For any space-like event you can find reference frames where things happen in different order. For the time-like situation you described the order indeed exists within the cone, which is to say that causality exists.
You can still order them with the spacetime interval compared to a reference event, even for space like separated events.
It allows for differing elements of the set to share the same value but so does using time alone. It just also allows every observer to agree on the ordering.
Bc Assigning a distance function to elements of a set is a common way to do that in fact. It doesn’t work with just a time coordinate or space coordinate, because that’s effectively a Euclidean metric.
You just have to contend with a few nonintuitive aspects but it’s not so bad.
I think you meant compared to a reference observer? Events are not really independent of observers. Consider the case in baseball where a runner and the baseman tag the base at the "same" time from opposite sides of the base. Assume they move at equal speeds. If the umpire is closer to the baseman then the baseman has tagged it first, if he is closer to the runner, then the runner has tagged it first. The "event" of "touching the base" has two possible outcomes depending on where the observer stands, and there is no "view from nowhere" or observer-free view that we can reference.
Enginerrrd|2 months ago
It allows for differing elements of the set to share the same value but so does using time alone. It just also allows every observer to agree on the ordering.
Bc Assigning a distance function to elements of a set is a common way to do that in fact. It doesn’t work with just a time coordinate or space coordinate, because that’s effectively a Euclidean metric.
You just have to contend with a few nonintuitive aspects but it’s not so bad.
shomp|1 month ago