Sorry for the delay. I had no idea that this may be out of bounds here, sorry if I may have bothered you, that wasn't my intention. My reference are the article by Roger Penrose about the negative dimensionality "Application of negative dimensional tensor", 1971, and of course the classic "Negative fractal dimensions and multifractals" by Benoit B. Mandelbrot, 1990. For the Von Neumann algebra about non integer degrees of freedom, between the others, "On the dimension theory of Von Neumann algebras", 2005. So I tried to mix both in a non integer Ricci tensor <0. This is effectively due my interest in Alain Connes fractional dimensionality theories, overall the article "The idea of Connes about inherent time evolutionof certain algebraic structures from TGD point of view", by Matt Pitkanen, 2021. Another interesting point to focus this is the article "Quantum Behavior Arises Because Our Universe is a Fractal", by Young Tao, 2017. It was just a gedankenexperiment, if there are problems with that I'll remove my question, no problems whatsoever.
Now, how would I proceed to describe fractal degrees of freedoms? Well, there are many approaches, one is following the Minkowski-Boulingand,
and the second option is using the Haussdorf dimension. This led me to think about the De Sitter invariant special relativity, and its symmeties. In this way of defining dimensionality, a curve can grow like the power of 1, of 2 (a square), of 3 (cube) and so on.
But what happens with the power of -1? Well, that leads to a dynamical behaviour for even and odd values, not a homogeneous curve or a line. So, every time you reach an integer number you have a sort of rythm. (1)
A curious behaviour indeed, and I just thought it could be expressed like this, and I wonder if it made some sense.
A negative dimension in the Minkowski-Boulingand, in the Haussdorf or even the Packing dimension model, the number of the "contiguous" D-dimensional balls of dimension ε should decrease for a diminishing value of ε. In other words, there is a diminishing quantity as you "squeeze in", or...move on. This lead me to think to the number of possible outcomes of a state tending towards zero, as with adiabatic systems and entropy. (2)
Plus, inductively, one can argue that dimensions are boundaries of suitable open sets. So, for instance, an infinite serie of 0 dimensional contiguous points you get a line, a infinite series of contiguous 1D lines you get a 2D surface and an infinite number of contiguous sheets you get a volume. But for every D you have two degrees of freedom, back and forth. So to have a line you shoud draw your zero dimensional point in one side and the other, or you will get a half-line. That could be calculated easily with Minkowski-Boulingand as having a fractal dimensionality of 1/2: as you "squeeze in" ε, you get a bigger number of "balls" only in one side, not both. (3)
This means that this "special" dimensionality it has a cardinality.
So, with this, with rhytm (1), a flowing number of possible states running towards zero (2) and cardinality (3), I just wanted to explore how would behave the EFE with this kind of definition of a very strange kinds of dimensionality and it sounded me a little bit like some sort of a weird bimetric gravity, and I wanted to reach out some help to see if anyone else could help me with that. Can anyone help me fix it, please?
I also tried that with Schroedinger and that was fun too, but that's for another post ahaha!
Sorry if I bothered you again, I could remove my post as soon as you don't consider it appropriate!
Best regards and thank you again for listening me...
Thomas