  We present the gauge-invariant formalism of cosmological weak lensing,
accounting for all the relativistic effects due to the scalar, vector, and
tensor perturbations at the linear order. While the light propagation is fully
described by the geodesic equation, the relation of the photon wavevector to
the physical quantities requires the specification of the frames, where they
are defined. By constructing the local tetrad bases at the observer and the
source positions, we clarify the relation of the weak lensing observables such
as the convergence, the shear, and the rotation to the physical size and shape
defined in the source rest-frame and the observed angle and redshift measured
in the observer rest-frame. Compared to the standard lensing formalism,
additional relativistic effects contribute to all the lensing observables. We
explicitly verify the gauge-invariance of the lensing observables and compare
our results to previous work. In particular, we demonstrate that even in the
presence of the vector and tensor perturbations, the physical rotation of the
lensing observables vanishes at the linear order, while the tetrad basis
rotates along the light propagation compared to a FRW coordinate. Though the
latter is often used as a probe of primordial gravitational waves, the rotation
of the tetrad basis is indeed not a physical observable. We further clarify its
relation to the E-B decomposition in weak lensing. Our formalism provides a
transparent and comprehensive perspective of cosmological weak lensing.
