Are Christoffel Symbols Considered Tensors?

[Karl G.]

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: G_a'b' = L^c_a'L^d_b'G_cd
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?

 

[cristo]

No, Christoffel symbols are not tensors.

 

[Karl G.]

Oh ... thought they were

 

[Mentz114]

Christoffel symbols make the covariant derivative a tensor. Try transforming one.

 

[dx]

I haven't seen a single book on GR which introduces the Christoffel symbols without immediately pointing out that they're not tensors.

 

[samalkhaiat]

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: G_a'b' = L^c_a'L^d_b'G_cd
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