  Modified Newtonian dynamics (MOND) is an alternative theory of gravity that
aims to explain large-scale dynamics without recourse to any form of dark
matter. However the theory is incomplete, lacking a relativistic counterpart,
and so makes no definite predictions about gravitational lensing. The most
obvious form that MONDian lensing might take is that photons experience twice
the deflection of massive particles moving at the speed of light, as in general
relativity (GR). In such a theory there is no general thin-lens approximation
(although one can be made for spherically-symmetric deflectors), but the
three-dimensional acceleration of photons is in the same direction as the
relativistic acceleration would be. In regimes where the deflector can
reasonably be approximated as a single point-mass (specifically low-optical
depth microlensing and weak galaxy-galaxy lensing), this naive formulation is
consistent with observations. Forthcoming galaxy-galaxy lensing data and the
possibility of cosmological microlensing have the potential to distinguish
unambiguously between GR and MOND. Some tests can also be performed with
extended deflectors, for example by using surface brightness measurements of
lens galaxies to model quasar lenses, although the breakdown of the thin-lens
approximation allows an extra degree of freedom. Nonetheless, it seems unlikely
that simple ellipsoidal galaxies can explain both constraints. Further, the
low-density universe implied by MOND must be completely dominated by the
cosmological constant (to fit microwave background observations), and such
models are at odds with the low frequency of quasar lenses. These conflicts
might be resolved by a fully consistent relativistic extension to MOND; the
alternative is that MOND is not an accurate description of the universe.
