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Christoffel connection did not enter gravity as an axiom of minimal lengthfor the free fall of particles (where anyway length action is not defined formassless particles), nor out of economy, but from the weak equivalenceprinciple (gravitational force is equivalent to acceleration according toEinstein) together with the identification of the local inertial frame with thelocal Lorentz one. This identification implies that the orbits of all particlesare given by the geodesics of the Christoffel connection. Here, we show that inthe presence of only massless particles (absence of massive particles) theabove identification is inconsistent and does not lead to any connection. Theproof is based on the existence of projectively equivalent connections and theabsence of proper time for null particles. If a connection derived by somekinematical principles for the particles is to be applied in the world, it isbetter these principles to be valid in all relevant spacetime rather thandifferent principles to give different connections in different spacetimeregions. Therefore, our result stated above may imply a conceptualinsufficiency of the use of the Christoffel connection in the early universewhere only massless particles are expected to be present (whenever at leastsome notions, like orbits, are meaningful), and thus of the total use of thisconnection. If in the early universe the notion of a massive particle, whichappears latter in time, cannot be used, in an analogous way in a causallydisconnected high-energy region (maybe deep interior of astrophysical objectsor black holes) the same conclusions could be extracted if only masslessparticles are present.

Is the use of Christoffel connection in gravity theories conceptually correct?