I am trying to reproduce the results from this paper. On page 2 of the paper, they have an equation: $$2 H=-\frac{\dot{r}^2}{g(r)}-L \dot{\phi }+E \dot{t}=\epsilon\tag{9}$$ where they make a comment that $\epsilon=1$ for the time-like particle on the geodesic and $\epsilon=0$ for the light-like particle.
I am unable to comprehend the fact that how the Hamiltonian can be 0 or 1 for the given particles.
Any help in this regard would be truly beneficial!
In the Hamiltonian here the dot in $\dot t$ etc. refers to the derivative with respect to the affine parameter. So the Hamiltonian represents a sort of conserved quantity with respect to the affine parameter. This conserved quantity turns out to be the norm of the tangent vector to the worldline.
For a timelike worldline the norm of the tangent vector is always 1. This is the source of the famous idea that is often taught in pop-sci sources where they describe all objects going through spacetime at c. It is not really a deep statement, just that the affine parameter makes the tangent vector a unit vector and in relativity the unit of speed is c.
For a lightlike worldline the norm of the tangent vector is, by definition, 0. Hence $\epsilon = 0$ indicates a null geodesic and $\epsilon = 1$ indicates a timelike geodesic.
