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]

See also

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

External links

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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