Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 13; Problem 13.1.
We’ve seen how to derive the equations of motion for a photon in a gravitational field, so we can now apply these equations to study the deflection of light as it passes a massive object. The equations of motion are
These equations describe a photon’s motion for , but when we’re discussing the deflection of light as it passes a star such as the sun, typically . For example, for the Sun, and the radius of the sun is so a photon grazing the surface of the Sun as it passed has a value of . In this limiting case, we can make a few approximations to get an idea of how much light is deflected as it passes a massive object.
Things are a bit easier if we work with the variable , so we can use the chain rule to write
Plugging this into the radial equation of motion above, we get
We can take the derivative of this with respect to and get
If (that is, space is flat), this equation reduces to
which has the general solution
If we orient the coordinate system so that is the angle of closest approach to the origin, then is a maximum at that angle so
and we can set where is the distance of closest approach to the origin:
This is, in fact, the equation of a straight line in polar coordinates as can be verified by drawing the right-angled triangle with sides (as the hypotenuse) and (the third side is just ). From the equation above, with , we have
Thus , which is the impact parameter, and gives the distance of closest approach.
Now let’s put the mass back in and apply the assumption . In this case, we expect that will be fairly close to the flat space solution, so we can try a solution of the form
where is some function that should be small for all values of . Plugging this into the equation of motion above:
Because of our assumption , as well, so if we’re saving only up to first-order terms in the ‘small’ quantities, only the first term on the RHS will contribute. We then get
This differential equation can be solved by assuming a solution of form :
from which we get
We saw above that in flat space, , the impact parameter. How about when there is a mass present? We can find out by substituting this equation into the equation of motion (above)
and saving only terms linear in . We have
We can now save only terms up to those involving :
Collecting terms, we get
Using the shorthand notation and , and the identities and , we get
Thus to this level of approximation, .
Finally, in our coordinate system the closest approach to the mass occurs at so from 24 this distance is
where the last line uses a Taylor expansion to first order. The closest approach is therefore less than the impact parameter .