4.51)


Note that 1\geq f_{k-1}(x)\geq0. Therefore from equation (4.51) we have

0\leq p^{2}\leq6. (eq. 4.52)


Finally, substituting from equation (4.51) into equations (4.17) - (4.20) and equations (4.24) and (4.25), we have

E_{k}=-\frac{p^{2}mc^{2}}{2n_{k}^{2}}\left(  1-\frac{p^{2}}{6n^{2}}\right)^{2} (eq. 4.53)


r_{k}=\frac{\pi^{2}kRn_{k}^{2}}{16p^{2}\left(  1-\frac{p^{2}}{6n^{2}}\right)} (eq. 4.54)


M_{k}=mk\left(  1-\frac{p^{2}}{6n^{2}}\right) (eq. 4.55)


R_{k}=kR\left(  1-\frac{p^{2}}{6n^{2}}\right) (eq. 4.56)


\alpha_{k}=p\left(  1-\frac{p^{2}}{6n^{2}}\right) (eq. 4.57)


\frac{R_{k}}{r_{k}}=\frac{16p^{2}}{\pi^{2}n_{k}^{2}}\left(  1-\frac{p^{2}}{6n^{2}}\right)  ^{2} (eq. 4.58)


The density of the Macro Holeum is given by

\rho_{k}=\frac{3072m_{p}p^{5}\left(  1-\frac{p^{2}}{6n^{2}}\right)  ^{4}}{\pi^{7}kR_{p}^{3}n_{k}^{6}} (eq. 4.59)


In equations (4.53) - (4.59), p=k\alpha_{g} satisfies equation (4.52); and R_{p} given by

R_{p}=\frac{2m_{p}G}{c^{2}} (eq. 4.60)


is the Schwarzschild radius of a microscopic black hole of Planck mass. Apart from the integers n and n_{k} the seven properties listed in equations (4.53) - (4.59) depend, in general, on two parameters m and k or their two suitable combinations. But \alpha_{k} and \frac{R_{k}}{r_{k}} depend only upon the combination p=k\alpha_{g}. As mentioned earlier, p may be regarded as an effective coupling strength that determines the formation of the Macro Holeum.

The Ground State of the Macro Holeum

The ground state of Macro Holeums is characterized by n=\infty and n_{k}=1. The Macro Holeum has maximum binding energy, minimum physical radius, maximum Schwarzschild radius and maximum mass in this state. Such a system can be thought of as consisting of a gas of k-1 free (n=\infty) micro black holes that is bounded and therefore isolated from the outside world by a solitary outermost micro black hole whose principal quantum number is n_{k}=1.

The Stability of the Macro Holeum

It can be seen from the equations (4.53) - (4.57) that the condition for the stability of Holeums is given by

\frac{p^{2}}{6n^{2}}<1 (eq. 4.61)


Substituting the relations p=k\alpha _{g} and \alpha _{g}=\frac{m^{2}}{m_{P}^{2}} into this inequality, we see that the condition for the stability of Holeums can be expressed as

m<m_{P}\left( 6\right) ^{\frac{1}{4}}\left( \frac{n}{k}\right) ^{\frac{1}{2}} (eq. 4.62)


The ground state of Holeums is characterized by n=\infty, which gives us

m<\infty (eq. 4.63)


as the condition for stability. Thus, the ground state of Holeums is guaranteed to be always stable.

References

[1] L. K. Chavda and Abhijit L. Chavda, Dark matter and stable bound states of primordial black holes, arXiv:gr-qc/0308054 (2002).

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