  Conventional theories of electromagnetic waves in a medium assume that the
energy propagating with the light pulse in the medium is entirely carried by
the field. Thus, the possibility that the optical force field of the light
pulse would drive forward an atomic mass density wave (MDW) and the related
kinetic and elastic energies is neglected. In this work, we present foundations
of a covariant theory of light propagation in a medium by considering a light
wave simultaneously with the dynamics of the medium atoms driven by optoelastic
forces between the induced dipoles and the electromagnetic field. We prove that
the transfer of mass as MDW associated with the light pulse, the photon mass
drag effect, gives an essential contribution to the total momentum of the light
pulse, which becomes equal to the Minkowski momentum. Thus, our theory also
resolves the centenary Abraham-Minkowski controversy of the momentum of light
in a nondispersive medium. We derive the photon mass drag effect using two
independent but complementary covariant models. In the mass-polariton (MP)
quasiparticle approach, we consider the light pulse as a coupled state between
the photon and matter, isolated from the rest of the medium. The momentum and
the transferred mass of MP follow unambiguously from the Lorentz invariance and
the fundamental conservation laws of nature. To enable the calculation of the
mass and momentum distribution of a light pulse, we have also generalized the
electrodynamics of continuous media to account for the optoelastic dynamics of
the medium. In this optoelastic continuum dynamics (OCD) approach, we obtain a
numerically accurate solution of the Newtonian continuum dynamics of the medium
when the light pulse is propagating in it. We finally discuss a possibility of
an optical waveguide setup for experimental measurement of the transferred mass
of the light pulse.
