Hindawi Publishing Corporation
Journal of Gravity
Volume 2013, Article ID 686950, 4 pages
http://dx.doi.org/10.1155/2013/686950
Research Article
First-Order Light Deflection by Einstein-Strauss
Vacuole Method
Debasish Saha,
1
Amarjit Tamang,
1
Ramil Izmailov,
2
Carlo Cattani,
3
and Kamal K. Nandi
1,2
1
Department of Mathematics, University of North Bengal, Siliguri 734 013, India
2
Ya.B. Zel’dovich International Center for Astrophysics, BSPU, Ufa 450000, Russia
3
Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy
Correspondence should be addressed to Kamal K. Nandi; kamalnandi1952@yahoo.co.in
Received 20 February 2013; Accepted 15 July 2013
Academic Editor: Jose Antonio De Freitas Pacheco
Copyright © 2013 Debasish Saha et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We resolve here an outstanding problem plaguing conformal gravity in its role in making consistent astrophysical predictions.
Tough its static spherically symmetric solution incorporates all the successes of Schwarzschild gravity, the ft to observed galactic
rotation curves requires >0, while the observed increase in the Schwarzschild light defection by galaxies appears to demand
<0. Here we show that, contrary to common knowledge, there is an increase in the Schwarzschild defection angle in the
vicinity of galaxies due purely to the efect of >0, 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 defection by
galaxies.
1. Introduction
Te metric exterior to a static spherically symmetric dis-
tribution in Weyl conformal gravity has been obtained by
Mannheim and Kazanas [1]. Recently, the solution has been
used to ft rotation curves of many galaxy samples [2] as well
as to predict the maximal size of galaxies [3]. Te metric,
which we call Mannheim-Kazanas-de Sitter (MKdS) metric,
reads (==1)
2
=−()
2
+
1
()
2
+
2
(
2
+ sin
2
2
), (1)
()=1−
2
+−
2
, (2)
where is the central mass and and are arbitrary
constants that could be appropriately fxed by using the ft
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
defection in the above spacetime. First, the conventional
calculations for light defection show that the constant does
not appear in the relevant equations, leading fnally to the two
way defection as [4]
2=
4
0
−
0
, (3)
where
0
is the distance of closest approach. Te difculty is
that the ft to observed rotation curve requires >0, and
for consistency all other astrophysical observations should
respect this sign. Now, the observed light defection by a
galaxy is always more than the Schwarzschild value 4/
0
,
and hence to avoid the negative contribution in (3), one must
demand <0 [4]. Tus there appears an inconsistency from
the usual method. Te second option is to use the Rindler-
Ishak method [5], which is based on the realization that
conventional methods do not apply to asymptotically nonfat
spacetimes as the limit →∞ makes no sense in it. Teir
original method of invariant angle is most appropriate in such
situations, but it has an as yet unnoticed difculty on the
galactic scales, as explained below. Ishak et al. [6] thereafer
improved the calculations using the Einstein-Strauss vacuole
model and this provides us with the third and best option in
our opinion. Te purpose of this paper is to use the vacuole
model to show that there is an increase in the Schwarzschild