Thin accretion-disc transfer functions in the q-metric spacetime: quadrupolar lensing signatures

Abstract

We study geometrically thin equatorial disc images in the static, axisymmetric q-metric spacetime using backward ray tracing. The Schwarzschild solution is recovered at $q=0$; for the nonzero deformations considered here, $r=2M$ is a curvature singularity rather than an event horizon. For a distant nearly face-on observer, we construct the first three equatorial transfer functions $r_m(b;q)$. In the {Schwarzschild} limit, the resolved third-crossing interval contains the critical impact parameter $b=3\sqrt{3}M$. As $q$ increases from $-0.3$ to $0.3$, the branch center shifts from $b_{3,c}/M\simeq3.716$ to $6.702$. In units of the physical monopole mass $\mathcal M=(1+q)M$, it instead decreases from $b_{3,c}/\mathcal M\simeq5.309$ to $5.156$. Much of the displacement in metric-parameter units therefore reflects the changing monopole scale. At fixed $\mathcal M$, the inclined first-intersection maps show that the contour dimensions increase with $q$, with larger relative changes near the inner disc. A prescribed power-law emissivity and relativistic frequency-shift weighting yield direct images whose integrated flux differs from the Schwarzschild value by at most $2.4\%$. Between the endpoint models, the horizontal median-flux position increases by $8.8\%$ and the interquartile width by $14.5\%$. Within this emission prescription, the residual deformation dependence appears mainly as a displacement and broadening of the brightness distribution. Higher-order radiative contributions are not included.