382 Gravitational lensing
effective refraction index, given by
n= 1 −2
c^2= 1 +
2
c^2||. (14.1)
The Newtonian potential is negative and vanishes asymptotically. As in
geometrical optics a refraction indexn>1 means that the light travels with a
speed which is lower compared with its speed in the vacuum. Thus the effective
speed of light in a gravitational field is given by
v=c
nc−2
c||. (14.2)
Since the effective speed of light is less in a gravitational field, the travel time
becomes longer compared to the propagation in empty space. The total time delay
tis obtained by integrating along the light trajectory from the source until the
observer, as follows
t=∫observersource2
c^3||dl. (14.3)This is also called the Shapiro delay.
The deflection angle for the light rays which pass through a gravitational
field is given by the integration of the gradient component ofnperpendicular to
the trajectory itself:
α=−∫
∇⊥ndl=2
c^2∫
∇⊥dl. (14.4)For all astrophysical applications of interest the deflection angle is always
extremely small, so that the computation can be substantially simplified by
integrating∇⊥nalong an unperturbed path, rather than the effective perturbed
path. The so induced error is of higher order and thus negligible.
As an example let us consider the deflection angle of a point-like lens of
massM. Its Newtonian potential is given by
(b,z)=−GM
(b^2 +z^2 )^1 /^2, (14.5)
wherebis the impact parameter of the unperturbed light ray andzdenotes
the position along the unperturbed path as measured from the point of minimal
distance from the lens. This way we obtain
∇⊥(b,z)=GMb
(b^2 +z^2 )^3 /^2, (14.6)
wherebis orthogonal to the unperturbed light trajectory and is directed towards
the point-like lens. Inserting equation (14.6) in equation (14.4) we find, for the
the deflection angle,
α=2
c^2∫
∇⊥dz=4 GM
c^2 bb
b