# Given a Spacetime in terms of Lorentzian distance values, how to determine which pairs of events were spacelike separated?

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"), $$\ell : \mathcal S \times \mathcal S \rightarrow [0 ... \infty];$$ especially also in cases where values of spacetime intervals, $s^2$, cannot be obtained (i.e. if the spacetime under consideration is not flat).

For any two distinct events $\varepsilon_{A J}, \varepsilon_{A K} \in \mathcal S$ which were timelike separated from each other, such that some participant ("material point", "observer") $A$ had taken part in both, holds

$$\ell[ \, \varepsilon_{A J}, \varepsilon_{A K} \, ] + \ell[ \, \varepsilon_{A K}, \varepsilon_{A J} \, ] \gt 0;$$

where in particular

• either $\varepsilon_{A J}$ chronologically preceded $\varepsilon_{A K}$, and therefore
$\ell[ \, \varepsilon_{A J}, \varepsilon_{A K} \, ] \gt 0, \qquad \ell[ \, \varepsilon_{A K}, \varepsilon_{A J} \, ] = 0$,

• or the other way around;

while any pair of two distinct events which were not timelike seprated from each other (but instead lightlike, or instead spacelike), $\varepsilon_{A B}, \varepsilon_{J K} \in \mathcal S$, satisfy

$$\ell[ \, \varepsilon_{A B}, \varepsilon_{J K} \, ] = \ell[ \, \varepsilon_{J K}, \varepsilon_{A B} \, ] = 0.$$

My question:
Given these $\ell$-values of all pairs of events in $\mathcal S$, is it possible to distinguish and to determine which pairs were spacelike separated from each other (and not lightlike, nor timelike) ?

(The characterization of two events as "lightlike separated from each other" means, in terms of their causal relations, that one of them strictly causally preceded the other, without explicitly prescribing which one "came first", and that neither preceded the other chronologically.)

• I do not understand well your notation regarding events $\epsilon_{AJ}$. What is the meaning of $A$ and $J$? Commented Mar 22, 2017 at 8:32