Let us consider Einstein’s basic equation [1] of General Relativity (GR) completed by a positive mathematical constant [2] , that has a priori nothing to do with Cosmology (with Riemanian metric, Einstein’s curvature tensor, stress-energy tensor, G gravitational constant and c light velocity):

In order to discover the physical meaning of this constant, let us simplify with the Equation (1) by introducing Minkowskian metric. We obtain a tensor of Minkowskian Vacuum (2):

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 and a negative pressure¹:

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 in proper system for all components except for purely temporal components

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 to a cosmological constant (CC Λ) in the second member of (1) in order to have a second contribution to: (with c = 1):

By associating stress-energy tensor to the one of a perfect relativistic fluid we usually obtain a fluid (black energy of “quantum vacuum”³) characterized by an unknown Riemanian metric (2-Riemann) whilst in (2) the metric is determined a priori Minkowskian. In standard Cosmology, Minkowskian limit can only be a trivial result of a very improbable compensation.

Basic condition (4) gives a new physical interpretation of Minkowskian metric as a fluid.

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 is a density of energy of fluid, we have by integration a finite volume V with a finite energy U:

By differentiation we obtain:

that seems to trivially return to (4-bis) with reduction of element of volume . Usually it is claimed that Minkowskian vacuum would be static. Let us consider, at flat Minkowskian limit, an Euclidean sphere of fluid:

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 pseudomass⁴) on the surface of the sphere. We have to introduce the gravitational

constant because. We suggest then to study a Newtonian model where the Minkowskian 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. 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 and the Minkowskian solution (4-bis) seems impossible. We rediscover in this way that standard immutable Minkowskian vacuum must be defined without gravitation.

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 coupled with a constant density

Let us now consider that the Newtonian law of gravitation is also variable with a temporal gravitational potential. Our test point at radial distance has therefore a positive radial velocity. Potential energy can be then now compensated by kinetics energy:

(disappears). This Pseudo-Newtonian model⁵ 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. Physical meaning of mathematical constant Λ is now clarified by Minkowskian solution that implies the introduction (from 13) of a GLOBAL SCALE FACTOR (with Minkowskian metric, 2 or see 19):

with a constant of integration that seems, at first sight, not depending on Λ.

We can also define a constant of expansion of Fluid (Vacuum) that we suggest to note (15 left):

together with a density inside the sphere (15 right). Our model supposes that there is no fluid outside the sphere of fluid. Given that the fluid simulates space-time continuum itself, there is nothing outside the sphere. Everything happens as if our sphere was a “Universe”. By introducing mathematical constant Λ in (1) we are thus naturally led to a theory of Universe, i.e. a cosmological interpretation:

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 (if see Remark 3), there would exist two global constants of Hubble. This would be a nonsense. In order to have Pseudo-Newtonian model of Pseudo-Euclidean fluid without contradiction, we must have (16)⁶.

From we can define from radial acceleration also a (stan- dard) parameter of deceleration that we suggest to note:

is here negative and implies thus an acceleration of expansion. Initial condition of (17) are:

Initial conditions mean that and define “horizon values” exactly on the same way that defines a “horizon value” (Remark 3). We obtain a basic minimal relativistic acceleration⁷.

We deduce a pseudoNewtonian scalar field of gravitational force with a global principle of equivalence “acceleration-gravitation”. Our dynamical Universe supposes then, with, an initial linear density (together with an initial force and power of expansion).

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. Here we do not have a material point but a point of fluid in the framework of GR. Escape velocity for such a point can be as large as we wish (not limited by c). Pseudo-Newtonian Equation (12) is a relativistic equation because velocity of light plays a basic role. In fact, the escape velocity, of a point of space itself, is limited by c but not in usual meaning with domain of variation. Indeed, in order to avoid a supplementary constant R_H

(14), if we admit for the initial velocity (13), the domain of variation of is

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 Universe. The latter is also in

exponential expansion⁸ with and acceleration. Lemaitre’ scale factor is introduced in de Sitter’s metric (23) [4] .

whilst our scale factor is globally induced from (12). In de Sitter’s model, the constant A(“A = 1”) in

is then not a global constant determined by initial conditions of the problem (like R_H in 16).

With condition of radiality we have respectively de Sitter’s metric and Minkowski’s metric:

that are both particular cases of non-static [5] radial Robertson-Walker’s metric

with local (in metric) parameter of Gaussian curvature.

Let us now introduce the limit of light velocity with first in the flat metric of de Sitter:

Usually one deduces from de Sitter model the following formula of Redshift by Doppler effect in GR

(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 is standard law of Hubble (with radial comobile distance r₀). Recall that (s for space)

is not the velocity between two galaxies (two material points) but a velocity between the “points” (elements of volume) of space itself occupied by galaxies. In de Sitter’s case we have thus:

Let us now follow the same reasoning for our spherical fluid. Optical property of our fluid is given by Minkowskian limit for velocity of light: (Minkowskianscale factor, R(t), (19))

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

) where z can be as large as we wish ⁹. Equation (24) involves then a “GR interpretation”

(velocity of point of space) of Einstein’s Doppler formula [7] . With (in 25) we have precisely (23) until the second order.

We suggest then the conjecture that the parameters of expanding universe is given by famous “Bondi’s factor” [8] at any order (25). An expanding sphere of light (§3) is then inseparable from an expanding sphere of space (§4). We have thus perhaps here a new way towards CBR.

For the coherence of our model of points of space without baryonic mass “”, we need for the photon a

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

term of four-velocity for “particle” (see note 2 Poincaré’s electron)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. We wonder if we can introduce such a scale factor in a finite interval in another paper [9] . From relativistic pseudoNewtonian

Equation (12), we deduce dynamical properties, and optical property (24) with Bondi’s factor reinterpreted as a Redshift in.

Dynamical (§1, §2, §3) and optical (§4) properties of our Minkowskian fluid (or Continuum) are thus compatible with the most recent cosmological observations ([10] -[12] ):

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).