ghost matter cosmology. 7: Confinement of spheroidal galaxies by surrounding ghost matter. (p2)
- The origin of Newton's law and the Poisson equation.
Newton's law is an hypothesis, a principle. It works. Proof: we can calculate the trajectories of the planets, quite well, and send satellites to great distances, with remarkable precision.
The Einstein field equation is an hypothesis, a principle.
(7)
S = c T
It works. Proof: we can calculate the displacement of the perihelion of a mass, a satellite orbiting in the field created by a heavier mass. If we lived near a neutron star and if this object had a companion, we should observe the path shown in figure 4.
Fig. 4: Precession of the perihelion of the trajectory of a companion, orbiting around a very massive body.
The measurement would confirm the theory, as we do in the case of Mercury. By the way, this phenomenon is compatible with the ghost matter model.
(8)
S = c (T - T*)
(9)
S* = c (T* - T)
We should live in a region of the universe where matter dominates ( T* << T ), so that the field equations system becomes:
(10)
S » c T
(11) S* = - c T
When Einstein introduced the new concept of field equation, it was checked whether this formalism was compatible with Newton's law. Classically, one considers the metric as close to one describing a homogeneous medium (r = constant). Then, a mass concentration is considered as a small perturbation:
(12)
g = go + e g
go refers to this constant density medium. e being a small parameter, the second term e g represents the perturbation. The second member of the field equation is assimilated to:
(13)
But, and this is very important, the two terms go and e g are chosen time-independent. Then, one computes the left hand of (7) through the expansion into a series (12) and finds:
(14)
which can be written
(15)
and is identified to Poisson equation, through:
(16)
From this we also define the gravitational potential:
(17)
goo being one of the metric potentials. But all this is performed in steady state conditions. We need it to define the first order term go, chosen Lorentzian:
(18)
ds² = c² dt² - dx² - dy² - dz²
This is a good approximation if we deal with:
A portion of the universe
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where a mass concentration is surrounded by void.
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where the velocities are small with respect to c
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where the local curvature is weak
Then, is it convenient to describe an infinite medium? No. To do that, to set up a Poisson equation, applicable to a constant density infinite medium, we need a non-steady zeroth order solution go, which cannot have a Lorentz form. It must be something like a Friedmann solution. If the medium is fully homogeneous, if the non-steady mass density is constant throughout all space, there is no perturbation term. go is simply a Robertson-Walker solution, giving Friedmann models (for classical general relativity).
Where is the gravitational potential Y, for such an infinite medium, with mass density constant in space? Nowhere. It does not exist and we cannot define such a scalar quantity.
Then, for an infinite constant density medium, whether it is constant in time (which should not be physical) or time-dependent (Friedmann), the Poisson equation becomes a pure theoretical fantasy. It simply does not exist. It has no physical meaning. We cannot invoke it.
Then, what is the gravitational field around an arbitrarily chosen point in space? Our answer: zero.
The reader will say: What about the screening effect in electrostatics?
Can you deal with an infinite, constant electric charge density medium? Not physical. Such a medium should expand immediately, at tremendous velocity, if the charge density deviates significantly from equilibrium (n⁺ = n⁻).
Another reader will argue:
- In 1934, Milne and Mc Crea rediscovered the Friedmann equation, starting only from the Euler and Poisson equations.
What does it mean? Simply that the collapse, or expansion, of a dust (zero pressure) ball obeys the same equation as a constant density universe, corresponding to the Friedmann model. Nothing else.
