Holeum

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 $n$ 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 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 $k$ 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 $k\gg 2$. The system is simplified by assuming that all the micro black holes in the core are in the same quantum state described by $n$, 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 $k$, $m$, $n$, 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

1. L. K. Chavda & Abhijit Chavda, Dark matter and stable bound states of primordial black holes
2. L. K. Chavda & Abhijit Chavda, Dark matter and stable bound states of primordial black holes
3. M. Yu. Khlopov, Primordial Black Holes
4. L. K. Chavda & Abhijit Chavda, Quantized Gravitational Radiation from Black Holes and other macro holeums in the Low Frequency Domain
5. S. Al Dallal, Holeums as potential candidates for some short-lived gamma ray bursts
6. S. Al Dallal, Primordial black holes and holeums as progenitors of galactic diffuse gamma-ray background
7. 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