  The mechanical response of naturally abundant amorphous solids such as gels,
jammed grains, and biological tissues are not described by the conventional
paradigm of broken symmetry that defines crystalline elasticity. In contrast,
the response of such athermal solids are governed by local conditions of
mechanical equilibrium, i.e., force and torque balance of its constituents.
Here we show that these constraints have the mathematical structure of a
generalized electromagnetism, where the electrostatic limit successfully
captures the anisotropic elasticity of amorphous solids. The emergence of
elasticity from local mechanical constraints offers a new paradigm for systems
with no broken symmetry, analogous to emergent gauge theories of quantum spin
liquids. Specifically, our $U(1)$ rank-2 symmetric tensor gauge theory of
elasticity translates to the electromagnetism of fractonic phases of matter
with the stress mapped to electric displacement and forces to vector charges.
We corroborate our theoretical results with numerical simulations of soft
frictionless disks in both two and three dimensions, and experiments on
frictional disks in two dimensions. We also present experimental evidence
indicating that force chains in granular media are sub-dimensional excitations
of amorphous elasticity similar to fractons.
