Space with complex distance

I am interested in a mathematical space with specific properties, but I am not sure if such a space can be consistently defined. I would appreciate any guidance or ideas. If this space is known, what is its name? If this space cannot be defined, then why? If it can be defined, then what would its mathematical definition look like? Finally, if it can exist only in a certain number of dimensions, then what is this number? (Mostly interested in 2 to 4 dimensions).

Target properties:

1. A distance from any point to the chosen unique center of coordinates is a real number (radius-vector).
2. An infinitesimal distance in any direction perpendicular to the radius vector is an imaginary number. (Perhaps better stated this way: the distance between the ends of two radius-vectors of the same length separated by an infinitesimal angle is an imaginary number.)

Example: the circumference of a circle with the radius $R$ and the center in the center of coordinates is $2\pi\cdot R \cdot i$ where $i$ is the imaginary unit.

1. Only one imaginary unit regardless of the number of dimensions (e.g. no quaternions).

EDIT: Per the comments below, the following bullet is not well defined or even necesary. Therefore it is here only as a visual illustration of the expected symmetry.

1. Rotational symmetry in any direction, but only around the center of coordinates.

I need your help to understand if this space is mathematically possible or not. I would appreciate any answers, comments, suggestions, or requests to clarify the question.

EDIT: Clarifications/updates based on responses. The bullets #2 and #3 above probably should be combined this way (with no imaginary numbers):

1. The space must be locally asymptotic to a hyperbolic metric space with the following metric:

$ds^2 = ds_1^2 -ds_2^2$

where $ds_1$ is the radial coordinate and $ds_2$ is the rotational Euclidian distance (e.g. $ds_2^2 = dx^2 + dy^2 + dz^2$ where $x$, $y$, and $z$ are rotational coordinates). Local (infinitesimal) geodesics can be defined in the usual way by the vanishing partial second order derivatives along the geodesic.

Global geodesics should be segments of logarithmic spirals (including degenerate by either coordinate, such as radius segments or circular segments around the origin). Something like this (see the @pregunton comment below):

$r = ae^{b\theta}$

Geodesics may not be unique, but may depend on the number of revolutions around the origin. This is OK and expected.

• how would the triangular inequality work ? – mercio Sep 24 '17 at 7:18
• @mercio: Well, I guess the same way as on the complex plane (using absolute values), unless you mean something else. I don't think it would be a problem locally (over a relatively small area), would it? While I don't see it being a problem globally either, it is not really required to work globally (unless it is mandatory in general). Am I missing your point? Do you have a particular case in mind? This space must be metric locally, but not necessarily globally (such as over areas including the center of coordinates or large angles). If this makes any sense :) – safesphere Sep 24 '17 at 7:35
• One thing that I think fulfils your requirements is having the distance between two points in the plane be the complex number whose modulus is the Euclidean length of the unique segment of (possibly degenerate) logarithmic spiral centered at the origin and passing through the two points, and whose argument is the characteristic angle of that logarithmic spiral (i.e., $\arctan(1/b)$ if the spiral's equation is $r=ae^{b\theta}$ in polar coordinates). – pregunton Sep 24 '17 at 7:52
• In this way, the distance between two points lying on a circle centered at the origin is the length of the arc of circle (degenerate spiral) joining them multiplied by $i$, and the distance between two points lying on a ray from the origin is the length of the segment connecting them. The distance is evidently invariant under rotations around the origin. You can easily extend this definition to more than two dimensions by taking the unique plane formed by the two points and the origin (the plane is not unique if the points are aligned, but then choosing any plane passing through them works). – pregunton Sep 24 '17 at 7:52
• @pregunton: This is interesting, because I already had this spiral defined there for a different reason. The fact that this spiral appears again shows that it relates to the properties of this space. I need to sleep on this, but it looks promising. Meanwhile, would there be any mathematical inconsistency or concern regarding this space? – safesphere Sep 24 '17 at 8:24

The update makes it much clearer what is wanted for the metric.

Let $d\Theta^2$ denote the spherical metric (induced by the Euclidean) on $S^{n-1}=\{ x\in {\Bbb R}^n: \|x\|=1\}$ and write $r=\sqrt{x_1^2+\cdots + x_n^2}$. The standard Euclidean metric on ${\Bbb R}^n$ may then be written as follows: $\sum_i dx_i^2 = dr^2 + r^2 d\Theta^2$.

The central idea in the current post is to change the sign on the angular part, i.e. we consider the punctured Euclidean space $X={\Bbb R}^n\setminus\{0\}$ equiped with the following $(1,n-1)$ pseudo-Riemannian metric: $$g= dr^2 - r^2 d\Theta^2 = 2 dr^2 - \sum_{i=1}^n dx_i^2$$

We wish to describe geodesics in $(X,g)$. In relativistic terminology, a tangent vector with $g(v,v)>0$ is time-like, $g(v,v)<0$ is space-like, while $g(v,v)=0$ corresponds to a ligth-cone vector.

It is of interest to note that the metric is invariant under the orthogonal group which implies that there is angular momentum conservation: A geodesic starting at some given position in space and in a given direction will always stay in the span of those two directions, i.e. it suffices to restrict our attention to those two dimensions. So let us write: $$g= dr^2 - r^2 d\phi^2$$ with $(r,\phi)$ being standard polar coordinates in the plane.

Geodesics in normal Riemannian geometry are paths between points that are extremal for the length $\int \sqrt{g(\dot{x},\dot{x})} dt$. Normalizing to constant speed it is equivalent to be extremal for the action functional: $$S = \int g(\dot{x},\dot{x}) dt = \int {\cal L} (r,\phi,\dot{r},\dot{\phi}) dt,$$ with the Lagrangian ${\cal L} = \dot{r}^2 - r^2 \dot{\phi}^2$. So we take extremality of this action to define geodesics in the present context. An extremal path verifies Lagrange's equations: $$2 \ddot{r} = \frac{d}{dt} \frac{\partial L}{\partial \dot{r}} = \frac{\partial L}{\partial r} = - 2 r \dot{\phi}^2 \;\; \mbox{and} \; \; \frac{d}{dt} \left( r^2 \dot{\phi} \right) = \frac{d}{dt} \frac{\partial L}{\partial \dot{r}} = \frac{\partial L}{\partial r} = 0.$$ The last implies the above-mentioned angular conservation: $r^2 \dot{\phi}=A$ for some constant $A$. Similarly, there is conservation of energy (Hamiltonian): $$E = \dot{r}\frac{\partial L}{\partial \dot{r}} + \dot{\phi} \frac{\partial L}{\partial \dot{\phi}} - L = 2 L - L = L$$ So $\dot{r}^2 - r^2 \dot{\phi}^2 = E$ for some constant $E$.

First case: If $\dot{\phi}=0$ at some instant of time then $A=0$ and $\phi$ is a constant of motion. We may solve to get: $\dot{r}=\pm\sqrt{E}$ which is just a linear motion in time, $r(t) = \pm\sqrt{E}(t-t_0)$ (the geodesic ceases to exist when $r(t_0)=0$).

Second case: When $\dot{\phi}\neq 0$, it has a constant sign (same as the sign of $A$). We may then by the implicit function theorem write $r=r(\phi)$ so that $\dot{r} = r'(\phi) \dot{\phi}$. Then $(r'^2-r^2) \dot{\phi}^2 = E$ and inserting the angular momentum conservation we deduce the following equation for the trajectories: $$r'^2 - r^2 = \frac{E}{A^2} r^4$$

Subcases:

a) $E=0$: We get $r'=\pm r$ or $r(\phi) = \exp (\pm (\phi-\phi_0))$.

b) $E<0$ (space-like trajectories): Write $r=1/u$ and solve the resulting ode for $u$. You end up with (modulo mistakes in my calculations): $$r(\phi) = \frac{A/\sqrt{-E}}{\cosh(\phi-\phi_0)}$$

c) $E>0$ (time-like trajectories): $$r(\phi) = \frac{A/\sqrt{E}}{\sinh(\phi-\phi_0)}$$

Symmetries: A part from the rotational symmetry I don't think that there are any other. The fact that $(r,\phi)$ is identified with $(r,\phi+ 2\pi)$ gives a topological constraint which prevents us from doing Lorentz-like transformations.

My answer to the original post:

A suggestion: In ${\Bbb R}^n$, write $r\cdot r'$ and $|r|$ for the Euclidean scalar product and length, respectively.

Define two infinitesimal (Riemannian) pseudo-metrics between infinitesimal close vectors $r$ and $r+dr$ (with $r\neq 0$):

$$ds_1 = \left| \frac{r}{|r|} \cdot dr \right| \; \; {\rm and} \; \; ds_2 = \left|dr - \frac{r}{|r|}\left( \frac{r}{|r|} \cdot dr\right) \right|$$ $ds_1$ measures the radial distance, $ds_2$ the rotational. If you want to represent them as complex numbers you may set $dz=ds_1+i ds_2$. Then, $|dz|$ (modulus of complex number) corresponds to the infinitesimal Euclidean distance.

You may then measure "complex" path lengths: If $r(t)$, $t\in [0,1]$ is a $C^1$ curve then $$L={\rm len}_{\Bbb C} (r,[0,1]) = \int_0^1 \left| \frac{r}{|r|} \cdot \dot{r} \right| dt + i \int_0^1 \left|\dot{r} - \frac{r}{|r|}\left( \frac{r}{|r|} \cdot \dot{r}\right) \right|dt$$ separates the usual length of the curve into the radial part (real) and the rotational part (imaginary). The modulus of $L$ is equivalent (though not necessarily equal) to the usual Euclidean length of the path.

Depending on your purpose with defining a complex distance, there might be an ambiguity as to the definition of the distance between two finite vectors $r_1$ and $r_2$ as you would have to specify what a geodesic is in this picture. A perhaps natural choice is to define a geodesic as a path that minimizes Euclidean distances. Then geodesics are straight line segments and the complex distance may be (partially) calculated as follows:

Let $r'$ be the point on the line segment $[r_1;r_2]$ closest to the origin (could be one of the end-points) and let $a\geq 0$ be the distance from the line through $r_1$, $r_2$ and the origin. For $0< u\leq v$ write $\Theta(u,v) = u\ln \frac{v+\sqrt{v^2-u^2}}{u}$. Then if $r'$ is not one of the end-points: $$d_{\Bbb C}(r_1,r_2) = \left(|r_1|+|r_2|-2|r'| \right) + i \left(\Theta(a,|r_1|+ \Theta(a,|r_2|)-2\Theta(a,|r'|)\right)$$ while if $r'$ is one of the end-points you get the simpler: $$d_{\Bbb C}(r_1,r_2) = \left||r_2|-|r_1|\right| + i \; a \left| \ln \frac{|r_2|+\sqrt{|r_2|^2-a^2}}{|r_1|+\sqrt{|r_1|^2-a^2}}\right|$$ separating into how much you move radially and rotationally along the geodesic. In the limit $a\rightarrow 0$ the imaginary part vanishes as wanted since $r_1$ and $r_2$ are proportional in that case.

• Hi Hans Henrik, thank you so much for working on my problem! Please don't be discouraged by my lack of response. I understand most of the math involved, but I'm not a mathematician, so understanding your solution will take me a bit of time :) To answer your question on the purpose of the complex distance, this space must be locally asymptotic with the metric space of the hyperbolic metric $ds_1^2 - ds_2^2$ where $ds_1$ is asymptotic to the radial coordinate and $ds_2$ is asymptotic to the angular linear coordinate. I'm a bit over my head here, please let me know if this makes no sense :) – safesphere Sep 25 '17 at 18:52
• Accordingly, local (asymptotic or infinitely small, whichever is the appropriate term) geodesics perhaps should be defined by the vanishing partial second derivatives along the geodesic. Globally geodesic are probably logarithmic spirals (including degenerate by either coordinate). This is what I thought and also mentioned by pregunton in the comments, but please do not hesitate to correct, if this seems wrong. This way a geodesic may not be unique, but may depend on the number of revolutions around the origin. This is OK and expected, unless it creates a logical inconsistency. – safesphere Sep 25 '17 at 19:10
• Clarification: By "angular linear" I meant "rotational". Please excuse my lack of familiarity with the conventional terminology :) – safesphere Sep 25 '17 at 19:16
• Ok, at least that gives more sense to the question. I don't have the time right now but will try to get back to this later on. You should perhaps write some more details regarding this in your question. It looks like seeking geodesics (or rather stationary paths) for the integral of ${\rm Re}\; dz^2$. In fact the representation as complex numbers is irrelevant. – H. H. Rugh Sep 25 '17 at 19:39
• Perfect. Thanks so much! I have updated the question as requested. – safesphere Sep 26 '17 at 8:17