Radial motion of a photon in Schwarzschild spacetime

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?

• Are there any constraints on the relativistic angular momentum of a null particle, when compared to the relativistic energy of that particle? Commented Jun 13, 2021 at 22:53

1 Answer

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.