Question

How do you derive the equation for proper distance, and is it a coincidence that [proper distance]=i*[proper time]?

Answer 1

One simply calls/defines this d^2 - t^2 as a metric in Minkowski space.
The formula itself in not derived - it simply expresses an Manifestly-Lorentz-invariant measure of separation of distinct events

Answer 2

@henpen ?

Answer 3

But why does that particular value equate to the distance between 2 events from the frame of reference of an observer seeing them as if they were simultaneous?

Is it that:
That observer sees the 'proper' distance^2 as= d^2-t^2=d^2-(0)^2
d^2-t^2 is invariant.
Thus d^2-t^2=d'^2-t'^2

Answer 4

What is the proof that it's invariant? I have read multiple 'proofs', but none felt very rigourous.

Answer 5

The "proofs" are so short ( about 1 line, max 2 lines) that they can only be rigorous.
One simply substitutes the formula for Lorentz transformation of the coords. and immediately obtains that \[d_0^2 - t_0^2 = d_1^2 - t_1^2\]

Answer 6

Your first Q. is unclear. Anyways , the meaning, significance and properties of this metric are about 40 to 60 pages long if it is explained in intuitive terms. In mathematical formulas - well about 2 to 3 pages of transformations.

Answer 7

Thanks. Now I understand that it was rigourous, OK.

Can you direct me to any of these mathematical proofs?