  In this paper we examine cosmological weak lensing on non-linear scales and
show that there are Newtonian and relativistic contributions and that the
latter can also be extracted from standard Newtonian simulations. We use the
post-Friedmann formalism, a post-Newtonian type framework for cosmology, to
derive the full weak-lensing deflection angle valid on non-linear scales for
any metric theory of gravity. We show that the only contributing term that is
quadratic in the first order deflection is the expected Born correction and
lens-lens coupling term. We use this deflection angle to analyse the vector and
tensor contributions to the E- and B- mode cosmic shear power spectra. In our
approach, once the gravitational theory has been specified, the metric
components are related to the matter content in a well-defined manner.
Specifying General Relativity, we write down a complete set of equations for a
GR$+\Lambda$CDM universe for computing all of the possible lensing terms from
Newtonian N-body simulations. We illustrate this with the vector potential and
show that, in a GR$+\Lambda$CDM universe, its contribution to the E-mode is
negligible with respect to that of the conventional Newtonian scalar potential,
even on non-linear scales. Thus, under the standard assumption that Newtonian
N-body simulations give a good approximation of the matter dynamics, we show
that the standard ray tracing approach gives a good description for a
$\Lambda$CDM cosmology.
