So what happens if we create metrics which can take negative values?

Question

I'm studying metric spaces at the moment, and at the level I'm doing at least, a metric always positive by definition.

But out of curiosity, what happens if we replace the first axiom $(M1)$ by $|d(a,b)| \geq 0$ where $d(a,b) = 0$ iff $a=b$?

I mean, we created the square roots of negative numbers, so someone must have tested this right?

Is there an area of mathematics that deals with that?

If it's always negative, then you really have not done anything. The more delicate question is what happens if it can take on both signs. — Ian, Sep 11 '15 at 18:22
I suppose you mean a metric that is sometimes positive, sometimes negative? — Cataline, Sep 11 '15 at 18:23
Well I would guess that $d$ taking both signs would contradict the triangle inequality unless you were really careful about how you defined $d$, so you'd lose a lot of structure. — Cameron Williams, Sep 11 '15 at 18:23
And what would $d(a,b) = -1$ even mean? You can come up with all sorts of crazy definitions, but the definition has to be a definition of something interesting, otherwise it's pointless. — Najib Idrissi, Sep 11 '15 at 18:24
This is not a bad question at all. My understanding is that the Minkowski metric may be negative and this is useful for modeling special relativity. Maybe someone who knows more about this can comment. — Jair Taylor, Sep 11 '15 at 18:34
@AlecTeal It’s no longer a metric per se, but a quadratic form that can take on both positive and negative values figures rather prominently in the geometry of special relativity. — amd, Sep 11 '15 at 18:35
@NajibIdrissi, if you think of $d(a,b)$ as the "cost" of going from $a$ to $b$, you think of a negative metric value as being paid for making the trip. — Barry Cipra, Sep 11 '15 at 18:39
Well, it's clear that when someone says "a metric that can take on negative values" they refer to a modification of the definition of metric. Just as a "manifold with boundary" is not a manifold, etc. — Jair Taylor, Sep 11 '15 at 18:41
@CameronWilliams I wouldn’t say that you lose structure, but that you get a different structure. E.g., in Minkowski space you have instead a “reverse triangle inequality” in which the length of one side of a triangle is greater than the sum of the lengths of the other sides. — amd, Sep 11 '15 at 18:45
@amd I think you only get a reverse triangle inequality if the sign is always negative. If it is allowed to take both values, I don't think you can say anything in general. — Cameron Williams, Sep 11 '15 at 18:59
@CameronWilliams Fair point. One mostly works with timelike intervals in Special Relativity, i.e., negative values of the metric. — amd, Sep 15 '15 at 18:09

score 9 · Answer 1 · answered Sep 11 '15 at 18:27

9

The nonnegativity of metrics already follows from the triangle inequality and the symmetry.

Observe:

$d(x, x) \le d(x, x) + d(x, x) \implies 0 \le d(x, x)$
$0 \le d(x, x) \le d(x, y) + d(y, x) = 2d(x, y) \implies 0 \le d(x, y)$

So if you want a function $d$ that also takes negative values, you have to omit the symmetry or the triangle inequality.

answered Sep 11 '15 at 18:27

Dominik

19,963

So I guess I was half right. You do contradict the triangle inequality.. but always if $d$ is allowed to take negative, not just sometimes. – Cameron Williams Sep 11 '15 at 18:29
I don't see how this answers the question. A negative metric isn't a metric. ALL of the stuff we define for metrics might break! This answer doesn't explore the structure of whatever the OP wants which is what the question probably wanted in spirit – Alec Teal Sep 11 '15 at 18:29
2

@AlecTeal I think this answers it perfectly. If you allow $d$ to take negative and positive values, then you must lose the triangle inequality. – Cameron Williams Sep 11 '15 at 18:30
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@AlecTeal OP asked about "replacing the first axiom with...", so it's at least a strong partial answer. If we want a whole new definition, we need more information to form an intuition about what we're trying to capture. – Sep 11 '15 at 18:30
Since you lose the triangle inequality, you can't really do anything with it. Half of analysis boils down to using the triangle inequality. – Cameron Williams Sep 11 '15 at 18:32
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@AlecTeal Of course a function that assumes negative values isn't a metric, but this is not the point. The question is what happens if we generalize the definition of a metric and also allow negative values. The answer is simple: Nothing happens, a symmetric function that satisfies the triangle inequality is nonnegative. If we want a notion of a "generalized metric" that actually assumes negative values, we either have to drop symmetry or the triangle inequality. – Dominik Sep 11 '15 at 18:35
So to answer this question all I had to do was say 'the triangle inequality fails' - slow day today guys? – Alec Teal Sep 11 '15 at 18:35
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@CameronWilliams (and Alec), I would say you lose either the triangle inequality or the symmetric property. (Personally, if I were going to look into what you could say in such a theory, I would drop the symmetric property, and allow the "cost" of going from $x$ to $y$ to be different from the corresponding "distance" from $y$ to $x$.) – Barry Cipra Sep 11 '15 at 19:14
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@AlecTeal And if the question was "what are the integer solutions of $x^n + y^n = z^n$ for $n \ge 3$" all you have to answer is "there aren't any". What's interesting is the proof. Besides this particular question wasn't very deep to begin with. – Najib Idrissi Sep 12 '15 at 07:22

score 3 · Answer 2 · edited Jun 08 '20 at 03:09

This is what I found at wikipedia....
https://en.m.wikipedia.org/wiki/Metric_(mathematics)

Consider taking a look at the section on Generalised Metrics.

The requirement that the metric take values in $[0,\infty)$ can even be relaxed to consider metrics with values in other directed sets. The reformulation of the axioms in this case leads to the construction of uniform spaces: topological spaces with an abstract structure enabling one to compare the local topologies of different points.

There is a notion of signed distance function too! Check it out here.. https://en.m.wikipedia.org/wiki/Signed_distance_function

+1 for the link for the Signed_distance_function page. This question couldn't have an accepted answer without a mention to it — Saul Berardo, May 18 '22 at 20:49

So what happens if we create metrics which can take negative values?

2 Answers2