# Does a spacetime manifold have the structure of a metric space?

My first introduction to spacetime physics was Wheeler and Taylor's book Spacetime Physics. This book gave me an appreciation for how important the spacetime interval was for giving the distortions to space and time meaning. You know, the classical examples with the light beams and the trains, etc. So I have always associated the "spacetime interval" with the idea of distance.

However, I recently learned that, because a pseudo-Riemannian manifold can have zero length curves, it therefore is not a metric space by the specific definition mathematicians use. Now I am very confused. Is the concept of distance still useful to spacetime? Obviously it is in some sense, but now I'm not sure what sense I mean. Before I thought I might mean the metric space definition. But it would seem that metric spaces are not the right way to think about spacetime distance if a pseudo-Riemannian manifold isn't a metric space.

My Questions:

(1) What is a spacetime interval $ds^2$ mathematically? That is, what type of mathematical object is it?

(2) is the concept of a metric space useful for spacetime physics?

As you note, a spacetime is a pseudo-Riemannian manifold which is a pair $(M,g)$ where $M$ is a smooth manifold, and $g$ is a non-degenerate (but not necessarily positive definite) metric on $M$. The spacetime interval is a way of encoding the information contained in the metric $g$. Mathematically, the metric is a special kind of object called a two-tensor. Unfortunately, the word metric is begin used in a different way here than in the context of metric spaces.
The way a physicist often productively thinks about this is that given a set of coordinates $(x^\mu) = (x^0, x^1, x^2, x^3)$ on a spacetime (4D manifold) $M$, the metric assigns sixteen numbers $g_{\mu\nu}(x)$ to each point $x$ in that spacetime, and these numbers encode how to compute the spacetime interval $ds^2$ between nearby events close to $x$ separated by small coordinate distances $(dx^\mu) = (dx^0, dx^1, dx^2, dx^3)$: \begin{align} ds^2 = g_{\mu\nu}dx^\mu dx^\nu. \end{align} As you note, this does not naturally yield a metric (distance function) in the sense of metric spaces as it would in the Riemannian case. In the Riemannian case, one could endow the manifold with a distance function by defining distance between points using the greatest lower bound of the lengths of curves joining them. This yields a metric in the sense of metric spaces, but in the pseudo-riemannian case, it does not because distinct points can be joined by zero length curves.