Does the metric of the manifold provide any way of computing the length of one part of a null geodesic given the length of another portion of the same geodesic. If not, how does the photon do so?
Is this a question for QM and GR unification? I can’t figure out how to ask the question without QM terms.
Two photons traversing nearly the same null geodesic will agree that two portions are the same length. Alternatively sameness is a property of the manifold.
One answer might be that the metric tensor provides the answer in the form of a family of non-null geodesics that approach the null geodesic. The arc length along each of the family is multiplied by a different scalar in such a way that the limit on the null geodesic provides a distance along the null geodesic. This idea suggests the idea that a photon goes just a bit slower than the speed of light!
Γijk = ½ gin(∂gkn/∂xj + ∂gjn/∂xk − ∂gjk/∂xn)
Recall the differential equation for a geodesic given an affine connection:
d2xi/dλ2 = −Γijk dxj/dλ dxk/dλ
“λ” is known as the “affine parameter” with which the photon keeps its phase. This equation works fine along a null geodesic! Indeed an affine connection knows nothing of nullity.