Analytic Solutions for Geodesic Motion in Static Axially Symmetric Spacetime

TL;DR

The paper develops a method to obtain static, axially symmetric vacuum solutions to the Einstein field equations, yielding two general families and five specific solutions. It then derives and analyzes the full set of geodesic equations for these spacetimes, identifying when circular or z-direction geodesics exist and presenting explicit or semi-analytic forms where possible. A notable result is that certain z-directed geodesics in the γ-metric (notably along the axis) exhibit jet-like, accelerated motion that the authors relate to spacetime geometry rather than nongeodesic forces. The work provides a detailed catalog of geodesic behaviors across multiple solutions, highlighting conditions under which circular, helical, or radial motions arise and their potential physical interpretations in relativistic jet contexts.

Abstract

A procedure to find static axially symmetric solutions to the Einstein field equations is presented. We obtained two general solutions and five particular solutions, which depend on the existence conditions for circular and $z$ direction motion. Our endeavour consists making a thoroughrowly analysis of all the possible geodesics solutions stemming from this spacetime.

Figures (6)

• Figure 1: These figures show the time evolution of the $z$-geodesics of the Solution 1 and its velocity and acceleration of $z(\tau)$ at the axis ${\rho_0}=0$. We assume in these figures that $c_1=-1$, $c_2=1$, $c_3=0$, $C_1=1$, $C_2=1$, $C_3=1$ and $C_4=1$.
• Figure 2: These figures show the time evolution of the $z$-geodesics of the Solution 3 and its velocity and acceleration of $z(\tau)$ at the axis ${\rho_0}=0$. We assume in these figures that $C_1=3$, $C_2=1$, $C_3=1$, $C_4=1$, $C_5=1$ and $C_6=1$.
• Figure 3: These figures show the time evolution of the $z$-geodesics of the Solution 5b and its velocity and acceleration of $z(\tau)$ at the axis ${\rho_0}=1$. We assume in these figures that $C_1=1$, $C_2=1$, $C_3=1$, $C_4=1$, $C_5=1$ and $C_6=1$.
• Figure 4: These figures show the time evolution of the $z-\phi$ geodesics of the Solution 3 at the cylinder ${\rho_0}=1$. We assume in these figures that $C_1=1$, $C_2=-1$, $C_3=1$, $C_4=1$, $C_5=1$, $C_6=1$, $C_7=1/10$ and $C_8=1$.
• Figure 5: These figures show the time evolution of the $z-\phi$ geodesics of the Solution 5b at the cylinder ${\rho_0}=1$. We assume in these figures that $C_1=1$, $C_2=1$, $C_3=1$, $C_4=1$, $C_5=1$, $C_6=1$, $C_7=1$ and $C_8=1$.