Tensors and Relativity: Chapter 8

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 tex2html_wrap_inline1309 . The differential equation we need to solve is therefore

equation1092

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

equation1094

The general solution can be written in the form

equation1096

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

  figure774
Figure 8.2: Deflection of light ray

This is the equation of a straight line as tex2html_wrap_inline1354 goes from tex2html_wrap_inline1356 to tex2html_wrap_inline1358 . 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:

equation1098

where we have taken tex2html_wrap_inline1360 for convenience. It follows that the equation for tex2html_wrap_inline1326 is:

equation1100

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

equation1102

This gives [ Assignment 7 ]

equation1104

so the full solution is

equation1106

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

  figure794
Figure 8.3: Diagram showing the total deflection

When tex2html_wrap_inline1364, tex2html_wrap_inline1366 , so

equation1108

at the asymptotes tex2html_wrap_inline1368 and tex2html_wrap_inline1370 , and taking tex2html_wrap_inline1372 we get:

eqnarray1110

The total deflection is tex2html_wrap_inline1374 :

equation1112

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. 

  figure815
Figure 8.4: Einstein ring.

The distance of closest approach corresponds to an angle

equation1112

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

equation1116

since both tex2html_wrap_inline1378 and tex2html_wrap_inline1380 are small. It follows that the impact parameter can be written as

equation1118

  figure832
Figure 8.5: Einstein ring lensing event

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

equation1120


next up previous index
Next: Index Up: Orbits in Schwartzschild spacetime Previous: Solution for timelike orbits