Holeums are hypothetical stable, quantized gravitational bound states of primordial or micro black holes.
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+\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}}\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 k1 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
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 ultrahighenergy 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 shortlived gamma ray bursts

^ S. Al Dallal, Primordial black holes and Holeums as progenitors of Galactic diffuse gammaray 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


Forms of dark matter




Hypothetical particles



Theories and objects



Search experiments

Direct detection



Indirect detection



Other projects




Potential dark galaxies



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