Conformal compactification of the Minkowski space and its double cover

In the previous note Constructing the conformal infinity we have started with one-point compactification of the Euclidean space and ended up with what is called the conformally compactified Minkowski space. In this note we will start with the end of the previous note and study the emerging structure in some detail. Therefore, let V denote R⁶ with Cartesian coordinates x,v,w,

x= (x¹,x²,x³,x⁴) ∈ R⁴,

endowed with the quadratic form

q(x,v,w) = x²+v²-w²,

where

x²=(x¹)²+ (x²)²+ (x³)2- (x⁴)².

We define the “cone” C as the set of all (x,v,w) satisfying

q(x,v,w)=0, (x,v,w) ≠ 0.

So we have removed the origin x=v=w=0.

C inherits its topology from R⁶: open sets in C are intersections of open sets in R⁶ with C.

There are two equivalence relations in C that are of interest, we will denote then R and R+. They are defined as follows

(x,v,w) R (x',v',w') if and only if (x',v',w') = a(x,v,w), a∈R, a≠ 0

(x,v,w) R+ (x',v',w') if and only if (x',v',w') = a(x,v,w), a∈R, a>0

We take the quotients of C by these equivalence relations and denote them

M^c and M^2c respectively:

M^c = C/R,

M^2c = C/R+

We call M^c the compactified Minkowski space, and M^2c the double compactified Minkowski space.

We endow both sets with the quotient topology. Thus open sets in M^c are defined as those of which the preimages by the natural map from C to C/R are open in C. Similarly with M^2c .

We have a natural covering map from M^2c to M^c . Let us denote by [(x,v,w)] the equivalence class of (xv,w) with respect to R, and by

[(x,v,w)]₊ its equivalence class with respect to R+. Then [(x,v,w)] is the set

[(x,v,w)] ={a(x,v,w): a ≠ 0}

while

[(x,v,w)]₊ ={a(x,v,w): a > 0}

We can say that [(x,v,w)]₊ is a half-ray through (x,v,w), while [(x,v,w)] is the whole ray (excluding 0) through (x,v,w). Every half-ray is contained in a unique ray, and half-rays [(x,v,w)]₊ and half-rays [(x,v,w)]₊ and [-(x,v,w)]₊ are contained in the same ray. Therefore the natural map M^2c to M^c is 2:1.

The group O(4,2) is the group of linear transformations of R⁶ leaving invariant the quadratic form q. Its subgroup SO(4,2) consist of transformations given by 6x6 matrices of determinant 1. Since the transformations are linear, they transform equivalence classes of R and of R+ into equivalence classes, therefore they define transformations of M^c and M^2c.

We now show that the action of SO(4,2) on M^2c is transitive and that M^2c is (pathways) connected.  

Let [x,v,w]₊ and [x',v',w']₊ be two different points in M^2c. We have q(x,v,w)=0 and q(x',v',w')=0. Therefore

(x¹)²+ (x²)²+ (x³)² + v² = w²+(x⁴)²

and the expression above must be >0 since the origin is excluded. We can now replace (x,v,w) by the unique element in the same equivalence class for which the right hand side =1. We do the same with the second point. So we have

(x¹)²+ (x²)²+ (x³)² + v² = w²+(x⁴)²=1

(x'¹)²+ (x'²)²+ (x'³)² + v'² = w'²+(x'⁴)²=1

We can now continuously rotate (x¹, x², x³, v) to (x'¹, x'², x'³, v') using a rotation from SO(4) and, at the same time, continuously rotate (w,x⁴) to (w',x'⁴) by an SO(2) rotation. The two rotations together, written as a block diagonal (4+2)x(4+2) matrix,  give us a rotation in SO(4,2). The path is continuous in M^2c , because the topology in the quotient space is defined so that the quotient map is continuous. Therefore SO(4,2) acts transitively on M^2c and also M^2c is pathways connected. By projection (which is continuous) we get the same results for  M^c .

Assuming only one space dimension, therefore neglecting x² and  x³, here is what I got for a graphic representation of  M^2c , and the double conformal infinity (v=w):

Note: The numbers on the graphics are irrelevant.

Mathematica code used for the graphics:

torus2[a_, b_][fi_,psi_] := {(a + b Cos[fi]) Cos[psi], (a + b Cos[fi]) Sin[psi], b Sin[fi]}

mc = ParametricPlot3D[torus2[8, 3][fi, psi], {fi, 0, 2 Pi}, {psi, 0, 2 Pi}, Mesh -> None, PlotStyle -> Directive[Opacity[0.6], LightGray, Specularity[White, 10]]]

inf1 = ParametricPlot3D[torus2[8, 3][fi, fi], {fi, 0, 2 Pi}, PlotStyle -> {Black, Thick}]

inf2 = ParametricPlot3D[torus2[8, 3][fi, Pi-fi], {fi, 0, 2 Pi}, PlotStyle -> {Black, Thick}]

Show[{mc, inf1, inf2}]

Notice that M^2c is, topologically, S¹x S³ rather than S¹x S¹ as on the picture, where space is one- instead of 3-dimensional.

In the next post we will draw images of the Minkowski space on the torus, and we will see how time translations are represented there.

P.S.1. Something else to think about, that Laura brought my attention to:

 By the way a very good example of how to give a talk!