Solving Differential Equations in General Relativity

In summary, the discussion revolves around solving differential equations in general relativity, which is not a specific topic within the field. The best way to learn about solving differential equations is to study a math textbook. If there is a particular case related to general relativity that one is struggling with, they can open a new thread asking for help with that specific case. Giving lessons on general mathematical techniques is outside the scope of the forum.
  • #1
Jianbing_Shao
102
2
TL;DR Summary
differentail equations, parallel transport equations
In genaral relativity, how to solve differential equations is seldom be discussed. I want to know how to sole the differential equations like this:
$$\partial_kv^i(x)+\Gamma^i_{jk}(x)v^j(x)=\partial_kA^i(x)$$
Here ##\Gamma^i_{jk}(x)## is connection field on a manifod and ##A^i(x)## is a vector field on the manifold. then how to get ##v^i(x)##, Perhapa we can not find a global vector field which can satisfy the differential equations, But we can still find a solution on a partilular curve. then how to find it?
 
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  • #2
Jianbing_Shao said:
In genaral relativity, how to solve differential equations is seldom be discussed.

That's because it's a general mathematical subject, not specific to general relativity. So it's discussed under the general math subject of solving differential equations.

Jianbing_Shao said:
I want to know how to sole the differential equations like this

This question is much, much too general for a forum discussion. You need to study a math textbook on solving differential equations. If there is a particular case that arises in a particular problem in general relativity and you're having trouble solving that particular case, you can open a new thread asking about that particular case (and showing your attempts at solving it and where you're getting stuck). But this thread is closed since giving classes in general math techniques is outside the scope of PF.
 

Related to Solving Differential Equations in General Relativity

1. What is the purpose of solving differential equations in general relativity?

The purpose of solving differential equations in general relativity is to understand and describe the behavior of matter and energy in the presence of strong gravitational fields. This is important in understanding the structure of the universe and predicting the motion of celestial bodies.

2. What are the main challenges in solving differential equations in general relativity?

The main challenges in solving differential equations in general relativity include the complexity of the equations, the need for specialized mathematical techniques, and the difficulty in obtaining exact solutions. Additionally, the equations can be highly non-linear and require advanced computational methods to solve.

3. How do differential equations in general relativity differ from those in classical mechanics?

Differential equations in general relativity differ from those in classical mechanics in that they take into account the effects of gravity and the curvature of space-time. This means that the equations are more complex and require a different approach to solving them.

4. What are some applications of solving differential equations in general relativity?

Solving differential equations in general relativity has many practical applications, such as predicting the motion of planets and other celestial bodies, understanding the behavior of black holes, and studying the evolution of the universe. It also has applications in the fields of astrophysics, cosmology, and gravitational wave detection.

5. What are some common techniques used to solve differential equations in general relativity?

Some common techniques used to solve differential equations in general relativity include numerical methods, perturbation theory, and the use of computer simulations. Other techniques such as the use of symmetries and special coordinate systems can also be helpful in simplifying the equations and finding solutions.

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