> That's expected when you realize that time and space are not different things [...] The only thing separating them is a sign which in turn is based purely on convention.
The fact that spatial and temporal dimensions require different signs in the metric signature (regardless of whether you take the +--- or the -+++ convention) should suggest that time and space dimensions are indeed, in some sense, different (in the sense of "not interchangeable").
There's a reason space-time is usually modelled as a 4-dimensional Lorentizan manifold, rather than a 4-dimensional Riemannian manifold.
What do you mean with usually? Spacetime (i.e. the object of discussion in relativity) is by definition not positive definite. You said it yourself that space and time require different signs. But they are still interchangeable - without changing physics. On top of that, all that a Lorentz transformation (at least for boosts) does is rotate (hyperbolically) space and time into each other. Seeing space and time as two different things rather than interchangeable components of a more fundamental object is not in line with relativity.
Agree. tx bivectors squares to 1 giving hyperbolic rotations, while xy bivectors squares to -1 giving euclidean rotations.
saying time and space aren't different because spacetime exists sounds like saying electric fields and magnetic fields aren't different because EM field exists.
>like saying electric fields and magnetic fields aren't different because EM field exists.
Funny you should say that, because this idea is well known as S-duality [1] in Quantum Field Theory (and more generalized in String Theory). In fact, if magnetic monopoles (read: charges) exist, you could already see that simply by looking at Maxwell's equations and replacing E->B and B->-E. If you also know a bit about the Maxwell bivector, it's easy to see how closely this duality mirrors how space and time are related.
The fact that spatial and temporal dimensions require different signs in the metric signature (regardless of whether you take the +--- or the -+++ convention) should suggest that time and space dimensions are indeed, in some sense, different (in the sense of "not interchangeable").
There's a reason space-time is usually modelled as a 4-dimensional Lorentizan manifold, rather than a 4-dimensional Riemannian manifold.