Introduction to Cosmology

Gravitational Lensing 69

A more recently discovered millisecond pulsar is PSR J0337+1715, a hierarchi-
cal triple system with two other stars. Strong gravitational interactions are appar-
ent and provide precision timing and multi-wavelength observations. The masses of
the pulsar and the two white dwarf companions are 1. 4378 ± 0. 0013 푀⊙, 0. 19751 ±
0. 0015 푀⊙and 0. 4101 ± 0. 0003 푀⊙, respectively. The unexpectedly coplanar and nearly
circular orbits indicate a complex and exotic evolutionary past that differs from those
of known stellar systems. The gravitational field of the outer white dwarf strongly
accelerates the inner binary containing the neutron star, and the system will thus pro-
vide an ideal laboratory in which to test the strong equivalence principle of general
relativity.

4.3 Gravitational Lensing

Recall from Equation (3.4) and Section 3.1 that the Strong Equivalence Principle
(SEP) causes a photon in a gravitational field to move as if it possessed mass. A particle
moving with velocity푣past a gravitational point potential or a spherically symmetric
potential푈will experience an acceleration in the transversal direction resulting in a
deflection, also predicted by Newtonian dynamics. The deflection angle훼can be cal-
culated from the (negative) potential푈by taking the line integral of the transversal
gravitational acceleration along the photon’s path.

Weak Lensing. In the thin-lens approximation the light ray propagates in a straight
line, and the deflection occurs discontinuously at the closest distance. The transversal
acceleration in the direction푦is then

d^2 푦

d푡^2

=−

(

1 +

푣^2

푐^2

)

d푈

d푦

. (4.1)

In Newtonian dynamics the factor in the brackets is just 1, as for velocities푣≪푐.
This is also true if one invokes SEP alone, which accounts only for the distortion of
time. However, the full theory of general relativity requires the particle to move along
a geodesic in a geometry where space is also distorted by the gravitational field. For
photons with velocity푐the factor in brackets is then 2, so that the total deflection due
to both types of distortion is doubled.
The gravitational distortion can be described as an effective refraction index,

푛= 1 −

2

푐^2

푈> 1 , (4.2)

so that the speed of light through the gravitational field is reduced to푣=푐∕푛. Different
paths suffer different time delays훥푡compared with undistorted paths:

훥푡=^1

푐∫

observer

source

2

푐^2

d푙. (4.3)

For a thin lens, deflection through the small bend angle훼in Figure 4.1 may be taken
to be instantaneous. The angles휃Iand휃Sspecify the observed and intrinsic positions