Are Christoffel Symbols Considered Tensors?

In summary, the conversation discusses the confusion surrounding Christofel symbols being called tensors. The speaker notes that to be considered a tensor, the tensor must obey the standard component transformation law, but the Christofel symbols do not. However, they do transform as tensors under linear coordinate transformations.
  • #1
Karl G.
40
0
I'm sort of confused about Christofel symbols being called tensors. I thought that to be considered a tensor, the tensor had to obey the standard component transformation law. For example: Ga'b' = Lca'Ldb'Gcd
But the Christofel symbols don't obey this transformation law; their components transform in a different way (I would like to show it, but I don't know how; see exercise 10.3 of MWT).
What gives?
 
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  • #2
No, Christoffel symbols are not tensors.
 
  • #3
Oh ... thought they were
 
  • #4
Christoffel symbols make the covariant derivative a tensor. Try transforming one.
 
  • #5
I haven't seen a single book on GR which introduces the Christoffel symbols without immediately pointing out that they're not tensors.
 
  • #6
Karl G. said:
I'm sort of confused about Christofel symbols being called tensors. I thought that to be considered a tensor, the tensor had to obey the standard component transformation law. For example: Ga'b' = Lca'Ldb'Gcd
But the Christofel symbols don't obey this transformation law; their components transform in a different way (I would like to show it, but I don't know how; see exercise 10.3 of MWT).
What gives?

They transform inhomogeneously under general coordinates transformations, i.e., not tensors. However, the inhomogeneous term in the transformation law vanishes if the coordinate transformations are LINEAR. So, they do transform as tensors with respect to all linear coordinate transformations.

sam
 

1. What are Christoffel symbols tensors?

Christoffel symbols tensors are mathematical objects used in differential geometry to describe the curvature of a manifold. They represent the connection between tangent vectors at different points on a manifold, and are also used to calculate the covariant derivative of a vector field.

2. How are Christoffel symbols tensors used in physics?

In physics, Christoffel symbols tensors are used to describe the curvature of spacetime in Einstein's theory of general relativity. They are also used in other areas of physics, such as fluid dynamics and electromagnetism, to describe the behavior of objects in curved spaces.

3. What is the relationship between Christoffel symbols tensors and the metric tensor?

The Christoffel symbols tensors are directly related to the metric tensor, which defines the distance and angle measurements on a manifold. They are used to calculate the components of the metric tensor in different coordinate systems, and can also be used to raise and lower indices on tensors.

4. How are Christoffel symbols tensors calculated?

The Christoffel symbols tensors are calculated using the metric tensor and its derivatives. They can be calculated analytically or numerically, and their values depend on the specific curvature of the manifold in question.

5. What is the significance of the Christoffel symbols tensors in mathematics?

In mathematics, Christoffel symbols tensors are important for understanding the geometric properties of manifolds. They are also used in differential geometry to study the behavior of curves and surfaces, and play a crucial role in the formulation of the fundamental theorems of Riemannian geometry.

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