Graduate Two Covariant Derivatives (Chain Rule)?

[berlinspeed]

TL;DR

Failed find information on the internet, really appreciate any help.

Summary: Failed find information on the internet, really appreciate any help.

Can someone tell me what is ∇_ϒ∇_δ𝒆_β? It seems to be equal to 𝒆_μΓ^μ_βδ,ϒ+(𝒆_νΓ^ν_μϒ)Γ^μ_βδ. Is this some sort of chain rule or is it by any means called anything?

 

[Orodruin]

What do you get when you try to compute it yourself?

 

[berlinspeed]

What do you get when you try to compute it yourself?

At first look I just get the second term, so still trying to figure out where went wrong..

 

[Orodruin]

Please show your work.

 

[berlinspeed]

Please show your work.

Okay I think where went wrong is I need to expand ∇_ϒ(𝒆_μΓ^μ_βδ), which now looks like something to apply chain rule to, and if I do that I get the answer except shouldn't the first term be 𝒆_μΓ^μ_βδ;ϒ? The "," stands for partial derivative?

 

[Orodruin]

Yes, the , stands for a regular partial derivative. You are here including the basis vector $e_\mu$ in your notation. When you do that you are using a notation where anything multiplying it is just a scalar function. It holds that $\nabla_a e_b = \Gamma_{ab}^c e_c$ by definition. The covariant derivative is linear and satisfies the product rule (this is not chain rule)
$$
\nabla_a (fV) = V \nabla_a f + f \nabla_a V,
$$
where $f$ is a scalar field and $V$ is a vector. Note that $\nabla_a f = \partial_a f$ for any scalar field. In your case, therefore
$$
\nabla_a(\Gamma^c_{bd}e_c) = \Gamma^c_{bd} \nabla_a e_c + e_c \nabla_a \Gamma^c_{bd}
= \Gamma^c_{bd} \Gamma^f_{ac} e_f + e_c \Gamma^c_{bd,a}.
$$

Do not confuse this notation with the usual notation without writing out the vector basis, where we typically use the convention
$$
\nabla_a V^b \equiv (\nabla_a V)^b,
$$
i.e., the LHS is defined as the $b$-component of $\nabla_a V$. Actually writing out the basis in the RHS gives you
$$
(\nabla_a V)^b = [\nabla_a (V^c e_c)]^b = [V^c \nabla_a e_c]^b + [e_c \nabla_a V^c]^b
= V^c [\Gamma^d_{ac} e_d]^b + [V^c_{,a} e_c]^b
= V^c \Gamma^b_{ac} + V^b_{,a},
$$
since $[e_c]^b = \delta^b_c$.

 

[berlinspeed]

Yes, the , stands for a regular partial derivative. You are here including the basis vector $e_\mu$ in your notation. When you do that you are using a notation where anything multiplying it is just a scalar function. It holds that $\nabla_a e_b = \Gamma_{ab}^c e_c$ by definition. The covariant derivative is linear and satisfies the product rule (this is not chain rule)
$$
\nabla_a (fV) = V \nabla_a f + f \nabla_a V,
$$
where $f$ is a scalar field and $V$ is a vector. Note that $\nabla_a f = \partial_a f$ for any scalar field. In your case, therefore
$$
\nabla_a(\Gamma^c_{bd}e_c) = \Gamma^c_{bd} \nabla_a e_c + e_c \nabla_a \Gamma^c_{bd}
= \Gamma^c_{bd} \Gamma^f_{ac} e_f + e_c \Gamma^c_{bd,a}.
$$

Do not confuse this notation with the usual notation without writing out the vector basis, where we typically use the convention
$$
\nabla_a V^b \equiv (\nabla_a V)^b,
$$
i.e., the LHS is defined as the $b$-component of $\nabla_a V$. Actually writing out the basis in the RHS gives you
$$
(\nabla_a V)^b = [\nabla_a (V^c e_c)]^b = [V^c \nabla_a e_c]^b + [e_c \nabla_a V^c]^b
= V^c [\Gamma^d_{ac} e_d]^b + [V^c_{,a} e_c]^b
= V^c \Gamma^b_{ac} + V^b_{,a},
$$
since $[e_c]^b = \delta^b_c$.

Okay cleared now, thanks much.

 

[berlinspeed]

Yes, the , stands for a regular partial derivative. You are here including the basis vector $e_\mu$ in your notation. When you do that you are using a notation where anything multiplying it is just a scalar function. It holds that $\nabla_a e_b = \Gamma_{ab}^c e_c$ by definition. The covariant derivative is linear and satisfies the product rule (this is not chain rule)
$$
\nabla_a (fV) = V \nabla_a f + f \nabla_a V,
$$
where $f$ is a scalar field and $V$ is a vector. Note that $\nabla_a f = \partial_a f$ for any scalar field. In your case, therefore
$$
\nabla_a(\Gamma^c_{bd}e_c) = \Gamma^c_{bd} \nabla_a e_c + e_c \nabla_a \Gamma^c_{bd}
= \Gamma^c_{bd} \Gamma^f_{ac} e_f + e_c \Gamma^c_{bd,a}.
$$

Do not confuse this notation with the usual notation without writing out the vector basis, where we typically use the convention
$$
\nabla_a V^b \equiv (\nabla_a V)^b,
$$
i.e., the LHS is defined as the $b$-component of $\nabla_a V$. Actually writing out the basis in the RHS gives you
$$
(\nabla_a V)^b = [\nabla_a (V^c e_c)]^b = [V^c \nabla_a e_c]^b + [e_c \nabla_a V^c]^b
= V^c [\Gamma^d_{ac} e_d]^b + [V^c_{,a} e_c]^b
= V^c \Gamma^b_{ac} + V^b_{,a},
$$
since $[e_c]^b = \delta^b_c$.

Btw oddly enough it is named "chain rule" in the Charles & Wheeler book..