In a Minkowski space diagram of ct vs x, with 2 stationary events in the same plane of spacetime (1 event located along a worldline, 1 not on a worldline) is this still considered euclidean space?

The definition of euclidean space is a space in any finite number of dimensions, in which points are designated by coordinates (one for each dimension) and the distance between two points is given by a distance formula. (Given by Encyclopedia Britannica)

In our proposed Minkowski diagram above, there are a finite number of dimensions, and a defined distance in spacetime between them, given by: $$ (\Delta s)^2 =(\Delta x)^2 -(\Delta ct)^2$$

The book I am following (J. Hartle, Gravity: An Introduction to Einstein’s General Relativity), states that, in Euclidean space, for the distance between two points we will always have $\Delta s >0$. It cannot be negative, or even 0, unless $\Delta x =0$.

Does this mean if I'm finding the distance in spacetime between 2 points like the ones stated in the scenario at the beginning, the events can never be time-like $(\Delta s < 0)$? Or does the fact that one event is not located along a wordline mean that this distance can be located outside the light cone?


In Minkowski spacetime the distance of two events is time-like if the future or past light cone of one event comprises the other event. Instead it is space-like if the future or past light cone of one event does not comprise the other event.
If the two events are in a $t = const$ section of spacetime the distance will be space-like. The section is an Euclidean space at a constant time.


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