Speaker
Description
VLBI maps of H$_2$O megamaser disks often display a curved (``warped-looking'') locus of maser spots on the sky. We show that such a morphology can arise as a purely geometric projection effect, without invoking an intrinsically warped disk. To this end we develop a non-equatorial, finite-distance Gauss--Bonnet construction for null-ray propagation in a general static, spherically symmetric spacetime with two metric potentials, $f(r)\neq g(r)$. The orbit plane of the masers and the photon plane of each emitter--observer ray are characterized by two intrinsic tilts: the orbital inclination $\iota$ and the photon-plane tilt $\eta$. We build the corresponding tilted photon-plane two-surface and establish that its Gaussian curvature depends only on $g(r)$, while the tilts enter the Gauss--Bonnet balance through boundary terms at the endpoints. This yields a compact bending relation connecting the local emission angle, the on-sky aperture angle $\Theta$, and one-dimensional curvature integrals, now valid beyond the equatorial approximation. Combined with a fit-friendly expression for the total frequency shift $z$ and an explicit formula for the frequency-shift rapidity $\dot z$, the framework provides an observable-first mapping from $(\Theta,z,\dot z)$ to the emission geometry and physical parameters. The key outcome is a simple, non-warped geometric mechanism that can reproduce warping-like sky distributions of maser spots in inclined disks, offering a natural baseline for modeling megamaser data.