Discussion Overview
The discussion revolves around finding geodesics in the dynamic Ellis orbits metric, specifically the metric given by ##ds^2=-dt^2+dp^2+(5p^2+4t^2)d\phi^2##. Participants explore various methods for solving the geodesic equations, including the geodesic Lagrangian method, and consider the implications of the metric's properties on these methods.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant inquires about the possibility of finding geodesics analytically in the given metric.
- Another participant questions the applicability of techniques discussed in a previous thread on geodesics.
- Several participants reference the Wikipedia article on Ellis Wormholes to clarify the metric in question.
- It is noted that ##\partial_\phi## is a Killing field, which may provide a conserved quantity useful for simplifying the geodesic equations.
- Participants discuss the geodesic Lagrangian method and its potential advantages over brute force methods, particularly due to the metric's structure.
- There is uncertainty about the existence of Killing fields for ##t## and ##p##, which complicates the analysis.
- One participant expresses a desire to find a way to integrate geodesics exactly rather than using numerical methods like Euler or Runge-Kutta.
- Concerns are raised about the feasibility of integrating certain differential equations exactly, emphasizing the need to analyze the equations themselves.
- Discussion includes whether the Lagrangian method is applicable to both timelike and null geodesics, with some technical complications noted for the latter.
- Participants debate the merits of different approaches to solving the geodesic equations, with suggestions to experiment with various methods.
Areas of Agreement / Disagreement
Participants express differing views on the applicability of previous techniques and the effectiveness of various methods for finding geodesics. No consensus is reached on the best approach or the solvability of the equations involved.
Contextual Notes
Participants highlight the complexity of the geodesic equations due to the specific form of the metric and the presence of only one function of the coordinates. There is also mention of the potential for certain Christoffel symbols to vanish, affecting the analysis.
Who May Find This Useful
This discussion may be of interest to those studying general relativity, particularly in the context of wormhole metrics and geodesic calculations.