3
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Do you have a link to a resource on the generalisation of Penrose's approach to GR? $\endgroup$ – JamalS Jun 9 '17 at 10:02
  • $\begingroup$ 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. $\endgroup$ – Merlin1896 Jun 9 '17 at 11:54
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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"? $\endgroup$ – ACuriousMind Jun 14 '17 at 17:15
  • 1
    $\begingroup$ 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. $\endgroup$ – Lawrence B. Crowell Jun 14 '17 at 19:07
  • $\begingroup$ 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. $\endgroup$ – ACuriousMind Jun 14 '17 at 23:27
  • 1
    $\begingroup$ Strange. my edition not only talks about light cones there are picture of them and on 52 the connection to projective geometry. $\endgroup$ – Lawrence B. Crowell Jun 15 '17 at 2:21
  • $\begingroup$ @ACuriousMind so can I say that light cones are projective spaces? $\endgroup$ – Ooker Mar 24 '18 at 4:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.