# Metric Connection from Geodesic Equation

• alex3
In summary, the conversation discusses the use of the Euler-Lagrange equations to derive equations of motion for a two-dimensional metric. These equations are shown to describe geodesics, and the non-zero terms of the metric connection are identified. The question is raised about the missing factor of two in one of the terms, which is clarified through the understanding that the geodesic equation sums over both theta and phi.
alex3
For the following two-dimensional metric

$$ds^2 = a^2(d\theta^2 + \sin^2{\theta}d\phi^2)$$

using the Euler-Lagrange equations reveal the following equations of motion

$$\ddot{\phi} + 2\frac{\cos{\theta}}{\sin{\theta}}\dot{\theta}\dot{\phi} = 0$$
$$\ddot{\theta} - \sin{\theta}\cos{\theta}\dot{\phi}^2 = 0$$

Using the general geodesic equation form $\ddot{x}^{\alpha} + \Gamma^{\alpha}_{\beta\gamma}\dot{x}^\beta\dot{x}^{\gamma}=0$, we infer that the equations derived describe geodesics. This shows that the only non-zero terms of the metric connection $\Gamma^{\alpha}_{\beta\gamma}$ are

$$\Gamma^{\theta}_{\phi\phi} = -\sin{\theta}\cos{\theta},\quad \Gamma^{\phi}_{\theta\phi} = \Gamma^{\phi}_{\phi\theta} = \frac{\cos{\theta}}{\sin{\theta}}$$

My problem is comprehending where the factor of two has gone for the $\Gamma^{\phi}_{\phi\theta}$ term. Is it due to fact that it's a coefficient of a mixed derivative, why is that?

You have to sum over theta and phi, so the geodesic equation contains the two terms

$$\Gamma^{\phi}_{\theta\phi}\dot{\theta}\dot{\phi} + \Gamma^{\phi}_{\phi\theta}\dot{\phi}\dot{\theta}$$

Of course! Thank you, it's very clear now.

## 1. What is the Metric Connection from Geodesic Equation?

The Metric Connection from Geodesic Equation is a mathematical relationship that describes how the curvature of a space-time manifold is related to the metric tensor, which defines the distance between any two points in the space-time. It is a fundamental concept in the theory of general relativity, which explains the force of gravity as a result of the curvature of space-time.

## 2. How is the Metric Connection from Geodesic Equation derived?

The Metric Connection from Geodesic Equation is derived from the Einstein Field Equations, which relate the curvature of space-time to the distribution of matter and energy. By solving these equations, the Metric Connection can be calculated, allowing us to understand how matter and energy affect the curvature of space-time.

## 3. What is the significance of the Metric Connection from Geodesic Equation?

The Metric Connection from Geodesic Equation is significant because it provides a mathematical framework for understanding the effects of gravity on the motion of objects in space-time. It allows us to make predictions about the behavior of particles, planets, and even light in the presence of massive objects, such as stars and black holes.

## 4. How does the Metric Connection from Geodesic Equation relate to the concept of geodesics?

The Metric Connection from Geodesic Equation is directly related to the concept of geodesics, which are the shortest paths between two points in a curved space-time. The equation tells us how these paths are affected by the curvature of space-time and the distribution of matter and energy.

## 5. Can the Metric Connection from Geodesic Equation be applied to other theories besides general relativity?

While the Metric Connection from Geodesic Equation is primarily used in the theory of general relativity, it can also be applied to other theories, such as gauge theories and string theory. In these theories, the equation is used to describe how the curvature of space-time is related to other fundamental forces, such as the strong and weak nuclear forces.

Replies
11
Views
811
Replies
8
Views
1K
Replies
14
Views
731
Replies
4
Views
1K
Replies
7
Views
874
Replies
10
Views
2K
Replies
3
Views
1K
Replies
10
Views
1K
Replies
9
Views
2K
Replies
36
Views
4K