Light Deflection and Gauss-Bonnet Theorem: Definition of Total
  Deflection Angle and its Applications
  In this paper, we re-examine the light deflection in the Schwarzschild and
the Schwarzschild-de Sitter spacetime. First, supposing a static and
spherically symmetric spacetime, we propose the definition of the total
deflection angle $\alpha$ of the light ray by constructing a quadrilateral
$\Sigma^4$ on the optical reference geometry ${\cal M}^{\rm opt}$ determined by
the optical metric $\bar{g}_{ij}$. On the basis of the definition of the total
deflection angle $\alpha$ and the Gauss-Bonnet theorem, we derive two formulas
to calculate the total deflection angle $\alpha$; (i) the angular formula that
uses four angles determined on the optical reference geometry ${\cal M}^{\rm
opt}$ or the curved $(r, \phi)$ subspace ${\cal M}^{\rm sub}$ being a slice of
constant time $t$ and (ii) the integral formula on the optical reference
geometry ${\cal M}^{\rm opt}$ which is the areal integral of the Gaussian
curvature $K$ in the area of a quadrilateral $\Sigma^4$ and the line integral
of the geodesic curvature $\kappa_g$ along the curve $C_{\Gamma}$. As the curve
$C_{\Gamma}$, we introduce the unperturbed reference line that is the null
geodesic $\Gamma$ on the background spacetime such as the Minkowski or the de
Sitter spacetime, and is obtained by projecting $\Gamma$ vertically onto the
curved $(r, \phi)$ subspace ${\cal M}^{\rm sub}$. We demonstrate that the two
formulas give the same total deflection angle $\alpha$ for the Schwarzschild
and the Schwarzschild--de Sitter spacetime. In particular, in the Schwarzschild
case, the result coincides with Epstein--Shapiro's formula when the source $S$
and the receiver $R$ of the light ray are located at infinity. In addition, in
the Schwarzschild--de Sitter case, there appear order ${\cal O}(\Lambda m)$
terms in addition to the Schwarzschild-like part, while order ${\cal
O}(\Lambda)$ terms disappear.
