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.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.