Description

how to get the geodesic equation ?

# How to Obtain the Geodesic Equation

The concept of the geodesic equation is fundamental in both mathematics and physics, particularly in general relativity. It describes the path that objects follow when they are not subject to any additional forces, essentially representing the “straightest” path in curved spacetime. Let’s dive into how to obtain the geodesic equation and its significance.

## What is the Geodesic Equation?

The geodesic equation is a differential equation that describes the trajectory of a particle moving freely in curved space. In the context of general relativity, these paths are those of particles with no proper acceleration, influenced only by the curvature of spacetime.

## The Mathematical Basis

The geodesic equation can be derived using the principle of least action. In essence, the path that a particle takes is the one that minimizes the Riemannian line element for a given metric. Mathematically, this is expressed as:

[ frac{d^2x^mu}{dlambda^2} + Gamma^mu_{nusigma} frac{dx^nu}{dlambda} frac{dx^sigma}{dlambda} = 0 ]

Where:
– ( frac{d^2x^mu}{dlambda^2} ) is the second derivative of the position vector with respect to an affine parameter (lambda),
– (Gamma^mu_{nusigma}) are the Christoffel symbols, which are required for describing the curvature of spacetime.

## Applications in General Relativity

In general relativity, the geodesic equation is essential for understanding the behavior of particles and light in curved spacetime. For instance, the Schwarzschild orbits and other celestial mechanics can be derived using this equation. The use of Christoffel symbols and the metric tensor ( g_{munu} ) is crucial in these calculations.

### Deriving the Geodesic Equation from the Principle of Least Action

One practical approach to deriving the geodesic equation is from the principle of least action. By minimizing the action integral, we can arrive at dynamic equations for the particle’s motion in spacetime. This derivatives give us the familiar form of the geodesic equation:

[ nabla_lambda T^mu = frac{dT^mu}{dlambda} + Gamma^mu_{nusigma} T^nu frac{dx^sigma}{dlambda} = 0 ]

This equation can be thought of as an extension of Newton’s First Law, adapted to curved spacetimes.

### Practical Examples

– **Riemannian Line Element**: The Riemannian line element, (ds^2 = g_{munu} dx^mu dx^nu), is minimized by the geodesic equation.
– **Schwarzschild Orbits**: Using the Schwarzschild metric, one can derive the orbit equations for planets and satellites moving around a massive body like the Sun.

## Resources for Further Study

To deepen your understanding, you can explore several sources:

– **Wikipedia’s “Solving the Geodesic Equations”**: Offers a general overview and foundational concepts.
[Solving the Geodesic Equations – Wikipedia](https://en.wikipedia.org/wiki/Solving_the_geodesic_equations)

– **Physics LibreTexts**:
– **Geodesic Equation**: Introductory explanation focusing on summation over indices and the free index concept.
[5.8: The Geodesic Equation – Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/05:_Curvature/5.08:_The_Geodesic_Equation)
– **Schwarzschild Orbits**: Application to specific astrophysical scenarios.
[3.1: The Geodesic Equation – Physics LibreTexts](https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/03:_Schwarzschild_Orbits/3.01:_The_Geodesic_Equation)

– **Geodesic Equation from the Principle of Least Action**: Detailed derivation and practical examples.
[Geodesic equation from the principle of least action](https://cadabra.science/qa/1331/how-to-obtain-the-geodesic-equation-from-the-action)

– **Visualization and Tutorials**:
– YouTube: Offers a video breakdown of how to derive the geodesic equation via the action approach.
[Geodesic Equation Derivation | General Relativity](https://www.youtube.com/watch?v=PmmU6i0SMic)

## Conclusion

The geodesic equation is a powerful tool in both theoretical physics and applied mathematics. By understanding how to derive and apply it, one can delve deeper into the principles of general relativity and the nature of spacetime. Whether through academic articles, textbooks, or video tutorials, there are many resources available to help you master this crucial concept.