  The theory of mechanical response and stress transmission in disordered,
jammed solids poses several open questions of how non-periodic networks --
apparently indistinguishable from a snapshot of a fluid -- sustain shear. We
present a stress-only theory of emergent elasticity for a non-thermal amorphous
assembly of grains in a jammed solid, where each grain is subjected to
mechanical constraints of force and torque balance. These grain-level
constraints lead to the Gauss's law of an emergent $U(1)$ tensor
electromagnetism, which then accounts for the mechanical response of such
solids. This formulation of amorphous elasticity has several immediate
consequences. The mechanical response maps exactly to the static, dielectric
response of this tensorial electromagnetism with the polarizability of the
medium mapping to emergent elastic moduli. External forces act as vector
electric charges whereas the tensorial magnetic fields are sourced by momentum
density. The dynamics in the electric and magnetic sectors, naturally translate
into the dynamics of the rigid jammed network and ballistic particle motion
respectively. The theoretical predictions for both stress-stress correlations
and responses are borne out by the results of numerical simulations of
frictionless granular packings in the static limit of the theory in both 2D and
3D.
