Can someone please explain to me what the Christoffel symbols symbols are?

In summary, General Relativity deals with the curvature of space-time and the Riemann curvature tensor is used to describe this curvature by parallel transporting a vector around a closed path. The Christoffel symbols are used to describe the infinitesimal rate of changes in the parallel transport direction. In layman's terms, this means that the coordinates on a curved manifold do not behave the same way as coordinates on a flat plane, and the curvature tensor captures this difference. The Christoffel symbols also play a role in defining the rule of parallel transport, which is important in General Relativity due to its metric compatibility. For further reading on these concepts, "Riemannian manifolds: an introduction to curvature" by John M. Lee is
  • #1
zeromodz
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I am trying to understand everything about general relativity. I know that they have to do with how the Riemann curvature tensor uses parallel transporting a vector around a closed path. I really just don't understand the mathematics behind it. Thank you. I prefer layman's terms.
 
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  • #2
You should start with the Euclidean plane in polar coordinates. You take a vector at some point (better not the origin) and transport it parallelly to some other point. Its polar coordinates will change (do this!). Christoffel symbols describe the infinitesimal rate of such changes for the parallel transport in different directions. In the case of the Euclidean plane parallel transport does not depend on the path. Everybody knows what "parallel" in this case. This fact is expressed in vanishing of the curvature tensor, which is expressed in terms of Christoffel symbols and their derivatives. But not always this the case. Sphere with its natural parallel transport law has a nonzero curvature tenor.
 
  • #3
The best place to read about these things (not in layman's terms) is "Riemannian manifolds: an introduction to curvature", by John M. Lee.
 
  • #4
zeromodz said:
I am trying to understand everything about general relativity. I know that they have to do with how the Riemann curvature tensor uses parallel transporting a vector around a closed path. I really just don't understand the mathematics behind it. Thank you. I prefer layman's terms.
I think of the Christoffel symbols describing how the coordinates are curved. This is opposed to the curvature tensor which describes how the manifold is curved. In a flat manifold it is still possible to use curved coordinates, but in a curved manifold it is not possible to use (globally) straight coordinates.
 
  • #5
In addition to the above, one can think of the Christoffel symbols in a specific coordinate system as defining the rule of parallel transport. In GR, the rule of parallel transport is defined by its metric compatibility.
 

1. What are the Christoffel symbols and what do they represent?

The Christoffel symbols, also known as the Christoffel symbols of the first kind, are a set of mathematical quantities that represent the connection between coordinates in a curved space. They are used in the study of differential geometry, specifically in the theory of connections on manifolds.

2. How are the Christoffel symbols calculated?

The Christoffel symbols can be calculated using the metric tensor, which describes the distance between points in a curved space. The formula for calculating the Christoffel symbols involves taking derivatives of the metric tensor and performing matrix operations.

3. What is the significance of the Christoffel symbols in general relativity?

In general relativity, the Christoffel symbols play a crucial role in the calculation of the geodesic equation, which describes the motion of objects in a curved space. They also appear in the Einstein field equations, which relate the curvature of space-time to the distribution of matter and energy.

4. Are the Christoffel symbols used in any other areas of science?

Yes, the Christoffel symbols are used in various fields of physics and mathematics, including differential geometry, cosmology, and mechanics. They are also used in the study of black holes, gravitational waves, and other phenomena in general relativity.

5. Can you provide a real-world example of the use of the Christoffel symbols?

One example of the use of the Christoffel symbols is in the calculation of the precession of Mercury's orbit. The curvature of space-time caused by the Sun's mass can be described using the Christoffel symbols, and this contributes to the observed precession of Mercury's orbit, which cannot be explained by Newtonian mechanics alone.

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