# The bending of light

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We next consider the case of the trajectory of a light ray in a spherically symmetric static gravitational field. The calculation is essentially the same as that given in the last section, except that light rays travel on null geodesics, so that . The differential equation we need to solve is therefore

In the limit of Special Relativity, the last term vanishes and the equation becomes

The general solution can be written in the form

where *b* is the closest approach to the origin [ or impact parameter, see Figure 8.2 ].

**Figure 8.2:** Deflection of light ray

This is the equation of a straight line as goes from to . The straight line motion is the same as predicted by Newtonian theory.

We again solve the General Relativity problem by taking the general solution to be a perturbation of the Newtonian solution:

where we have taken for convenience. It follows that the equation for is:

This equation can be solved by trying a particular integral of the form

This gives [ Assignment 7 ]

so the full solution is

Let us now calculate the deflection of a light ray from a star which just grazes the sun [ see Figure 8.3 ].

**Figure 8.3:** Diagram showing the total deflection

When , , so

at the asymptotes and , and taking we get:

The total deflection is :

This works out to be about 1.75'' and was confirmed by Eddington in 1919 during a solar eclipse.

Another beautiful example of the bending of light is the gravitational lens. Take the example of a Quasar directly behind a galaxy in our line of sight.

The distance of closest approach corresponds to an angle

Now from the diagram [ see Figure 8.4 ] above we have

since both and are small. It follows that the impact parameter can be written as

**Figure 8.5:** Einstein ring lensing event

So the image of the quasar appears as a ring which subtends an angle

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