twin universe geminal cosmology
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But what about Planck time, in all this?
...It varies... like t, meaning it shrinks as we go further back into the past. The Planck barrier recedes like a mirage. As for the Planck length, it varies like R.
...Of course, this model does not account for "the rest of physics." To make it complete, one would need to adjoin arbitrary variations of constants related to the other interactions—strong, weak. Let us consider this another idea to debate (which is possible; we do so immediately. For the impossible, we request a delay...)
...The details of this model can be found in the article [On this site: Geometrical Physics A, 6, 1998]. For reference, we will give the variations of physical constants as functions of the chronological variable t.
Time in the second universe.
...In what precedes, we started from purely geometric assumptions, which led us to propose a system of two coupled field equations. We saw that this system is equivalent to reversing the sign of the masses of the second population, even though the masses m* remain positive.
...When solving these equations, we assign particular forms to the two metrics, which merely reflect different assumptions. We assume that Special Relativity "works" in both sheets. This leads us to choose a particular form of Riemannian metric, called "with signature (+ - - -)." Then we assume that these two universes are homogeneous (that parameters such as pressure and density are the same at every point in space) and isotropic (that the universe appears the same regardless of the direction we face). Using these special metrics, we can express the tensors S and S*, then solve the equations, ultimately obtaining coupled differential equations that define the evolutions of R and R*, the "scale factors" of the two universes.
...We do the same in standard theory, except that we have a single field equation—the Einstein equation—only one metric, and ultimately arrive at a single differential equation. This is the famous Friedmann equation:
Immediate remark: this equation is invariant when changing t into -t; it is "time-reversible."
...In fact, nothing in our physics allows us to distinguish between past and future. No matter what we do, we always end up with a subjective conception of time. Only our senses allow us to differentiate past from future.
...A surface has geodesics, but there is no inherent direction for them. The choice of time's direction is arbitrary.
...The coupled differential equations (equations (37-a) and (37-b) in the paper [Geometrical Physics A, 6, 1998]) are also invariant under the transformation t → -t.
...Going upstream, we know we can identify two conjugate points M and M* on our two hypersurfaces using the same coordinate set. Let us call these coordinates (t, z, x, h). We can then carry out the calculation to completion and obtain the final coupled differential equations (let us write them):
which remain invariant if I change t into -t.
At this stage, I can just as well decide that: t = t, t* = t
or that: t = t, t* = -t
...The equations do not define any temporal orientation a priori, any more than the Friedmann equation did. But then, what do these variables t and t* actually mean?
Addendum dated February 2000:
Between the time I composed this text and now, a whole range of new works on black holes (or rather, pointing toward their nonexistence) have emerged. In light of these developments, I would now say that the quantities t and t* are merely coordinates, nothing more. The fact that we decide, for example, that t* = -t does not at all imply that moving from sheet F to its geminal sheet F* would mean living "backwards in time." In these newly discussed works, particular attention is paid to how the two sheets might be brought into communication (for a very brief instant, the time of a hyperspatial transfer of matter from sheet F to sheet F*). What then happens to matter escaping toward "the retrograde side of our universe"? Does it travel backwards in time?
...It evolves within sheet F*, where the time coordinate is inverted. But during the transition from one sheet to the other, a test mass follows a geodesic. Its "onboard clock" (i.e., its proper time) continues progressing toward the future. Moreover, this test particle could theoretically re-emerge into F after traversing the "corridors of the twin." Does this mean the test particle could reappear before it even left?
...No. At no point was its path "retrograde." But then, what is the ontological nature of this time inversion? Beware: this refers only to the inversion of the time coordinate, not of proper time. Drawing inspiration from Souriau's work (Structure of Dynamical Systems, 1974, Dunod, p. 198, equation 14.67), we know that inversion of the time coordinate and inversion of mass (and energy) are joint phenomena. Time inversion results from the action of the "anticlockwise components of the Poincaré group." As for the inversion of mass and energy, it stems from the group's action on its momentum space.
...Thus, "traveling for a time in a sheet where the time coordinate t* is opposite to ours" simply means that during the time it is "submerged in the twin," a test mass m contributes negatively to the gravitational field (relative to particles remaining in its original sheet).
Inverting time is equivalent to inverting energy and mass.
We have seen that our ghost matter particles behave as if they possessed negative mass. We can say that if two interacting particles have positive masses but opposite time arrows, they gravitationally repel each other. In the paper: J.P. Petit and P. Midy: Geometrization of antimatter through coadjoint action of a group on its momentum space. 3: Twin group. Matter-antimatter duality in the ghost space. Reinterpretation of the CPT theorem.** **[On this site: *Geometrical Physics B, 3, 1998.] we attempted to uncover the group structure underlying this geminal geometry. We concluded that the two sheets are linked by symmetry relations, and in particular that their time arrows are opposite. This brings us back to André Sakharov's initial idea and his theory of twin universes.
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