Holeum

Properties

The binding energy E n of a holeum that consists of two identical micro black holes of mass m is given by

E n = − m c 2 α g 2 4 n 2

in which n is the principal quantum number, n = 1 , 2 , . . . , ∞ and α g is the gravitational counterpart of the fine structure constant. The latter is given by

α g = m 2 G ℏ c = m 2 m P 2

where:

ℏ is the Planck constant divided by 2 π ; 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 = 2 m + 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 = ( n 2 R α g 2 ) ( π 2 8 )

where:

R = ( 2 m G c 2 ) 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.

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

E k = − p 2 m c 2 2 n k 2 ( 1 − p 2 6 n 2 ) 2

where p = k α g and k ≫ 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 t h 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 = π 2 k R n k 2 16 p 2 ( 1 − p 2 6 n 2 )

The mass of the macro holeum is given by

M k = m k ( 1 − p 2 6 n 2 )

The Schwarzschild radius of the macro holeum is given by

R k = k R ( 1 − p 2 6 n 2 )

The entropy of the system is given by

S k = k 2 S ( 1 − p 2 6 n 2 )

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 = ∞ 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 = ∞ ) 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

p 2 6 n 2 < 1

Substituting the relations p = k α g and α g = m 2 m P 2 into this inequality, the condition for the stability of holeums can be expressed as

m < m P ( 6 ) 1 4 ( n k ) 1 2

The ground state of holeums is characterized by n = ∞ , which gives us m < ∞ 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 ⩽ 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 ⩾ m P 2 ( π n k k ) 1 2

For the ground state, which is characterized by n k = 1 , this reduces to

m ⩾ m P 2 ( π k ) 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. It is also speculated that holeums give rise to cosmic rays of all energies, including ultra-high-energy cosmic rays.