Take the spacetime $(M,g)$ that satisfies Einstein's Field Equations exactly where $g$ is locally:
$$g= - c^2 dt^2 + d \rho^2 + (\kappa^2 \rho^2 - a^2) d \phi^2 - 2 ac d\phi dt + dz^2 \ $$
in the cylindrical coordinate system where $z$ is called the axis of rotation. Where the $\kappa$ is the related to string tension and $a= c/\omega$ is inversely related to the angular frequency. Ostensibly this metric is not strongly causal since it contains CTC (closed timelike curve).
This metric is due to the rotating Cosmic String which is a particular type of topological defect that appears in the classical (topologically nontrivial) solutions in field theory, which in our case is a $U(1)$-Higgsed Yang-Mills in $3+1$ dimensions (flux tube stretched along its centre from $-\infty$ to $+\infty$ on the $z$ axis, and rotating around the same axis).
The question:
What is the Lebesgue covering dimension of the interval topology induced by the Lorentzian metric $g$?
Does this dimension change in the vicinity of the $z$ axis? In case is it a fractal, what are the necessary and sufficient conditions for it to be so? Is it finite? What is the upper bound on its dimension?
The regimes of interest are:
- $0< \rho |\kappa| / a \ll 1 $
- $0< \rho /a \ll |\kappa| \ll 1$
- $0< \rho /a \ll 1 \ll |\kappa|$
- $0< |\kappa| \ll \rho /a \ll 1$
I'd be interested in an analysis of the interval topology of the metric in this regimes.
I already know that this topology is "not locally Euclidean" and "non-Hausdorff".
PS:
Interval topology:
Take a time-oriented Lorentzian manifold that is weakly causal(aka not strongly causal), fix a point $p\in M$ and consider all (smooth) timelike future-directed curves through $p$ and denote by $L(\gamma)$ the Lorentzian length of $\gamma= \gamma(\xi)$, $\xi \in [a,b]$. $$L(\gamma) = \int_a^b \sqrt{|g(\dot{\gamma},\dot{\gamma})|} d \xi$$ If $q \in M$ define the so called Lorentzian distance of $q$ from $p$ as $$\tau(q,p) := \sup \{L(\gamma) \:|\: \mbox{$\gamma$ timelike future-directed from $p$ to $q$}\}$$ If no timelike future-directed from $p$ to $q$ exists, $\tau(q,p) :=0$.
Next define $I^+(p) := \{q \in M \:|\: \tau(q,p)>0\}$ and $I^-(p) := \{q \in M \:|\: \tau(p,q)>0\}$.
It is possible to prove that the family of sets $I(p,q):= I^+(p) \cap I^-(q)$ (with the suitable chronological order of the arguments) is a basis of the interval topology on $M$ which for weakly causal spacetimes is strictly coarser than the manifold topology $\tau_{\text{Manifold}}$.
