SPHERICAL STARS WITHOUT SCHWARSZCHILD SINGULARITY
Francis MATHE. 44 La Clairière, 78830 BULLION, France
CONVENTIONS AND ABBREVIATIONS
The sign conventions for the metric and curvature tensors are (-, +, +) in the terminology of Mismer, Thorne & Wheeler . 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
Dμ 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 .
The basic hypothesis in GR is that spacetime is a four dimensional differentiable Riemannian manifold U whose the metric has the signature (+, -, -, -).
ds² = gλμ dxλdxμ (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 Tλμ into Einstein’s equations to give the geometry of U.
Rλμ – ½ R gλμ = 8π Tλμ (1.2)
Where Rλμ is the Ricci tensor and R = gλμRλμ the Riemannian curvature.
In the external case i.e. when Tλμ = 0, the trajectories of particles are geodesics of the metric ds². This property has not been always verified for interior case.
2 – HOLONOMIC MEDIUMS .
If we assume that U contains a material distribution such that the stress-energy tensor can be written:
Tλμ = r uλ uμ - θλμ (2.1)
r is a positive scalar.
uλ is the 4- velocity of the medium.
θλμ isa symmetrical covariant tensor.
Then the distribution described by Tλμ can be said to be a holonomic medium if and only if the vector K defined by:
r Kμ = Dλ θλμ (2.2)
is a gradient. Well then we take:
Kλ = ∂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 = γλμ dxλ dxμ (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 (Fuμ) - ∂μ (Fuλ) (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:
Tλμ = (ρ + p) uλ uμ – p gλμ (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 , , , .
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) uλ = δλ0 / √(g00) uλ = g0λ / √(g00) (3.2)
Theorem 1. Let a holonomic medium there exist a comoving coordinate system such we have:
dσ² = (dx0)² + 2 γ0i dx0dxi + γij dxidxj (3.3)
∂0 γ0i = 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 Γλμν the Christoffel symbol of dσ², the geodesic equation of dσ² is:
d²xλ/dσ² + Γλμν (dxμ/dσ)(dxν/dσ) = 0 (3.5)
The coordinates are comoving, hence the curves (x1, x2, x3) =cst are geodesic i.e.
dxμ/dσ = δμ0
(3.5) gives Γλ00 = 0 hence
Γi 00 = ½ γiλ (∂0 γ0λ + ∂0 γλ0 - ∂λ γ00) = 0
γij ∂0 γ0j = 0
∂0 γ0i = 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 - ηij dxidxj (3.6)
ds² = dτ² / F² - hij dxidxj (3.7)
Where hij is definite positive.
2) r √(h) / F = f (x1, x2, x3 ) (3.8)
Where h = det(hij ).
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 (Fuμ) - ∂μ (Fuλ)
Ω λμ = ∂λ (F²g0μ) - ∂μ (F²g0λ)
Ω λμ = ∂λ γ0μ - ∂μ γ0λ
The movement is without vorticity hence:
Ω λμ = 0
Hence with (3.4)
∂i γ0j = ∂j γ0i ∂0 γ0i = 0
Hence it exits a numerical function f = f (x1, x2, x3) such as:
γ0i = ∂i f
Let τ = x0 + f
dτ = dx0 + ∂i f dxi = dx0 + γ0i dxi
dτ² = (dx0)² + 2 γ0i dx0dxi + γ0i γ0j dxidxj
(dx0)² + 2 γ0i dx0dxi = dτ² - γ0i γ0j dxidxj
We put in (3.3)
dσ² = dτ² + ( γij - γ0i γ0j )dxidxj
Let ηij = γ0i γ0j - γij
We obtain (3.6)
dσ² = dτ² - ηij dxidxj
Lastly with hij = ηij / F² we are:
ds² = dσ² / F² = dτ² / F² - hij dxidxj
Secondly, we write the conservation identities.
Dλ Tλμ = 0
Dλ ( r uλ uμ ) - Dλ θλμ = 0
Dλ ( r uλ uμ ) – r ∂λ F / F = 0
We utilize a classical expression of the divergence of a symmetric tensor and the components of the 4-velocity.
uλ = F δλ0 & uλ = δ0λ /F
∂λ ( r δλ0 δ0μ √(-g) ) / √(-g) - ½ ( ∂μ gαβ ) ( r δα0 δβ0 F² ) - r ∂μ F / F = 0
Where g = det ( gλμ ) = h / F²
∂λ ( r δλ0 δ0μ √(h) / F )F / √(h) - ½ ( ∂μ g00 ) ( r F² ) - r ∂μ F / F = 0
But g00 = 1/ F²
∂0 ( r δ0μ √(h) / F )F / √(h) - ½ ( -2 ∂μ F / F3 ) ( r F² ) - r ∂μ F / F = 0
∂0 ( r δ0μ √(h) / F )F / √(h) = 0
∂0 ( r δ0μ √(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 .
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 - e2b dy² - r² (dθ² + sin²θ dφ²) (4.1)
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)
b,τ = r,τ , y / r,y
e2b = 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
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. 
(4.4) don’t give anything moreover, we set in (4.2) and we have:
A,y = 8πρr² r,y (4.8)
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
(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
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 .
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 - e2b dy² - r² (dθ² + sin²θ dφ²) (5.1)
We consider the function L defined by:
L = dτ²/ds² - e2b dy²/ds² - r² (dθ²/ds² + sin²θ dφ²/ds²) (5.2)
We note ′ the derivation d /ds.
L = τ′ ²- e2b 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 e2b y′ ² - r,τ r (θ′ ² + sin²θ φ′ ²) = 0 (5.5)
(e2b y′ )′ + b,y e2b 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.
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