f3213 Twin Universes cosmology (p 13)
Technical Comments:
Section 2: A "zoom" is performed on a region where the density of "twin matter" is assumed to be higher, since, as in the previous paper, we assume this emulsion system extends across both universes, both having equal average densities r and r*. However, in paper 6 (radiative era), it will be shown that a subsequent cosmological evolution model was considered where these two densities could strongly differ on the scale of the entire two universes, with the scale factors R(t) and R*(t) then exhibiting different, yet jointly evolving, behaviors.
Section 4: When calculating the exterior Schwarzschild solution (3), a parameter m appears, which is a length. Classically, it is represented by this letter. In fact, since it is merely a constant of integration, it can take positive or negative values. With a positive value m > 0, one obtains the geometry of a stationary spacetime with spherical symmetry outside a mass M. The didactic image of this 4D exterior Schwarzschild solution is the "side of a posicone," presented earlier, despite the obvious crudeness of such an image. With a negative value m < 0, one obtains a different geometry, with a completely different geodesic system (no more elliptical or quasi-elliptical trajectories). This would correspond to empty space surrounding a negative mass M < 0. The geodesic equations are given ((10) and (11)) for arbitrary m. In both cases, photons are assumed to follow null geodesics. When m < 0, a negative gravitational lensing effect arises, as mentioned in Figure 10 (referring to the text of paper 2). In this paper, twin matter is called "antipodal matter."
We attempt to explain strong effects associated with galaxies (multiple quasar images) and clusters (arcs) using this negative gravitational lensing effect, attributing them not to the presence of dark matter within these objects, but to the focusing effect of this surrounding invisible matter.
Section 5: In Einstein's equation appears a constant c. Classically, we are led to identify it as:
(1)
by expanding the metric in series (12) starting from a Lorentzian solution at zeroth order. However, what had not been noticed previously is that this zeroth-order solution and the perturbation term are fundamentally stationary. The absolute constancy of c follows from the assumption of conservation of matter energy. The tensor S is by construction divergence-free. Taking the divergence of Einstein's equation yields:
(2)
...That is, a conservation equation, which yields the Euler equations in the Newtonian approximation. However, one should note that identifying c with (23) does not automatically imply that G and c are absolute constants. It merely provides the current value of c, based on the current values of G and c. If these two quantities were capable of varying during cosmic evolution, the absolute constancy of c would imply only that:
(3) (ga32128)
...The idea of allowing the speed of light to vary may initially seem shocking. However, one should note that numerous studies have been published where G was considered to vary over time while c remained constant. It should be noted in passing that this would eliminate the conservation of matter energy, since c would no longer be an absolute constant under these conditions.
...Additionally, several studies have considered variations in various physical constants. In fact, the introduction of most of these constants is a relatively recent development. Prior to the beginning of this century, the existence of Planck's constant or the electron charge was unknown, since neither quanta nor the electron had yet been discovered. When these constants were established, physicists questioned whether they were absolute constants. Since they appeared not to vary from day to day or from one point on Earth to another, and treating them as absolute constants seemed to yield interesting results, this hypothesis was adopted. Only Milne, in the 1930s, felt this was moving too quickly.
...More recently, researchers took these constants, one after another, and considered what might happen if they had varied throughout cosmic evolution. Each time one touched upon one of these constants, everything fell apart: atoms could no longer form, life could not emerge, stars could not function, etc...
...All these arguments were perfectly sound and irrefutable. But no one had considered varying all these constants simultaneously and in a coordinated manner.
...Since no local variation could be detected in the laboratory, the model had to account for this fact. But what are laboratory instruments, measuring devices? They are apparatuses constructed and designed based on physical equations, which themselves contain all these "constants." To give an image: we try to determine whether an iron table expands or not by measuring it with a ruler made of the same metal.
If the measurement always gives the same value, this could mean two things:
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Either the table has an invariant length.
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Or the table and the ruler expand or contract "in parallel," for example depending on the room's temperature.
...We searched for variations in the constants that leave all physical equations invariant. Under these conditions, it is clear that no measurement could ever detect any variation, since measuring instruments themselves evolve alongside the quantities they are supposed to measure, "in parallel." We acknowledge that this property of the entire set of available equations is somewhat disconcerting, but it is a fact.
...The recipe is, after all, quite simple. Students from top engineering schools and physics students routinely perform dimensional analysis. Take, for example, equations of fluid mechanics. Variables such as pressure, density, temperature, etc., appear. One can then set
pressure p = po p
temperature T = To t
introducing characteristic quantities and dimensionless variables p, t, etc.
Then, the equations are put into dimensionless form, simultaneously revealing characteristic numbers (Prandtl number, Reynolds number), etc.
...Take all the equations you can find (they are not all independent) and vary everything. Not only the usual variables, but also those assumed not to vary (the "physical constants"). You will find, at random:
R, characteristic length, derived from the variables (x, y, z)
T, characteristic time, derived from the temporal variable t
G: gravitational constant
Masses: m, mn, mp, me
h: Planck's constant
c: speed of light
Velocities (orbital, thermal agitation): v
e: electron charge
A characteristic value of the electric field: E
A characteristic value...