Lie derivative
The Lie derivative of a vector field V ⃗ \vec V V γ \gamma γ A ⃗ \vec A A γ \gamma γ
Let u ( λ ) u(\lambda) u ( λ ) γ \gamma γ λ \lambda λ V ⃗ \vec V V A A A x α x^\alpha x α B B B ( x + d x ) α = ( x ′ ) α (x+dx)^\alpha = (x')^\alpha ( x + d x ) α = ( x ′ ) α
V LT α ( A → B ) = ∂ ( x ′ ) α ∂ x β V β ( A ) = ( δ β α + ( ∂ β u α ) d λ ) V β ( A ) = V α ( A ) + ( ∂ β u α ) V α ( A ) d λ .
\begin{align*}
V^\alpha_\text{LT}(A \to B) &= \frac{\partial (x')^{\alpha}}{\partial
x^{\beta}} V^\beta(A) \\
&= (\delta^\alpha_\beta + (\partial_\beta u^\alpha)d\lambda)
V^\beta(A) \\
&= V^\alpha(A) + (\partial_\beta u^\alpha) V^\alpha(A) d\lambda.
\end{align*}
V LT α ( A → B ) = ∂ x β ∂ ( x ′ ) α V β ( A ) = ( δ β α + ( ∂ β u α ) d λ ) V β ( A ) = V α ( A ) + ( ∂ β u α ) V α ( A ) d λ . Then we define the Lie derivative L u ⃗ \mathcal L_{\vec u} L u
L u ⃗ V α = V α ( B ) − V LT α ( A → B ) d λ = u β ( ∂ β V α ) − V β ( ∂ β u α ) .
\begin{align*}
\mathcal L_{\vec u} V^\alpha &= \frac{V^\alpha(B) - V^\alpha_\text{LT}(A\to
B)}{d\lambda} \\
&= u^\beta (\partial_\beta V^\alpha) - V^\beta(\partial_\beta u^\alpha).
\end{align*}
L u V α = d λ V α ( B ) − V LT α ( A → B ) = u β ( ∂ β V α ) − V β ( ∂ β u α ) . By the equivalence principle, we can just as well write the Lie
derivative in terms of the covariant derivative ∇ β \nabla_\beta ∇ β