Flows and Lie Derivatives

A vector field compares a tensor with itself at nearby points by generating a local flow and pulling the tensor back to the original point. The derivative of that comparison is the Lie derivative. It needs no metric or connection: it records change under an infinitesimal diffeomorphism and therefore provides the natural language for continuous geometric symmetry.

Required background. Tensor Fields and Natural Operations supplies pointwise tensor products, contractions, and symmetry operations; Smooth Maps, Pushforwards, and Pullbacks supplies differentials and pullbacks, and Vector Fields, Flows, and Lie Brackets supplies local flows and brackets. Affine Connections and Covariant Derivatives supplies the general connection framework used for the comparison below.

Helpful background. Calculus and Linear Algebra for Relativity supplies directional differentiation and linear maps; the commutator of vector-field derivations is developed here. Affine and Variational Geodesics provides the curve and affine-geodesic language used in later applications.

Flow-generated comparison

Let $X$ be a smooth vector field. Near any point, its integral curves define a local one-parameter family of diffeomorphisms $\Phi_s$ satisfying $\Phi_0=\mathrm{id}$ and

$$\frac{d}{ds}\Phi_s(p)=X_{\Phi_s(p)}.$$

For a tensor field $T$, the Lie derivative is

$$\mathcal L_XT=\left.\frac{d}{ds}\right|_{s=0}\Phi_s^*T.$$

The pullback returns the value at $\Phi_s(p)$ to $p$ before differentiating. For contravariant indices, $\Phi_s^*$ means the natural inverse pushforward, so the definition applies to every tensor type. Only a local flow near $s=0$ is required. A complete vector field, whose flow exists for every real $s$, is needed only when one wants a global one-parameter group. This distinction matters: on $\mathbb R$, $X=x^2\partial_x$ has flow $x(s)=x_0/(1-sx_0)$ and is not complete.

Carroll develops this pullback construction and its tensorial consequences in Carroll 1997, §5, pp. 135–138, eqs. (5.18)–(5.33).

Scalars, vectors, forms, and tensors

The definition gives familiar operations without introducing parallel transport:

$$\mathcal L_X f=X(f), \qquad \mathcal L_XY=[X,Y].$$

The second identity compares the two orders of flowing along $X$ and $Y$. On a one-form $\alpha$,

$$(\mathcal L_X\alpha)(Y) =X\bigl(\alpha(Y)\bigr)-\alpha([X,Y]).$$

Lie differentiation is a degree-zero derivation: it obeys the Leibniz rule, commutes with contractions, and therefore extends uniquely from functions, vector fields, and one-forms to arbitrary tensors. For differential forms, Cartan’s identity gives the especially useful formula

$$\mathcal L_X\alpha=d(\iota_X\alpha)+\iota_X(d\alpha).$$

It follows immediately that $[\mathcal L_X,d]=0$. The Jacobi identity for vector fields also yields

$$[\mathcal L_X,\mathcal L_Y]=\mathcal L_{[X,Y]},$$

which is the infinitesimal form of composing flows.

These identities also express naturality. If $F:M\to N$ is a diffeomorphism, $X$ is $F$-related to $\widetilde X$, and $T$ is a tensor on $N$, then

$$F^*(\mathcal L_{\widetilde X}T)=\mathcal L_X(F^*T).$$

The statement says that Lie differentiation does not depend on a chart chosen before or after the calculation. In coordinates it is still concrete. For a type-$(0,2)$ tensor,

$$(\mathcal L_XT)_{ab} =X^c\partial_cT_{ab} +T_{cb}\partial_aX^c +T_{ac}\partial_bX^c.$$

The last two terms are not optional corrections: they describe how the covector slots themselves are pulled back. Omitting them would differentiate only the component functions and would not produce a tensor.

Relation to a connection

Although the definition is connection-free, a torsion-free connection rewrites it in components. For a vector and a covector,

$$(\mathcal L_XY)^a=X^b\nabla_bY^a-Y^b\nabla_bX^a,$$ $$(\mathcal L_X\alpha)_a=X^b\nabla_b\alpha_a+\alpha_b\nabla_aX^b.$$

The connection terms cancel by torsion freedom, consistent with the connection-free definition. The Lie derivative produces a tensor field, but it is a differential operator, not a $C^\infty$-linear map in the differentiated field: $\mathcal L_X(fT)=X(f)T+f\mathcal L_XT$. The connection itself is not a tensor, but its change under the flow is. Define

$$(\mathcal L_X\nabla)_YZ =\mathcal L_X(\nabla_YZ)-\nabla_{\mathcal L_XY}Z-\nabla_Y(\mathcal L_XZ).$$

For a torsion-free connection and the site’s curvature convention, this becomes

$$(\mathcal L_X\nabla)_YZ =(\nabla^2X)(Y,Z)+R(X,Y)Z,$$

where $(\nabla^2X)(Y,Z)=\nabla_Y\nabla_ZX-\nabla_{\nabla_YZ}X$. In flat Cartesian coordinates the curvature and connection vanish, leaving $\partial_b\partial_cX^a$, a useful sign and tensor-type check.

Another discriminating check uses a nowhere-vanishing volume form $\varepsilon$. Its Lie derivative must be proportional to itself, and the coefficient defines divergence:

$$\mathcal L_X\varepsilon=(\nabla_aX^a)\varepsilon$$

for a metric volume form and its Levi-Civita connection. Thus a flow preserves oriented volume precisely when $\nabla_aX^a=0$. This does not imply that it preserves the metric: a shear can preserve volume while changing lengths and angles.

What the derivative detects

The condition $\mathcal L_XT=0$ says that the local flow preserves $T$. For a metric this becomes Killing’s equation; for a volume form it becomes vanishing divergence; for a connection it characterizes affine symmetries. These conditions are distinct. Preserving a metric automatically preserves its Levi-Civita connection, but preserving a connection need not preserve any chosen metric.

A frequent error is to read $\mathcal L_XT$ as a derivative “along $X$” with all other arguments parallel transported. That is $\nabla_XT$, which depends on a connection. The Lie derivative instead transports every tensor slot by the same flow, and the derivatives of $X$ in the formulas above are precisely the record of that transport.

Completeness is a second common confusion. Every smooth vector field has a local flow, so every Lie derivative above is well defined. Completeness becomes relevant only when finite transformations must exist for arbitrary parameter values. On a compact manifold without boundary, every smooth vector field is complete. With a boundary, tangency matters: $\partial_x$ on the compact interval $[0,1]$ reaches the boundary in finite parameter time and does not define a flow $[0,1]\to[0,1]$ for all times, whereas a vector field tangent to the boundary has a global flow on the compact manifold. Noncompact manifolds require a separate argument as well.

Exercises

On $\mathbb R^2$ with coordinates $(x,y)$, take $X=x\partial_y$ and $Y=\partial_x$. Compute $\mathcal L_XY$, then verify the result directly from the failure of their flows to commute.

Solution

The bracket is

$$[X,Y] =\left(X^b\partial_bY^a-Y^b\partial_bX^a\right)\partial_a =-\partial_y,$$

so $\mathcal L_XY=-\partial_y$. The $X$-flow is $\Phi_s(x,y)=(x,y+sx)$ and the $Y$-flow is $\Psi_t(x,y)=(x+t,y)$. Follow the four successive flows $+X,+Y,-X,-Y$:

$$\Psi_{-t}\circ\Phi_{-s}\circ\Psi_t\circ\Phi_s(x,y) =(x,y-st).$$

The leading displacement is $st[X,Y]=-st\,\partial_y$, with no unspecified orientation sign. Alternatively, the inverse differential of $\Phi_s$ sends $\partial_x$ back to $\partial_x-s\partial_y$, whose derivative at $s=0$ is the same Lie derivative.

References

• Sean M. Carroll, Lecture Notes on General Relativity, arXiv:gr-qc/9712019 (1997), arXiv.