# Relation of specific angular momentum $h$ with velocity and impact parameter for massive particle?

I am computing the path of an incoming massive (!!) particle with speed $$v$$ far from the black hole in the schwarzchild metric. To determine its path, i need to input the specific angular momentum parameter $$h$$, defined as $$h = r^2 \dot{\phi}$$. What is the function $$h = h(b,v)$$ that i am looking for? (where $$b$$ is the impact parameter)

Please ask further questions if i am not being clear enough.

## 1 Answer

Angular momentum is $$\mathbf{L} = \mathbf{r} \times \mathbf{p}$$. Thus by definition, the impact parameter is a ratio of the magnitudes of angular momentum to linear momentum. $$b = \frac{L}{p}$$ From the energy-momentum 4-vector (assuming units with $$c = 1$$), $$E^2 - p^2= m^2$$ $$p = \sqrt{E^2-m^2}$$ $$b = \frac{L}{\sqrt{E^2-m^2}}=\frac{L/m}{\sqrt{(E/m)^2-1}}$$ For a particle with initial velocity, $$v_f$$, starting far from a Schwarzschild black hole $$\gamma_f = \frac{1}{\sqrt{1-v_f^2}}$$ $$E^2 = m^2 +p^2=m^2(1+\gamma_f^2 v_f^2)$$ $$\frac{E^2}{m^2} = 1+\frac{v_f^2}{1-v_f^2}=\frac{1}{1-v_f^2}$$ $$\frac{E}{m} = \frac{1}{\sqrt{1-v_f^2}}=\gamma_f$$ $$\frac{L}{m} = b \sqrt{\gamma_f^2-1}$$ Both $$E/m$$ and $$L/m$$, as measured by the far-away observer, are conserved on the particle's trajectory.

• This is exactly what I've been looking for, i cannot thank you enough!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! – user11339690 Sep 27 '19 at 21:50
• @user11339690, You are welcome! A good elementary reference for simple black hole calculations, orbits, etc., is Taylor and Wheeler, Exploring Black Holes. An updated pdf version is avaliable. – amateurAstro Sep 27 '19 at 22:14
• do u also have the relation of h with b for the case of a photon? – user11339690 Sep 28 '19 at 17:15
• @user11339690, The impact parameter is still $b=L/p$, but for light you must use the momentum of a photon (hint, what happens as $m \rightarrow 0$?). The impact parameter for massive particles and light is discussed in Chapter 11 of the reference I gave in a previous comment. It was discussed in Chapter 5 in the original book. – amateurAstro Sep 28 '19 at 18:17