Fermat potentials for nonperturbative gravitational lensing

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.

On bending angles by gravitational lenses in motion

“…The relationship between the two can be found, in this case, by defining a new integration variable for , which we here denote by q : In terms of this variable we have where γ ⊥ is essentially a constant, as is x in . The differential form of is With this change of variable, becomes In terms of the Doppler shift of the deflector z d , since points towards the observer, we have , so the bending angle by a moving deflector in terms of the same deflector at rest is This result is in agreement with Frittelli et al (2002), where a significantly different method was used, and the deflector was assumed not to move across the line of sight. An important point to emphasize is the fact that the transverse motion of the deflector has no effect on the bending angle, as long as it is approximately uniform, as noted by Pyne & Birkinshaw (1993).…”

“…Our method uses elements of both of the calculations in Pyne & Birkinshaw (1993) and Capozziello et al (1999), as well as new elements, and thus functions as an independent cross‐check. We find full agreement with Pyne & Birkinshaw (1993) and Frittelli et al (2002), and we back up our calculation with a check of consistency with the leading correction to the time delay, which, to our knowledge, has not appeared before except in Frittelli et al (2002). The geodesic equation for linearized perturbations corresponding to a non‐stationary deflector of certain generality is written down in , where the notation and approximations are defined.…”

“…More recently, in Frittelli, Kling & Newman (2002), we use a completely different method based on envelopes of null surfaces (as opposed to null geodesics) to rederive the lens equation, and we find α= (1 + z d )α s for the case of a lens moving along the line of sight.…”

Aberration by gravitational lenses in motion

“…The v / c correction of first order to the bending angle was predicted (to the best of my knowledge) by Pyne & Birkinshaw (1993), and subsequently confirmed by Kopeikin & Schäfer (1999), Frittelli, Kling & Newman (2002) and Frittelli (2003). The recent observational verification strongly suggests that such a correction needs to be incorporated into the fitting of gravitational lensing events.…”

The Bianchi identity and weak gravitational lensing

“…These papers also wrote out integral relations that showed how image distortion grew continuously along the pencil of rays connected a source and observer. A generalization of the Fermat principle of least time was present in Frittelli et al [17].…”