_{1}

^{*}

Within the thermodynamic model of gravity the dark energy is identified with the energy of collective gravitational interactions of all particles in the universe, which is missing in the standard treatments. For the model-universe we estimate the radiation, baryon and dark energy densities and obtain the values which are close to the current observations. It is shown that total gravitational potential of a particle from the world ensemble is a scale-dependent quantity and its value is twice Newtonian potential. The Einstein-Infeld-Hoffmann approximation to general relativity was used to show that the acceleration of a particle from the world ensemble can be considered as a relative quantity when the universe is described by the flat cosmological model.

To construct a physical theory it is convenient to use inertial reference frames. The crucial question is: What are the inertial frames? How are they found? The precise acceleration of the Earth relative to the universe as a whole is quite difficult to measure. However, now it is possible to find quite accurately the absolute velocity of the Earth with respect to the distant stars or cosmic microwave background. Such measurements reveal that “universal” reference frames, measured by these two different methods, coincide and constant velocities with respect to the universe seem to correspond to inertial frames [

The possibility of identifying the absolute reference frame of the universe can be understood as an observational verification of Mach’s principle [

Machian models usually assume that inertia of a particle with the mass m is determined by the gravitational field of the whole universe and the particles total energy (inertial + gravitational) at rest with respect to the universe is zero [

where

It is known that the usual interpretation of Mach’s principle that inertia is a relative quantity and is not attributed to an object, leads to the anisotropy of the rest mass of particles due to the influence of nearby massive objects (like the Galaxy) [

As an attempt to address these problems a thermodynamic model of gravity, where the universe is considered as the statistical ensemble of all gravitationally interacting particles inside the horizon, was proposed in [

According to the thermodynamic model [

Within the model [

were

where

If we interpret

To avoid variations of fundamental physical constants, c or G, which is ruled out by experiments [

where we introduced the total energy density of the flat cosmological model:

which is the only cosmological model for which

should take into account the total matter content of the universe and not only the ordinary matter for which the classical Newton’s law is written. This point will be discussed in details in Section 3.

In Section 4 we demonstrate that the inertia of a particle can be related to the total energy content of the universe. In standard physics acceleration is absolute in origin and all forces arise from close sources. We want to show that the acceleration can be considered as a relative quantity and Newton’s second law can be written in the form:

In this formula the reactive acceleration:

appears due to the non-local forces of the surrounding universe and

At first let us show that realistic values of cosmological parameters can be obtained for the simplest model-universe of radius

were

Correspondingly, according to the relation like (3), the gravitational mass of the world ensemble is a quadratic function of the number of particles:

In the model [

Indeed, under the above identification the energy balance condition (1), written for all N particles in the universe, is equivalent to the equation of state of the dark energy:

In our model the appearance of the “exotic negative pressure

Relations (2), (10), (12) and (13) allow us to estimate the total action of the universe,

and the number of typical particles in it:

This number is known to have appeared in a different context in the Dirac’s “large numbers” hypothesis [

Using the estimation (16) and the formulae (2), (3), (10) and (12), from (13) we can express the value of the dark energy density in our model-universe in terms of the fundamental physical parameters:

which is very close to the observed value [

Now, let us consider a little bit more realistic model-universe assuming that a part of particles of the world ensemble is charged. The universe as a whole is neutral, i.e. a half of charged particles carries positive charge

Then, according to (12), for the total gravitational energy of the baryon component of matter we obtain:

It is natural to expect that the ratio (13) of the gravitational and total energy is valid also for the corresponding contributions of the baryon component:

where

Further, let us estimate the total electromagnetic energy of all

where

Using this formula and the observed value of the radiation energy density [

we can estimate the number of baryons in the universe:

which turns out to be only one order of magnitude less than the estimated total number of particles (16) in our model-universe.

Finally, Equations (19), (20), (21) and (23) yield for the ratio of the radiation and baryon densities in the universe

From (2), (13) and (10) we find

whence it follows:

which is also very close to the observed value.

In this section, using different arguments, we show that the total gravitational potential of an object in the universe is scale dependent quantity and obtain twice of its Newtonian value. Only half of this total energy can be transformed to the kinetic energy, while the other half is needed to compensate the negative vacuum energy, and thus does not affect local physics. This idea is not quite new. It was already noticed that the general relativistic deflection for a test particle with an arbitrary velocity

In the case of photons the inertial mass

So if we consider a free photon with the energy:

and apply to it the so-called Tolman’s formula for active gravitational mass [

i.e. two times bigger value than the expected one. To restore the Einstein’s standard relation one needs to consider interaction of the photon with other bodies. If a photon is confined in a box with mirrors, then we have a composite body at rest. In this case, as shown in [

Consider a particle with the energy

Assuming the translation invariance of the energy, one should write:

If our universe is close to the black hole state, i.e. it obeys (3), we find:

Thus, as the particle reaches the horizon, its energy and mass are doubled. Half of the particles energy at the horizon,

Thus the total change in potential energy of a particle of mass

and not to

Half of (36) is spent to the change of the particle’s internal energy:

Also, the total change in the effective gravitational potential

For small non-relativistic systems the Newtonian potential does not lead to mistake, since only a half of the total gravitational energy transforms to the kinetic energy:

However, for the relativistic cases, or cosmological distances, the Newtonian theory gives wrong results.

Let us recall how relativistic formulae appear in the thermodynamic model of gravity [

The velocity dependent parameter of inertia of this particle is defined as:

where, according to (3),

The number of particles in the world ensemble is conserved. Thus the Machian energy

or

Then, using the Hamilton’s definition of the velocity:

Consequently, the quantity

where

is the standard Lorentz factor, is constant, and hence it can be interpreted as a mass parameter of a particle (also known as the rest mass) which is valid in any inertial frame.

Returning to the main question about the total energy of a object, let us consider a generalization of the energy balance Equation (41):

where we had added the term contained

In the Newtonian approximation

which leads to the standard conservation of energy in classical physics:

where

In more general case, when we introduce the standard definition of the rest mass:

from (49) it is clear that the rest energy of the particle is not constant, but

This means that some part of the gravitational energy of the object

let us consider the non-relativistic case:

and for the total change of the Machian energy of a particle,

which is twice the Newtonian value. The Newtonian value is restored if one assumes

Let us study the appearance of the factor two in the expressions of the gravitational potentials of the ensemble of massive particles (3) in the thermodynamic language. In the thermodynamic approach the source of the rest energy of a particle,

where

In the thermodynamic model of gravity the inertial frames correspond to the thermodynamic equilibrium when

This situation, when the temperature is constant in spite of the heat transfer (i.e. we neglect the energy of vacuum heating) corresponds to the Newtonian approximation, and we can write:

On the other hand, one can take into account the energy transfer for the whole ensemble and can use the relations from the Schwarzschild black hole thermodynamics:

where the constant

Thus the expression (57) arises when

In the case of Einstein’s gravity the thermodynamic expression similar to (61), in the context of a general horizon, was considered in [

In particle physics the vacuum energy itself is unobservable, only the quantum fluctuations have a physical meaning. As first suggested in [

where

where

where the integration constant

Equation (64) tells us that for small scales

and at the horizon scales the mass is twice of this value:

We want to show that within the thermodynamic model [

were

The equation of motion for a particle from the world ensemble, which we label by 1, is given by the Euler-La- grange equation:

where the generalized momentum can be found from (67) to have the form:

For simplicity we have neglected the term

We see from (67) that, while the forces arising from

Assume now that a selected particle is at the origin in a homogeneous isotropic expanding universe of density

were

where

We then replace the sum in (70) by the integral:

We need to calculate the integrals in this expression for the spherical volume,

Without the loss of generality we assume:

Since

The terms involving

Inserting these results into the expression of the momentum (73), we find:

Here we have inserted the cosmological density parameter of the flat cosmological model

The Formula (77) differs by the factor two from the result of [

Now note that for the considered Einstein-Infeld-Hoffmann ensemble of

to the actual mass of the particle

Returning to (70) we note that, as it is seen from the relations (77) and (79), if

then one can explain the inertia of a particle by its gravitational interactions with the whole universe:

and conclude that the acceleration in Newton’s second law can be considered as a relative quantity with respect to the universe.

In this paper we assumed the existence of the relations between the fundamental physical constants and the cosmological parameters using the thermodynamic approach of [

This research is supported by the grant of Shota Rustaveli National Science Foundation