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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\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 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]