This post is a continuation of Floating in the Einstein Universe.
Let us recall the relevant definition from the previous post.
![\[L(p) = \bigcup\{\phi\in\text{Pho}^{n,1}: p\in\phi \}\]](https://arkadiusz-jadczyk.eu/blog/wp-content/ql-cache/quicklatex.com-40b095430c619d3e351ce61de1fc7187_l3.png "Rendered by QuickLaTeX.com")
.
So 
is the union of all light rays through p. For
we have only two light rays, one “going to the right”, and one “going to the left”. In fact, we have an even more elegant description of , valid for any 
:
![\[L(p)=p^\perp.\]](https://arkadiusz-jadczyk.eu/blog/wp-content/ql-cache/quicklatex.com-5fb4daf16f76790a17c25a17b5b838b6_l3.png "Rendered by QuickLaTeX.com")
Light cones are important as they uniquely determine and are determined by the conformal structure that we will discuss in details later on. But first we need to address the next object: “Minkowski patch”. It is much like stereographic projection that maps the sphere minus the infinity point onto the plane, preserving angles. Here we remove 
, and map the rest onto the flat Minkowski space. In fact, in 
we map only half of the rest — for the oriented Grassmannian we have two Minkowski patches to cover the complement of p⟘. The “Primer” defines the two Minkowski patches by
![\[\text{Min}^+(\hat{p}):=\{\hat{q}\in\text{Ein}^{n,1}|<p,q>>0\,\forall p,q\in\mathbb{R}^{n+1,2}\text{ representing }\hat{p},\hat{q}\}.\]](https://arkadiusz-jadczyk.eu/blog/wp-content/ql-cache/quicklatex.com-56c8e16f72c858296a055e504f55ed09_l3.png "Rendered by QuickLaTeX.com")
![\[\text{Min}^-(\hat{p}):=\{\hat{q}\in\text{Ein}^{n,1}|<p,q> <0\,\forall p,q\in\mathbb{R}^{n+1,2}\text{ representing }\hat{p},\hat{q}\}.\]](https://arkadiusz-jadczyk.eu/blog/wp-content/ql-cache/quicklatex.com-32041674211d98e75352ed0123ab39e7_l3.png "Rendered by QuickLaTeX.com")
For these are the two regions, blue and red, in the figure below
Interesting that Mathematica graphics shows some penetration of red into blue. Here it is a computational artefact, but perhaps it reflects what happens in reality – if there are indeed two Minkowski patches in our universe.
The Minkowski patch, according to the definition in the paper is just an open set that has, a priori, no relation to the Minkowski space
Interesting that Mathematica graphics shows some penetration of red into blue. Here it is a computational artefact, but perhaps it reflects what happens in reality – if there are indeed two Minkowski patches in our universe.
The Minkowski patch, according to the definition in the paper is just an open set that has, a priori, no relation to the Minkowski space. We will discuss this relation in the next post
To be continued…
