How to define *dynamical* dimensions?

Question

I'm considering a simple toy model. The spacetime is flat with $d$ space dimensions. Using cartesian coordinates, the spacetime metric is Minkowskian : $$\tag{1} ds^2 = dt^2 - dx_1^2 - dx_2^2 - dx_3^2 - \ldots - dx_d^2. $$ A massless scalar field $\Phi$ is propagating in that spacetime according to the following PDE : $$\tag{2} \frac{\partial^2 \Phi}{\partial \, t^2} - \frac{\partial^2 \Phi}{\partial \, x_1^2} - \frac{\partial^2 \Phi}{\partial \, x_2^2} - \frac{\partial^2 \Phi}{\partial \, x_3^2} - \ldots - \frac{\partial^2 \Phi}{\partial \, x_d^2} = 0. $$ Of course, $d \ge 1$ is just any integer. But what if it becomes a dynamical variable, dependant on the value of $\Phi$? I'm not considering an extension to real values ; $d$ would still be an integer, but that could vary from place to place in spacetime : $d = 1$ in some patch of spacetime while $d = 2$ or $d = 3$ in other parts of spacetime. Matching the patches boundaries could be done a bit like a thickless string ($d = 1$) smoothly getting a thickness (cylindrical surface ; $d = 2$), then getting a bulk ($d = 3$), etc, but I'm not sure this is actually making any sense.

How could we make sense of this idea of a dynamical spacetime dimension, and define the dynamics of $d$ now considered as a kind of a new scalar field? (constrainted on integers ?).

Introducing some variable scale factors, I was thinking about an infinite metric like this : $$\tag{3} ds^2 = dt^2 - a_1^2(t, x)\, dx_1^2 - a_2^2(t, x)\, dx_2^2 - a_3^2(t, x)\, dx_3^2 - a_4^2(t, x)\, dx_4^2 - \ldots, $$ where $a_i(t, x)$ are new dynamical variables and $i = 1, 2, 3, \dots, \infty$. In a one dimensional space ($d = 1$), $a_i = 0$ for all $i = 2, 3, \dots, \infty$. If space is two dimensional ($d = 2$), then $a_i = 0$ for all $i = 3, 4, \dots, \infty$, etc.

I'm "visualizing" these dynamical dimensions as an infinity of closed loops ; $0 \le x_i < 2 \pi \ell$, all fluctuating at a tiny scale ($a_i$ randomly pulsating and oscillating, such that $a_i \approx 0$ for most $i > 1$), then suddenly blowing up if the scalar field $\Phi$ gets a special high value, locally or globally. When the spacetime is getting (or losing) some new (old) local space dimension, it would define a dimensional phase transition.

How could we obtain the Einstein (or FLRW) equations for such an infinite metric like (3) above, especially if $a_i$ depends on $t$ only? Was this idea already considered before?

Answer 1

I can tell you what you need to ask on Math SE in order to resolve this question, it is whether an "impure manifold" is always the "disjoint union" of "pure manifolds." To my knowledge there is no smooth way to glue a line into a sphere for example.

I would strongly suspect it is, because I think one could not get the local coordinate charts to cross over from one space to the other, which I think would mean that considering all of the open balls of coordinate-domains on the sphere one has the sphere as an open set, and considering the open balls of coordinate-domains on the line one has the line as an open set, and thus the space is disconnected as a union of two open sets.

With that said, if all you care about is field theories, why not instead try to mock up something where the space is higher dimensional but the field simply has regions where it does not vary with $x_1$?