Null geodesics and gravitational lensing

Let E and L be the conserved photon energy and angular momentum defined using the asymptotic time and rotational symmetries. The impact parameter b is L times c divided by E, with length units and a sign that can encode orientation; use its positive magnitude when discussing capture. With inverse radius u and orbital angle phi, eliminating the affine parameter gives the nonlinear orbit equation below. Its curvature correction is proportional to mass length and vanishes in the flat limit.

For an incoming ray whose closest approach is far from the mass scale, treat that correction perturbatively around the straight-line orbit. The leading net deflection between asymptotically distant directions is four m divided by b. This approximation requires b much greater than m and endpoints far enough away for the asymptotic interpretation. A Newtonian particle-like light estimate using only temporal attraction gives half the result; spatial metric curvature contributes at the same order and cannot be omitted consistently.

A thin-lens model approximates the bend as occurring in one plane and uses small angular separations. For an isolated spherical point lens, beta is the unlensed angular source position and theta is an image position relative to the lens. Angular-diameter distances connect observer, lens, and source. In the elementary nearly Euclidean setup they reduce to the corresponding ordinary distances, while cosmological applications require the actual distance-redshift model. The lens equation then balances straight propagation and the weak deflection at the lens plane.

At exact alignment the point-lens equation predicts an Einstein ring with angular radius theta E. Its square is proportional to lens mass at fixed distances, so quadrupling the mass doubles the ring angle. A ring is therefore not a direct angular measurement of the event horizon. Finite source size, extended lens structure, multiple lens planes, wave effects, and strong deflection can all invalidate parts of this reduction. A measured angle constrains a combination of mass and geometry rather than independently revealing every distance.

For null radial motion, the angular effective potential is proportional to f divided by radius squared. Its maximum occurs at three m. That circular null orbit is unstable: a small radial departure can lead toward escape or inward capture rather than bounded oscillation. Evaluating the ratio of radius squared to f there gives the critical impact parameter three square root of three times m. This impact parameter is larger than the photon orbit’s areal radius because it is defined by asymptotic conserved quantities.

For a ray arriving from infinity at an ideal Schwarzschild black hole, an impact parameter below the critical value yields capture and one above yields scattering, with the threshold ray asymptotically approaching the unstable orbit. A material surface can intercept rays and alter the observable fate; the exterior trajectory alone does not establish a black-hole completion. Near the threshold, winding paths and large deflections replace the simple weak-lens formula. A planar quadratic instrument tests the square-root mass scaling only and traces no null geodesic.