Abstract
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.
Highlights
Christoffel connection is undoubtedly the dominant affine connection in gravitational physics
Christoffel connection did not enter gravity as an axiom of minimal length for the free fall of particles, nor out of economy, but from the weak equivalence principle together with the identification of the local inertial frame with the local Lorentz one
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, the same conclusions could be extracted if only massless particles are present
Summary
Christoffel connection ( known as Levi-Civita connection) is undoubtedly the dominant affine connection in gravitational physics. The Christoffel connection in this regime is not the result of physical arguments on particle kinematics, probably losing its overall reliability (in this same spirit, even the axiom of continuation to the purely massless case of the minimal length geodesics of the massive particles appeared later in time, is insufficient). The existence of the notion of orbit (either classical or semiclassical) seems still reasonable around the TeV scale since the full strong quantum gravity regime is normally expected at much higher energies Another case where the insufficiency of the Christoffel connection may occur is in spacetime regions of very high energy and temperature where only massless particles may exist, which regions are causally disconnected from the rest of spacetime where massive particles are present (deep interior of supermassive astrophysical objects or even black holes are not impossible to be such regions). The above thoughts could be relevant in the presence of extra dimensions with some fundamental mass scale
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