If the ratio of the Schwarzschild radius of a Macro Holeum to its bound state radius , namely, 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:

(eq. 4.53) |

(eq. 4.54) |

(eq. 4.55) |

(eq. 4.56) |

(eq. 4.57) |

Substituting the expressions for and into the Black Holeum condition , we obtain the condition for a Holeum to be a Black Holeum to be

(eq. 4.64) |

It is interesting to note that this depends only on and ; the quantity does not matter here. For the ground state, which is characterized by , this reduces to

(eq. 4.65) |

Now, the entropy of a black hole is given by

(eq. 4.66) |

where is the Boltzmann constant, is the black hole’s surface area, and is the Planck length.

The Black Holeum’s entropy is given by

(eq. 4.67) |

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

(eq. 4.68) |

where

(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 , , and . 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).