"Regular polytopes" in Minkowski spacetime

Question

Is there an analogue of regular "polytopes" (hyperbolic honeycombs?) in the 4D Minkowski spacetime of special relativity, just as there are six regular polytopes in Euclidean 4D space?

If so, what is their classification?

Answer 1

I don't think that's possible.

First of all one have of course know how you generalize some concepts. The concepts of flatness must be the same, even if you defines flats in terms of distance you will end up with a linear equation anyway. This means that convex polytopes are the same as for euclidean metric (since convexity is defined in terms of line segments between points inside the polytope).

In addition we must define the concept of angles and that could be defined by the bilinear form that defines the metric $d(x,y)^2 = \langle x-y, x-y\rangle$ as if it were a inner product.

Now we can see that regular polygons only can exist in planes that has only space like lines (or time like if you have more than one time dimension). This is because if for example there's both time and space like lines we can find a coordinate system for the space where $\langle u, v\rangle = u_xv_x-u_yv_y$ and if we consider the vectors for the edges of $u_n = r_n(\cos\varphi_n, \sin\varphi_n)$, $r_n>0$ we have that without loss of generality:

$$\langle u_n, u_n \rangle=r_n^2 \cos^2\varphi_n - r_n^2\sin^2\varphi_n = r_n^2\cos(2\varphi_n) = 1$$ $$\langle u_n, u_{n+1}\rangle = r_nr_{n+1}\cos(\varphi_n + \varphi_{n+1}) = C$$

in addition due to convexity $\varphi_n$ must be strictly monotone and $r_n$ and $\varphi_n$ must repeat the same values after one turn. In addition if the polygon is to be closed it must have $\varphi$ both in the intervals $(-\pi/4,\pi/4)$ and $(3\pi/4, 5\pi/4)$. Now that means we have one in the first interval followed by one in the second and we can assume these are $\varphi_0$ and $\varphi_1$, this makes $C<0$ and $\varphi_n$ must alter between the intervals and since this makes $\varphi_3$ is in the second after more than one turn it must have repeated and $\varphi_3=\varphi_1$, that is we have only two edges.

The case where $\cos(2\varphi_n)=0$ has to be handled separately. There's also the case where the plane contains light like lines.

For regular polyhedra we can in a similar way see that it's required that they are in a hyperplanes with only space like lines. For example if the hyperplane contains perpendicular spacelike lines and timelike line (ie we can form a coordinate system with two space axes and one time axes). Since it's faces will be in space like planes their normal has to be time like, if we place the coordinate system around the center (in some sense) of the polygon we can see that it has to have faces with normals to the positive and negative time axis and two of these has to be adjecent which means that all adjecent faces must have time like normals pointing in opposite directions. This rules out more than two faces.

Same reasoning will rule out regular polychora entirely since the space contains both time and space like lines.