The Lie Derivative: A Fundamental Definition

1. The Intuition: Flows and Lie Dragging

Imagine a fluid flowing on the surface of our manifold. The velocity of the fluid at each point defines a vector field, let's call it $V$. The path a small particle follows in this fluid is an integral curve of $V$. The collection of all these integral curves defines the flow of the vector field, which we can represent by a one-parameter family of diffeomorphisms, ${\varphi }_t$. For a given time $t$, the map ${\varphi }_t$ takes every point $p$ on the manifold and moves it to the point ${\varphi }_t(p)$ by following the integral curve of $V$ for a parameter distance $t$.

Now, suppose there is another field on the manifold, say a temperature field (a scalar field) or an electric field (a vector field). How does this second field change from the perspective of an observer being carried along by the flow of $V$? To answer this, we need a way to compare the field at point $p$ with the field at the nearby point ${\varphi }_t(p)$. But these fields live in different tangent spaces.

The solution is to "drag" the field from ${\varphi }_t(p)$ back to $p$ using the inverse flow map, ${\varphi }_{-t}$. This process is called Lie dragging. Once we have dragged the field back to the original point $p$, we can compare it to the original field at $p$ and see how it has changed.

The following text formalizes this idea of "dragging" objects—points, functions, and vector fields—along the flow of a vector field. This is the essential machinery for defining the Lie derivative.

2. The Formal Definition of the Lie Derivative

With the concept of Lie dragging established, the formal definition of the Lie derivative becomes a natural extension of the standard definition of a derivative from calculus. We want to find the rate of change of a tensor field. We do this by dragging the field at a nearby point ${\varphi }_t(p)$ back to $p$, finding the difference from the original field at $p$, dividing by the parameter $t$, and taking the limit as $t\to 0$.

Definition: The Lie Derivative

The Lie derivative of a tensor field $T$ with respect to a vector field $V$ is the tensor field ${\mathcal{L}}_{V}T$ defined by:
$({\mathcal{L}}_{V}T{)}_p={\lim }_{t\to 0}\frac{({\varphi }_t^{\ast }T{)}_{{\varphi }_t(p)}-T_p}{t}$
Here, ${\varphi }_t^{\ast }T$ denotes the pullback of the tensor field $T$ by the flow map ${\varphi }_t$. A more explicit way to write this for a vector field $Y$ is:
$({\mathcal{L}}_{V}Y{)}_p={\lim }_{t\to 0}\frac{({\varphi }_{-t}{)}_{\ast }(Y_{{\varphi }_t(p)})-Y_p}{t}$
where $({\varphi }_{-t}{)}_{\ast }$ is the pushforward map that does the "dragging" from $T_{{\varphi }_t(p)}M$ back to $T_{p}M$.

Let's see this definition in action, starting with the simplest case.

The result ${\mathcal{L}}_{V}f=V(f)$ is very satisfying. It confirms that the Lie derivative is a sensible generalization of the directional derivative you already know. The vector field $V$, which we defined as a derivation on functions, is precisely the Lie derivative operator for scalar fields.

3. The Lie Bracket: A Geometric and Computational Shortcut

While the limit definition is conceptually fundamental, it is cumbersome for calculations. Fortunately, for vector fields, there is a remarkably simple and powerful equivalent formulation.

Theorem: The Lie derivative of a vector field $W$ with respect to a vector field $V$ is their Lie bracket (or commutator):
${\mathcal{L}}_{V}W=[V,W]\equiv VW-WV$
Since vector fields are differential operators, $VW$ means "first apply $W$, then apply $V$". This expression makes it clear that ${\mathcal{L}}_{V}W$ is also a vector field.

Geometric Interpretation of the Lie Bracket

The Lie bracket has a beautiful geometric meaning. It measures the failure of infinitesimal flows to commute. Imagine starting at a point $p$ and trying to trace out an infinitesimal parallelogram.

1. Flow along $V$ by $ϵ$.
2. Then, flow along $W$ by $ϵ$.

Now, compare this to the path where you flow in the opposite order:

1. Flow along $W$ by $ϵ$.
2. Then, flow along $V$ by $ϵ$.

If you end up at the same point, the flows commute, and $[V,W]=0$. If you don't, the vector connecting the two endpoints is proportional to $[V,W]$.

Proving the Equivalence

Connecting the intuitive limit definition to the elegant Lie bracket formula is a cornerstone of understanding the Lie derivative. Given your interest in formal proofs, the following reading provides a detailed derivation.

4. Generalization and Component Formulas

The Lie derivative can be defined for any $(r,s)$ tensor field. The principle remains the same: use the flow to drag the tensor and its arguments back to the starting point.

• For a (0,1) tensor (a one-form) $\omega$, its Lie derivative ${\mathcal{L}}_V\omega$ is defined such that for any vector field $W$:
  $({\mathcal{L}}_V\omega )(W)={\mathcal{L}}_V(\omega (W))-\omega ({\mathcal{L}}_{V}W)$
  This is a direct consequence of the Leibniz rule, ${\mathcal{L}}_V(T_1\otimes T_2)=({\mathcal{L}}_V{T}_1)\otimes T_2+T_1\otimes ({\mathcal{L}}_V{T}_2)$, applied to the scalar $\omega (W)$.

The following video provides a clear derivation of the component formulas for the Lie derivatives of scalar fields, vector fields, and one-forms.

For a general $(r,s)$ tensor $T$, the component formula can be constructed by applying the Leibniz rule. For example, for a $(1,1)$ tensor with components $T_{\nu }^{\mu }$, the Lie derivative has components:
$({\mathcal{L}}_{V}T{)}_{\nu }^{\mu }=V^{\rho }{\partial }_{\rho }{T}_{\nu }^{\mu }-T_{\nu }^{\rho }{\partial }_{\rho }{V}^{\mu }+T_{\rho }^{\mu }{\partial }_{\nu }{V}^{\rho }$
Notice the pattern:

1. The first term is the directional derivative of the tensor's components along $V$.
2. Then, there is a "correction" term for each index. It involves the derivative of the vector field $V$ and the tensor components. A contravariant index gets a minus sign, and a covariant index gets a plus sign.

Conclusion

In this lesson, we have developed a powerful and intuitive notion of differentiation on a manifold that requires no extra structure like a metric.

Key Takeaways:

• The Lie derivative ${\mathcal{L}}_{V}T$ measures the change of a tensor field $T$ along the flow of a vector field $V$.
• It is conceptually defined by "dragging" the tensor field from a nearby point back to the original point and taking a limit.
• For vector fields, the Lie derivative is equivalent to the Lie bracket, ${\mathcal{L}}_{V}W=[V,W]$, which geometrically measures the failure of their flows to commute.
• The Lie derivative obeys the Leibniz rule and can be applied to any tensor field, with component formulas that follow a consistent pattern.
• Crucially, this entire construction is metric-independent. It relies only on the differentiable structure of the manifold.

We now have one way to differentiate tensors. However, the Lie derivative has a limitation for physics: the Lie derivative of a tensor of type $(r,s)$ is another tensor of type $(r,s)$. It doesn't allow for the kinds of derivatives you're familiar with from vector calculus, like gradient ($f\to \nabla f$), divergence ($\vec{F}\to \nabla \cdot \vec{F}$), or curl. For that, we need a different kind of derivative.

In our next lesson, we will explore differential forms and the exterior derivative. The exterior derivative is another metric-independent derivative, but one that specifically acts on a special class of antisymmetric tensors called differential forms. It naturally generalizes the gradient, curl, and divergence operations into a unified framework.