  Christoffel connection did not enter gravity as an axiom of minimal length
for the free fall of particles (where anyway length action is not defined for
massless particles), nor out of economy, but from the weak equivalence
principle (gravitational force is equivalent to acceleration according to
Einstein) together with the identification of the local inertial frame with the
local Lorentz one. This identification implies that the orbits of all particles
are given by the geodesics of the Christoffel connection. Here, we show that in
the presence of only massless particles (absence of massive particles) the
above identification is inconsistent and does not lead to any connection. The
proof is based on the existence of projectively equivalent connections and the
absence of proper time for null particles. If a connection derived by some
kinematical principles for the particles is to be applied in the world, it is
better these principles to be valid in all relevant spacetime rather than
different principles to give different connections in different spacetime
regions. Therefore, our result stated above may imply a conceptual
insufficiency of the use of the Christoffel connection in the early universe
where only massless particles are expected to be present (whenever at least
some notions, like orbits, are meaningful), and thus of the total use of this
connection. If in the early universe the notion of a massive particle, which
appears latter in time, cannot be used, in an analogous way in a causally
disconnected high-energy region (maybe deep interior of astrophysical objects
or black holes) the same conclusions could be extracted if only massless
particles are present.

General Physics;High Energy Physics - Theory