# Escape, bound and capture geodesics in local static coordinates in Schwarzschild spacetime

## TL;DR

This paper analyzes the conditions for escape, bound, and capture geodesics of particles in Schwarzschild spacetime based on initial radius, velocity, and angle, revealing critical thresholds and cone structures affecting particle trajectories.

## Contribution

It provides a detailed parameter space analysis of geodesic behaviors in Schwarzschild spacetime, including the identification of critical velocities and radii for different orbit types.

## Key findings

- Bound orbits appear only beyond radius 4M or below velocity c/√2.
- Critical velocities and radii determine the transition between capture, bound, and escape regions.
- Cone structures in velocity and angle space delineate different orbital regimes.

## Abstract

The classical geodesics of timelike particles in Schwarzschild spacetime is analyzed according to the particle starting radius $r$, velocity $v$ and angle $\alpha$ against the radial outward direction in the reference system of an local static observer. The region of escape, bound and capture orbits in the parameter space of $(r,~v,~\alpha)$ are solved using the three cases of the effective potential. It is found that generally for radius smaller than $4M$ or velocity larger than $c/\sqrt{2}$ there will be no bound orbits. While for fixed radius larger than $4M$ (or velocity smaller than $c/\sqrt{2}$), as velocity (or radius) increase from zero (or $2M$), the particle is always captured until a critical value $v_{\mathrm{crit1}}$ (or $r_{\mathrm{crit1}}$) when the bound orbit start to appear around $\alpha=\pi/2$ between a double-napped cone structure. As the velocity (or radius) increases to another critical value $v_{\mathrm{crit2}}$ (or $r_{\mathrm{crit2}}$) then the bound directions and escape directions in the outward cone become escape directions, leaving only the inward cone separating the capture and bound directions. The angle of this cone will increase to its asymptotic value as velocity (or radius) increases to its asymptotic value. The implication of these results in shadow of black holes formed by massive particles, in black hole accretion and in spacecraft navigation is briefly discussed.