Hello Jose,bda said:By the way, this brings me to one of my strong points of divergence with Rob. Rob says that photons travel on null geodesics of [itex]X_4[/itex] but null geodesics need mixed signature spaces; as far as I can understand Rob's spaces have all plus signatures and so they cannot have null geodesics. Maybe Rob will like to clarify this point.
Excuse me for reacting so late but I only now returned from work.
As I see it, the "null geodesic" is defined as the path along which the tangent vector has norm 0 in a geometry with metric (-+++), hence the word "null". The norm 0 results from the [itex]ds^2=0[/itex] in the geodesic.
When this is translated to Euclidean relativity with metric (++++), [itex]ds^2[/itex] is not zero (it then equals [itex](cdt)^2[/itex] for the photon which >0) but the displacement in the [itex]\tau[/itex] dimension is zero. So strictly spoken one could not speak of a "null geodesic" any more; it is a timelike geodesic according to the original definition from Minkowski space-time. I have maintained the use of the familiar term in Euclidean relativity because it is generally associated with the path of massless particles and referring to that as "timelike" in Euclidean relativity would likely cause a lot of confusion.
It would probably be best however to introduce a whole new term in Euclidean relativity for the path of massless particles and I am inclined to suggest something like "3D geodesics" versus "4D geodesics" of mass carrying particles.