Twin Universes Cosmology (p. 13)
Technical Comments:
Section 2: We perform a "zoom" on a region where the density of "twin matter" is supposed to be higher, since, as in the previous paper, we assume that this emulsion system extends across both universes, both having equal average densities r and r*. (However, in paper 6 (radiative era), we will see that a subsequent cosmological evolution model was considered where these two densities could differ significantly at the scale of the entire two universes, with the scale factors R(t) and R*(t) then exhibiting different, joint evolutions.)
Section 4: When calculating the exterior Schwarzschild solution (3), a parameter m appears, which is a length. But it is classically 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, we obtain 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, modulo the inherent crudeness of such an image, of course. With a negative value m < 0, we obtain a different geometry, with a completely different geodesic system (there are no longer 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 any value of m. In both cases, photons are assumed to follow null geodesics. When m < 0, a negative gravitational lensing effect is obtained, as evoked 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 related to galaxies (multiple images of quasars) and clusters (arcs) using this negative gravitational lensing effect, attributing them not to the presence of dark matter within these objects, but rather to the focusing effect of this surrounding invisible matter.
Section 5: In Einstein's equation appears a constant c. We are led to identify it classically as:
(1)
by expanding the metric in series (12) from a Lorentzian solution at zeroth order. However, what had not been noted 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, we then obtain:
(2)
...
That is, a conservation equation, which yields the Euler equations in the Newtonian approximation. But we then note that identifying c with (23) does not automatically imply that G and c are absolute constants. It only provides the current value of c, based on the current values of G and c. If these two quantities were susceptible to varying over cosmic evolution, the absolute constancy of c would imply only that:
(3)
(ga32128)
...
The idea of allowing the speed of light to vary may seem, at first glance, shocking. However, we note that numerous studies have been published where G was considered to vary over time while c remained constant. Incidentally, this would eliminate the conservation of matter energy, since c would no longer be an absolute constant under these conditions.
...
There have also been several studies where variations in the various constants of physics were considered. Indeed, the invention of most of these constants is relatively recent. 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 since treating them as absolute constants seemed to yield interesting results, this hypothesis was adopted. Only Milne, in the 1930s, felt that this conclusion was reached too hastily.
...
More recently, researchers took these constants, one after another, and considered what might happen if they had varied over cosmic evolution. Each time a constant was altered, 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, in a coordinated way.
...
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 yields the same value, this could mean two things:
-
Either the table has an invariant length.
-
Or the table and the meter stick expand or contract "in parallel," for example depending on the temperature of the room.
...
We searched for variations in the constants that leave all physical equations invariant. Under these conditions, it is clear that no measurement could ever reveal any variation, since measuring instruments themselves evolve in parallel with the quantities they are supposed to measure. We acknowledge that this property of the 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 practice what is called dimensional analysis. Take, for example, equations of fluid mechanics. Variables such as pressure, density, temperature, etc., appear. We can then set:
pressure p = p₀ p
temperature T = T₀ t
introducing characteristic quantities and dimensionless variables p, t, etc.
We then rewrite the equations in dimensionless form, thereby simultaneously revealing characteristic numbers (Prandtl number, Reynolds number), etc.
...
Take all the equations you can find (they are not all independent) and let everything vary. Not only the usual variables, but also those that are supposed to remain constant (the "physical constants"). You will find, at random:
R, characteristic length, derived from variables (x, y, z)
T, characteristic time, derived from the temporal variable t
G: gravitational constant
...