1.2 – A stable Holeum

For the sake of simplicity we consider two identical black holes of mass $m$. The gravitational potential between them is given by

 $V(r)=-\frac{m^{2}G}{r}=-\frac{\alpha_{g}\hbar c}{r}$ (eq. 1.25)

where $r$ is the distance between them. $\hbar$, $c$ and $G$ are Planck’s constant reduced by $2\pi$, the speed of light in vacuum and Newton’s universal gravitational constant respectively. $\alpha_{g}$ is the gravitational analogue of the fine structure constant, given by

 $\alpha_{g}=\frac{m^{2}G}{\hbar c}=\frac{m^{2}}{m_{P}^{2}}$ (eq. 1.26)

where

 $m_{P}=\left( \frac{\hbar c}{G}\right) ^{\frac{1}{2}}$ (eq. 1.27)

is the Planck mass. The Schrödinger equation is exactly solvable for the $r^{-1}$ potential and the energy eigenvalues, formally identical with those of the hydrogen atom, are given by [2]

 $E_{n}=-\frac{\mu c^{2}\alpha_{g}^{2}}{2n^{2}}$ (eq. 1.28)

where $n$ is the principal quantum number, n=1,2,…$\infty$ and $\mu=m/2$ is the reduced mass. In the following we will consider, for simplicity, only the $l=0$, $s$-states. The eigenfunction for an $ns$ state is given by [1]

 $\Psi_{ns}=A_{n}L_{n-1}^{1}(t)e^{-t/2}$ (eq. 1.29)

where

 $t=2\chi r$ (eq. 1.30)

 $\chi=\frac{\alpha_{g}^{2}}{nR}$ (eq. 1.31)

 $R=\frac{2mG}{c^{2}}$ (eq. 1.32)

where $R$ is the Schwarzschild radius of the black hole. Here $L_{n}^{m}(x)$ is the associated Laguerre polynomial and

 $A_{n}^{2}=\frac{4\chi^{3}}{n^{2}(n!)^{2}}$ (eq. 1.33)

The maxima of the probability density

 $g(r)=r^{2}|\Psi_{ns}|^{2}$ (eq. 1.34)

give us the radii of the stable orbits. For the 1s state the radius of the orbit is given by

 $r_{_{1}}=\frac{R}{\alpha_{g}^{2}}$ (eq. 1.35)

For the 2s state there are two orbits with radii

 $r_{2\pm}=\frac{\left(3\pm\sqrt{5}\right)R}{\alpha_{g}^{2}}$ (eq. 1.36)

For $m\gg1$ we have [3],

 $L_{m}^{\alpha}(x)\cong\pi^{-\frac{1}{2}}\left(m+\alpha\right)!x^{-\frac{\alpha}{2}-\frac{1}{4}}m^{\frac{\alpha}{2}-\frac{1}{4}}e^{\frac{x}{2}}\cos\left[2(mx)^{\frac{1}{2}}-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right]$ (eq. 1.37)

Using this we can show that for $n$, $n^{^{\prime}}\gg1$, the radii of the stable orbits are given by

 $r_{n}=\frac{\pi^{2}n^{^{\prime}2}R}{8\alpha_{g}^{2}}$ (eq. 1.38)

where $n^{^{\prime}}$=1,2,…,n. The maxima of the probability density are given by

 $g_{\max}\cong\frac{n^{^{\prime}}\alpha_{g}^{2}}{2n^{3}R}$ (eq. 1.39)

Because of the large factor $n^{3}$ in the denominator this is appreciable only for $n^{^{\prime}}=n$. Thus we take $n^{^{\prime}}=n$ in equations (1.38) and (1.39) to get the equation

 $r_{n}\cong\left(\frac{n^{2}R}{\alpha_{g}^{2}}\right)\left(\frac{\pi^{2}}{8}\right)$ (eq. 1.40)

It is interesting to note that the value given by the semiclassical Bohr theory is

 $r_{n}\cong\frac{n^{2}R}{\alpha_{g}^{2}}$ (eq. 1.41)

Since $\pi^{2}/8$ in equation (1.40) is of the order of unity the two results, equations (1.40) and (1.41), are comparable. Since the area of a black hole never decreases and since the black holes in a stable Holeum must not overlap, all bound state radii $r_{n}$ must exceed twice the black hole horizon radius in appropriate coordinates. Naïvely one might be tempted to say $r_{n}>2R$. Strictly speaking, we are prevented from extracting such a precise inequality, however, by the fact that our analysis is only valid for values of $r\gg R$.

A true “nonoverlap” condition for the black holes can only really be extracted in the strong field regime, where our analysis is not valid. In the strong field case care must be taken, since the position of the black hole horizon is coordinate-dependent, and in terms of an isotropic radial coordinate for which $r^{2}=x^{2}+y^{2}+z^{2}$ in terms of asymptotically Euclidean coordinates $x$, $y$, $z$ (and which, therefore, is the radial coordinate appropriate to the weak field limit used here), the position of the horizon is actually at $r=R/4$.

However, black holes which are very close together can really only be analysed by a full general relativistic solution to the appropriate two-body problem. Coordinate distances cannot be expected to be simply additive in this case. We nonetheless find that it is useful to extract a rough dividing line between stable and unstable Holeums. We will, therefore, define the gravitational radius of a black hole in a purely Newtonian sense as the radius for which the escape velocity from a spherical body of mass m is equal to the velocity of light. This singles out the Schwarzschild radius, equation (1.32), as the “Newtonian black hole radius”.

As outlined above, using $2R$ as the minimum possible separation of Newtonian “black holes”, we are led to conclude that

 $r_{n}>2R$ (eq. 1.42)

for all $n$ and at all times. Now we consider two cases:

\alpha_{g}^{2}<\frac{\pi^{2}}{16}[/latex] (eq. 1.43)
and
 [latex size="2"]\alpha_{g}^{2}>\frac{\pi^{2}}{16}
(eq. 1.44)

From equations (1.40) and (1.43) we see that the condition for nonoverlap, equation (1.42), can be satisfied for all $n$, including $n=1$, the ground state. In this case, the Holeum is as stable as a hydrogen atom. On the other hand when $\alpha_{g}$ is given by equation (1.44) the nonoverlap condition, equation (1.42) can be satisfied only if

 $n^{2}>\frac{16\alpha_{g}^{2}}{\pi^{2}}>1$ (eq. 1.45)

In this case, the Holeum can exist only in excited states and the system is denied the stable ground state $n=1$. This will eventually result in the coalescence of the constituent black holes and the destruction of the Holeum.

Now we would like to discuss the validity of equation (1.40). It is derived in the framework of Newtonian Gravity which applies for $r\gg R$. A Holeum of atomic size has a ground state radius of about $10^{-10}$ m whereas $R<10^{-35}[/latex] m. Thus, the condition $r\gg R$ is eminently satisfied. Similar considerations apply to the nuclear-sized Holeum. In fact, even if we take the black hole mass as big as $m=0.1m_{P}$, we get the Holeum radius  $r_{n}=10^{4}n^{2}R\left(\frac{\pi^{2}}{8}\right)$ (eq. 1.46) And this, too, eminently satisfies the NG condition, $r_{n}\gg R$. Now consider the dividing line  $\alpha_{g}^{2}=\frac{\pi^{2}}{16}$ (eq. 1.47) between the unstable and stable Holeums. This corresponds to the mass of the black holes $m=0.8862m_{P}$. For this case the NG breaks down as expected from our discussion above. Nevertheless, we note that the potential for $r\gg R$ is still $r^{-1}$ and in view of the BSW and the POTHA presented in the introduction, we might still hope to get reasonable order of magnitude values of the bound state parameters in this strong field regime. In summary, equation (1.40) is to be regarded as an asymptotic expression in the strong field case and exact elsewhere, whereas the inequalities in equations (1.42) - (1.45) are to be regarded as probably only valid to within an order of magnitude. This caveat must be borne in mind in what follows. The mass of the bound state is given by  $M_{n}=2m+\frac{E_{n}}{c^{2}}$ (eq. 1.48) Substituting equation (1.28) into equation (1.48) we obtain  $M_{n}=2m\left( 1-\frac{\alpha_{g}^{2}}{8n^{2}}\right)$ (eq. 1.49) From equations (1.31), (1.40) and (1.49) we get  $\frac{M_{n}}{r_{n}}=\left(\frac{16\alpha_{g}^{2}}{\pi^{2}n^{2}}\right)\left(\frac{c^{2}}{2G}\right)\left(1-\frac{\alpha_{g}^{2}}{8n^{2}}\right)$ (eq. 1.50) In view of equation (1.43), we see that for a stable Holeum \frac{M_{n}}{r_{n}}<\frac{c^{2}}{2G}$ (eq. 1.51)
for all n. This shows that a stable Holeum satisfying equation (1.43) is not a black hole. With the help of equation (1.26) we can rewrite the condition for a stable Holeum, equation (1.43) as
 [latex size="2"]m<\left(\frac{\pi^{\frac{1}{2}}}{2}\right) m_{P}\equiv m_{c}[/latex] (eq. 1.52)

where [latex]m_{c}
will be called the cosmic limit for the formation of a stable Holeum. The numerical value of $m_{c}$, equation (1.52), is

 $m_{c}=0.8862m_{P}$ (eq. 1.53)

whereas the semiclassical Bohr result, equation (1.41), gives a slightly different value $m_{c}=2^{-\frac{1}{4}}m_{P}=0.8409m_{P}$. Thus, we find that if each of the masses of two identical black holes is less than $m_{c}$ then they will form a stable Holeum. We note that if the black holes have unequal masses $m_{1}$ and $m_{2}$ then the condition for a stable Holeum would be

\left(m_{1}m_{2}\right)^{\frac{1}{2}} (eq. 1.54)
Equation (1.52) is both the necessary and sufficient condition for the nonoverlap of the constituent black holes of a stable Holeum embodied in equation (1.42). Not only that, it guarantees that the Holeum will be stable. Equation (1.52) implies equation (1.42) which, in turn, implies that a Holeum occupies space just like ordinary matter. Its size cannot be reduced below [latex]2R
. This nonoverlap property is similar to the Pauli exclusion principle and reminds us of the following result from the second quantized field theory.

If we try to second-quantize a spinor field using a commutation rule rather than an anticommutation one, then there is no lower bound on the energy of the bound state and there will be no stable fermions in the universe. On the other hand, if we quantize a spinor field using an anticommutation rule then there is a lower bound and the system is stable.

The anticommutation rule leads to the exclusion property. This is a spin-dependent property. In our case we have derived equation (1.40) from the maxima of the probability density which has no classical analogue. This is a purely
quantum mechanical property except that it is a mass-dependent one. If the mass of the black holes is less than $m_{c}$, there is no overlap and the system also has the ground state $n=1$. If the mass is greater than $m_{c}$, they overlap and the ground state $n=1$ is unavailable to them. They would annihilate.

References

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

[2] Merzbacher E, Quantum Mechanics (New York: Wiley) p 190 (1961).

[3] Gradshteyn I S and Ryzhik I W, Table of Integrals, Series and Products (New York: Academic) p 1039 (1965).

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