Il existe plusieurs types de temps ou plutôt plusieurs conceptions du temps. Le temps de la physique classique, linéaire et infini dans le passé et l’avenir. Le temps des grecs anciens ou de Nietzsche, un temps circulaire, celui de l’éternel retour ; les notions de passé ou d’avenir sont alors beaucoup plus floues.

Je ne crois ni à l’un ni à l’autre, je préfère le concept de Jean-Marie Guyau, un philosophe un peu méconnu du XIXième siècle ; selon lui, je traduis sa pensée en des termes imagés, le temps est analogue à une collection de tableaux, ceux du passé sont quasi terminés, ceux du présent sont en cours d’achèvement et  ceux de l’avenir sont à l’état d’esquisse ;  on peut les regarder dans n’importe quel ordre. Pour moi, ou plutôt pour mon Moi, une scène de mon adolescence peut être plus proche qu’un événement de ce matin.

Le problème des humains c’est que, outre une pinacothèque temporelle commune, ils ont chacun leur galerie, différentes par leurs contenus et par leurs classements. Chez l’un, un tableau peut occuper la place centrale, tandis que chez l’autre, le même tableau est rangé au grenier ou à la cave.

Bien évidemment, cette théorie implique que la notion de temps est une création du cerveau humain, peut-être même animal. En utilisant les termes de la scholastique (la philosophie du Moyen-âge), le temps serait un universel c’est-à-dire un concept sans existence intrinsèque, tout au moins pour les nominalistes.

Si cela est, les concepts de voyage dans le temps ou de prédiction n'ont évidemment aucun sens.

 

Ma nouvelle adresse mail: francismathe@orange.fr

Papier exposé à Londres en 1992 à la conférence PIRT.

pirt1992

Papier présenté en 2002 à Londres à la conférence PIRT.

pirt2002

Ce papier a été présenté à Londres en 2000 à la conférence PIRT.

pirt2000

L'article que je viens de publier sur ce blog date de 2006. J'y montre, en particulier, que les trous noirs ne sont pas une nécessité... loin de là !

                                 SPHERICAL STARS WITHOUT SCHWARSZCHILD SINGULARITY  

 

                                                  Francis MATHE.  44 La Clairière,  78830 BULLION,  France

 

                                     E-MAIL:   mailto:francismathe@orange.fr

 

 

 

CONVENTIONS AND ABBREVIATIONS

 

The sign conventions for the metric and curvature tensors are (-, +, +) in the terminology of Mismer, Thorne & Wheeler [1]. That is, the metric signature is  ( +, -, -, - ).

For this paper we use units in which c = G =1.

The following symbols and abbreviations are used throughout:

 

∂_μ or _,μ                  partial derivative

D_μ or _;μ                 covariant derivative

ln                          natural logarithm

i, j, k,….               Latin indices equal to 1, 2 & 3

λ, μ, ν,…              Greek indices equal to 0, 1, 2, & 3

cst                         constant quantity

 

 

 

 

1 – INTRODUCTION  [1].

 

The basic hypothesis in GR is that spacetime is a four dimensional differentiable Riemannian manifold U whose the metric has the signature (+, -, -, -).

 

ds² = g_λμ dx^λdx^μ                                                                                                                             (1.1)

 

 

There is no point in retracing the history of what led to this hypothesis; it is enough to know that essentially its source is the Mach’s Principe of local equivalence which identifies the fields of gravitation and acceleration. It will later be seen that the definitions of time and space are linked to the metric ds² (but not as directly as is normally supposed).

 

To this framework must be added a description the material content of U; this done by bringing the stress-energy tensor T_λμ into Einstein’s equations to give the geometry of U.

 

R_λμ – ½ R g_λμ = 8π T_λμ                                                                                                  (1.2)

 

Where R_λμ is the Ricci tensor and R = g^λμR_λμ  the Riemannian curvature.

 

In the external case i.e. when T_λμ = 0, the trajectories of particles are geodesics of the metric ds². This property has not been always verified for interior case.

 

 

2 – HOLONOMIC MEDIUMS  [2].

 

If we assume that U contains a material distribution such that the stress-energy tensor can be written:

 

T_λμ = r u_λ u_μ - θ_λμ                                                                                                                      (2.1)

 

Where:

 

r is a positive scalar.

u_λ is the 4- velocity of the medium.

θ_λμ isa symmetrical covariant tensor.

 

Then the distribution described by T_λμ can be said to be a holonomic medium if and only if the vector K defined by:

 

r K_μ =  D_λ θ^λ_μ                                                                                                                (2.2)

 

is a gradient. Well then we take:

 

 K_λ = ∂_λ lnF                                                                                                                   (2.3)

 

 r being the pseudo-density and F the index of the distribution.

In that case the flow lines of the medium are geodesics of the conformal metric:

 

 dσ² = F² ds² = γ_λμ dx_λ dx_μ                                                                                              (2.4)

 

The metric dσ² is thus the only one having physical reality. Consequently, the notions of time and space must be deduced from it.

 

We define the vorticity tensor of the medium by:

 

 Ω _λμ = ∂_λ (Fu_μ) - ∂_μ (Fu_λ)                                                                                              (2.5)

 

A. Lichnerowicz says that the motion of a holonomic medium is without vortices or irrotational if and only if:

 

Ω _λμ = 0                                                                                                                         (2.6) 

 

It is well to remember that a perfect fluid of density ρ and pressure p has a stress-energy tensor:

 

T_λμ = (ρ + p) u_λ u_μ – p g_λμ                                                                                              (2.7)

 

If an equation of state  ρ = φ(p)  exits  the perfect fluid is a holonomic medium with:

 

r = ρ + p                F = exp ( ∫ dp /( ρ + p))                                                                   (2.8)

 

 

3 – COMOVING COORDINATE SYSTEMS AND ABSOLUTE TIME  [3], [4], [6], [7].

 

Definition. It is said that a coordinate system of U is comoving if and only if:       

 

uⁱ = 0                                                                                                                             (3.1) 

 

Hence, we have:

 

u⁰ = 1/ √(g₀₀)          u^λ = δ^λ₀ / √(g₀₀)        u_λ = g_0λ / √(g₀₀)                                              (3.2)

 

Theorem 1.  Let a holonomic medium there exist a comoving coordinate system such we have:

 

dσ² = (dx⁰)² + 2 γ_0i dx⁰dxⁱ  +  γ_ij dxⁱdx^j                                                                         (3.3)

 

with

 

∂₀ γ_0i = 0                                                                                                                        (3.4)

 

Proof.    With the possible coordinate transformations we can choose the value of four quantities, hence it exist a comoving coordinate system such that  γ₀₀ = 1  i.e.

 

u¹ = u² = u³ = 0  &  γ₀₀ = 1 

 

We note Γ^λ_μν the Christoffel symbol of  dσ², the geodesic equation of  dσ² is:

 

d²x^λ/dσ²  +  Γ^λ_μν (dx^μ/dσ)(dx^ν/dσ)  =  0                                                                         (3.5)

 

The coordinates are comoving, hence the curves (x¹, x², x³) =cst  are geodesic i.e.

 

 dx^μ/dσ = δ^μ₀  

 

(3.5) gives Γ^λ₀₀ = 0 hence

 

Γⁱ ₀₀ = ½ γ^iλ (∂₀ γ_0λ + ∂₀ γ_λ0  - ∂_λ γ₀₀) = 0

 

Hence

 

γ^ij ∂₀ γ_0j = 0

 

And

 

∂₀ γ_0i = 0                                                                                                                        

 

That completes the proof.

 

Theorem 2.   Let a holonomic medium where the motion is without vortices then:

 

1)  It exists a comoving coordinate system such that:

 

dσ² = dτ²  -  η_ij dxⁱdx^j                                                                                                     (3.6)

 

ds² = dτ² / F²  -  h_ij dxⁱdx^j                                                                                               (3.7)

 

Where h_ij is definite positive.

 

2)  r √(h) / F  = f (x¹, x², x³ )                                                                                         (3.8)

 

Where h = det(h_ij ).

 

 

Proof.

 

Firstly, we apply the theorem 1 and  we utilize a comoving coordinate system satisfying to (3.3) & (3.4).

 

F²g₀₀ = γ₀₀ = 1

 

g₀₀ = 1/ F²

 

 

We consider the vorticity tensor:

 

Ω _λμ = ∂_λ (Fu_μ) - ∂_μ (Fu_λ) 

 

Ω _λμ = ∂_λ (F²g_0μ) - ∂_μ (F²g_0λ) 

 

Ω _λμ = ∂_λ γ_0μ  - ∂_μ γ_0λ

 

The movement is without vorticity hence:

 

Ω _λμ = 0

 

Hence with (3.4)

 

∂_i γ_0j = ∂_j γ_0i               ∂₀ γ_0i = 0

 

Hence it exits a numerical function f = f (x¹, x², x³)  such as:

 

γ_0i = ∂_i f

 

Let τ = x⁰ + f

 

dτ = dx⁰ + ∂_i f dxⁱ = dx⁰ + γ_0i dxⁱ

 

dτ² = (dx⁰)²  + 2 γ_0i dx⁰dxⁱ +  γ_0i γ_0j dxⁱdx^j 

 

(dx⁰)²  + 2 γ_0i dx⁰dxⁱ = dτ² -  γ_0i γ_0j dxⁱdx^j

 

We put in (3.3)

 

dσ² = dτ² + ( γ_ij  - γ_0i γ_0j )dxⁱdx^j  

 

Let  η_ij = γ_0i γ_{0j -} γ_ij

 

We obtain (3.6)

 

dσ² = dτ² - η_ij dxⁱdx^j

 

Lastly with  h_ij = η_ij / F² we are:

 

ds² = dσ² / F² = dτ² / F² - h_ij dxⁱdx^j   

 

Secondly, we write the conservation identities.

 

D_λ T^λ_μ = 0

 

D_λ ( r u^λ u_μ ) - D_λ θ^λ_μ = 0

 

D_λ ( r u^λ u_μ ) – r ∂_λ F / F = 0

 

We utilize a classical expression of the divergence of a symmetric tensor and the components of the 4-velocity.

 

u^λ  = F δ^λ₀             &           u_λ = δ⁰_λ /F 

 

∂_λ ( r δ^λ₀ δ⁰_μ √(-g) ) / √(-g)  -  ½ ( ∂_μ g_αβ ) ( r δ^α₀ δ^β₀ F² ) - r ∂_μ F / F = 0

 

Where g = det ( g_λμ ) = h / F²

 

Therefore

 

∂_λ ( r δ^λ₀ δ⁰_μ √(h) / F )F / √(h)  -  ½ ( ∂_μ g₀₀ ) ( r F² ) - r ∂_μ F / F = 0

 

But g₀₀ = 1/ F²

 

∂₀ ( r δ⁰_μ √(h) / F )F / √(h)  -  ½ ( -2 ∂_μ F / F³ ) ( r F² ) - r ∂_μ F / F = 0

 

∂₀ ( r δ⁰_μ √(h) / F )F / √(h) = 0

 

∂₀ ( r δ⁰_μ √(h) / F ) = 0

 

∂₀ ( r √(h) / F ) = 0

 

That completes the proof.

 

The two theorems preceding have a important consequence.

 

The time τ is the same for all points of U in relative rest. Therefore this is an absolute time defined with a univocal manner.

 

These results are particularly applicable to a perfect fluid having an equation of state linking ρ and p, but they can be used in other cases i.e. for a scalar field interacting with dust [5].

 

 

4 – GRAVITATION FIELD WITH A SPHERICAL SYMMETRY

 

We consider a gravitational field having a spherical symmetry generated by an arbitrary material distribution of dust (zero pressure), having the same symmetry.

 

 

SETTING IN EQUATION

 

In this case we are:

 

F = 1      &        ds² = dσ²

 

Hence there is a comoving coordinate system such as:

 

ds² = dτ² - e^2b dy² - r² (dθ² + sin²θ dφ²)                                                                         (4.1)

 

Where

 

b = b(τ, y)    &    r = r(τ, y)

 

The Einstein field equations are:

 

r_,τ² / r²  + 2 b_,τ r_,τ / r – e ^-2b ( 2 r_{,y ,y} + r_,y² / r² - 2 b_,y r_,y / r ) + 1/ r² = 8πρ                        (4.2)

 

2 r_{,τ ,τ} / r  + r_,τ² / r² - e ^-2b r_,y² / r²  + 1 / r² = 0                                                                  (4.3)

 

b_{,τ ,τ} + r_{,τ ,τ} / r  + b_,τ r_,τ / r  + b_,τ² - e ^-2b (r_{,y ,y} / r - b_,y r_,y / r ) = 0                                       (4.4)

 

b_,τ r_,y / r - r_{,τ ,y} / r = 0                                                                                                      (4.5)

 

 

RESOLUTION

 

(4.5) gives:

 

b_,τ = r_{,τ , y} / r_,y

 

e^2b = r_,y² / (1 + a)                                                                                                           (4.6)

 

Where a = a(y)

 

We set in (4.3) and we obtain:

 

2 r r_{,τ ,τ} + r_,τ² - a = 0

 

Thus

 

r_,τ² = a + A / r                                                                                                                (4.7)

 

Where A = A(y)

 

It is easy to integrate this differential equation and we obtain the Tolman’s solutions. [8]

 

(4.4) don’t give anything moreover, we set in (4.2) and we have:

 

A_,y = 8πρr² r_,y                                                                                                                (4.8)

 

 

DISCUSSION

 

We apply the preceding result to a spherical star with a mass M and we consider the exterior case i.e. ρ = 0 hence (4.8) gives:

 

A_,y = 0  

 

A =cst

 

(4.7) gives by derivation:

 

r_{,τ ,τ} = - A / (2r²)                                                                                                            (4.9)

 

If we consider a point at rest (the coordinate system is comoving ), we have y = cst hence:

 

r_{,τ, τ} = d²r / dτ²

 

We obtain:

 

d²r / dτ² = - A / (2r²)

 

It is enough to pose A = 2M et we obtain the Newton law.

 

d²r / dτ² = - M / r²                                                                                                        (4.10)

 

A test particle of mass m is subjected to a central force being worth:   -mM / r² .

 

                           

 CONCLUSION

 

   The metric generated by a star of mass M, with spherical symmetry, is written by using the theorems 1 & 2:

 

ds² = dτ² -  r_,y² dy² / (1 + a)  - r² (dθ² + sin²θ dφ²)                                                       (4.10)

 

With:  r_,y² = a + 2M /r     &      a = a(y)

 

The preceding results show that the Schwarzschild solution is inadequate to describe the field creates by a spherical star. The time of Schwarzschild is not an absolute time and its use is not satisfactory. Moreover several solutions (4.10) do not have others solutions that the origin, the singularity of Schwarzschild is thus a fictitious singularity. For example if a = 0 we have the solution of Lemaître:

 

ds² = dτ² - (2M / r) dy² - r² (dθ² + sin²θ dφ²)                                                               (4.11)

 

With   r = (9M / 2)^⅓ (y + τ)^⅔

 

 

5 -MOTION IN A FIELD WITH SPHERICAL SYMMETRY

 

To determine the geodesics of the metric:

 

ds² = dτ² - e^2b dy² - r² (dθ² + sin²θ dφ²)                                                                         (5.1)

 

We consider the function L defined by:

 

L = dτ²/ds² - e^2b dy²/ds² - r² (dθ²/ds² + sin²θ dφ²/ds²)                                                    (5.2)

 

We note ′ the derivation d /ds.

 

L = τ′ ²- e^2b y′ ² - r² (θ′ ² + sin²θ φ′ ²)                                                                             (5.3)

 

We write the Lagrange equations.

 

(∂L/∂q′ ) ′  - ∂L/∂q = 0                                                                                                  (5.4)

 

With q = τ, y, θ, φ.

 

τ″ - b_,τ e^2b y′ ² - r_,τ r (θ′ ² + sin²θ φ′ ²) = 0                                                                      (5.5)

 

(e^2b y′ )′ + b_,y e^2b y′ ² + r_,y r (θ′ ² + sin²θ φ′ ²) = 0                                                          (5.6)

 

(r² θ′ )′ + r² sinθ cosθ φ′ ² = 0                                                                                        (5.7)

 

(r²sin²θ φ′ )′ = 0                                                                                                             (5.8)

 

(5.7) admits θ = π/2 as particular solution, what corresponds to the motions around the star in the equatorial plane.

 

(5.8) gives then:

 

(r² φ′ )′ = 0                                                                                                                     (5.9)

 

r² φ′  = cst                                                                                                                    (5.10)

 

We obtain the generalization of the law of areas.

 

Besides by using the Schwarzschild coordinates we have:

 

ds² = (1 – M/r) dt² - dr²/(1 – M/r) - r² (dθ² + sin²θ dφ²)                                              (5.11)

 

ds² = dτ² -  r_,y² dy² / (1 + a)  - r² (dθ² + sin²θ dφ²)

 

Therefore for θ = π/2 the differential equation connecting r and φ is valid for the two coordinate systems.  

Consequently the considerations on the advance of the perihelion of planets and on the deviation of luminous rays are valid in the coordinate system (τ, y, θ, φ).

 

 

 

 6 – CONCLUSION

 

In short we obtained the following results:

 

1)                     In GR, for the large class of the holonomic mediums (perfect fluids, scalar fields, etc…) we can define an absolute time in univocal way.

2)                     In the case of a star with spherical symmetry we find by using absolute time:

a)      The Newtonian gravitation law.

b)      The law of the areas.

c)      The advance of the perihelion of planets.

d)      The deviation of the luminous rays.

e)      The disappearance of the Schwarzschild singularity. Hence the black holes are dreams which have not any physical reality.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

REFERENCES

 

 

 

1- MISMER (Charles) and all. - Gravitation. - San Francisco: Freeman, 1973.

 

2- LICHNEROWICZ (A.).– Théories relativistes de la gravitation et de l’electromgnétisme. - Paris : Masson, 1955, p. 71.

 

3- MATHE (Francis). – “ Une autre échelle de temps en cosmologie”. – C.R. de l’Académie des Sciences de Paris, t. 303, série II, n°5, 1986, p. 369 – 374.

 

4- MATHE (Francis). – “Study of an equation of state for cosmological fluid in a new time scale”, in: Proceedings “Physical Interpretations of Relativity Theory I” edited by M. C. Duffy.-London, 1988.

 

5- MATHE (Francis). – “A scalar field in cosmology considered from a new time scale”, in: Proceedings “Physical Interpretations of Relativity Theory II” edited by M. C. Duffy.-London, 1990.

 

6- MATHE (Francis). – “A star without singularity”, in: Proceedings “Physical Interpretations of Relativity Theory VI” edited by M. C. Duffy.-London, 1998.

 

7- MATHE (Francis). – “Problems of time scale in cosmology”. , p.107 – 110 in: Recent advances in relativity theory / M.C. Duffy and M. Wegener. – Florida: Hadronic Press, 2000.

 

8- TOLMAN (R.C.). – “Effect of inhomogeneity on cosmological models”, Proc. Nat. Acad. Sc., 1934, 20, 169.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                  ANOTHER THEORY OF GRAVITATION 

 

                                          Francis MATHE

                                            E.MAIL : francismathe@orange.fr

                                                          newgrav

 

 

ABSTRACT

 

For four centuries, Galileo and Eötvös have shown in several experiments the equality of the gravitational mass and the inertial mass. Einstein has explained this equality by the equivalence principle (EQ), but other possibilities exist.

 The reason of the equality of the gravitational mass and the inertial mass could be much deeper hence the introduction of the EQ is not necessary.

To show it, we consider a test particle P of inertial mass m and gravitational mass M in a Newtonian gravitational field U. If the ratio K=M/m is variable the mass M is a function of U.

Consequently the internal energy of the particle E=Mc² depends on U. However a variation of E implies a correlated variation of the fundamental constants.

But the existence of a stable universe (allowing the construction of complex and durable systems) claims, on the contrary, that these quantities remain constant. Therefore it is not need to resort to the principle of equivalence.

The theory developed in this paper makes the distinction between inertial field and gravitational field but with K=1. As the RG it explains the advance of the Mercury perihelion, the deviation of the light by the Sun and the Mössbauer effect.

         But contrary to the RG this theory authorizes the existence of stable celestial bodies whose mass is not limited.

         The metric one thus obtained does not present any singularity (in RG, Schwarzschild singularity). In this theory the concept of black hole disappears.

 

 

 

 

 

CONVENTIONS AND ABBREVIATIONS

 

The sign conventions for the metric and curvature tensors are (-, +, +) in the terminology of Mismer, Thorne & Wheeler [1]. That is, the metric signature is  ( +, -, -, - ).

For this paper we use geometric units in which c = G =1. (Except the § 4.22)

The following symbols and abbreviations are used throughout:

 

∂_μ or _,μ                  partial derivative

D_μ or _;μ                 covariant derivative

ln                          natural logarithm

i, j, k,….               Latin indices equal to 1, 2 & 3

λ, μ, ν,…              Greek indices equal to 0, 1, 2, & 3

cst                        constant quantity

ð                                                      laplacian on a four dimensional manifold

[ , ]_L                      Lie’s brackets

         ≈                           Asymptotically equal to

         #                           Approximately equal to

 

 

 

1 – INTRODUCTION AND HYPOTHESIS [5].

 

 

The study of the movement of the non charged matter lead to consider that the space-time is a four dimensional differentiable Riemannian manifold U whose the metric tensor g has the signature (+, -, -, -).

 

ds² = g_λμ dx^λdx^μ                                                                                           (1.1)

 

For example the space-time of the rotating disk is not flat.

 

On the other hand we reject, with J. L. Synge, the weak equivalence principle. We utilise, for describe the non charged matter, three fields on the manifold U:

-         The inertial field who is a field of symmetric connection G.

-         The matter field who is a field q taking its values in a three dimensional manifold.

-         The gravitational field who is a real scalar field F.

 

         From now we suppose that the matter is a perfect fluid with an equation of state r = j(p) where r is the density and p the pressure of the fluid, the habitual hypothesis of approximation lead to the lagrangians :

 

         L_inert = g^λμR_λμ Ö(-g)

         L_grav = 2F_{, l} F^{, l} Ö(-g)

         L_mat  = 16pr Ö(-g)

         L_{mat + grav} = 16pr f(F)Ö(-g)

         L = L_inert + L_grav + L_{mat + grav}

         L ={ g^λμR_λμ + 2F_{, l} F^{, l} + 16pr f(F) } Ö(-g)                                                       (1.2)                                                          

           

             Where:

-         R_λμ is the Ricci tensor of the connection G.

-         r = ρ(q^j, det(q^j_{, l}q^{k, l})).

-         f is a function describing the interaction between the matter and the gravitational field, with f(0) =1. 

 

The constants in (1.2) are done by choice of the units. 

The eulerian equations for G show that G is the riemannian connection of U [9 p. 338 to 345].

The other equations are (we don’t write the equations for the q^j ):

 

ðF = 4pf ¢(F)r                                                                                                             (1.3)

R_λμ – ½ R g_λμ = 8π T_λμ                                                                                                  (1.4)

                                                                                                 

         Where R = g^λμR_λμ  is the Riemannian curvature of U, ðF = F^{, l} _{; l} and :

 

T_λμ = T_λμ( L_{mat + grav}) + T_λμ( L_grav)

 

But  f(F) is independent of  g_lm  hence we have :

 

T_λμ( L_{mat + grav}) = f(F) T_λμ( L_mat)

 

T_λμ = f(F)((r +p)u_lu_m - p g_λμ ) - {F_{, l} F_{, m}   - ½ g_λμ F_{, l} F^{, l}}/4p                               (1.5)

 

 

 

2 – HOLONOMIC MEDIUMS [2].

 

If we assume that U contains a material distribution (in interaction with a gravitational field or no) such as the stress-energy tensor can be written:

 

T_λμ = r u_λ u_μ - θ_λμ                                                                                                    (2.1)

 

Where:

 

r is a positive scalar.

u_λ is the 4- velocity of the medium.

θ_λμ isa symmetrical covariant tensor.

 

Then the distribution described by T_λμ can be called a holonomic medium if and only if the vector K defined by:

 

r K_μ =  D_λ θ^λ_μ                                                                                                     (2.2)

 

is a gradient. So we take:

 

 K_λ = ∂_λ lnF                                                                                                           (2.3)

 

 r being the pseudo-density and F the index of the distribution.

In that case the flow lines of the medium are geodesics of the conformal metric:

 

 dσ² = F² ds² = γ_λμ dx^λdx^μ                                                                                    (2.4)

 

The tensor metric γ is thus the only one having physical reality. Consequently, the notions of time and space must be deduced from it.

 

We define the vortex tensor of the medium by:

 

 Ω _λμ = ∂_λ (Fu_μ) - ∂_μ (Fu_λ)                                                                            (2.5)

 

A. Lichnerowicz says that the motion of a holonomic medium is without vortex or irrotational if and only if:

 

Ω _λμ = 0                                                                                                              (2.6) 

 

It is important to remember that a perfect fluid of density ρ and pressure p has a stress-energy tensor:

 

T_λμ = (ρ + p) u_λ u_μ – p g_λμ                                                                                 (2.7)

 

If an equation of state  ρ = φ(p)  exits  the perfect fluid is a holonomic medium with:

 

r = ρ + p                F = exp ( ∫ dp /( ρ + p))                                                (2.8)

 

 

 

3 – COMOVING COORDINATE SYSTEMS AND ABSOLUTE TIME [3], [4], [6], [7].

 

Definition. It is said that a coordinate system of U is comoving if and only if:       

 

uⁱ = 0                                                                                                                  (3.1) 

 

Hence, we have:

 

u⁰ = 1/ √(g₀₀)          u^λ = δ^λ₀ / √(g₀₀)        u_λ = g_0λ / √(g₀₀)                         (3.2)

 

Theorem 3.1  Let a holonomic medium then it exists a comoving coordinate system such we have:

 

dσ² = (dx⁰)² + 2 γ_0i dx⁰dxⁱ  +  γ_ij dxⁱdx^j                                                            (3.3)

 

with

 

∂₀ γ_0i = 0                                                                                                        (3.4)

 

Proof.    With the possible coordinate transformations we can choose the value of four quantities, hence it exists a comoving coordinate system such that  γ₀₀ = 1  i.e.

 

u¹ = u² = u³ = 0  &  γ₀₀ = 1 

 

We note Γ^λ_μν the Christoffel symbol of  dσ², the geodesic equation of  dσ² is:

 

d²x^λ/dσ²  +  Γ^λ_μν (dx^μ/dσ)(dx^ν/dσ)  =  0                                                                         (3.5)

 

The coordinates are comoving, hence the curves (x¹, x², x³) =cst  are geodesic i.e.

 

 dx^μ/dσ = δ^μ₀  

 

(3.5) gives Γ^λ₀₀ = 0 hence

 

Γⁱ ₀₀ = ½ γ^iλ (∂₀ γ_0λ + ∂₀ γ_λ0  - ∂_λ γ₀₀) = 0

 

Hence

 

γ^ij ∂₀ γ_0j = 0

 

And

 

∂₀ γ_0i = 0                                                                                                                        

 

That completes the proof.

 

Theorem 3.2   Let a holonomic medium where the motion is without vortex:

 

1)  It exists a comoving coordinate system such that:

 

dσ² = dt²  -  η_ij dxⁱdx^j                                                                                                     (3.6)

 

ds² = dt² / F²  -  h_ij dxⁱdx^j                                                                                               (3.7)

 

Where h_ij is definite positive.

 

2)  r √(h) / F  = C(x¹, x², x³ )                                                                                         (3.8)

 

Where h = det(h_ij ).

 

 

Proof.

 

Firstly, we apply the theorem 1 and we utilize a comoving coordinate system satisfying to (3.3) & (3.4).

 

F²g₀₀ = γ₀₀ = 1

 

g₀₀ = 1/ F²

 

 

We consider the vorticity tensor:

 

Ω _λμ = ∂_λ (Fu_μ) - ∂_μ (Fu_λ) 

 

Ω _λμ = ∂_λ (F²g_0μ) - ∂_μ (F²g_0λ) 

 

Ω _λμ = ∂_λ γ_0μ  - ∂_μ γ_0λ

 

The movement is without vortex hence:

 

Ω _λμ = 0

 

Hence with (3.4)

 

∂_i γ_0j = ∂_j γ_0i               ∂₀ γ_0i = 0

 

Hence it exits a numerical function C = C(x¹, x², x³)  such as:

 

γ_0i = ∂_i f

 

Let t = x⁰ + f

 

dt = dx⁰ + ∂_i f dxⁱ = dx⁰ + γ_0i dxⁱ

 

dt² = (dx⁰)²  + 2 γ_0i dx⁰dxⁱ +  γ_0i γ_0j dxⁱdx^j 

 

(dx⁰)²  + 2 γ_0i dx⁰dxⁱ = dτ² -  γ_0i γ_0j dxⁱdx^j

 

We put in (3.3)

 

dσ² = dt² + ( γ_ij  - γ_0i γ_0j )dxⁱdx^j  

 

Let  η_ij = γ_0i γ_{0j -} γ_ij

 

We obtain (3.6)

 

dσ² = dt² - η_ij dxⁱdx^j

 

Lastly with  h_ij = η_ij / F² we are:

 

ds² = dσ² / F² = dτ² / F² - h_ij dxⁱdx^j   

 

Secondly, we write the conservation identities.

 

D_λ T^λ_μ = 0

 

D_λ ( r u^λ u_μ ) - D_λ θ^λ_μ = 0

 

D_λ ( r u^λ u_μ ) – r ∂_λ F / F = 0

 

We use a classical expression of the divergence of a symmetric tensor and the components of the 4-velocity.

 

u^λ  = F δ^λ₀             &           u_λ = δ⁰_λ /F 

 

∂_λ ( r δ^λ₀ δ⁰_μ √(-g) ) / √(-g)  -  ½ ( ∂_μ g_αβ ) ( r δ^α₀ δ^β₀ F² ) - r ∂_μ F / F = 0

 

Where g = det ( g_λμ) = h / F²

 

Therefore

 

∂_λ ( r δ^λ₀ δ⁰_μ √(h) / F )F / √(h)  -  ½ ( ∂_μ g₀₀ ) ( r F² ) - r ∂_μ F / F = 0

 

But g₀₀ = 1/ F²

 

∂₀ ( r δ⁰_μ √(h) / F )F / √(h)  -  ½ ( -2 ∂_μ F / F³ ) ( r F² ) - r ∂_μ F / F = 0

 

∂₀ ( r δ⁰_μ √(h) / F )F / √(h) = 0

 

∂₀ ( r δ⁰_μ √(h) / F ) = 0

 

∂₀ ( r √(h) / F ) = 0

 

That completes the proof.

 

The two theorems preceding have an important consequence.

 

The time t is the same for all points of U in relative rest. Therefore this is an absolute time defined with a univocal manner.

 

 

 

4 – FUNDAMENTAL PROPERTIES OF THE GRAVITATIONAL FIELD

 

 

4.1 - TRAJECTORIES IN A GRAVITATIONAL FIELD

 

         We consider a gravitational field interacting with a perfect fluid; we have with the notations of the paragraph 1:

 

         T_λμ = f(F)((r +p)u_lu_m - p g_λμ ) - {F_{, l} F_{, m}   - ½ g_λμ F_{, l} F^{, l}}/4p                               (4.1)

 

         Theorem 4.1 A gravitational field interacting with a perfect fluid is a holonomic medium with a pseudo-density:

 

          r = (r + p) f(F)                                                                                                            (4.2)

 

and an index :

 

          F = f(F)F₀                                                                                                                    (4.3)                 

 

where F₀ = exp ( ∫ dp /( ρ + p)) is the index of the fluid only.

          More over the trajectory of a test-body in a gravitational field is a geodesic of the conformal metric : 

         

          dσ² = (f(F)F₀ )² ds²                                                                                                       (4.4)

 

         Proof.

 

         Necessary we have r = (r + p) f(F) and :

         T_λμ = f(F)((r +p)u_lu_m - p g_λμ ) - {F_{, l} F_{, m}   - ½ g_λμ F_{, l} F^{, l}}/4p

         T^λ_μ = f(F)((r +p)u^lu_m - p g^λ_μ ) - {F^{, l} F_{, m}   - ½ g^λ_μ F_{, l} F^{, l}}/4p

         T^λ_μ = r u^lu_m - q^l_m

         Where :

         q^l_m = f(F)pg^λ_μ  + {F^{, l} F_{, m}   - ½ g^λ_μ F_{, l} F^{, l}}/4p

         D_lq^l_m = ¶_m(f(F)p) + {D_l(¶^lF) ¶_mF + ¶^lF D_l(¶_mF)  - D_m(¶_lF) ¶^lF}/4p

         D_lq^l_m = ¶_m(f(F)p) + {D_l(¶^lF) ¶_mF + ¶^lF [D_l(¶_mF)  - D_m(¶_lF)]}/4p

         D_lq^l_m = ¶_m(f(F)p) + {D_l(¶^lF) ¶_mF + ¶^lF [¶_l ,¶_m]_LF}/4p

         D_lq^l_m = ¶_m(f(F)p) + ðF ¶_mF/4p

         D_lq^l_m = ¶_m(f(F)p) + f ¢(F)r ¶_mF

         D_lq^l_m = (r + p) f ¢(F)¶_mF +f(F)¶_mp

         D_lq^l_m = (r + p) f(F)[f ¢(F)¶_mF/f(F) + ¶_mp/(r + p)]

         D_lq^l_m = (r + p) f(F)[¶_m ln f(F) + ¶_m ln F₀]

         D_lq^l_m = r ¶_m ln (f(F)F₀)

    

         Hence, by definition, the medium is holonomic and, by virtue of the paragraph 2, the trajectory of a test-body in a gravitational field is a geodesic of the conformal metric: 

 

         dσ² = (f(F)F₀ )² ds²

  

         For the determination of the function f see § 4.31.

 

4.2 - THE GRAVITATIONAL FIELD IN VACUUM

 

4.21 -EQUATIONS WITH SPHERICAL SYMMETRY

 

         In vacuum we have r = p = 0 and the trajectory of a test- body is a geodesic of the metric ds².

 

         We write the metric ds² with a spherical symmetry:

 

         ds² = e^2adt² - e^2b (dr² + r² (dθ² + sin²θ dφ²))                                                                  (4.5)

         

         where a and b are some functions of r.

         We have r = 0 and  F is a function of r, the Einstein's equations give :

 

         4 b¢ + r b¢ ² - r F¢ ² + 2 r b² = 0                                                                                     (4.6)

 

         2 a¢ + 2 b¢  + 2 r a¢ b¢ + r b¢ ² + r F¢ ² = 0                                                                      (4.7)

 

         a¢ + r a¢ ² + b¢ - r F¢ ² + r a² + r b² = 0                                                                          (4.8)

 

         We can add the field equation for F :

 

         (2/r + a¢ + b¢) F¢ + F ² = 0                                                                                           (4.9)

 

         The complete integration of these equations is easy, but we have a particular important solution:

       

         b = - a = F = m / r                                                                                                       (4.10)           

 

         We see that F is similar to the Newtonian potential and we have:

 

         ds² = e^-2Fdt² - e^2F (dr² + r² (dθ² + sin²θ dφ²))                                                              (4.11)

 

 

 

4.22 – MOTION IN A STATIC FIELD WITH A SPHERICAL SYMMETRY

 

We determine the geodesics of the metric (We use physic units):

 

ds² = A c²dt² - B (dr² + r² (dθ² + sin²θ dφ²))                                                                (4.12)

                                                   

Where A and B are function of r with A ≈ 1 and B ≈ 1.

We consider the function L defined by:

 

L = A c²dt²/ds² - B dr²/ds² - r² B (dθ²/ds² + sin²θ dφ²/ds²)                                          (4.13)

 

We note ′ the derivation d /ds.

 

L =A c² t′ ²- B r′ ² - r² B (θ′ ² + sin²θ φ′ ²)                                                                   (4.14)

 

We write the Lagrange equations.

 

(∂L/∂q′ ) ′  - ∂L/∂q = 0                                                                                                (4.15)

 

with q = t, θ, φ.

 

(A t¢)¢ = 0                                                                                                                    (4.16)

 

(B r² θ′)′ - r² B sinθ cosθ φ′ ² = 0                                                                                 (4.17)

 

(r² B sin²θ φ′ )′ = 0                                                                                                      (4.18)

 

(4.16) gives :

 

A t¢ = k/c                        

 

dt = k ds / Ac                                                                                                              (4.19)

 

where k = cst and k #1, (4.17) admits θ = π/2 as particular solution, that corresponds to the motions around the star in the equatorial plane.

 

(4.17) gives then:

 

(r² B φ′ )′ = 0                                                                                                               (4.20)

 

r² B φ′  = h/c 

 

ds = r²c B dφ / h                                                                                                          (4.21)

 

where h = cst.                  

 

In (4.12) we replace dt by it value in (4.19) and with θ = π/2, we obtains:

 

B dr²  +  r² dφ² = (k² /A - 1) ds²                                                                                  (4.22)

 

Now we substitute for ds with (4.21):

 

B (dr²  +  r² dφ²) = (k² /A - 1) r⁴ c² B² dφ² / h²                                                            (4.23)

 

(d(1/r) / dφ)² = (k² /A – 1) B c²/ h² - 1 / r²                                                                   (4.24)                                                              

 

We put B = 1/A = e^2mG/rc² and u = 1/r in (4.24) and then we expand in series to the third order. We obtain:

 

(du/dφ)² = P(u) = c²(k²-1)/h² + 2G(2k²-1)mu/h² -u² +

                             2G²m²(4k²-1) u²/c²h² +  4G³(8k²-1)m³u³/(3c⁴h²)                           (4.25)

 

         With this expression we can compute the advance of the perihelion of Mercury (see for example [10], pages 115 to 117), we obtain (with k = 1):

 

          δω = 6G²m²π /c²h²                                                                                                     (4.26)

 

          It is the value usually accepted. 

 

 

 

4.3 – THE INTERIOR CASE

 

4.31– DETERMINATION OF THE FUNCTION f

 

         We consider a material distribution without pressure (pure matter or dust) interacting with a gravitational field Φ by virtue of the theorem 4.1 its index F is:

 

          F = f(Φ)                                                                                                                      (4.27)

 

by virtue of the theorem (3.1) it exists a comoving coordinates system such as, if g is the metric tensor, γ =  F²g, we have:

 

          γ₀₀ = F² g₀₀ = f(Φ)² g₀₀ = 1                                                                                         (4.28)              

 

          f(Φ) = 1/√(g₀₀)                                                                                                           (4.29)

 

          If on the analogy of  (4.10) we want g₀₀ = e^-2Φ then we must have:

 

          f(Φ) = e^Φ                                                                                                                   (4.30)

 

          These considerations determine, in general, the function f. The equations of the theory become:

 

 ðF = 4pr e^Φ                                                                                                              (4.31)

                                                                                               

 R_λμ – ½ R g_λμ = 8πe^Φ((r +p)u_lu_m - p g_λμ ) – 2(F_{, l} F_{, m}   - ½ g_λμ F_{, l} F^{, l})           (4.32)                                                                                         

         

          It is important to observe that the quantities appearing in these equations, in particular ρ and p are measured in the Riemannian manifold (U, ds²), a contrario the real values must be measured in (U, dσ²); we have for example, with evident notations:

 

          ρ_real = dm/dv_real = dm/(F³dv) = ρ/ F³                                                                          (4.33)

         

          In the same way we have:

 

          p_real =p/ F³                                                                                                                  (4.34)

 

 

 

4.32– EQUATIONS WITH SPHERICAL SYMMETRY IN COMOVING

          COORDINATES SYSTEM

 

          We utilize the metric (4.5), we have p = 0 and ρ ≠ 0.

 

          ds² = e^2adt² - e^2b (dr² + r² (dθ² + sin²θ dφ²))                                                               (4.35)

         

          a, b, ρ and the gravitational field Φ are some functions of r, the Einstein's equations (1.4)   give:

 

          (4b′ + r b′ ² + 2r b″) /r e^2b = - 8π ρ e^Φ + Φ′ ² / e^2b                                                       (4.36)

 

              

          (2a′ + 2b′ + 2r a′ b′ + r b′ ²) /r e^2b = - Φ′ ² / e^2b                                                           (4.37)

 

 

          (a′ + b′ + r a′ ² + r a″ + r b″) /r e^2b = Φ′ ² / e^2b                                                            (4.38)

 

          and the field equation for Φ:

 

          ðF = - (2Φ′ + r a′ Φ′ + r b′ Φ′ + r Φ″)/ r e^2b = 4π ρ e^Φ                                              (4.39)

 

          With (4.29) we obtain:

 

          Φ = -a                                                                                                                        (4.40)

 

          We replace Φ by -a in (4.36 to 39):

 

          (4b′ + r b′ ² + 2r b″) /r e^2b = - 8π ρ e^-a + a′ ² / e^2b                                                       (4.41)

   

          (2a′ + 2b′ + 2r a′ b′ + r b′ ²) /r e^2b = - a′ ² / e^2b                                                            (4.42)

 

          (a′ + b′ + r a′ ² + r a″ + r b″) /r e^2b = a′ ² / e^2b                                                             (4.43)

 

          (2a′ + r a′ ² + r a′ b′ + r a″)/ r e^2b = 4π ρ e^-a                                                                (4.44)

 

          In (4.44) we replace ρ by it value in (4.41):

 

          (4b′ + r b′ ² + 2r b″) /r e^2b = - 2(2a′ + r a′ ² + r a′ b′ + r a″)/ r e^2b + a′ ² / e^2b               (4.45)

 

          We simplify (4.45) then (4.42) and (4.43), we obtain:

 

          4a′ + 4b′ + r a′ ² + r b′ ² + 2r a′ b′  +  2r a″ + 2r b″ = 0                                               (4.46)

 

          2a′ + 2b′ + r a′ ² + r b′ ² + 2r a′ b′ = 0                                                                         (4.47)

 

          a′ + b′  +  r a″ + r b″ = 0                                                                                             (4.48)

 

          In (4.46 to 48) we put y = (a + b), we are:

 

          4y′ + r y′ ² + 2r y″ = 0                                                                                                (4.49)

 

          y′ (2 + r y′) = 0                                                                                                           (4.50)

 

          y′ + r y″ = 0                                                                                                                (4.51)      

 

          The solutions are evident:

 

1)      y′ = 0  ó  y = a + b = K = cst.                                                                             (4.52)

Using a change of variable (r → αr) we can choose K = 0, we obtain:

 

          b = -a  = Φ                                                                                                                 (4.53)

 

          The equation (4.44) becomes:

          

          (2F′ + r F″)/r = - 4π ρ e^3F                                                                                         (4.54)

 

          This equation permits, knowing ρ, the determination of the field F, this situation is the same one as in classic mechanics, it is not the case in RG. For the metrics we have:                                           

          

          ds² = e^-2Fdt² - e^2F (dr² + r² (dθ² + sin²θ dφ²))                                                             (4.55)

 

          dσ² = e^2F ds² = dt² - e^4F (dr² + r² (dθ² + sin²θ dφ²))                                                   (4.56)

 

          The metric dσ² is the frame of the physics and all the measures must be done with its.

 

2)      2 + r y′ = 0 ó a′ + b′ = -2/r ó b = -a – ln r² ó b = F – ln r² ó e^b = e^F/r². We have:

 

ds² = e^-2Fdt² - e^2F (dr² + r² (dθ² + sin²θ dφ²))/r⁴                                                         (4.57)

 

ds² = e^-2Fdt² - e^2F (dr²/r⁴ + (dθ² + sin²θ dφ²)/r²)                                                         (4.58)

 

We u = 1/r and we obtain:

 

          ds² = e^-2Fdt² - e^2F (du² + u² (dθ² + sin²θ dφ²))                                                            (4.59)

 

          We return to the first case.

 

 

5– APPLICATIONS

 

 

         For example we remember Einstein, in the year 1917, wanted to build a static hyper-spherical universe filled up pure matter, and with this intention, he has introduced the cosmological constant. In our theory that constant is not necessary. The metric of the static hyper-spherical universe is:

 

         ds² = dt² - (dr² + r² (dθ² + sin²θ dφ²))/ (1+r²/4a²)²                                                         (5.1)

 

        where a is a constant strictly positive. The comparison between (5.1) and (4.56)  gives:

 

        Φ = -ln(1 + r²/4a²)/2                                                                                                       (5.2)

 

        Then the equation (4.54) gives:

 

        ρ = - e^-3Φ (2F′ + r F″)/4πr                                                                                             (5.3)

 

        The relations (4.30) and (4.33) give:

 

        ρ_real = ρ e^-3Φ = - e^-6Φ (2F′ + r F″)/4πr = (4a² + r²)(12a² +r²)/ 256π a⁶                            (5.4)

 

       We can compute the mass of that universe, it is infinite.

       Now that universe has only a historic interest but one never knows.

 

                                                           

     

6 – CONCLUSION

 

 

       The equality of the inertial mass and the gravitational mass do not imply necessarily the weak principle of equivalence. The theory presented in this paper makes the distinction between the gravitational field and the inertial field. It gives the correct value for the advance of the perihelion of Mercury but on the over hand it presents several interesting and innovative points.

        Firstly the analogue of the Schwarzschild solution does not present a singularity except the origin.

        Secondly it is possible to build a stable mass of matter as large as one wants.

        These last considerations show the possibility to re-examine the theory of the black holes.  

  

 

REFERENCES

 

 

1- MISMER (Charles) and all. - Gravitation. - San Francisco: Freeman, 1973.

 

2- LICHNEROWICZ (A.).– Théories relativistes de la gravitation et de l’electromgnétisme. - Paris : Masson, 1955, p. 71.

 

3- MATHE (Francis). – “ Une autre échelle de temps en cosmologie”. – C.R. de l’Académie des Sciences de Paris, t. 303, série II, n°5, 1986, p. 369 – 374.

 

4- MATHE (Francis). – “Study of an equation of state for cosmological fluid in a new time scale”, in: Proceedings “Physical Interpretations of Relativity Theory I” edited by M. C. Duffy.-London, 1988.

 

5- MATHE (Francis). – “A scalar field in cosmology considered from a new time scale”, in: Proceedings “Physical Interpretations of Relativity Theory II” edited by M. C. Duffy.-London, 1990.

 

6- MATHE (Francis). – “A star without singularity”, in: Proceedings “Physical Interpretations of Relativity Theory VI” edited by M. C. Duffy.-London, 1998.

 

7- MATHE (Francis). – “Problems of time scale in cosmology”. , p.107 – 110 in: Recent advances in relativity theory / M.C. Duffy and M. Wegener. – Florida: Hadronic Press, 2000.

 

8- TOLMAN (R.C.). – “Effect of inhomogeneity on cosmological models”, Proc. Nat. Acad. Sc., 1934, 20, 169

 

9- SOURIAU (J.M.). − Géométrie et Relativité. –Paris : Hermann 1964.

 

10- CHAZY (Jean). − Mécanique céleste. – Paris : PUF 1953.