twin universe cosmology Matter ghost-matter astrophysics. 1. Geometrical framework. The matter era and the Newtonian approximation. (p4)
3) Typical scenario of matter ghost matter evolution :
...We can express this with dimensional quantities R, R*, t, r, r*. T and T* are the temperatures (not the characteristic times T and T*). See figure 3.
.
**Fig. 3 ** :The evolution of the scale parameters of the universe and ghost universe.
...As evoked in the previous paper this enlarges the estimated age of our universe, based on the Hubble constant measurement. The ghost matter plays the role of a "cosmological constant", for it gives a positive acceleration R" in our fold.
...As we can see, the system is not symmetrical. One universe (supposed to be ours) expands faster. In the universe of matter the Hubble constant is Ho. But we get a different one Ho in the ghost universe (that cannot be measured, for we cannot observe it optically). In this coupled evolutions of the two worlds, the world of matter and the ghost matter world, there are two stages. During the radiative stage, we have assumed that the scale factors R(t) and R(t) would be "initially equal". Same assumption for the two radiation temperatures Tr and Tr. But these are only assumptions. As a consequence the density rm and the temperature Tm* later become higher in the ghost universe, (in the twin fold F*). We will use it for a future paper, devoted to Very Large Structure.
4) The Newton law and the Poisson equation.
...Let us point out something. In classical General Relativity the Newton law and the Poisson equation can be derived from the field equation, but only through steady state solution (zeroth order plus a perturbation term).
...From our field equations (24) and (25) we may consider a steady Lorentzian solution and add to the metrics some perturbation terms :
(38)
(39) Write the geodesic systems :
(40)
(41)
With low velocities conditions :
(42)
(43)
With w and (w - w*) << 1 (small curvature) the field equations give :
(44)
(45)
Whence
(46)
Introducing the adimensional gravitational potential :
(47)
we get the Poisson equation, written in the system {z i} :
(48)
where
(49)
Similarly, in fold F :
(50)
in fold F* (51)
which corresponds to the Newton law, and justifies our initial assumption about the dynamics of the two folds. All masses are positive. A m = +1 test particle, located in the fold F, gives a gravitational potential :
(52)
In the fold F the Newton law gives :
(53)
i.e. an attractive force. Conversely it repels a test particle located in the fold F*. It justifies our initial assumption :
-
m and m' (both located in the fold F) mutually attract, through Newton's law.
-
m* and m*' (both located in the fold F) mutually attract, through Newton's law.
-
m and m* repel each other, through an "anti-Newton's law".
...All equations can be expressed in any system of coordinates, with the subsequent set of constants. The Newton law gives :
(54)
With :
(55)
(56)
...Similarly, all equation or systems of equations can be phrased in a given set of coordinates, with adequate values of the constants of physics. For an example :
(57)
gives :
(58)
with :
(59)
we get the Poisson equation, in a more familiar form :
(60) ΔY = 4πG (ρ - ρ*)
that can be phrased in a similar way in the second set of coordinates too, with different expressions for Laplacian, mass-densities and gravitational constant's value. With the compatibility condition :
(70)
We take G = G* (as we take c = c*). We get the coordinate-invariant equations :
(71)
S = c ( T - T*)
(72) S* = c ( T* - T)
** ** Matter and ghost matter are self-attractive, but repel each other.
** ** 