Spring quarter has just started up; I am planning on taking it easy this term, enrolling only in the two required astronomy courses and one physics course.  Even so, I will probably be busy enough to continue to neglect writing here more often.  So, I will try to crank out at least this post before things pick up.

   Our theory group has been devoting half of each of our weekly meetings to reviewing classic papers.  My turn to present is at the end of the month, but I’d like to start throwing something together – hence, this post.  I’ll keep adding things in bits and pieces as I go.

   The paper I chose to review is a classic by Blandford and Narayan: Fermat’s Principle, Caustics, and the Classification of Gravitational Lens Images.  I have a fond spot for this paper, since it is the first paper I was asked to read when I began doing research.  I was a naïve freshman back then, and I emailed pretty much every professor in the physics department looking for someone who would give me a project.  I got a surprising number of replies, mostly saying things like “We don’t have any open projects, but would be glad to help you find one.”  I received two positive responses: one from a professor in the astrophysics department, asking me if I was interested in working on gravitational lenses, and another from a professor in the theoretical physics department who researched string theory (again, I was a naïve freshman; I thought I was only years away from finding a grand unified theory just because I read Brian Greene’s The Elegant Universe). 

   I met with the string theory professor, and he asked me to read this paper to see if I could write some code to replicate and improve on the results.  It’s been four years and I still have no idea what the hell the paper is talking about.  The astrophysics professor asked me to read Blandford and Narayan, and I ended up working with him for the rest of my undergraduate years.

   So basically, the important results of this paper should be accessible to anybody who is smarter than I was when I was a lowly freshman, which should be nearly everybody.  All you will need to know is some simple multivariable calculus, and be willing to take one or two results from general relativity on faith; then you should be able to understand Blandford and Narayan’s elegant presentation of gravitational lensing.

Gravitational Deflection of Light by a Point Mass

   One of the first important tests of general relativity was confirming that the deflection of light by mass was predicted correctly by the theory.  This was confirmed (albeit with extremely large error bars) by the Eddington expedition, which measured the angle of deflection of starlight passing by the sun; such an observation could only be made during a solar eclipse.  Subsequent observations have verified that indeed the angle of deflection predicted by general relativity (exactly twice the angle predicted by a Newtonian calculation) is observed.

   So what is the expression for the angle of deflection (which we will denote by \alpha)?  Despite the importance of this calculation in the history of general relativity, we can actually find the answer quite simply using dimensional analysis.  Consider the figure above (taken from here).  The light passes by the star with a given impact parameter, or distance of closest approach of the light ray, which is denoted by \xi (not exactly the distance of closest approach, as drawn in the figure, but since the angles involved are small the figure should suffice to get the idea across).  Clearly, this impact parameter \xi should enter into the expression for the deflection angle \alpha; the closer the light ray passes, the more it should be deflected.  The other important physical variables in the problem are the mass of the deflecting object M (the smaller the mass, the smaller the deflection angle), strength of the gravitational constant G (the weaker the strength of gravity, the smaller the deflection angle), and the speed of light c (the faster light goes, the less it should be deflected).

   We now have four physical variables M, G, c, and \xi, which we must somehow combine to form a dimensionless quantity \alpha.  By using dimensional analysis (more exactly, the Buckingham \Pi Theorem), we see that the only way this can be done is: \alpha = f\left(\frac{GM}{c^2\xi}\right) .  That is, the deflection angle is some function of the dimensionless quantity q \equiv \frac{GM}{c^2\xi} .  We can easily see this quantity is dimensionless; \frac{GM}{\xi} has units of (gravitational potential) energy/mass, and c^2 also has units of (kinetic) energy/mass.  Physical intuition (along with our arguments above on how changing each variable should affect the size of the angle) tells us that we should have f(q) \sim q, and that the deflection angle is thus given by:

     \alpha \sim \frac{GM}{c^2\xi} .

   We have found roughly the right answer just from doing dimensional analysis!  The Newtonian calculation simply gives a correction f(q)=2q; this results from integrating the deflecting component of the gravitational force along the light ray, i.e.

     \alpha = \lim \limits_{v\to c} \frac{\Delta v}{v} = \frac{1}{c} \int \limits_{-\infty}^{\infty} a_{\perp} \,dt = \frac{1}{c} \int \limits_{-\infty}^{\infty} \frac{dv_{\perp}}{dt} \frac{dt}{dx} \,dx.

Using \frac{dt}{dx} = \frac{1}{c} , we then have:

     \alpha = \frac{1}{c^2} \int \limits_{-\infty}^{\infty} \frac{GM}{x^2+\xi^2} \frac{\xi}{\sqrt{x^2+\xi^2}} \,dx = \frac{2GM}{c^2 \xi}.

   To find the exact answer using general relativity, the calculation is a little more involved.  We still must add up the deflections using an integral along the trajectory; however, now we cannot simply write F_{\perp} = \frac{dp_{\perp}}{dt} for the gravitational force.  Instead, we must use the geodesic equation p^{\mu} \nabla_{\mu} p^{\nu} = 0, using the Christoffel symbols of the Schwarzschild metric in the weak-field \Phi to calculate the covariant derivative.  This gives relations between the p^{\mu}; the relation of interest is \partial_x p^y = 2 p^x \partial_y \Phi = p^x \frac{2GM\xi}{c^2\left(x^2+\xi^2\right)^{3/2}} (assuming p^0 \approx p^x \gg p^y \equiv p_{\perp}).  Thus, we have:

     \alpha = \lim \limits_{t \to \infty} \frac{p_{\perp}}{p^x} = \frac{1}{p^x} \int \limits_{-\infty}^{\infty} \frac{dp^y}{dx} \,dx = \int \limits_{-\infty}^{\infty} \frac{2GM\xi}{c^2\left(x^2+\xi^2\right)^{3/2}} \,dx = \frac{4GM}{c^2\xi} .

Thus, the general relativistic calculation simply gives f(q)=4q , an extra factor of two greater than the Newtonian result.  Finally, we have found that the deflection angle given by general relativity is \alpha = \frac{4GM}{c^2\xi} .