# Is projective geometry important for general relativity?

Today someone claimed that projective geometry is one of the building blocks for general relativity. I really do not know much about projective geometry and I only had one lecture about GR, so I fail to see a connection between the two subjects. Sure, differential geometry is important, but is there a connection to projective geometry?

To which extend and in which sense is the above statement correct?

• Mar 24, 2018 at 11:20

Importance is in the eye of the beholder, but e.g. Penrose famously reformulated SR using projective twistor space $P_3(\mathbb{C})$. This can be generalized to GR. Perhaps OP's someone is referring to this?

Projective geometry also appears in many modern physics theories, such as, e.g., twistor formulation of ${\cal N}=4$ SYM, or Calabi-Yau compactifications in ST, etc.

• Do you have a link to a resource on the generalisation of Penrose's approach to GR? Jun 9, 2017 at 10:02
• My "someone" has no real background in GR or physics but is more like a layman who just learned about projective geometry. I assume that he has just heard in a lecture that there is a connection between GR and projective geometry and this triggered the statement. Jun 9, 2017 at 11:54

Light cones are projective spaces. The null rays satisfy the condition of being a set of rays modulo scale. This is because then have no length! Light cones then form a projective Lorentz group, sometimes denoted by $PSL(2,~\mathbb Z)$. In general relativity sets of null rays form similar structures, where an event horizon is a congruence of null rays.

• Take a light cone in 2d Minkowski space. It's just the crossing of two straight lines. How is that a projective space? What does "Light cones then form a projective Lorentz group" mean - what is the group operation on "light cones"? Jun 14, 2017 at 17:15
• See Ward and Wells "Twistor Geometry and Field Theory" pages 50-52. That light cones are projective rays is discussed in a number of sources. Jun 14, 2017 at 19:07
• The relevant pages in my edition of Ward & Wells do not talk about projective spaces at all, they talk about conformal equivalences and conformal compactification, and a conformal space is something entirely different from a projective space. Also, please specify how "light cones" can form a group, let alone a "projective Lorentz group", given that there is no obvious group operation on them. Jun 14, 2017 at 23:27
• Strange. my edition not only talks about light cones there are picture of them and on 52 the connection to projective geometry. Jun 15, 2017 at 2:21
• @ACuriousMind so can I say that light cones are projective spaces? Mar 24, 2018 at 4:38