# Schwarzschild affine connection

• alle.fabbri
In summary, the Schwarzschild affine connection is a mathematical concept in general relativity that describes the curvature of spacetime around a non-rotating spherical mass, such as a black hole. It is derived from the metric tensor and is used to calculate the geodesics of particles in curved spacetime. The main properties of the Schwarzschild affine connection include symmetry, torsion-freeness, and divergence-freeness. It differs from other affine connections in its applicability to spherically symmetric masses and its relationship to the Kerr affine connection. Practical applications of the Schwarzschild affine connection include studying black holes, cosmology, and astrophysics.
alle.fabbri
Hi all!
Anyone knows where to find, online or on a book, the affine connection for the schwarzschild metric?
Thanks

Do you mean the Christoffel symbols?

Yep...

http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html#Section5.2

Last edited by a moderator:
thanks all

## 1. What is the Schwarzschild affine connection?

The Schwarzschild affine connection is a mathematical concept used in general relativity to describe the curvature of spacetime around a non-rotating spherical mass, such as a black hole. It is named after the German physicist Karl Schwarzschild, who first studied this phenomenon in 1916.

## 2. How is the Schwarzschild affine connection related to general relativity?

In general relativity, the curvature of spacetime is described by a mathematical object called the metric tensor. The Schwarzschild affine connection is derived from this metric tensor and is used to calculate the geodesics, or shortest paths, that particles will follow in the curved spacetime around a massive object.

## 3. What are the main properties of the Schwarzschild affine connection?

The Schwarzschild affine connection is symmetric, meaning that it is the same in both directions along a geodesic. It is also torsion-free, which means that the path followed by a particle is not affected by its rotation or spin. Additionally, the Schwarzschild affine connection is divergence-free, meaning that the flow of matter or energy does not change the curvature of spacetime.

## 4. How does the Schwarzschild affine connection differ from other affine connections?

The Schwarzschild affine connection is unique in that it describes the curvature of spacetime around a spherically symmetric mass, while other affine connections may describe different types of curvature or be applicable in different scenarios. Additionally, the Schwarzschild affine connection is a special case of the more general Kerr affine connection, which describes the curvature around a rotating mass.

## 5. What are some practical applications of the Schwarzschild affine connection?

The Schwarzschild affine connection is primarily used in theoretical physics, particularly in the study of black holes and the behavior of matter and light in their vicinity. It has also been applied in other fields, such as cosmology and astrophysics, to better understand the behavior of massive objects and their effects on surrounding spacetime.

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