Two occurences of "convenien..."! Infinitesimal calculations are also convenient but subject to physical interpretation and full of physical signifiances.
Infinitessimal calculations also describe things that ordinary calculations cannot; they make mathematical powers of expression
more powerful.
The use of ds is simply to unite two similar formulae to reduce the work we need to do. ds is always an observable; over a temporal duration, ds is simply the speed of light times the elapsed proper time. Over a spatial dislpacement, ds is simply
i times the proper length.
Anyways, if you're
really fishing for a physical interpretation, you should look into other geometries. One interesting one in particular is the
hyperbolic plane, whose oddities are intimately related to the reasons
i appears in that one formula, and the geometry of Special Relativity in general. (note: I
think the hyperbolic plane as I'm going to use it has some relation to hyperbolic geometry, but I don't know for sure)
The Euclidean plane is usually described as the x-y plane, but I'm sure you know of polar coordinates (r, θ); the conversion between them is, of course:
(x, y) = (r cos θ, r sin θ)
We imagine the r-coordinate being the size of a circle and the θ-coordinate being the angular position along that circle.
Well, we can do a similar kind of thing with the hyperbolic trig functions, though it's not quite so nice! We have to imagine three standard kinds of hyperbolas; ones that open left-right, ones that open up-down, and degenerate ones that are just two diagonal lines. The formula for this class of hyberbolas is:
x
2 - y
2 = a
If a is positive, then we have left-right opening hyperbolas, if a is negative we have up-down opening hyperbolas, and if a is zero we have a pair of diagonal lines.
Just like with circles, any point of the plane is on one of these hyperbolas, and we can write hyperbolic coordinates for a point of the plane. Because we have 3 different kinds of hyperbolas, there are three kinds of conversions:
(x, y) = (r cosh φ, r sinh φ)
(x, y) = r (+/-1, 1)
(x, y) = (r sinh φ, r cosh φ)
(here r can be both positive and negative)
This plane can be described, IIRC, in a way similar to how complex numbers work. Any number of the hyperbolic plane can be written as (x + h y) where h
2 = 1 (not -1), and you can derive properties for these hyperbolic numbers that are similar to those of the complex numbers. These numbers even have some
utility; I've seen them used in a short derivation of the cubic formula (though I don't remember how to do it)... but be careful because I don't think you can, in general, divide by a hyperbolic number.
The connection between this and the oddity with the metric goes back to the defining equation for the class of hyperbolas:
x
2 - y
2 = a
I wrote it like that for simplicity of exposition; I'm sure you're aware that when writing the equations for standard conics, the parameter a should have been squared. The equations for the hyperbolas should be:
x
2 - y
2 = r
2
or
y
2 - x
2 = r
2
But if you don't want to have this proliferation of equations, you can instead write it as:
x
2 - y
2 = r
2 where r is either purely real or purely imaginary.
r here is analogous to the metric ds just like the
r in polar coordinates is analogous to the ordinary Euclidean metric.
The quaternions may be something interesting to study as well.
