Floating in the Einstein Universe

Posted on 2025/11/24 by arkajad

This post is the third chapter of the “Primer” saga, based on   “A primer on the (2 + 1) Einstein universe” – a paper is written by five mathematicians, T. Barbot, V. Charette, T. Drumm, M. Goldmann, K. Melnick, while they were visiting the Schrodinger Institute in Vienna and published in print in “Recent Developments in Pseudo-Riemannian Geometry”, (ESI Lectures in Mathematics and Physics), Dmitri V. Alekseevsky and Helga Baum eds.,  EMS – Publishing House, Zürich 2008, pp. 179-229. You can download it from arxiv here. The first two chapters: “Perplexed by AI” and “A primer on the Universe” had no math. Now it is time to get serious and to learn how to swim in the Einstein Universe. We will start with learning how to float freely on our backs.

The main arena of “A primer” is the space denoted by \text{Ein}^{n,1}, and we will describe this space now. It is where “geometrical objects” under study live. Then we will start discussing these objects and relate them to the concepts that we have already met in my previous posts, in particular for n=1. What we have done in our sandbox here is a particular case of a universal machine that embeds space-time with n-dimensional space and one-dimensional time. Much of this machine works also with more general, m-dimensional “time”, but the case of one-dimensional time is, in a sense, “special”. So, here, I will stay with this special case. The paper has “2+1” in the title. This is even a more special case, but a large part of the paper deals with more general n+1 objects. In my recent series of posts we were playing the toy case of 1+1, but we have also given some attention to the 3+1 case relevant to “adult physics”. Personally I think that all (n,m,r) cases are pertinent to physics (r is the number of zeros in the signature). But even that will be not enough. Quite probably we will find important uses for ternary, not just quadratic, forms, and complex instead of real structures and geometries. But here let us stay with the synthetic geometry as presented in the “primer”. The notion in the paper is sometimes rather original, so we will need a translation between the notation there and the one I was using so far.

Minkowski space E^{n,1}

We start with the Minkowski space. It is denoted E^{n,1}. It is defined as an affine space whose underlying vector space is \mathbb{R}^{n,1} – the space \mathbb{R}^{n+1} endowed with the quadratic form

    \[q(x) = (x^1)^2\cdots + (x^n)^2 -(x^{n+1})^2.\]

The only difference between E^{n,1} and \mathbb{R}^{n+1} is that in E^{n,1} there is no distinguished “origin”. Any point cab be selected as the origin, and then  E^{n,1} can be identified with  \mathbb{R}^{n,1}.

Möbius extension \mathbb{R}^{n+1,2}

The paper does not call it so, but that what it is. We start with \mathbb{R}^{n,1} and add two extra dimensions, one with signature +, and one with signature -.  Thus our quadratic form, written as a scalar product, becomes:

(v,v) = (v1)2 + …. + (vn)2 – (vn+1)2 + (vn+2)2 – (vn+3)2.

The null cone \mathfrak{N}^{n+1,2}.

The paper is using the symbol \mathfrak{N}^{n+1,2}. We have been using just N:

    \[\mathfrak{N}^{n+1,2}= \{v\in \mathbb{R}^{n+1,2}: (v,v)=0\}.\]

The Einstein universe E^{n,1} and \widehat{E^{n,1}}

These are the same that we have denoted PN and PN_+ in the case of n=1 that we have discussed here. We take the null cone, remove the origin, and identify proportional vectors with proportionality constant \lambda being non-zero for E^{n,1},  and \lambda>0  for \widehat{E^{n,1}}. The authors notice that \widehat{E^{n,1}} is a double covering for E^{n,1} ,  and that

    \[\widehat{E^{n,1}} \approx S^n \times S^1.\]

For n=1 we recover our torus S^1 \times S^1.

The term “Einstein Universe”, used by the authors, is not very fortunate. In Einstein Universe time is linear instead of circular. Yes, it is true, the authors consider also the universal covering space with \widetilde{E^{n,1}} \approx S^n \times \mathbb{R}^1, but it is not the main subject of the paper. Secondly, the true Einstein Universe comes with a metric – a solution of Einstein field equations for empty space and with a cosmological constant. But E^{n,1}, carries no natural metric tensor, only a conformal structure. It is true, that we can always endow E^{n,1} with a metric compatible with the conformal structure, but such choice depends on which point p we select as the “infinity point”. So, let us keep these comments in mind while learning how to float.

The objects

Here is the list of important objects in this synthetic geometry of E^{n,1}:

  • Photons
  • Lightcones
  • The Minkowski patch

We start with “photons”. 

The space of photons \text{Pho}^{n,1}

Here are is an illustration showing two “photons” in \widehat{E^{n,1}}  from our previous discussion in Perpendicular light:

These are two light rays forming the infinity p^\perp.. 

 In E^{n,1} these two light rays intersect in a different way, with just one common point.

For n>1 the infinity p^\perp consists of a whole family of “photons”, so a more general definition is needed. In the paper “photons” are defined as “projectivizations of totally isotropic 2-planes. But there is another, equivalent, definition.  Namely En,1 is automatically endowed with a conformal structure. Therefore the concept of null geodesics is well defined in E^{n,1}. Photons are just null geodesics. We will return to this subject in the future. The space of photons is denoted \text{Pho}^{n,1} in the paper.

Next come null cones.

Nullcones L(p)

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