# 4.2 – The Black Holeum

If the ratio of the Schwarzschild radius $R_{k}$ of a Macro Holeum to its bound state radius $r_{k}$, namely, $\frac{R_{k}}{r_{k}}$ is greater than or equal to unity, then the Schwarzschild radius of the Macro Holeum is greater than its physical radius. Therefore the Macro Holeum is a black hole. We call such a Macro Holeum a Black Holeum and denote it as BH.

Recall that the properties of a Macro Holeum are given by the following equations:

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

Substituting the expressions for $r_{k}$ and $R_{k}$ into the Black Holeum condition $R_{k}\geqslant r_{k}$, we obtain the condition for a Holeum to be a Black Holeum to be

 $m\geqslant \frac{m_{P}}{2}\left( \frac{\pi n_{k}}{k}\right) ^{\frac{1}{2}}$ (eq. 4.64)

It is interesting to note that this depends only on $k$ and $n_{k}$; the quantity $n$ does not matter here. For the ground state, which is characterized by $n_{k}=1$, this reduces to

 $m\geqslant \frac{m_{P}}{2}\left( \frac{\pi }{k}\right) ^{\frac{1}{2}}$ (eq. 4.65)

Now, the entropy of a black hole is given by

 $S=\frac{k_{B}A}{4l_{P}^{2}}$ (eq. 4.66)

where $k_{B}$ is the Boltzmann constant, $A=4\pi R^{2}$ is the black hole’s surface area, and $l_{P}=\sqrt{G\hbar /c^{3}}$ is the Planck length.

The Black Holeum’s entropy is given by

 $S_{k}=\frac{k_{B}c^{3}\pi }{\hbar G}R_{k}^{2}=\frac{k_{B}c^{3}\pi }{\hbar G}k^{2}R^{2}\left( 1-\frac{p^{2}}{6n^{2}}\right) ^{2}$ (eq. 4.67)

Thus, the expression for the entropy of a Black Holeum is

 $S_{k}=k^{2}\left( 1-\frac{p^{2}}{6n^{2}}\right) ^{2}S$ (eq. 4.68)

where

 $S=\frac{k_{B}c^{3}\pi }{\hbar G}R^{2}$ (eq. 4.69)

is the entropy of the individual MBHs that constitute the Macro Holeum.

Black Holeums are quantum black holes whose internal structure can be predicted by means of the quantities $k$, $m$, $n$ and $n_{k}$. Black Holeums are an example of black holes with internal structure. Per the black hole no-hair theorem, one can argue that all macroscopic (non-MBH) black holes are Black Holeums. This has interesting implications for black hole physics. More on this later.

### 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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