Bending of light

Suppose a particle with mass $m$ and velocity $v$ flies with impact parameter $b$ past an object with mass $M$. The gravitational pull of $M$ will deflect $m$, and give it a component $v_\perp$ perpendicular to its initial $v$. For the case $m=M$, this is in fact the situation we envisaged when deriving the relaxation time in Section 8.1. There we found that (see Eq. 8.3) when the deflection angle $\alpha$ is small,

$$v_\perp = {2GM\over b\,v}\,.$$ (12.1)

Convince yourself that this is the correct result also in case $m\ne M$. Finally, since $\tan(\alpha)=v_\perp/v$ and when $\alpha$ is small so that $\tan(\alpha)\approx \alpha$,

$$\alpha = {2GM\over b\,v^2}\,.$$ (12.2)

Now here's the trick: the expression for $\alpha$ does not depend on the mass $m$ of the particle being reflected, only on the mass $M$ of the deflector. So we might be tempted to put $m=0$, $v=c$ and claim this is the angle by which light will be deflected as well.

Applying this to starlight passing close to the sun, hence using $M=\hbox{$M_\odot$}$, $R=\hbox{$R_\odot$}=7\times 10^5{\hbox{\rm km}}$, we get $\alpha=0.83{\hbox{\rm arcsec}}$ - so $\alpha\ll 1$ is certainly satisfied. This was actually the value Einstein found in 1911, though one Johann Soldner already obtained it almost a century earlier.

Curiously, the answer is wrong! Once Einstein formulated his theory of General Relativity, he did the problem again, and found that the deflection angle is twice our `Newtonian' value. So the correct deflection angle for light is

$$\alpha = {4GM\over b\,c^2}\,.$$ (12.3)

and it was of course a great vindication of the theory when Eddington verified this during a solar eclipse in 1920. Note that the deflection angle increases with $M$ and decreases with $b$ - the closer you pass to the more massive an object, the bigger is the deflection.

In the case of the Sun, the deflection is so small that you might think it if of little use in Astronomy. This was actually true for a long time, but recently it has really taken off as a new way of looking at the Universe. Before I show you some applications, we need to do a little bit more maths first.