Abstract:
Interactions incorporating the vacuum polarization effects in curved backgrounds modify the null cone structure in such a way that the photon trajectories would not be the space-time geodesics anymore. The gravitational birefringence introduced as a direct consequence of these effects, will allow shifts in the photon velocities leading to polarization dependent superluminal propagation. Taking these effects into account we study Fermat's principle in the context of the 1+3 (threading) formulation of the space-time decomposition. We find an expression for the modified spacetime refractive index and show it is proportional to the light cone correction to the first order. Consequences of this modification on polarization sum rules and spatial light paths are considered.

Abstract:
The current standard time delay formula (CSTD) in gravitational lensing and its claimed relation to the lens equation through Fermat's principle (least time principle) have been puzzling to the author for some time. We find that the so-called geometric path difference term of the CSTD is an error, and it causes a double counting of the correct time delay. We examined the deflection angle and the time delay of a photon trajectory in the Schwarzschild metric that allows exact perturbative calculations in the gravitational parameter $GM$ in two coordinate systems -- the standard Schwarzschild coordinate system and the isotropic Schwarzschild coordinate system. We identify a coordinate dependent term in the time delay which becomes irrelevant for the arrival time difference of two images. It deems necessary to sort out unambiguously what is what we measure. We calculate the second order corrections for the deflection angle and time delay. The CSTD does generate correct lens equations including multiple scattering lens equations under the variations and may be best understood as a generating function. It is presently unclear what the significance is. We call to reanalyze the existing strong lensing data with time delays.

Abstract:
We present a framework, based on the null-surface formulation of general relativity, for discussing the dynamics of Fermat potentials for gravitational lensing in a generic situation without approximations of any kind. Additionally, we derive two lens equations: one for the case of thick compact lenses and the other one for lensing by gravitational waves. These equations in principle generalize the astrophysical scheme for lensing by removing the thin-lens approximation while retaining the weak fields.

Abstract:
The images of many distant galaxies are displaced, distorted and often multiplied by the presence of foreground massive galaxies near the line of sight; the foreground galaxies act as gravitational lenses. Commonly, the lens equation, which relates the placement and distortion of the images to the real source position in the thin-lens scenario, is obtained by extremizing the time of arrival among all the null paths from the source to the observer (Fermat's principle). We show that the construction of envelopes of certain families of null surfaces consitutes an alternative variational principle or version of Fermat's principle that leads naturally to a lens equation in a generic spacetime with any given metric. We illustrate the construction by deriving the lens equation for static asymptotically flat thin lens spacetimes. As an application of the approach, we find the bending angle for moving thin lenses in terms of the bending angle for the same deflector at rest. Finally we apply this construction to cosmological spacetimes (FRW) by using the fact they are all conformally related to Minkowski space.

Abstract:
The contribution of the cosmological constant to the deflection angle and the time delays are derived from the integration of the gravitational potential as well as from Fermat's Principle. The findings are in agreement with recent results using exact solutions to Einstein's equations and reproduce precisely the new $\Lambda$-term in the bending angle and the lens equation. The consequences on time delay expressions are explored. While it is known that $\Lambda$ contributes to the gravitational time delay, it is shown here that a new $\Lambda$-term appears in the geometrical time delay as well. Although these newly derived terms are perhaps small for current observations, they do not cancel out as previously claimed. Moreover, as shown before, at galaxy cluster scale, the $\Lambda$ contribution can be larger than the second-order term in the Einstein deflection angle for several cluster lens systems.

Abstract:
Using Fermat's principle, we analyze the effects of very long wavelength gravitational waves upon the images of a gravitationally lensed quasar. We show that the lens equation in the presence of gravity waves is equivalent to that of a lens with different alignment between source, deflector, and observer in the absence of gravity waves. Contrary to a recent claim, we conclude that measurements of time delays in gravitational lenses cannot serve as a method to detect or constrain a stochastic background of gravitational waves of cosmological wavelengths, because the wave-induced time delay is observationally indistinguishable from an intrinsic time delay due to the lens geometry.

Abstract:
In this paper we use a general version of Fermat's principle for light rays in General Relativity and a curve shortening method to write the Morse relations for light rays joining an event with a smooth timelike curve in a Lorentzian manifold with boundary. As a physical meaning, one can apply the Morse relations to have a mathematical description of the "gravitational lens effect" in a very general context.

Abstract:
The effect of currents of mass on bending of light rays is considered in the weak field regime. Following Fermat's principle and the standard theory of gravitational lensing, we derive the gravitomagnetic correction to time delay function and deflection angle caused by a geometrically-thin lens. The cases of both rotating and shifting deflectors are discussed.

Abstract:
Mimicking the description of spinning particles in General Relativity, the Fermat Principle is extended to spinning photons. Linearization of the resulting Papapetrou-Souriau type equations yields the semiclassical model used recently to derive the ``Optical Hall Effect'' (alias the ``Optical Magnus Effect'') for polarized light.

Abstract:
The purpose of these lecture notes is to describe the gravitational lens effects in different astrophysical contexts. These notes are voluntarily focused on the fundamental mechanisms and the basic concepts that are useful to describe these effects. The observational consequences are presented in more details in accompanying notes by Y. Mellier. The content of these notes is the following. In the first section describe of the basic mechanisms of gravitational lenses, techniques and approximations that are usually employed. The second section is devoted to the case of a very simple deflector, a point-like mass distribution. This corresponds to microlensing events in which the deflectors are compact objects of a fraction of a solar mass that may populate the halo of our Galaxy. The last two sections are devoted to cosmological applications. After a presentation of the geometrical quantities that are specific to cosmology, I will present the various phenomena that can be observed in this context. Finally I describe the weak lensing regime. This is a rapidly developing area that should eventually allow us to map the mass distribution in the Universe.