4.2 – The Black Holeum

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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