What you use for measuring

distance in

Riemann'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 t_{0} to t_{1}) F(z(t)) (dz/dt) dt

(the curve goes from z(t

_{0}) to z(t

_{1})). F(z) is required to be

positive definite so that distance do not become negative.

**Example**: in Euclidean 3-space, ds^{2} = dx^{2}+dy^{2}+dz^{2}, in agreement with Pythagoras's Theorem.

In General Relativity the metric is expressed in terms of the metric tensor, ds^{2} = g_{a}dx^{a}dx^{b}. It is techincally a pseudo-Riemannan metric, since the metric tensor is found not to be positive definite.