The surface geometry and shadow of a Schwarzschild black hole with halo

TL;DR

This paper investigates how quadrupolar and octopolar halo terms influence the surface geometry and shadows of Schwarzschild black holes, revealing distinctive deformation patterns and potential observational signatures.

Contribution

It introduces a detailed analysis of halo-induced multipolar effects on black hole shadows, highlighting observable geometric distortions.

Findings

Quadrupole term causes prolate or oblate deformation of the black hole surface.

Octopole term results in upward or downward stretching of the shadow.

Observable shadow parameters can determine the halo's multipolar strengths.

Abstract

We have studied the surface geometry and shadows of Schwarzschild black hole with a halo containing the quadrupolar and octopolar terms. We found the quadrupole term makes the Schwarzschild black hole prolate for the quadrupole strength $\mathcal{Q}<0$ and oblate for $\mathcal{Q}>0$, and the octopole term makes shadow stretch upward for the octopolar strength $\mathcal{O}<0$ and stretch downward for $\mathcal{O}>0$. The shadow of Schwarzschild black hole with halo stretches and squeezes along the horizontal direction for $\mathcal{Q}<0$ and $\mathcal{Q}>0$ respectively. Meanwhile, black hole shadow shifts upward for $\mathcal{O}<0$ and shifts downward for $\mathcal{O}>0$. We exhibited the light rays that form the shadow boundary to explain the emergence of the extraordinary patterns of black hole shadow with the quadrupole and octopole terms. From the observable width $W$, height $H$, oblateness $K$ and distortion parameter $\delta_{c}$ of black hole shadow, one can determine the quadrupole and octopolar strengths of Schwarzschild black hole with halo. Our results show that the quadrupolar and octopolar terms yield a series of interesting patterns for the shadow of a Schwarzschild black hole with halo.