  We present an analytic and fully relativistic framework for studying the
self-intersection of tidal disruption event (TDE) streams, restricting
ourselves to the Schwarzschild spacetime. By taking advantage of the
closed-form solution to the geodesic equations in the Schwarzschild metric, we
calculate properties of the self-intersection without numerically evaluating
the geodesic equations or making any post-Newtonian approximations. Our
analytic treatment also facilitates geometric definitions of the orbital
semi-major axis and eccentricity, as opposed to Newtonian formulas which lead
to unphysical results for highly-relativistic orbits. Combined with assumptions
about energy dissipation during the self-intersection shock, our framework
enables the calculation of quantities such as the fraction of material unbound
during the self-intersection shock, and the characteristic semi-major axes and
eccentricities of the material which remains in orbit after the collision. As
an example, we calculate grids of post-intersection properties in stellar and
supermassive black hole (SMBH) masses for disruptions of main sequence stars,
identifying regions where no material is ejected during self intersection (e.g.
SMBH mass $\lesssim 5\times10^6\, {\rm M_\odot}$ for $1\,{\rm M_\odot}$ stars
disrupted at the tidal radius), potentially explaining the TDEs observed by
SGR/eROSITA which are visible in X-rays but not optical wavelengths. We also
identify parameters for which the post-intersection accretion flow has low
eccentricity ($e\lesssim0.6$), and find that the luminosity generated by
self-intersection shocks only agrees with observed trends in the relationship
between light curve decay timescales and peak luminosities over a narrow range
of SMBH masses.
