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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?

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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.

Despite this, you can still think of the interval as a sort of "distance" as long as you abandon some of your intuition about this sort of distance, e.g. the idea that distinct points cannot be zero distance apart.

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  • $\begingroup$ Thanks for answering. I guess what I am trying to understand is, how drastic are those relaxed constraints you mention in the last sentence? In other words, I had assumed that a pseudo-Riemannian manifold, although not quite the same as a Riemannian manifold, it would be fairly similar. But I am reading this book called Riemannian Manifolds by John Lee and he notes in Chapter 6 many properties he discusses in that chapter don't apply to pseudo Riemannian manifolds. This sounds like the two then have very diffrent mathematical properties if you can't do the same mathmatics on them. $\endgroup$ – Stan Shunpike Jan 14 '15 at 6:59
  • $\begingroup$ The differences are drastic in some senses, but not so drastic in others. Some salient similarities: in both cases, one can define notions of Levi-Civita connection, covariant derivative, and various curvatures like the Riemann curvature tensor. We've already listed one difference (lack of natural metric space structure), and there are others. I'd highly recommend that you look at *Semi-riemannian Geometry" by O'Neill for more. In short, these two beasts are identical in their properties as manifolds, but their differential-geometric properties diverge. $\endgroup$ – joshphysics Jan 14 '15 at 7:06
  • $\begingroup$ Great! Thanks for the reference. I will check it out. That sounds like exactly what I'm looking for. $\endgroup$ – Stan Shunpike Jan 14 '15 at 7:07
  • $\begingroup$ Yeah it's the definitive reference. It's pretty darn complete and well-written imo. Have fun! $\endgroup$ – joshphysics Jan 14 '15 at 7:08

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