Neither am I. I can only try to answer this question on the basis of my hypothesis. A GR and CMB specialist most probably would give a completely different answer, and would consider what I present here as nonsense.

I’m wondering if your hypothesis can predict the real curvature or the radius (depending on the shape) of our Universe at a time. I didn’t make any calculation, but it will be no surprises to me if it can imply some kind of c(t) dependence.

Ah, Riccardo, a most interesting but vexing question, to which I have not figured out yet a satisfactory answer.

There are two considerations here. The first: In Figure 11 (where T=10) let us imagine that at T=8 a new Photon was emitted by the Stationary Body. If we assume a constant c=1, and that the Photon remains at the time coordinate at which it was emitted, then this Photon at T=10 would be at x=2, and so it will not coincide with the space dimension as depicted there (which for T=8 is at x=6). This would violate the Lorentz transformation. The only way for this not to happen would be (I think, but I am not sure if I remember correctly) if the space dimension was not circular or elliptic but a straight line that met the t axis at such an angle that it would give us the known speed of light. (Before I realized that it violated the Lorentz transformation, I did make this calculation (it is a simple application of the equation of the circle), and the result was that, if the “shape” of the Universe is completely spherical in respect to time, for the speed of light to have the value we observe today (299,792,458 m/s) the age of the Universe would have to be 1,425 billion years, which is off the accepted value approximately by a factor of 10. Intriguing.)

The second consideration puts a completely different spin on things. If we calculate the proper time of the Photon through Equation 1b, we see that it is always 0. So, according to this, a Photon emitted at T=0 will remain at T=0 (its proper time would be 0). Furthermore, a Photon emitted, for instance, at T=8, would still remain at T=0 (its proper time again would be 0).

Now, this is mandated by the Lorentz transformation. If you try to calculate the proper time of photons, you will get 0. Any other result would violate the Lorentz transformation. Here you can say that the Lorentz transformation breaks down for photons, but this is rather difficult to do because you have neither infinities nor divisions by zero, just a straightforward calculation that results in 0. Or you can accept this result as it is, that is, accept that photons are situated at time moment zero, which would have to be just a moment before the start of time in the universe (the moment of the “Big Bang” would have to be the first moment that was actually greater than zero).

Now, this opens up a whole series of new issues (which, to be frank, almost lead us into science fiction or phantasy ). It would mean that when we have a light producing event at T=10, for instance, light starts at x=0, T=0, the origin of the universe (the locus of the Big Bang?) and propagates moving across past time moments. (These may by the longtitudinal waves that Carl Brannen was talking about?). An observer sees this light coming from a specific point of space, but the light has always the same origin, the common origin of all the points of space, the “time center” of the universe.

(See, that’s why mainstream physicists don’t work with fundamental issues, so that they don’t have to make a fool of themselves. They leave that to us crackpots . And when they do, they coach it in such obscure mathematical language so that no one will be able to undestand them, except for their colleagues.)

Anyway, that is the only interpretation I can come up with about the proper time of photons. If there is any other I would be interested to know.

At this point, I’m wondering if your hypothesis can predict the real curvature or the radius (depending on the shape) of our Universe at a time. I didn’t make any calculation, but it will be no surprises to me if it can imply some kind of c(t) dependence.

In the first case (the one depicted in figure 12), the curvature of the space dimension in relation to time as depicted affects the worldlines of the objects within the “pointed concave funnel”, and this in its turn affects the distances among the worldlines, making masses approach each other. So here, the curvature in respect to time creates “curvature” in respect to space, in the sense that the Pythagorean theorem would not hold true for such a region.

On the other hand, in Figure 9, for instance, we have a depiction of the global geometry of the universe, without taking into consideration the mass of the two bodies. You will notice that the Moving Body traverses the same distance in its “adjusted” and in its “unadjusted” position (in the specific case depicted, x=6 for both). The difference between them is their time. Here the space dimension curves within time only, wordlines are not affected by any local gravitational distortion, so distances are not affected also, and therefore the Pythagorean Theorem would hold true for such a region. That is, this region would appear flat, because it curves solely in respect to time. (This is where the extrinsic description of curvature is helpful.)

Perhaps I should point out also that what we term “proper” time is the time that we (being in the position of the Stationary Body) attribute to the Moving Body because we use a linear coordinate system. In a polar coordinate system, both bodies are at the same moment of time (they measure the same age for the “universe”, in the case depicted, 10 seconds).

no massless charged particle has ever been observed … I’m still looking for a clear geometric interpretation of the fact

The above just offer a way to arrive at such a clear geometric interpretation. I suspect that what will prove (in this “theory”) to determine whether a particle can exist or not (and for how long it will exist) is the stability of the photon trap configuration of the space dimension that constitutes the particle. And of course this will have to be formulated mathematically in order to be able to talk meaninfully about the possible configurations that can exist, their stability and their duration.

I am not sure I understand the comment in your second paragraph. Do you mean “one thing that does not work”, meaning that the observation of a flat universe contradicts the curved expanding universe of GR described here?

To me, one thing that does work (at the moment) is the contrast with the experimental observation of a flat universe with \Omega ~ 1. I don’t want to say that your idea is wrong, but just that there might be something very deep behind.

There is no Stationary Body in nature.

Yes, of course. “Stationary” always relatively speaking. In the Galileo and the Lorentz transformation, we have a “Stationary” and a Moving Body or observer or reference system (for instance, the train embankement and the train), and the transformation allows us to compute the coordinates measured for an event by the latter given the coordinates measured for the event by the former (and vice versa).

where t’ is the proper time of the Moving Body and T is the time of the Stationary Body.

I object in principle to the assumption of a “Stationary Body.” There is no Stationary Body in nature. Are you using this in a non-precise so to speak way? I am sure that there is a good explanation but I thought I mention it.