Going Further 10.3: Angle of Light Deflection

Here we will derive the angle that light is deflected by the Sun, using the weak equivalence principle.

According to the Newtonian view, the photon will always fall toward the center of the Sun, and by the equivalence principle we know that it will always follow a straight-line path according to a freely falling observer in the same region near the Sun. Since the gravity of the Sun is not constant but increases as 1/r² as we approach the Sun, we cannot easily find the change in velocity of the photon along its entire path. However, we know that the photon will experience its greatest deflection where gravity is the strongest and where the acceleration is mostly perpendicular to the photon's path. This happens when the photon’s distance from the Sun is smallest, at the point labeled “b” in Figure B.10.3.

We also know that the photon will experience this maximum acceleration for a time roughly equal to the time required for the photon to travel a distance equal to b: at greater distances the strength of gravity drops rapidly. In addition, at large distances the gravitational acceleration is nearly along the photon’s direction of motion, so it does not tend to cause much deflection.

The time over which the photon experiences the maximum deflection is roughly t ~ b/c. The amount of acceleration the photon feels over this period is roughly a ~ GM/b², where M is the mass of the Sun. Combining these, the amount by which the velocity of a free-falling observer’s speed will increase over this time is v ~ GM/cb. The photon must also experience this change in velocity, transverse to its original direction of motion: it must fall freely toward the Sun’s center just as the observer does.

Using the photon’s original velocity and its change in velocity, we can estimate the change in direction of the photon as it travels from a great distance to its closest approach to the Sun—and note again, it is not the speed of the photon that changes, but only the direction. The speed is always c in a vacuum. In radians, the angle of deflection is given as θ = v/c = GM/bc².

Of course, the path of the photon must be completely symmetric on the way out and the way in, so an equal amount of deflection occurs as the photon leaves the vicinity of the Sun and moves off to infinity again. Thus the total amount of deflection the photon experiences is twice the angle we have found above: θ = 2GM/bc².

Figure B.10.3: As the particle approaches the Sun, its path curves. The Sun is represented by the black dot. The distance of closest approach is given by b, and it deflects through the angle θ. Credit: NASA/SSU/Aurore Simonnet.