What you use for measuring distance
's scheme of geometry
. If z is your coordinate, then you would write the metric in the following from:
What is meant by this is that if you have some curve z(t), you can measure the distance along it as:
distance = (integral) ds = (integral) F(z)dz = (integral from t0 to t1) F(z(t)) (dz/dt) dt
(the curve goes from z(t0
) to z(t1
)). F(z) is required to be positive definite
so that distance do not become negative.
Example: in Euclidean 3-space, ds2 = dx2+dy2+dz2, in agreement with Pythagoras's Theorem.
In General Relativity the metric is expressed in terms of the metric tensor, ds2 = gadxadxb. It is techincally a pseudo-Riemannan metric, since the metric tensor is found not to be positive definite.