- #1
Hunterc2429
- 3
- 0
- TL;DR Summary
- Hi all, I am currently working through Sean Carroll's "An Introduction to General Relativity," in the process, I am trying to numerically integrate and plot test particles using the geodesic equations provided. However, I want to parameterize them such that the azimuthal angle is the independent variable, but I am unsure of my results.
Hi all,
I am working through Sean Carroll's Textbook, particularly Chapter 5 regarding the Schwarzschild Solution. In this chapter, Energy and Angular Momentum are defined as follows:
Upon substitution into the four-norm one can derive that,
Using the chain rule,
And so, we have the following system of first-order geodesic equations,
1.
2.
3.
4.
Then, by the chain rule, I can divide each of these equations by (2.) to find,
1.
2.
3.
Now, what I am having trouble with is,
1. Is setting epsilon=0 for null trajectories enough of a modification to the system above to describe null trajectories?
2. How can I find an expression for ? It is my understanding that if a particle is timelike then so then and
So then, if timelike,
1.
2.
3.
4.
So then, if lightlike,
1.
2.
3.
Any advice would be greatly appreciated!
I am working through Sean Carroll's Textbook, particularly Chapter 5 regarding the Schwarzschild Solution. In this chapter, Energy and Angular Momentum are defined as follows:
Upon substitution into the four-norm
Using the chain rule,
And so, we have the following system of first-order geodesic equations,
1.
2.
3.
4.
Then, by the chain rule, I can divide each of these equations by (2.) to find,
1.
2.
3.
Now, what I am having trouble with is,
1. Is setting epsilon=0 for null trajectories enough of a modification to the system above to describe null trajectories?
2. How can I find an expression for
So then, if timelike,
1.
2.
3.
4.
So then, if lightlike,
1.
2.
3.
Any advice would be greatly appreciated!
Last edited: