Radial motion of a photon in Schwarzschild spacetime

Question

For photon worldlines in the equatorial plane of the Schwarzschild coordinate system ($\theta=\frac{\pi}{2}$) in Schwarzschild space-time, the metric equation is given by,

$$-\Big(1-\frac{2GM}{r}\Big)dt^2+\Big(1-\frac{2GM}{r}\Big)^{-1}dr^2+r^2d\phi^2=0$$

$$\Rightarrow -\Big(1-\frac{2GM}{r}\Big)+\Big(1-\frac{2GM}{r}\Big)^{-1}\Big(\frac{dr}{dt}\Big)^2+r^2\Big(\frac{d\phi}{dt}\Big)^2=0$$

Now we define the impact parameter $b$ of a body to be,

$$b=\frac{l}{e}=\frac{r^2\frac{d\phi}{d\tau}}{(1-\frac{2GM}{r})\frac{dt}{d\tau}}=r^2\Big(1-\frac{2GM}{r}\Big)^{-1}\frac{d\phi}{dt}$$

where $l$ is the relativistic angular momentum per unit mass and $e$ is the relativistic energy per unit mass at infinity. This expression of the impact parameter is also valid for photons as the limit $\lim_{m\to 0}b$ is defined.

If we substitute for $\frac{d\phi}{dt}$ in the metric equation, do a bit of algebraic manipulation, we get:

$$\frac{dr}{dt}=\Big(1-\frac{2GM}{r}\Big)\sqrt{1-\frac{b^2}{r^2}\Big({1-\frac{2GM}{r}}\Big)}$$

The problem here is that $\frac{dr}{dt}$ will be a complex number if,

$$b>\frac{r}{\sqrt{1-\frac{2GM}{r}}}$$

What is stopping the impact parameter from obtaining these 'forbidden' values? What does it mean for the radial motion i.e. $\frac{dr}{dt}$ of a photon to be a complex number?

Answer 1

What stops the impact parameter from being too high is that at a given radius $r$, there is a maximum angular momentum coming from the speed of light. $l$ will be highest when the velocity is purely tangential; substituting $dr/dt=0$ in your first equation will show that in that case we have

$$\frac{d\phi}{dt} = \frac{\sqrt{1 - 2M/r}}{r},$$

which means that $b$ will achieve its maximum value. If the photon doesn't point in the purely tangential direction, the angular momentum (and hence the angular velocity and impact parameter) will be lower.