_{1}

^{*}

A Minkowskian solution of the equation of General Relativity (as written by Einstein in 1915) is trivial because it simply means that both members of the equation are equal to zero. However, if alternatively, one considers the complete equation with a non-zero constant Λ (Einstein 1917), a Minkowskian solution is no longer trivial because it amounts to impose a constraint on the right hand side of the equation ( i. e. a non-null stress-energy tensor). If furthermore one identifies (as usual) this tensor to the one of a perfect fluid, one finds that this fluid has a positive energy density and a negative pressure that depend on the three constants of the equation ( i. e. gravitational constant G, cosmological constant Λ and velocity of light c). When doing that (§1), one has to consider the “Minkowskian Vacuum” as a physical object of GR (an enigmatic non-baryonic Minkowskian fluid). Can one build a model of this object on the basis of a dynamical equilibrium between the effective gravitational attraction due to the positive energy density versus the negative pressure repulsion? We propose to study such a model, where the (enigmatic) fluid is assumed to exist only in a limited sphere whose surface acts like a “test body” sensitive to the gravitational field created by the fluid. No static equilibrium exists, but a pseudoNewtonian “dynamical equilibrium” (§2) can be reached if the pseudoEuclidean fluid is in state of expansion. Up to there, we have simply constructed a model of an “abstract Universe” ( i. e. the limited sphere: There is no fluid outside this sphere!) that gives to a (purely mathematical) constant Λ a concrete physical meaning. We discover finally that our expanding fluid has not only dynamical (gravitational) properties (§3) but also optical properties that are connected with Doppler Redshift (§4). Remembering that recent observations in Cosmology indicate that the “real Universe” seems to be “Flat” and in “Accelerated Expansion”; remembering also (after all) that the archetypal Flat Universe is simply a Minkowskian Universe, we logically wonder if the unexpected Minkowskian global solution, could not be also a significant cosmological model (conclusion).

Let us consider Einstein’s basic equation [

In order to discover the physical meaning of this constant

Let us now associate to this tensor (2) the one of a perfect relativistic fluid:

Minkowskian Vacuum is then simulated by an enigmatic fluid with a positive density of energy ^{1}:

Our enigmatic Minkowskian fluid becomes a physical object in the framework of (complete) GR. Before the examination of the physical properties of our fluid determined by three basic constants (Λ, G and c) (§2), let us formulate two remarks.

REMARK 1 Our non-usual fluid (4) cannot be confused with usual perfect fluid (4-SR) in the framework of standard Special Relativity (SR). In this case, we have

Any relativistic usual perfect fluid has a positive pressure ^{2}. In this way our enigmatic fluid is

no longer a usual fluid in “immutable Minkowskian Vacuum” (standard SR) but it is the (Classical) Minkowskian Continuum itself.

REMARK 2 Our non-usual (classical) fluid (2) cannot be confused with usual (quantum) black energy (2bis) in the framework of Cosmology. Standard method in Cosmology consists in associating a supplementary stress-tensor

By associating ^{3}) characterized by an unknown Riemanian metric

Basic condition

The geodesic of a material point is usually determined in Minkowskian space-time as a straight line. But here we have a point of space-time continuum itself. In order to discover physical properties of our enigmatic fluid the only possible point of departure is local thermodynamical properties given by (4-bis): where h is null density of enthalpy. Given that

By differentiation we obtain:

that seems to trivially return to (4-bis) with reduction of element of volume

At Minkowskian limit we have also to take into account Einstein’s relation of “materialization” of energy:

How can we test the behavior (static or not static) of such a Euclidean Sphere of fluid? Let us consider a test point (infinitesimal pseudomass^{4}) on the surface of the sphere. We have to introduce the gravitational

constant because

is assumed to exist only in a limited sphere whose surface acts like a “test body” sensitive to the gravitational field created by the fluid. The surface is submitted to gravitational attractive potential:

Then the surface of the fluid will collapse towards the center of the sphere given that we have only attractive potential energy. So a static finite sphere of our fluid is unstable

The existence of our fluid is directly connected with Minkowskian (Pseudo-Euclidean) space-time, where basically the time is not separated from space (2). Let us thus consider that thermodynamical differential dV variation of volume of fluid is a temporal variation

In this way, Equation (6) is no longer trivial. We have a variable volume

Let us now consider that the Newtonian law of gravitation is also variable with a temporal gravitational potential

(^{5} of Pseudo-Euclidean fluid is based on a dynamical equilibrium “sphere-test body” between attraction and repulsion. We obtain in this way a stability of expanding sphere with a radial enigmatic (Remark 3) “escape velocity”

If we suppose a finite spherical volume of fluid in dynamical equilibrium then it is in exponential expanding (11). Escape velocity (13) disappears if and only if

with a constant of integration

We can also define a constant of expansion of Fluid (Vacuum) that we suggest to note

together with a density inside the sphere (15 right). Our model supposes that there is no fluid

Our model explains then why Hubble’s expansion is necessarily a global expansion (no local observed effect of expansion). If the constant of integration R_{H} (15)-(16), i.e. a global constant, is not equal to ^{6}.

From

Initial conditions mean that ^{7}.

We deduce a pseudoNewtonian scalar field of gravitational force with a global principle of equivalence “acceleration-gravitation”

REMARK 3 An important objection could be formulated at this stage: Our Pseudo-Newtonian model would not be a Pseudo-Euclidean model because our basic Equation (12) uses a non-relativistic form of energy.

Everybody knows how to write kinetics energy for a material particle_{H}

(14), if we admit for the initial velocity (13)

limited by c. We rediscover in this way a basic tachyonic Pseudo-Euclidean “light-space-time” structure. We have to expect then optical properties of fluid.

In Cosmology our model is very near the model of de Sitter’s empty

exponential expansion^{8}

whilst our scale factor

_{H} in 16).

With condition of radiality

that are both particular cases of non-static [

with local (in metric) parameter of Gaussian curvature

Let us now introduce the limit of light velocity with

Usually one deduces from de Sitter model the following formula of Redshift

(with standard notations of the time of emission of radial photon from a remote galaxy towards the time of reception in our galaxy). Moreover with two usual cosmological measurable parameters H and q, we obtain the following standard development into series:

where _{0}). Recall that

is not the velocity between two galaxies (two material

Let us now follow the same reasoning for our spherical fluid. Optical property of our fluid is given by Minkowskian limit

in contrast with (21). Let us recall that Einstein’s standard SR Doppler radial factor for material point is

^{9}. Equation (24) involves then a “GR interpretation”

(velocity of point of space) of Einstein’s Doppler formula [

We suggest then the conjecture that the parameters of expanding universe is given by famous “Bondi’s factor” [

For the coherence of our model of points of space without baryonic mass “

null rest mass m = 0 in such a way that in the perfect fluid (in 3), we would have “

term of four-velocity ^{10}.

We showed the existence of a simple unexpected global Minkowskian solution of Einstein’s complete (with CC) equation of GR. The logical sequence from Pseudo-Euclidean solution (2) towards the Pseudo-Newtonian Fluid (12) is the following

Minkowskian metric (infinitesimal interval) involves (with CC) then a global scale factor

Equation (12), we deduce dynamical properties

Dynamical (§1, §2, §3) and optical (§4) properties of our Minkowskian fluid (or Continuum) are thus compatible with the most recent cosmological observations ([

1) Hubble’s Redshift,

2) Parameter of curvature near zero

3) Density near “critical density”,

4) Parameter of acceleration near

5) The Dark energy connected with a non-null CC (note 6).

Einstein’s SR in 1905 consisted in dissolving a ghost: The old electromagnetic ether. Our relativistic approach involves also the dissolution of a ghost: the Dark Energy. This new cosmological ether becomes a pure relativistic effect of Minkowskian solution with CC. Unlike usual Quantum approach of Vacuum (Lemaître) our approach consists in simulating properties of Vacuum with a “Classical (apparently at the departure) Continuum”. With quantum representation of light (note 10), our model becomes compatible, for example, with a continuum spectrum of a “black body” in Universal Vacuum.

I would like first to thank Jean Reignier (ULB). I thank also Laurent Favart (IIHE, ULB), Jan de Bruyne (IIHE, ULB), Nicolas Vansteenkiste (ESI-heb), Frédéric Servais (ESI-heb) and Eytan Levy (ESI-heb).