# Causal concept for black hole shadows

## TL;DR

This paper introduces a causal framework for understanding black hole shadows, focusing on wandering null geodesics and their accumulation, rather than traditional photon sphere concepts, to better describe shadow formation in general spacetimes.

## Contribution

It defines wandering null geodesics and sets, providing a new causal perspective on black hole shadows beyond stationary spacetime assumptions.

## Key findings

- Wandering null geodesics are linked to shadow formation in general black hole spacetimes.
- The accumulation of null geodesics, not stationary photon orbits, underpins shadow appearance.
- Weyl focusing influences the existence of wandering null geodesics.

## Abstract

Causal concept for the general black hole shadow is investigated, instead of the photon sphere. We define several `wandering null geodesics' as complete null geodesics accompanied by repetitive conjugate points, which would correspond to null geodesics on the photon sphere in Schwarzschild spacetime. We also define a `wandering set', that is, a set of totally wandering null geodesics as a counterpart of the photon sphere, and moreover, a truncated wandering null geodesic to symbolically discuss its formation. Then we examine the existence of a wandering null geodesic in general black hole spacetimes mainly in terms of Weyl focusing. We will see the essence of the black hole shadow is not the stationary cycling of the photon orbits which is the concept only available in a stationary spacetime, but their accumulation. A wandering null geodesic implies that this accumulation will be occur somewhere in an asymptotically flat spacetime.