  The rich structure of solid state physics provides us with Dirac materials
the effective theory of which enjoys the Lorentz symmetry. In non-symmorphic
lattices, the Lorentz symmetry will be deformed in a way that the null
energy-momentum vectors will correspond to on-shell condition for tilted Dirac
cone dispersion. In this sense, tilted Dirac/Weyl materials can be viewed as
solid state systems where the effective spacetime is non-Minkowski. In this
work, we show that the polarization tensor for tilted Dirac cone systems
acquires a covariant from only when the spacetime is considered to be an
appropriate deformation of the Minkowski spacetime. As a unique consequence of
the deformation of the geometry of the spacetime felt by the electrons in
tilted Dirac cone materials, the Coulomb density-density interactions will
generate corrections in both longitudinal and transverse channels. Therefore
the transverse photons also participate in mediating the Coulomb forces,
implying emergent Amperean forces associated with the tilt of the spacetime.
