[simiones]

> One could still argue that time is just an ordinary dimension and the structure of the metric is just a property of how we measure distances.

The math doesn't work unless time is different from the 3 spatial dimensions. In particular, distances in space-time can be negative, unlike distances in space.

[edna314]

Well, I'm just arguing that it isn't clear whether this prefactor (that shows up in the metric in front of time) is a property of time or a property of the metric. It is pretty clear that the math wouldn't work if the prefactor was different, though.

[simiones]

Given that there is a physical difference between events separated by positive and negative distances in spacetime, I think it's pretty clear that the prefactor is a property of spacetime.

[edna314]

Yes, it is mostly semantics but the question really is what you think the space-time metric is. In order to attribute the prefactor to time you would need to argue that the space-time metric really is Euclidean, but because time is different from space there is this negative prefactor showing up in the metric. But you can also claim that time and space are the same and the space time metric is hyperbolic. Since there is there is no good argument why metrics necessarily need to be Euclidean, I find the second option more convincing.

[dfilppi]

Implying that there is space separate from time denies spacetime

[simiones]

Not at all - spacetime is real, but it is not a 4-dimensional Cartesian coordinate system where all 4 dimensions are of the same kind. For example, for any two events that are time-like separated, any two observers will agree on the order in which they happen, whereas for two events that are space-like separated, there will be observers that see these two events happen in either order. Time is different from space, though they are related.