OK - I think I have an answer. Tell me what you think. I have two metrics that I can define:
dx^2+dy^2
and
dx^2 + sin^2( x) dy^2.
These are intrinsically different in some way in that I don't think there is a coordinate transformation that can take one to the other.
You are agreeing with me. We choose the Lorentzian metric because it is physically interesting, but we could equally well choose the Euclidean metric - it just wouldn't be as interesting. My actual question is "What is the significance of a change of signature in general relativity"
Thank you, but I don't think this is the correct answer. I measure coordinates, and their infinitesimal changes. I am free to plug those coordinates into any scalar function I choose.
I'm a bit confused about the idea of "Change of Signature in Classical Relativity". As I see it, a metric is just a scalar function that I make up. For example, in the x,y plane I can define the functions x^2+y^2 and x^2-y^2 simultaneously. What, then, is the significance of "changing" the...