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The geometric relations between pairs of events of a spacetime $\mathcal S$ can generally be characterized by values of Lorentzian distance (cmp. J.K.Beem, P. … Ehrlich, K.Easley, "Global Lorentzian Geometry"), $$\ell : \mathcal S \times \mathcal S \rightarrow [0 ... …

Mathematically one might pose lots of opposition since they fix the topology before assigning a metric to the topological manifold and in case of Lorentzian geometry the topology induced by the Lorentzian … casual, the manifold topology is strictly finer than the Lorentzian distance induced topology just like in the case of spinning Cosmic String). …

non-diffeomorphic differential structures just like that of $\mathbb{R}^4$) can be derived by demanding invariance of the gravity action under the change of differential structure in 3+1 dimensional Lorentzian … Personally (not only) do not assume strong causality (but also assume the opposite), where the manifold topology matches the interval topology introduced by the Lorentzian distance, but any answer or comment …

The lorentzian (or, if you prefer, minkoskian) distance from the blue point to $P$ is 1.25, meaning that a clock traveling along the green worldline will record 1.25 ticks between those points. … The lorentzian distance from the blue point to the pink point is 1, meaning that a "stationary" clock (i.e. one with the red worldline) will record 1 tick between those points. …

Indeed there is, for instance, the "Lorentzian distance $\ell : \mathcal S \times \mathcal S \rightarrow [0, \infty]$". (Cmp. Beem, Ehrlich, Easley, "Global Lorentzian Geometry".) … Values of Lorentzian distance obey the reverse triange inequality, $$ \forall \varepsilon_{A B}, \varepsilon_{G H}, \varepsilon_{J K} \in \mathcal S : \ell[~\varepsilon_{A B}, \varepsilon_{G H}~] + \ell …

The main idea is seen from the abstract: Let $(M,g)$ be a time oriented Lorentzian manifold and $d$ the Lorentzian distance on $M$. … consequences: (1) It forces $(M,g)$ to be globally hyperbolic, (2) every point of $(M,g)$ can be connected to the initial singularity by a rest curve (i.e., a timelike geodesic ray that maximizes the distance …

In the context of this Lorentzian distance $\ell$, a point $q$ is spacelike separated from $p$ if $$q\in \text{int(}\ell^{-1}_p(0)).$$ This should in principle be known if knowledge of $\ell$-values …

\ll \varepsilon_{A P_{(n)}} \equiv \varepsilon_{A Q} \, \right\} \, \right]$ ... provided that suitable distance values $\ell$ (so-called Lorentzian distances) are given (as a corresponding pseudo-metric … In turn, values of Lorentzian distance $\ell$ might "simply" be expressed in terms of the durations (path lengths) of any and all participants; for example the specific value $$\ell[ \, \varepsilon_{A …

distance between the respective pairs of events in which $A$ took part. … Note that the infimum is to be evaluated of all sums (as opposed to evaluating the supremum when determining arc lengths of spatial path segements) because Lorentzian distances are superadditive by definition …

(a spacetime-filling) family $\mathfrak C \equiv \{ \mathcal P_n \}$ of world lines which are each timelike, each geodesic (i.e. each straight in terms of Lorentzian distance, resp. …

$$L(\gamma) = \int_a^b \sqrt{|g(\dot{\gamma},\dot{\gamma})|} d \xi$$ If $q \in M$ define the so called Lorentzian distance of $q$ from $p$ as $$\tau(q,p) := \sup \{L(\gamma) \:|\: \mbox{$\gamma$ timelike … This distance evidently depends on the choice or the Minkowkian reference frame. …

$ 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 …

It is the so-called Lorentzian distance between pairs of events $e,e'$. …

The four-dimensional case is no different provided we concern ourselves only with vector spaces, but the idea of a Lorentzian distance is of course different from the idea of a Euclidean distance. …

In Minkowski spacetime, the distance $d_S$ between two space-like separated events $x$ and $y$ [...] "Distance" $d_S$ ?? … In GR "we have" (in a manner of speaking) values $\ell$ of (asymmetric) Lorentzian distance for each pair of events (with non-zero values for pairs of time-like separated events; depending on the order …