Doug is funny and right, but you may find him irritating. Personally i find him interesting.
So doug explains very simply what a "metric" is.
Well a metric is so commonplace that it is no wonder we do not know what it means. We do not use metric in everyday language therefore it is a clear signal that a mathematical physicist is talking or writing!
We tend to use measure, measuring tool, tape measure,measuring instrument,carpenters measure.
The Ancients used a carpenters square as the first fundamental measure. It was so important that it was used as a symbol of power and rulership for the Pharaohs!
The Carpenters' square was generalised to the Gnomon by the greeks, and they collected and organised the worlds geometrical knowledge into a theoretical and logical system derived empirically. One of the general relations that was known and idealized was the relationship of areas around a rectangular triangle.
This was a conservation of area relationship, and from this other conserved measures could be derived.It extended to a conservation of volume relationship for a parallepiped.
This conservation of magnitudes has to be dynamic, as magniudes are generally dynamic. However we do have equilibrium states where static or dynamic states exist as well as explosive. For each of these there is a conservation of volume, et al that exists. We can therefore find a relationship between magnitudes related by a rectangule trianglular form which conserves magnitude.
This property made the right angled relationships a special measuring relationship, and consequently any tool designed in this way a special measuring tool called a "metric" to highlight it.
Euclid had an even more general theorem from which the pythagoras theorem drops out as a special case. Therefore there are an infinite number of metrics and pythgoras in many ways is the simplest to remember.
So this measuring tool is more general than a plane right angled triangle, because we exist in 3d space (and some want to describe it as a space-time "space" of 4 "dimensions").
Now time is not a space dimension, it is thought, however it is in my opinion. And, it constitutes a closed boundary dimension.
We can have any number of "open" directed dimensions, called generalised coordinate reference frames, and any number of closed boundary dimensions which i will call a generalised "polar" coordinate systems.
I use the idea of a polar coordinate system to dimensionalise "time", as it is familiar enough to attach the notion of a closed boundaried dimension.
Every dimension needs parameters to measure it, and the most used parameters are edges of a solid form fractalised into a scale, by choosing a suitable unit solid. We aggregate these to measure.
The fractal scale we use for a closed boundedaried dimension are based on solid sectors as units, the sectors have a common centre, or origin.
Now do we have a relationship for closed boundaried dimensions that conserves area, volume et al?
I will give it some thought, because it will represent a metric for closed boundaried dimensions in the form of a sector or solid cone-like sector.
My first guess would be cos^2(a)+cos^2(b)cos^2(c)=1, where a, b, c are the arc length in radians to the 3 orthogonal radii of a spherical reference frame.
Trigonometry is the consumation of years of theory and pragmatic experience in calculating area and from these finding edge lengths or Arc lengths. Arc lengths are the basis of the notion of angle measure, but the word "angle" derives from the notion of bending into a hook shape.
So trigonometry of the plane or the sphere provides a rich area to find the relationships we need to measure everything.
I extend trigonometry by Napiers Logarithms, and because of this i point to Euler, Wallis,Newton, Hamilton, De Moivre, and Roger Cotes as individuals who derived advances in mathematics by this extension.
I have also analysed the twisted notions in complex numbers down to their core relationship to spherical trig. Thus having a geometry, a dynamic spaciometry and the relationships between unities, and a summetria i am able to, through trigonometry, calculate between dynamic magnitudes.
The trig denotes, connotes and supports the metrics we utilise,founded on dynamic geometric relationships.