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 are not black holes, even though they are made up of black holes.

Properties

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 {\displaystyle n} 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 c is the speed of light in vacuum;
G is the gravitational constant.

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

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

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

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

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

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 {\displaystyle k} k identical micro black holes of mass m {\displaystyle m} 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 $$k\gg 2$$. The system is simplified by assuming that all the micro black holes in the core are in the same quantum state described by n {\displaystyle n} 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

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 termed 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 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]

Micro black hole
Black hole electron
Planck particle
Dark matter

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

Acta Physica: Chronicles the development of the theory of holeums
A Stable Holeum
Gravitational Radiation from Holeums
Constructing a Macro Holeum from the Inside Out
The Black Holeum

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