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I propose first a simple model for quantum black holes based on a harmonic oscillator representing the black hole horizon covered by Planck length sized squares carrying soft hair. Secondly, I redefine the partition function sum over horizon squares by a sum over black hole stretched horizon constituents which are black holes themselves. From this partition function Hawking radiation and Bekenstein-Hawking entropy law have been predicted. Based on this model I propose that the structures of black holes and spacetime atoms are the same. Requiring consistent quantization for both spacetime and matter, black hole structure is considered for matter particles using a composite model for quarks and leptons.

Thermodynamical properties, like entropy and temperature, of black holes have been established about four decades ago [

I describe first a warmup model for the structure of quantum black holes. The black hole horizon is a spherical membrane covered with

Secondly, I redefine the membrane partition function as a sum over black hole stretched horizon constituents based on [

Large amounts of results of research by the various schools of thought towards quantum gravity are widely scattered around in various journals in the literature, see e.g. [

This note is organized as follows. The presentation is very concise in all sections. After the Introduction the simple oscillator model for black holes is described in Section 2. In Section 3 I present the main points of the stretched horizon black hole model. An attempt for consistent quantization of spacetime and matter is discussed in Section 4 in connection with a model for composite quarks and leptons. Finally in Section 5 I give a brief discussion of results and conclusions.

As the first model for black holes of any size I assume the standard picture of a hole as a spherical horizon covered with

where

The vibrations of an oscillator can be calculated in normal way, with certain frequency restrictions due to the grid. The geometry is a two dimensional sphere

The energy eigenvalues for a square are given

The partition function Z is (to be used in section 5)

where

I consider a micro black hole dressed by a (virtual reality [

where n is the number of constituents, the

To calculate the partition function of a Schwarzschild black hole one needs to know the energy levels of the system. The energy of the hole from the point of view of an observer on its stretched horizon is called Brown- York energy [

where a is the constant proper acceleration of an observer on the stretched horizon and A is the area of the horizon. The possible energy values of a black hole are, from the point of view of an observer located on its stretched horizon, in terms of the acceleration

where

It depends on n and N only^{1}, and it gives the degeneracy function

whenever

If

The resulting partition function

where

where

The lengthy calculation yields a simple expression [

and the temperature

From the partition function one can calculate the average energy

of the hole at temperature

The average energy per constituent is

and one gets for large N

where

where

It is shown in [

and that

where

The most important implication of the observed phase transition at the characteristic temperature

With some more effort one can obtain the Bekenstein-Hawking entropy law for the Schwarzschild black hole from its partition function which, in turn, followed from the specific microscopic model of its stretched horizon [

When

It is interesting that, up to an unimportant numerical factor

How do we interpret the stretched horizon model? It applies both to black holes and atoms of spacetime. The next question is: are matter particles pointlike in this spacetime or do particles have some kind of substructure? In a unified model both spacetime and matter particles have the same type of substructure as we see in the next section.

Statistical methods of section 3 for spacetime offer a possibility to study the matter sector from a novel point of view for an attempt to build a consistent, unified statistical picture of both spacetime and matter. This goal would imply some internal structure at scale of the order of

The basic idea in [

The construction (23) on maxon level is matter-antimatter symmetric and 'color' singlet, which is desirable for early cosmology.

The maxons are black holes [

The gauge bosons and the Higgs would be elementary (but their composite nature is not ruled out). The three generations would be due to a gravitational mechanism or a new symmetry. Missing at the moment are calculational methods for the bound states. Numerical methods can usually be developed at some level of accuracy.

If a non-inertial observer perceives a horizon, he will attribute to it the Davies-Unruh temperature

where

It has turned out that horizons have profound importance in gravity both on thermodynamical and statistical levels. There are interesting questions of heat as inertial effect in a quantum equivalence principle and static observer’s virtual reality in [

In the UV black holes cannot be probed deeper than

The regime of real quantum gravity is limited to the vicinity of mini black holes and very early universe. Otherwise classical theory is accurate.

A model of decay and radiation of black holes has been proposed in [

There are at present a number competing theoretical schemes for quantum gravity like string theory, loop quantum gravity, causal dynamical triangulation, and others. The model of Section 3 goes deep into the structure of the physical universe and can be considered as a promising candidate. In that scenario the horizon properties of black holes and local Rindler frames are the origin of gravity and general relativity is its IR limit. These ideas have also been applied to the matter sector in connection of a preon model in Section 4.

The results of this simple model can be considered encouraging for the development of a more realistic model. A more involved model for the stretched horizon should include gravitons and is expected to produce a smoother transition for the system in getting down from Planck scale down to standard model particle mass scale. Finally, this note is a program definition for future work with a number of model results as guidelines.

Risto Raitio, (2016) A Statistical Model of Spacetime, Black Holes and Matter. Open Access Library Journal,03,1-6. doi: 10.4236/oalib.1102487