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:
- A distance from any point to the chosen unique center of coordinates is a real number (radius-vector).
- 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.
- 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.
- 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):
-
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.
Best Answer
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.