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The Wikipedia article for metrics mentions several generalizations of metric spaces, but all of them seem to have the property that the metric must be non-negative for all x and y. To me it seems like a space where distances don't have to be non-negative would be an obvious generalization (removing the symmetry property would also be necessary). For example, consider the real line with the "distance function"


under which "distances" to numbers on one side of y would be negative while "distances" to numbers on the other side would be positive.

Does this generalization have a name, or is it useless enough not to have been studied?

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What properties would you want the distance functions to have, to distinguish them among arbitrary functions $X\times X\to\Bbb R$? – anon Aug 6 '12 at 20:42
$d(x,y) = 0$ iff $x = y$ would still work. I am unsure if an analog to the triangle inequality would work. – Aqwis Aug 6 '12 at 20:44
We call that distance function subtraction! That is, we don't call it a distance function at all. Are subtractions the only kind of examples you care about? – Qiaochu Yuan Aug 6 '12 at 20:53
Not sure if I asked the question properly. Basically I'm wondering whether spaces where non-negativity and the "identity of indiscernibles" don't hold (of which subtraction of real numbers is an example) have been studied or are worth studying at all. – Aqwis Aug 6 '12 at 20:59
Look up Minkowski space and Lorentz manifolds in general. – Will Jagy Aug 6 '12 at 21:02
up vote 4 down vote accepted

Will Jagy's comment deserves to be an answer:

Look up Minkowski space and Lorentz manifolds in general.

Minkowski space in $n+1$ dimensions is $\mathbb R^{n+1}$ with the "distance function" $$ d(\langle x_1,\ldots,x_n, t\rangle, \langle y_1,\ldots,y_n,u\rangle) = \sqrt{(t-u)^2-(x_1-y_1)^2-(x_2-y_2)^2\cdots-(x_n-y_n)^2}$$

Here $d(x,y)=d(y,x)$ and distances cannot be negative, but they can be null or purely imaginary! (For mathematical sanity, one usually considers the square of this distance function, such as not to be troubled with the multi-valuedness of the square root, though).

Minkowski space is the basic fabric of relativity. Indeed the fundamental postulate of the Special Theory of Relativity could be phrased as:

The stage on which physics plays out can be given the structure of $3+1$-dimensional Minkowski space, such that all fundamental laws of nature are preserved by every Minkowski isometry. (Or at least by every Minkowski isometry that can be "smoothly" turned into the identity).

(And, by the way, light rays connect points whose mutual Minkowski distance is 0).

Beware, however, that the Minkowski distance is not used to define the topology of Minkowski space. One uses the ordinary Euclidean topology on $\mathbb R^{n+1}$.

Minkowski isometries are better known as Lorentz transformations, though often that name is only used for isometries that fix the point $\langle 0,\ldots,0,0\rangle$.

Lorentzian manifolds generalize Minkowski space to curved spacetimes for General Relativity.

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