160.39.193.45 at 05:24, 29 April 2012

2012-04-29T05:24:56Z

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'''Holeums''' are hypothetical stable, quantized gravitational bound states of [[Primordial black hole|primordial]] or [[micro black holes]].

==Introduction==

Holeums were proposed by L.K. Chavda and Abhijit Chavda in 2002.<ref>L.K. Chavda & Abhijit Chavda, [http://iopscience.iop.org/0264-9381/19/11/311 Dark matter and stable bound states of primordial black holes]</ref> 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]] <math>E_{n}</math> of a Holeum that consists of two identical micro black holes of mass <math>m</math> is given by<ref>L.K. Chavda & Abhijit Chavda, [http://arxiv.org/abs/gr-qc/0308054 Dark matter and stable bound states of primordial black holes]</ref>

: <math>E_{n}=-\frac{mc^{2}\alpha_{g}^{2}}{4n^{2}}</math>

in which <math>n</math> is the principal [[quantum number]], <math>n=1,2,...\infty</math> and <math>\alpha_{g}</math> is the gravitational counterpart of the [[fine structure constant]]. The latter is given by

: <math>\alpha_{g}=\frac{m^{2}G}{\hbar c}=\frac{m^{2}}{m_{P}^{2}}</math>

where:

: <math>\hbar</math> is the [[Planck constant]] divided by <math>2\pi</math>;
: <math>c</math> is the [[speed of light]] in vacuum;
: <math>G</math> is the [[gravitational constant]].

The <math>n^{\text{th}}</math> excited state of a Holeum then has a mass that is given by

: <math>m_{H}=2m+\frac{E_{n}}{c^{2}}</math>

The Holeum's atomic transitions cause it to emit [[gravitational radiation]].

The radius of the <math>n^{\text{th}}</math> excited state of a Holeum is given by

: <math>r_{n}=\left( \frac{n^{2}R}{\alpha_{g}^{2}}\right) \left( \frac{\pi^{2}}{{8}}\right)</math>

where:

: <math>R=\left( \frac{2mG}{c^{2}}\right)</math> 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.<ref>M. Yu. Khlopov, [http://arxiv.org/abs/arXiv:0801.0116 Primordial Black Holes]</ref>

==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 <math>k</math> identical micro black holes of mass <math>m</math> are given by<ref>L.K. Chavda & Abhijit Chavda, [http://arxiv.org/abs/0903.0703 Quantized Gravitational Radiation from Black Holes and other Macro Holeums in the Low Frequency Domain]</ref>

: <math>E_{k}=-\frac{p^{2}mc^{2}}{2n_{k}^{2}}\left( 1-\frac{p^{2}}{6n^{2}}\right)^{2}</math>

where <math>p=k\alpha_{g}</math> and <math>k\gg2</math>. The system is simplified by assuming that all the micro black holes in the core are in the same quantum state described by <math>n</math>, and that the outermost, <math>k^{th}</math> micro black hole is in an arbitrary quantum state described by the principal quantum number <math>n_{k}</math>.

The ''physical radius'' of the bound state is given by

: <math>r_{k}=\frac{\pi^{2}kRn_{k}^{2}}{16p^{2}\left( 1-\frac{p^{2}}{6n^{2}}\right)}</math>

The mass of the Macro Holeum is given by

: <math>M_{k}=mk\left( 1-\frac{p^{2}}{6n^{2}}\right)</math>

The Schwarzschild radius of the Macro Holeum is given by

: <math>R_{k}=kR\left( 1-\frac{p^{2}}{6n^{2}}\right)</math>

The [[entropy]] of the system is given by

: <math>S_{k}=k^{2}S\left( 1-\frac{p^{2}}{6n^{2}}\right)</math>

where <math>S</math> 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 <math>n=\infty</math> and <math>n_{k}=1</math>. 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 <math>k-1</math> free (<math>n=\infty</math>) 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 <math>n_{k}=1</math>.

== Stability of Holeums ==

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

<math>\frac{p^{2}}{6n^{2}}<1</math>

Substituting the relations <math>p=k\alpha _{g}</math> and <math>\alpha _{g}=\frac{m^{2}}{m_{P}^{2}}</math> into this inequality, the condition for the stability of Holeums can be expressed as

<math>m<m_{P}\left( 6\right) ^{\frac{1}{4}}\left( \frac{n}{k}\right) ^{\frac{1}{2}}</math>

The ground state of Holeums is characterized by <math>n=\infty</math>, which gives us <math>m<\infty</math> 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

<math>r_{k}\leqslant R_{k}</math>

Such Holeums are termed Black Holeums. Substituting the expressions for <math>r_{k}</math> and <math>R_{k}</math>, and simplifying, we obtain the condition for a Holeum to be a Black Holeum to be

<math>m\geqslant \frac{m_{P}}{2}\left( \frac{\pi n_{k}}{k}\right) ^{\frac{1}{2}}</math>

For the ground state, which is characterized by <math>n_{k}=1</math>, this reduces to

<math>m\geqslant \frac{m_{P}}{2}\left( \frac{\pi}{k}\right) ^{\frac{1}{2}}</math>

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 <math>k</math>, <math>m</math>, <math>n</math>, and <math>n_{k}</math>.

== Holeums and cosmology ==

Holeums are speculated to be the progenitors of a class of short duration [[gamma ray bursts]].<ref>S. Al Dallal, [http://adsabs.harvard.edu/abs/2007AdSpR..40.1236A Holeums as potential candidates for some short-lived gamma ray bursts]</ref><ref>S. Al Dallal, [http://adsabs.harvard.edu/abs/2010AdSpR..46..468D Primordial black holes and Holeums as progenitors of Galactic diffuse gamma-ray background]</ref> It is also speculated that Holeums give rise to [[cosmic rays]] of all energies, including [[ultra-high-energy cosmic rays]].<ref>L.K. Chavda & Abhijit Chavda, [http://arxiv.org/abs/0806.0454 Ultra High Energy Cosmic Rays from decays of Holeums in Galactic Halos]</ref>

== References ==

{{Reflist}}

==External links==
* [http://actaphysica.com Acta Physica: Chronicles the development of the theory of Holeums]
* [http://actaphysica.com/black-holes/a-stable-holeum/ A Stable Holeum]
* [http://actaphysica.com/black-holes/gravitational-radiation/ Gravitational Radiation from Holeums]
* [http://actaphysica.com/holeum/constructing-a-macro-holeum-from-the-inside-out/ Constructing a Macro Holeum from the Inside Out]
* [http://actaphysica.com/holeum/the-black-holeum/ The Black Holeum]

{{Dark matter}}

[[Category:Physical cosmology]]
[[Category:Dark matter]]
[[Category:Large-scale structure of the cosmos]]
[[Category:Astroparticle physics]]
[[Category:Exotic matter]]
[[Category:Unsolved problems in physics]]
[[Category:Particle physics]]
[[Category:Theoretical physics]]
[[Category:Hypothetical particles]]

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160.39.193.45 at 05:24, 29 April 2012