The Critical Impact Parameter
For a non-rotating black hole, the shadow diameter is determined by the critical impact parameter \(b_c\). Light rays with an impact parameter smaller than \(b_c = \frac{3\sqrt{3}}{2} r_s\) fall into the black hole. This results in a shadow diameter of approximately \(5.2 r_s\), where \(r_s\) is the Schwarzschild radius.
Gravitational Lensing Effect
In flat space, the geometric cross-section would simply be the area of the event horizon. However, in curved spacetime, light paths are bent inward. Photons that would normally miss the black hole in Newtonian physics are captured if they pass within the photon sphere. This effectively enlarges the 'target' presented by the black hole to incoming light.
Key Distinction
Remember: The bright ring you see in EHT images is the 'photon ring,' formed by light that orbits the black hole multiple times before escaping to your eye. The dark region inside is the shadow.