In history there was an attempt to reach (+, +, +, +) by replacing "ct" with "ict", still employed today in form of the "Wick rotation". Wick rotation supposes that time is imaginary. I wonder if there is another way without need to have recourse to imaginary numbers.
Minkowski spacetime is based on the signature (-, +, +, +). In a Minkowski diagram we get the equation: δt2−δx2=τ2
By replacing time with proper time on the y-axis of the Minkowski diagram, the equation would be δx2+τ2=δt2
I am aware of the fact that the signature (-, +, +, +) is necessary for the majority of physical calculations and applications (especially Lorentz transforms), and thus the (+, +, +, +) signature would absolutely not be practicable.( Edit: In contrast to some authors on the website about Euclidian spacetime mentioned in alemi’s comment below)
But I wonder if there are some few physical calculations/ applications where this signature is useful in physics (especially when studying the nature of time and of proper time).
Edit (drawing added): Both diagrams (time/space and proper time/space) are observer's views, even if, as it has been pointed out by John Rennie, dt is frame dependent and τ is not.
Answer
The significance of the metric:
dτ2=dt2−dx2
is that dτ2 is an invarient i.e. every observer in every frame, even accelerated frames, will agree on the value of dτ2. In contrast dt and dx are coordinate dependant and different observers will disagree about the relative values of dt and dx.
So while it is certainly true that:
dt2=dτ2+dx2
this is not (usually) a useful equation because dt2 is frame dependant.
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