Holeum

Holeums are stable, quantized gravitational bound states of primordial or micro black holes.

1 Introduction
2 Properties of Holeums
3 Macro Holeums and their properties
4 The ground state of Macro Holeums
5 Stability of Holeums
6 Black Holeums
7 Holeums and Cosmology
8 References

Introduction

Holeums were proposed by L.K. Chavda and Abhijit Chavda in 2002.^[1] They have all the properties associated with cold dark matter. Holeums are not black holes, even though they are made up of black holes.

Properties of Holeums

The binding energy $E_{n}$ of a Holeum that consists of two identical micro black holes of mass $m$ is given by^[2]

$E_{n}=-\frac{mc^{2}\alpha_{g}^{2}}{4n^{2}}$

in which $n$ is the principal quantum number, $n=1,2,...\infty$ and $\alpha_{g}$ is the gravitational counterpart of the fine structure constant. The latter is given by

$\alpha_{g}=\frac{m^{2}G}{\hbar c}=\frac{m^{2}}{m_{P}^{2}}$

where:

$\hbar$ is the Planck constant divided by $2\pi$ ;

$c$ is the speed of light in vacuum;

$G$ is the gravitational constant.

The $n^{\text{th}}$ excited state of a Holeum then has a mass that is given by

$m_{H}=2m%2B\frac{E_{n}}{c^{2}}$

The Holeum's atomic transitions cause it to emit gravitational radiation.

The radius of the $n^{\text{th}}$ excited state of a Holeum is given by

$r_{n}=\left( \frac{n^{2}R}{\alpha_{g}^{2}}\right) \left( \frac{\pi^{2}}{{8}}\right)$

where:

$R=\left( \frac{2mG}{c^{2}}\right)$ is the Schwarzschild radius of the two identical micro black holes that constitute the Holeum.

The Holeum is a stable particle. It is the gravitational analogue of the hydrogen atom. It occupies space. Although it is made up of black holes, it itself is not a black hole. As the Holeum is a purely gravitational system, it emits only gravitational radiation and no electromagnetic radiation. The Holeum can therefore be considered to be a dark matter particle.^[3]

Macro Holeums and their properties

A Macro Holeum is a quantized gravitational bound state of a large number of micro black holes. The energy eigenvalues of a Macro Holeum consisting of $k$ identical micro black holes of mass $m$ are given by^[4]

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

where $p=k\alpha_{g}$ and $k\gg2$ . The system is simplified by assuming that all the micro black holes in the core are in the same quantum state described by $n$ , and that the outermost, $k^{th}$ micro black hole is in an arbitrary quantum state described by the principal quantum number $n_{k}$ .

The physical radius of the bound state is given by

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

The mass of the Macro Holeum is given by

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

The Schwarzschild radius of the Macro Holeum is given by

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

The entropy of the system is given by

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

where $S$ is the entropy of the individual micro black holes that constitute the Macro Holeum.

The ground state of Macro Holeums

The ground state of Macro Holeums is characterized by $n=\infty$ and $n_{k}=1$ . The Holeum has maximum binding energy, minimum physical radius, maximum Schwarzschild radius, maximum mass, and maximum entropy 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 $n_{k}=1$ .

Stability of Holeums

It can be seen from the above equations that the condition for the stability of Holeums is given by

$\frac{p^{2}}{6n^{2}}<1$

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

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

The ground state of Holeums is characterized by $n=\infty$ , which gives us $m<\infty$ as the condition for stability. Thus, the ground state of Holeums is guaranteed to be always stable.

Black Holeums

A Holeum is a black hole if its physical radius is less than or equal to its Schwarzschild radius, i.e. if

$r_{k}\leqslant R_{k}$

Such Holeums are known as Black Holeums. Substituting the expressions for $r_{k}$ and $R_{k}$ , and simplifying, we obtain the condition for a Holeum to be a Black Holeum to be

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

For the ground state, which is characterized by $n_{k}=1$ , this reduces to

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

Black Holeums are an example of black holes with internal structure. Black Holeums are quantum black holes whose internal structure can be fully predicted by means of the quantities $k$ , $m$ , $n$ , and $n_{k}$ .

Holeums and Cosmology

Holeums are speculated to be the progenitors of a class of short duration gamma ray bursts.^[5]^[6] It is also speculated that Holeums give rise to cosmic rays of all energies, including ultra-high-energy cosmic rays.^[7]

References

^ L.K. Chavda & Abhijit Chavda, Dark matter and stable bound states of primordial black holes
^ L.K. Chavda & Abhijit Chavda, Dark matter and stable bound states of primordial black holes
^ M. Yu. Khlopov, Primordial Black Holes
^ L.K. Chavda & Abhijit Chavda, Quantized Gravitational Radiation from Black Holes and other Macro Holeums in the Low Frequency Domain
^ S. Al Dallal, Holeums as potential candidates for some short-lived gamma ray bursts
^ S. Al Dallal, Primordial black holes and Holeums as progenitors of Galactic diffuse gamma-ray background
^ L.K. Chavda & Abhijit Chavda, Ultra High Energy Cosmic Rays from decays of Holeums in Galactic Halos