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.
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.
