I've been day dreaming about this idea for a month or two. There are many hypothesises that describe the universe in some higher number of dimensions. As far as I know, this is always an integer number of dimensions. I've been wondering if it is possible for the universe to have a fractional number of dimensions, say 3.139 dimensions, where there are 3 spatial dimensions plus the 0.139. The 0.139 could be time. It's fractionality would explain why it seems to behave in a different manner than the other dimensions. The flow of time could actually be the dimensionality of the universe increasing towards four dimensions. In other words, time is inflating into a full dimension. This would also explain why the universe is expanding because [tex] distance = \sqrt{x^2+y^2+z^2+t^2} > \sqrt{x^2+y^2+z^2+(t-\alpha)^2} [/tex] and because t is increasing. In reality, time would have to increase at a rate of (if I did my math right) [tex]t(\tau) = \frac{1}{2}\tau^2[/tex] to account for linearly expanding space. This leads to a time-dependent distance formula given by [tex] distance(\tau) = \sqrt{x^2+y^2+z^2+\frac{1}{2}\tau^{2}w^{2}} [/tex] But this is more in the finer details. Conversely, it could be that time is actually moving backwards and the dimensionality of the universe is decreasing. This would easily explain the relatively even distribution of matter in the universe. In this case, the perceived expansion of the universe could be some kind of "conservation of space." I just thought that this was interesting because it unifies space and time, and it also explains why the universe is expanding. I'm not posting this to the personal research forum because it is largely speculative, and I'm really looking more for comments and criticism.
What you've written down wouldn't mathematically represent a non-integer number of dimensions. If you want to learn about things with non-integer dimensions, read up on fractals.
The expansion of the universe results in a change in the spatial separation between two points: [tex]d = \sqrt{x(t+\Delta t)^2 + y(t+\Delta t)^2 + z(t+\Delta t)^2} > \sqrt{x(t)^2 + y(t)^2 + z(t)^2}[/tex] not the distance you've given.
Sorry. What I meant was (and this is slightly different than my earlier notation): [tex]d = \sqrt{ {\Delta x}^2 + {\Delta y}^2 + {\Delta z}^2 + \tau {\Delta w}^2} > \sqrt{{\Delta x}^2 + {\Delta y}^2 + {\Delta z}^2 + (\tau-h) {\Delta w}^2}[/tex] where [tex]0 \leq \tau \leq 1[/tex] This is intended to give the distance between two stationary points as a function of time, where the universe is expanding and the number of spatial dimensions n is [tex]3 < n < 4[/tex] or [tex] n = 3 + \tau[/tex] EDIT: and the dimensions are x, y, z, and w.
What you have written down are three spatial and one time dimension. Those equations do not represent a change in the number of dimensions.
Fractal dimensions require that one redefine the term dimension. I'm trying to use a more classical definition.
I may have been editing my post when you wrote this. It is intended to [tex]3+\tau[/tex] spatial dimensions. When [tex]\tau = 1[/tex], there are then 4 spatial dimensions and every molecule and every atom in the universe is ripped apart.
But because there are a fractional number of spatial dimensions, there are [tex]3 +\tau[/tex] dimensions being projected on to three. Only when there are four full dimensions x, y, z, and w, will a point have four spatial coordinates.
Except your line element includes the w coordinate, and thus there are always four spatial coordinates in this scheme.
If you want non integer dimensions look at fractals. But I do not see what non integer dimensions have to do with time.