On spacetime geometry above the electroweak symmetry breaking scale

Abstract: While it is generally agreed that the nature of spacetime must be drastically different at the Planck scale, it has been a common practice to assume that spacetime is endowed with a full pseudo-Riemannian geometry regardless of the physical fields present or the length scale at which the geometry is probed, and to adopt this assumption in theories of cosmology, particle physics, quantum gravity, etc. Following Einstein's view that the mathematical description of spacetime ought to be physically motivated, we initiate a discussion on the validity of this assumption, and propose that the full pseudo-Riemannian geometry of spacetime could emerge as late as the time of electroweak phase transition when spacetime acquires the projective structure necessary to describe the motions of massive particles.