# Holeum

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Holeums are hypothetical stable, quantized gravitational bound states of primordial or micro black holes. Holeums were proposed by L. K. Chavda and Abhijit Chavda in 2002.[1] They have all the properties associated with cold dark matter. Holeums do not have to be black holes, even though they are made up of black holes.

## Properties

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

${\displaystyle E_{n}=-{\frac {mc^{2}\alpha _{g}^{2}}{4n^{2}}}}$

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

${\displaystyle \alpha _{g}={\frac {m^{2}G}{\hbar c}}={\frac {m^{2}}{m_{P}^{2}}}}$

where:

${\displaystyle \hbar }$ is the Planck constant divided by ${\displaystyle 2\pi }$;
${\displaystyle c}$ is the speed of light in vacuum;
${\displaystyle G}$ is the gravitational constant.

The nth excited state of a holeum then has a mass that is given by

${\displaystyle m_{H}=2m+{\frac {E_{n}}{c^{2}}}}$

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

The radius of the nth excited state of a holeum is given by

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

where:

${\displaystyle R=\left({\frac {2mG}{c^{2}}}\right)}$ 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 ${\displaystyle k}$ identical micro black holes of mass ${\displaystyle m}$ are given by[4]

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

where ${\displaystyle p=k\alpha _{g}}$ and ${\displaystyle k\gg 2}$. The system is simplified by assuming that all the micro black holes in the core are in the same quantum state described by ${\displaystyle n}$, and that the outermost, ${\displaystyle k^{th}}$ micro black hole is in an arbitrary quantum state described by the principal quantum number ${\displaystyle n_{k}}$.

The physical radius of the bound state is given by

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

The mass of the macro holeum is given by

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

The Schwarzschild radius of the macro holeum is given by

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

The entropy of the system is given by

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

where ${\displaystyle S}$ 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 ${\displaystyle n=\infty }$ and ${\displaystyle n_{k}=1}$. 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 ${\displaystyle k-1}$ free (${\displaystyle n=\infty }$) 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 ${\displaystyle n_{k}=1}$.

## Stability

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

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

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

${\displaystyle m

The ground state of holeums is characterized by ${\displaystyle n=\infty }$, which gives us ${\displaystyle m<\infty }$ 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

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

Such holeums are termed black holeums. Substituting the expressions for ${\displaystyle r_{k}}$ and ${\displaystyle R_{k}}$, and simplifying, we obtain the condition for a holeum to be a black holeum to be

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

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

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

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 ${\displaystyle k}$, ${\displaystyle m}$, ${\displaystyle n}$, and ${\displaystyle n_{k}}$.

## 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]