# Lorentz transformations in non-Euclidean geometries

I have an assessment/investigation coming up in my math class and I plan to investigate Lorentz transformations in geometries other than Euclidean such as spherical or hyperbolic. For spherical/polar, my preexisting knowledge tells me that I can convert coordinates in spacetime diagrams from cartesian to polar and derive the equations for Lorentz transformations in that case.

How feasible do you guys think this topic is and what is your general opinion about it? I'm a high school senior.

## 1 Answer

Since Lorentz boosts mix time and space, to study non-euclidean generalization of Lorentz symmetries you should consider spacetimes and not just spaces. And it turns out that there are two special space-times that have natural generalizations of Lorentz symmetries: de Sitter spacetime and anti-de Sitter spacetime for two possible signs of a constant $\Lambda$ parametrizing curvature, so I would suggest the amended topic of the investigation to be isometries of (anti-)de Sitter spacetimes as generalizations of inhomogeneous Lorentz symmetries. If OP is familiar with matrices and is reasonably proficient with hyperbolic functions the investigation should be feasible for a motivated high school student. This should be especially simple for the de Sitter spacetime, since its isometries is a group of homogeneous Lorentz transformations of 5D spacetime (one time dimension and 4 space dimensions).

As a motivation/historical context for this potential project I would recommend reading section 4 (and maybe also section 3) of F. Dyson's talk and this Wikipedia page.

• Thank you for the response. No, I'm not quite familiar with matrices and hyperbolic functions yet but I'm willing to learn it if needed. Thank you again. – Baalateja Kataru Aug 12 '18 at 18:44
• @BaalatejaKataru: As an introductory exercise try figuring out how (homogeneous) Lorentz transformations in 2+1 spacetime (just two spatial coordinates) act on hyperboloid $-T^2+X^2+Y^2=\ell^2$. This hyperboloid is the canonical embedding of ${\rm dS}_2$ spacetime and is a Lorentzian version of a sphere $X^2+Y^2+Z^2=\ell^2$. – A.V.S. Aug 12 '18 at 19:30
• This answer may jump past basic points that the OP is not aware of in order to make more esoteric points. The basic thing to understand is that in a semi-Riemannian geometry, things like Lorentz boosts and Lorentz symmetry are local things, not global things. – Ben Crowell Aug 12 '18 at 23:37
• @BenCrowell: answer may jump past basic points that is the purpose of an investigation: it's for OP to discover them. things like Lorentz boosts … are local I have to disagree here, this in not GR, the beauty of dS & AdS (and Minkowski space) is that their isometries are global. – A.V.S. Aug 13 '18 at 4:30