The Lie Derivative

The Lie derivative is not a museum piece; it runs inside every numerical-relativity code that models colliding black holes. In the $3+1$ (ADM/BSSN) split of spacetime that these simulations use, the geometry is evolved slice by slice, and the time-evolution operator is not a plain $\partial_t$ but $\partial_t-\mathcal{L}_\beta$, where $\beta$ is the shift vector that reroutes coordinates across the slice. Every evolution equation for the spatial metric and extrinsic curvature carries a Lie-derivative-along-the-shift term, and getting it right is essential to the stability of the run — the same runs that produce the gravitational-waveform templates matched-filtered against LIGO and Virgo data. Elsewhere the Lie derivative organizes the search for symmetry itself: "approximate Killing vectors" are found by minimizing $\|\mathcal{L}_\xi g\|$ over vector fields on a horizon to define a black hole's spin, and the hidden Killing tensor behind the Carter constant makes Kerr geodesics integrable enough to model extreme-mass-ratio inspirals for the future LISA mission. From gauge transformations in cosmological perturbation theory ($\delta T=\mathcal{L}_\xi T$) to the very definition of a stationary spacetime, the operator Lie could not have imagined applied to gravity is now indispensable to it.

Picture a fluid whose velocity field is $X$. Its flow $\phi_t$ carries every fluid element downstream, and with it every tensor painted on the fluid. To differentiate a tensor field $T$ at a point $p$, photograph $T$ a little way downstream at $\phi_t(p)$, then "un-flow" that photograph back to $p$ using the same flow that took it there. If the dragged-back picture lies exactly on top of the original, the tensor is Lie-constant — the flow carries it rigidly. Any mismatch, divided by the elapsed time, is $\mathcal{L}_X T$. Nothing in this recipe mentions distance, angle, or parallel transport; the flow is doing all the work, and the flow is nothing but the vector field integrated.

Why is $\mathcal{L}_X Y$ the same as the commutator $[X,Y]$? Flow a short step along $X$, then along $Y$; separately, flow first along $Y$, then along $X$. You generally land at two different points, and the gap between them — to second order in the step — is precisely $[X,Y]$. So the Lie derivative of one field along another measures the failure of their flows to commute, a purely kinematic fact that never once invokes a metric. The extra $\partial X$ terms that distinguish $\mathcal{L}_X$ from a covariant derivative are exactly the bookkeeping for how the flow deforms the frame: where flow lines converge or shear, even a "constant-looking" tensor drifts relative to the moving grid, and $\mathcal{L}_X$ captures that drift.

Because $\mathcal{L}_X$ uses no metric, it sees only the smooth structure — and that is its power for symmetry analysis. Asking whether $\mathcal{L}_\xi g=0$ is asking a clean, non-circular question: does the flow of $\xi$ leave every distance and angle unchanged? A metric-blind operator is exactly the right tool to interrogate the metric, because it cannot smuggle the answer in through the back door.

The starting point is the geometric definition. Let $\phi_t$ be the flow of $X$, so $\phi_0=\mathrm{id}$ and $\tfrac{\mathrm{d}}{\mathrm{d}t}\phi_t(p)|_{t=0}=X(p)$. The pullback $\phi_t^{*}$ drags a tensor field from $\phi_t(p)$ back to $p$; the Lie derivative is the $t$-derivative of that dragged-back field at the origin.

where: $X$ — vector field generating the flow; $\phi_t$ — flow (one-parameter group of diffeomorphisms) of $X$, with $\phi_0=\mathrm{id}$; $\phi_t^{*}$ — pullback by $\phi_t$, dragging the tensor back to the base point; $T$ — tensor field of arbitrary type $(p,q)$. Physical reading: the diffeomorphism, not a connection, identifies the two tensor spaces, so the subtraction is legal and $\mathcal{L}_X T$ is again a type-$(p,q)$ tensor. On a scalar there are no indices to correct, and the pullback is just composition with the flow. The definition collapses to the ordinary directional derivative.

where: $f$ — smooth scalar field; $X^\mu$ — components of $X$ in a coordinate basis; $\partial_\mu$ — coordinate partial derivative $\partial/\partial x^\mu$. Physical reading: rate of change of $f$ as you drift along the flow, no more and no less. On a vector field the flow both moves the base point and shears the arrow, producing two terms. The result is the commutator of $X$ and $Y$ regarded as first-order operators — the Lie bracket.

where: $Y$ — the vector field being differentiated; $[X,Y]$ — commutator $XY-YX$ of the two fields acting as derivations. Physical reading: the first term transports $Y$ downstream; the second, carrying $\partial X$, accounts for the flow stretching the arrow. Manifestly antisymmetric under $X\leftrightarrow Y$. For a one-form the correction flips sign, because a covector transforms with the inverse Jacobian. The sign is pinned by demanding the Leibniz rule on the scalar $\omega(Y)$.

where: $\omega_\mu$ — components of a one-form (covector) field. Physical reading: the lower index earns a $+\partial X$ term, exactly opposite in sign to the upper-index case of equation (3). Equations (2)–(4) are special cases of one master rule. Every upper index contributes a $-\partial X$ term, every lower index a $+\partial X$ term, all riding on a single transport term.

where: $\mu_a$ — each contravariant (upper) index gets a $-\partial X$ term; $\nu_b$ — each covariant (lower) index gets a $+\partial X$ term; $\sigma$ — summed (dummy) index. Physical reading: transport plus one deformation term per slot, mirroring how each index type responds to the flow's Jacobian. Although (5) is written with ordinary derivatives, the non-tensorial pieces cancel and the result is a true tensor. One may replace every $\partial$ by any torsion-free connection $\nabla$ without changing a thing — the Christoffel additions are antisymmetric and vanish for zero torsion.

where: $\nabla$ — any torsion-free connection (e.g. Levi-Civita); all Christoffel terms cancel identically. Physical reading: the covariant form makes tensoriality manifest and is the shortest road to the Killing equation. On differential forms the Lie derivative factorizes through the exterior derivative $\mathrm{d}$ and the interior product $\iota_X$ (contraction of $X$ into the first slot). This is Cartan's "magic" formula, a computational workhorse for forms.

where: $\alpha$ — differential $p$-form; $\mathrm{d}$ — exterior derivative; $\iota_X$ — interior product. Physical reading: because $\mathrm{d}^2=0$, this immediately gives $\mathcal{L}_X\,\mathrm{d}=\mathrm{d}\,\mathcal{L}_X$ — the Lie derivative commutes with $\mathrm{d}$. Finally, the Lie derivatives themselves close into a Lie algebra: the commutator of two Lie derivatives is the Lie derivative along the bracket. This is what makes the symmetry vector fields of a spacetime a Lie algebra.

where: $[\cdot,\cdot]$ — commutator of operators on the left, Lie bracket of fields on the right. Physical reading: the map $X\mapsto\mathcal{L}_X$ is a Lie-algebra homomorphism, so the Killing vectors of a metric form a finite-dimensional Lie algebra.

The Lie derivative specializes to familiar operations as the rank of the tensor drops, and each row below is a consistency check on the master formula (5).

1 · Rotation and translation do not commute

On the plane with Cartesian coordinates $(x,y)$, take $X=\partial_\phi=x\,\partial_y-y\,\partial_x$ (the generator of rotations) and $Y=\partial_x$ (the generator of translations). Compute $\mathcal{L}_X Y=[X,Y]$ and interpret the result geometrically, then evaluate it at the point $(1,0)$.

Step 1. Read off the components. In the basis $(\partial_x,\partial_y)$ the fields are $X^\mu=(-y,\,x)$ and $Y^\mu=(1,\,0)$. The bracket formula (3) is $(\mathcal{L}_X Y)^\mu=X^\nu\partial_\nu Y^\mu-Y^\nu\partial_\nu X^\mu$.

Step 2. The first term vanishes because $Y^\mu=(1,0)$ is constant, so $\partial_\nu Y^\mu=0$. Only the second term survives: $-Y^\nu\partial_\nu X^\mu=-\partial_x X^\mu$, since $Y^\nu\partial_\nu=\partial_x$.

Step 3. Differentiate $X^\mu=(-y,x)$ with respect to $x$: $\partial_x X^x=\partial_x(-y)=0$ and $\partial_x X^y=\partial_x(x)=1$. Hence $-\partial_x X^\mu=(0,\,-1)$, i.e. $\mathbf{[\,\partial_\phi,\,\partial_x\,]=-\partial_y}$.

Step 4. Interpret. The bracket is nonzero, so rotating-then-translating differs from translating-then-rotating — the two flows fail to commute, and the mismatch points in the $-\partial_y$ direction. At the point $(1,0)$ the rotation field is $X=(0,1)=\partial_y$ and the answer $[X,Y]=(0,-1)=-\partial_y$ is comparable in size to the fields themselves, so the non-commuting is a leading-order effect, not a tiny correction. This is precisely why $\partial_\phi$ and $\partial_x$ cannot serve together as a coordinate basis: a coordinate basis always has vanishing brackets.

2 · Which flows are isometries of the round sphere?

On the unit 2-sphere with metric $g_{\mu\nu}=\mathrm{diag}(1,\sin^2\theta)$ in coordinates $(\theta,\phi)$, test whether $\xi=\partial_\phi$ and whether $\eta=\partial_\theta$ are Killing vectors by computing $\mathcal{L}_\xi g$ and $\mathcal{L}_\eta g$ directly from the coordinate formula.

Step 1. For a $(0,2)$ tensor with generator $V$, formula (5) reads $(\mathcal{L}_V g)_{\mu\nu}=V^\sigma\partial_\sigma g_{\mu\nu}+g_{\sigma\nu}\partial_\mu V^\sigma+g_{\mu\sigma}\partial_\nu V^\sigma$. The only non-constant metric component is $g_{\phi\phi}=\sin^2\theta$, with $\partial_\theta g_{\phi\phi}=2\sin\theta\cos\theta$.

Step 2. Take $\xi=\partial_\phi$, so $\xi^\sigma=(0,1)$ is constant and $\partial_\mu\xi^\sigma=0$. Every derivative term drops except the transport term $\xi^\sigma\partial_\sigma g_{\mu\nu}=\partial_\phi g_{\mu\nu}$. But no metric component depends on $\phi$, so $\partial_\phi g_{\mu\nu}=0$ for all $\mu,\nu$. Therefore $\mathbf{\mathcal{L}_{\partial_\phi} g=0}$: $\partial_\phi$ is a Killing vector, the axial symmetry of the sphere.

Step 3. Take $\eta=\partial_\theta$, so $\eta^\sigma=(1,0)$ is constant and again $\partial_\mu\eta^\sigma=0$. Only the transport term survives: $(\mathcal{L}_\eta g)_{\mu\nu}=\partial_\theta g_{\mu\nu}$. The $\phi\phi$ component is nonzero: $(\mathcal{L}_\eta g)_{\phi\phi}=\partial_\theta\sin^2\theta=\mathbf{2\sin\theta\cos\theta}=\sin 2\theta$.

Step 4. Interpret. Because $(\mathcal{L}_{\partial_\theta}g)_{\phi\phi}=\sin2\theta\neq0$ except on the equator, $\partial_\theta$ is not a Killing vector: pushing the sphere along lines of constant longitude stretches the latitude circles (they grow from pole to equator), so distances are not preserved. The contrast is the whole lesson — the metric being $\phi$-independent makes $\partial_\phi$ an isometry, while its $\theta$-dependence spoils $\partial_\theta$. The sphere in fact has three independent Killing vectors (the $\mathfrak{so}(3)$ rotations); $\partial_\phi$ is only the most obvious one.

The Lie derivative carries a name that reaches back to the nineteenth-century mathematician who first understood that a differential equation's symmetries are continuous flows — and that those flows can be differentiated.

References. Misner, Thorne & Wheeler, Gravitation, §3.5–3.6 (Lie dragging) and Ch. 25 (Killing vectors) · Wald, General Relativity, Appendix C.2–C.3 and §2.2 · Carroll, Spacetime and Geometry, §5.3 · Schutz, A First Course in General Relativity, Ch. 5 · Hartle, Gravity, Ch. 8 (symmetries and Killing vectors). The key exact identities in this article are: the pullback expansion giving $[X,Y]^\mu=X^\nu\partial_\nu Y^\mu-Y^\nu\partial_\nu X^\mu$; the cancellation of Christoffel terms under the torsion-free replacement $\partial\to\nabla$; Cartan's formula on a one-form by component cancellation; and the concrete results $[\partial_\phi,\partial_x]=-\partial_y$, $(\mathcal{L}_{\partial_\theta}g)_{\phi\phi}=\sin2\theta$, and $\mathcal{L}_{\partial_\phi}g=0$. No numerical values are asserted beyond these exact tensor identities.

Builds on: Vector Fields and Integral Curves · Tangent Vectors and the Tangent Space · Cotangent Space and One-Forms · General Tensors and the Tensor Product · Smooth Manifolds and Coordinate Charts. Leads to: Killing Vectors and Isometries · The Metric Tensor and the Line Element · Tensor Algebra: Contraction, Symmetrization and Antisymmetrization · Differential Forms and Exterior Calculus.

Ask the tutor: Want to see how the two Killing vectors $\partial_t$ and $\partial_\phi$ of Schwarzschild fall straight out of $\mathcal{L}_\xi g=0$ and hand you the conserved energy and angular momentum of an orbit?