Exact Null Geodesics of the Interior Schwarzschild Spacetime

The geodesic equation of general relativity—comprised of coupled nonlinear ordinary differential equations—is rarely solvable in closed form. For the interior Schwarzschild spacetime describing a static, spherically symmetric body of uniform density, an exact analytical solution for null geodesics parameterized by coordinate time with general initial conditions is derived using conserved quantities associated with Killing symmetries. Although this method is standard, it is uncommon for the resulting integrals to admit closed-form solutions and inversions as achieved here, and significant effort was required to reconcile multiple solution branches into a continuous global trajectory.

 

To facilitate geometric interpretation, the solution is extended beyond the bounding surface to the theoretical event horizon, revealing complete light orbits through the interior. Expressions are obtained for quantities including angular momentum, energy, minimum and maximum radial approach, orbital period, precession rate, and emission angle. Interestingly, precession arises only for trajectories that reach the theoretical event horizon, analogous to the precession of bound Newtonian orbits within uniform density bodies when the orbit exceeds the surface. Rotating reference frames are explored to eliminate precession, and preliminary evidence suggests that families of geodesics may lie on common surfaces—potentially enabling a geometric reformulation of the problem akin to conic sections in Keplerian dynamics.

 

This solution provides a rare analytical tool for modeling light propagation in curved interiors. Its initial application is to anisotropic neutrino emission in neutron stars, where curved null trajectories tend to align directionally, generating a net momentum transfer. Beyond its immediate relevance to pulsar kicks, the solution offers a foundation for deeper exploration of relativistic transport phenomena in compact astrophysical objects.