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In a pseudometric space (a metric space with signature $(1,1)$, for example), the "distance" between two points can be negative. That happens (in this example) when $d^2 = x^2 - y^2$ is negative because $y > x$.

But people don't like when I talk about "negative length" because apparently, the words "length" and "distance" are defined as always being positive.

In the case of 4-dimensional spacetime (which is Riemannian), the word to use is "interval." But what's the word to use for negative distance in the general case of any pseudometric space?

Dan Uznanski
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Duce ex Machina
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  • This is not an answer because I don't know if there is a standard terminology, however I think that one could feel free to use the word "interval" in that context, accompanied with a brief explanation of its origins in spacetime physics. – Lee Mosher Dec 17 '17 at 15:01
  • Thanx, Lee, but when I explain stuff to non-Jedi, I can't use words like "interval" because it confuses them. I usually say "4D distance," which is true, but when I refer to its pseudometric nature, I have to distinguish between time distance and space distance. I guess there's no word for "negative distance," even though there shoukd be, because that's what time is. -- Thanx agin, Luxine – Duce ex Machina Aug 21 '19 at 06:13

2 Answers2

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In your example $d^2$ is unfortunate common notation (since d doesn't exist as a real number when $d^2$ is negative ) for what you should call a quadratic form which could be used for 1 dimensional space time where x (say in feet) is the space coordinate and y (say in seconds)is the time coordinate . It is not a pseudometric in the (x,y) plane .the metric used is the usual one D((x,y),(x',y')) = $((x-x')^2+(y-y')^2)^{1\over2}$ which is the 2 dimensional euclidean metric .

In general a pseudometric D on a set M satisfies D(a,b)=0 ,D(a,b)$\ge$0 ,symmetry and the triangle inequality where a,b are in M ;but leaves out that D(a,b)=o implies a=b .

user439545
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  • Thanx, Stuart! But consider a Euclidean metric like the Pythagorean theorem. If you increase x², you must DECREASE y² by the same amount if you want to keep the hypotenuse the same length. Now consider the (pseudometric) spacetime interval. If you increase the space distance (x,y,z) and keep d the same, you must INCREASE the time distance. == Elapsed time (often called imaginary distance) behaves like negative real distance. == My question is, is their another name, because "distance" is defined to be positive. Also, see next comment (if I can make two in a row). – Duce ex Machina Aug 21 '19 at 05:56
  • Also, in your example, you talk about the x,t plane (which is complex), but suddenly switch to talking about the x,y (real) plane. Pls dont confuse the vertical t axis with a ct² axis, which is real. – Duce ex Machina Aug 21 '19 at 06:01
  • The definition of pseudometric distance (the triangle inequality in the complex plane) is "rigged" to use absolute values of z1 and z2. See http://mathworld.wolfram.com/TriangleInequality.html That's okay, but then there has to be a be a definition for a new term to describe the distance when z1 and/or z2 are negative and not "absoluted." – Duce ex Machina Aug 21 '19 at 06:20
  • Example: It takes light one second to reach the moon. The interval between a point on Earth and a point on the moon at the same time, is 250,000 miles. But the 4D distance (the interval) between the earth now and the moon one second from now is zero, because the space distance equals the time distance, and both events are on the null cone. And the interval between earth now and the moon next year is imaginary. That's what I mean by "negative distance" because time is treated as a negative real number because, in the equation, ct² it is subtracted. – Duce ex Machina Aug 21 '19 at 06:39
  • i don't know another name for elapsed time ,sorry . – user439545 Aug 25 '19 at 01:08
  • I'm not asking for another name for elapsed time, I'm asking about the name of elapsed time times c -- that is, expressed as a negative spatial distance (because the term -ct² is ALWAYS negative. – Duce ex Machina Oct 31 '20 at 13:27
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[...] the "distance" between two points can be negative. That happens (in this example) when $d^2 = x^2 − y^2$ is negative because $y \gt x$.

If $d$ is supposed to denote the distance value under consideration then the example you gave is not applicable because: the square root of a negative real number, i.e. for all cases $x^2 - y^2 \lt 0$ is not a negative real number but of course imaginary.

(And whether there may be objections to calling and considering values of imaginary or of complex numbers as values of "distance" or of "length" is a separate issue.)

An applicable example, somewhat based on the above, is instead:

$$d : \mathbb R^2 \rightarrow \mathbb R, \qquad d \mapsto \text{Sgn}[ \, x^2 - y^2 \, ] \, \sqrt{ \text{Sgn}[ \, x^2 - y^2 \, ] \, (x^2 - y^2) },$$

where $\text{Sgn}$ denotes the sign function.

But people don't like when I talk about "negative length" because apparently, the words "length" and "distance" are defined as always being positive.

Clearly, the (applicable) example function $d$ above does not satisfy all defining properties of a (plain) metric space.

Still, at least it shares one property of (plain) metric spaces, namely "Indiscernability of the Identical": $\forall x : d[ \, x, x \, ] = 0$.

(This illustrates by the way that even in the plain notion of distance its values are not always defined as strictly positive.)

However, obviously there are generalizations of the "plain" notion of distance.

In a pseudometric space

Example function $d$ defined above is (obviously) not an instance of pseudometric distance either. Therefore the relevant "larger" questions would be, to which type of generalization belongs the above example function, or how to call such a generalization of "plain" metric space.

For lack of knowing of any suitable precedence (on this website, in Wikipedia, or elsewhere), I have already proposed here to call the particular type of generalization examplified by the function $d$ above as quasihypometric distance.

In the case of 4-dimensional spacetime (which is Riemannian),

... actually, pseudo-Riemannian and even more specifically Lorentzian ...

the word to use is "interval."

Yes values of spacetime interval, $s^2$ can have negative real values (as well as non-negative values). But (alledgedly), the notion of spacetime interval is not applicable to Lorentzian manifolds in general, but only to flat manifolds (such as Minkowski space). A more general notion would be Synge's world function, $\sigma$.

But what's the word to use for negative distance in the general case

A notion for referring to the separation between timelike (or lightlike) separated events in general is Lorentzian distance.

(Apparently, terminology for naming the quantities $\text{Sgn}[ \, \sigma \, ] \, \sqrt{ \text{Sgn}[ \, \sigma \, ] \, \sigma }$ still needs to be proposed.)

Martin Sleziak
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user12262
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