First-Order Light Deflection by Einstein-Strauss Vacuole Method

We resolve here an outstanding problem plaguing conformal gravity in its role in making consistent astrophysical predictions. Though its static spherically symmetric solution incorporates all the successes of Schwarzschild gravity, the fit to observed galactic rotation curves requires , while the observed increase in the Schwarzschild light deflection by galaxies appears to demand . Here we show that, contrary to common knowledge, there is an increase in the Schwarzschild deflection angle in the vicinity of galaxies due purely to the effect of , when the idea of the Einstein-Strauss vacuole model is employed. With the inconsistency now out of the way, conformal gravity should be regarded as a good theory explaining light deflection by galaxies. 1. Introduction The metric exterior to a static spherically symmetric distribution in Weyl conformal gravity has been obtained by Mannheim and Kazanas [1]. Recently, the solution has been used to fit rotation curves of many galaxy samples [2] as well as to predict the maximal size of galaxies [3]. The metric, which we call Mannheim-Kazanas-de Sitter (MKdS) metric, reads ( ) where is the central mass and and are arbitrary constants that could be appropriately fixed by using the fit to rotation curves. For distances neither too small nor too large, the above mentioned metric is a good approximation. Now, there could be three possible ways to calculate light deflection in the above spacetime. First, the conventional calculations for light deflection show that the constant does not appear in the relevant equations, leading finally to the two way deflection as [4] where is the distance of closest approach. The difficulty is that the fit to observed rotation curve requires , and for consistency all other astrophysical observations should respect this sign. Now, the observed light deflection by a galaxy is always more than the Schwarzschild value , and hence to avoid the negative contribution in (3), one must demand [4]. Thus there appears an inconsistency from the usual method. The second option is to use the Rindler-Ishak method [5], which is based on the realization that conventional methods do not apply to asymptotically nonflat spacetimes as the limit makes no sense in it. Their original method of invariant angle is most appropriate in such situations, but it has an as yet unnoticed difficulty on the galactic scales, as explained below. Ishak et al. [6] thereafter improved the calculations using the Einstein-Strauss vacuole model and this provides us with the third and best option in our opinion. The purpose of