The Null Geodesic

As a photon traverses a null geodesic it tracks its own phase. This is the way that two coherent photons know how to interfere if they should meet. This phase information is a bit like a distance along the null geodesic but the metric of the underlying manifold provides no information for such a distance. There is no objective energy or wavelength for a photon; indeed the energy of a photon is relative. What is absolute however is that one portion of the null geodesic is as many wavelengths as some other portion of the same geodesic.

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.

Hidden assumption:
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!

Answer

The answer is that a geodesic is defined by an affine connection alone where Γⁱ_jk are given but no metric tensor is known. The affine connection supports comparing vectors along a path and in particular along a geodesic. In a space-time manifold the metric tensor produces an affine connection which allows comparison of vectors along even a null geodesic.

Γⁱ_jk = ½ gⁱⁿ(∂g_kn/∂x^j + ∂g_jn/∂x^k − ∂g_jk/∂xⁿ)

Recall the differential equation for a geodesic given an affine connection:
d²xⁱ/dλ² = −Γⁱ_jk dx^j/dλ dx^k/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.

Ramifications

If you think of a geodesic as a curve and a curve as a set of points from the manifold, then the geodesic defined by x^j = f^j(λ) is the same geodesic as defined by the curve x^j = f^j(2λ); it is the same set of points. It seems that the difference is noticed by physics, however. Just now I don’t know the units for λ.