[tsimionescu]

Well, even in this metric, there is an obvious separation between the 3 spacial dimensions and time. For example, a negative ds^2 indicates a fundamentally different kind of distance than a positive ds^2. In fact, events separated by positive distances would be considered entirely independent, while those separated by negative distances can have influenced each other; events for which ds^2 = 0 are simultaneous in any frame of reference.

Further, if two events have are separated by a negative ds^2, then all observers will agree on the order in which they happened, though they will not agree on the length of time that passed between them, or the relative positions.

Note that I'm using your version of the equation for the definition of ds^2 > or <0, though in general I've seen it expressed the other way around, ds^2 = dt^2 - (dx^2+dy^2+dz^2).

[gizmo686]

There is a clear assymetry between the temporal dimension and the spatial dimension; but it is not clear to me that this implies a seperation.

We can distances with ds^2 < 0 timelike, and ds^2 > 0 spacelike. We say that events with a spacelike cannot influence each other; but there is nothing in relativity that requires that (unless you introduce causality as an additional assumption, which we generally do).

This gets more messy in general relativity when you allow for large masses (read. Black holes).

In the case of a non spining black hole, we spacetime is described by the Schwarzschild metric. Expressing this metric in spherical cordinates, you find that when you pass the event horizon the sign of dt^2 and dr^2 flip, where r is distance from the singularity. This means a "timelike" seperation means that events are closer to each other in the time dimension; and events with a "spacelike" seperation are farther from each other in time. That is to say, "time" behaves in the way we think of as "space", and "space" behaves the way we think of as "time".