Lie Derivatives and Killing Vectors

Ask how a tensor field changes as you slide along a flow, and you hit a subtlety ordinary calculus hides: the field at a nearby point lives in a different vector space than the field where you stand, so you cannot subtract the two. The covariant derivative fixes this by importing a connection — an extra rule for carrying tensors between tangent spaces. The Lie derivative fixes it with nothing beyond the flow itself. A vector field $X$ generates a family of diffeomorphisms $\phi_t$ that drag every geometric object along; to differentiate $T$ , you let that drag carry the value from the future point back to where you are, compare, and take the rate at $t=0$ . No connection, no metric, no coordinates enter — which is why $\mathcal{L}_X$ is the natural derivative of bare differential topology, while $\nabla$ belongs to Riemannian geometry. Two special cases fix everything. On a function the Lie derivative is the directional derivative $X^\mu\partial_\mu f$ ; on a vector field it is the commutator $[X,Y]$ , which measures how badly the flows of $X$ and $Y$ fail to commute. Demand that $\mathcal{L}_X$ obey the Leibniz rule and commute with contraction, and its action on any tensor is forced. The payoff is physical. A Killing vector is a direction along which the drag leaves the metric fixed, $\mathcal{L}_\xi g=0$ — its flow is an isometry, a continuous symmetry of the geometry. Killing's equation $\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu=0$ turns "the metric has a symmetry" into a solvable first-order system, and every solution buys a conserved quantity $\xi_\mu u^\mu$ along geodesics — the constants that make orbits, light-bending, and black-hole shadows computable.

Key takeaways

The Lie derivative $\mathcal{L}_X T$ measures how a tensor field changes as the flow of $X$ drags it, comparing $T$ at a point with its push-forward from a neighbouring point — a metric-free, connection-free operation.
On a scalar it is the directional derivative $X^\mu\partial_\mu f$ ; on a vector it is the Lie bracket $[X,Y]$ ; the Leibniz rule then fixes it on every tensor.
The bracket $[X,Y]$ is the failure of flows to close: run along $X$ then $Y$ then back and you miss the start by $\epsilon^2[X,Y]$ .
A Killing vector satisfies $\mathcal{L}_\xi g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu=0$ — its flow is an isometry, encoding a continuous symmetry of the metric.
Each Killing vector gives a conserved charge $\xi_\mu u^\mu$ along any geodesic; Schwarzschild's $\partial_t$ and $\partial_\phi$ yield conserved energy and angular momentum.
A maximally symmetric $n$ -manifold admits exactly $\tfrac12 n(n+1)$ Killing vectors — $10$ for 4D Minkowski (the Poincaré algebra), $3$ for the sphere $S^2$ .
Key idea: The Lie derivative differentiates a tensor field by dragging it with the flow of $X$ — a metric-free, connection-free operation; a Killing vector is one whose drag leaves the metric fixed, so $\mathcal{L}_\xi g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu=0$ , encoding a continuous isometry and hence a conserved quantity.

Where this lives today

Killing vectors are the daily currency of relativistic astrophysics. Every stationary, axisymmetric black-hole spacetime carries $\partial_t$ and $\partial_\phi$ , and the conserved $E$ and $L$ they generate reduce the geodesic equations to a one-dimensional radial problem — the machinery behind the Event Horizon Telescope's images of M87 and Sgr A, whose shadow radius is the $3\sqrt3\,GM/c^2$ impact parameter computed above. In cosmology, the observed homogeneity and isotropy of the universe is the statement that space is maximally symmetric — six Killing vectors per spatial slice — forcing the FLRW metric of the concordance model, while de Sitter and anti-de Sitter space, each with the full $10$ isometries, anchor inflation and the AdS/CFT correspondence. Closer to Earth, Cartan's formula drives the transport identities of continuum mechanics — the vorticity two-form's vanishing Lie derivative is Helmholtz's frozen-in theorem — and the "3+1" codes of numerical relativity.

The physical picture

Think of $X$ as the velocity field of a fluid whose flow $\phi_t$ carries every particle — and every field painted on it — downstream. To Lie-differentiate a tensor you do not compare its values at two points directly; you let the flow carry the value from $\phi_t(p)$ back to $p$ , lay it beside the value already there, and read off the rate of mismatch. A quantity advected without distortion has zero Lie derivative; a nonzero $\mathcal{L}_X T$ is the rate at which the flow stretches, shears, or rotates $T$ . The bracket $[X,Y]$ is the sharpest instance: what is left over when two flows refuse to close a little coordinate box.

Figure

Flow along X = ∂ₓ then Y = x∂_y then back along each does not return to the start. The order-ε² gap drawn in red is the Lie bracket of X and Y, here equal to ∂_y, so the loop misses by ε²∂_y — vector fields commute exactly when this gap vanishes

A Killing vector is the opposite of a distorting flow: a direction in which the geometry looks exactly the same before and after. Slide the entire metric along $\xi$ and it superimposes on itself perfectly, every length and angle preserved. On a round sphere the rotations do this; on a generic bumpy surface no rigid motion exists and there are no Killing vectors at all. Killing's equation $\nabla_{(\mu}\xi_{\nu)}=0$ says the flow has no shear and no expansion, only rotation — the symmetric part of $\nabla\xi$ , which encodes stretching, is forced to vanish. The count of independent Killing vectors is a hard invariant of the space, maxing out at $\tfrac12 n(n+1)$ .

Figure

A round 2-sphere looks identical after any rotation, so each rotation axis furnishes a Killing vector: n(n+1)/2 = 3 in all, the generators of SO(3). Dragging the metric along ξ = ∂_φ leaves ds² unchanged, so its Lie derivative vanishes; a bumpy surface has none

The core equations

The definition is a limit built from the flow. The tensor at $\phi_t(p)$ cannot be subtracted from the tensor at $p$ directly, so the flow's own push-forward $(\phi_{-t})_*$ carries it back to $p$ first; the rate of the resulting comparison at $t=0$ is the Lie derivative — using nothing but the manifold's differentiable structure.

\mathcal{L}_X T \;=\; \lim_{t\to 0} \frac{(\phi_{-t})_* T_{\phi_t(p)} - T_p}{t} \;=\; \left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0} (\phi_{-t})_* T \qquad (1)

where: $X$ — the vector field generating the flow; $\phi_t$ — its one-parameter group of diffeomorphisms; $(\phi_{-t})_*$ — the push-forward that drags a tensor from $\phi_t(p)$ back to $p$ ; $T$ — an arbitrary $(p,q)$ tensor field. Read it as "compare $T$ with its dragged-back self and take the rate" — the drag-back is what makes the subtraction legal. On a scalar there is nothing to drag but the value itself, so the Lie derivative collapses to the plain directional derivative — the rate of change of $f$ along $X$ .

\mathcal{L}_X f = X^\mu \partial_\mu f = X(f) \qquad (2)

where: $f$ — a smooth function; $X^\mu$ — the components of $X$ in a coordinate basis. This is the one case where $\mathcal{L}_X$ and $\nabla_X$ agree identically — a scalar has no indices for a connection to act on. On a vector field the flow drags $Y$ as well as evaluates it, and the extra term $-Y^\nu\partial_\nu X^\mu$ is exactly the commutator of the two fields — the Lie bracket.

(\mathcal{L}_X Y)^\mu = X^\nu \partial_\nu Y^\mu - Y^\nu \partial_\nu X^\mu = [X,Y]^\mu \qquad (3)

where: $Y^\mu$ — the components of the field being differentiated; $[X,Y]$ — the Lie bracket, the commutator of $X$ and $Y$ as first-order operators. It is antisymmetric and Jacobi-satisfying, so vector fields form a Lie algebra under it. A one-form is dual to a vector, and the Leibniz rule fixes its Lie derivative once the scalar and vector cases are known — the $X$ -derivative term switching sign because a one-form carries a lower index.

(\mathcal{L}_X \omega)_\mu = X^\nu \partial_\nu \omega_\mu + \omega_\nu \partial_\mu X^\nu \qquad (4)

where: $\omega_\mu$ — the components of the one-form (a lower, covariant index); the second term derives $X$ rather than $\omega$ . The rule generalizes: every upper index contributes a $-(\partial X)$ term and every lower index a $+(\partial X)$ term. Assembling these, the Lie derivative of a general $(p,q)$ tensor is a transport term plus one correction per index. This single formula reproduces (2), (3), and (4) as special cases.

\mathcal{L}_X T^{\mu_1\cdots}{}_{\nu_1\cdots} = X^\lambda \partial_\lambda T^{\mu_1\cdots}{}_{\nu_1\cdots} - \sum_i T^{\cdots\lambda\cdots}{}_{\nu\cdots}\,\partial_\lambda X^{\mu_i} + \sum_j T^{\mu\cdots}{}_{\cdots\lambda\cdots}\,\partial_{\nu_j} X^{\lambda} \qquad (5)

where: $T^{\mu\cdots}{}_{\nu\cdots}$ — the tensor field; $\partial_\lambda X^\mu$ — the derivatives of the generator that source every correction term. Because only $\partial X$ appears, the Lie derivative depends on $X$ throughout a neighbourhood, not just at a point. Remarkably, the ordinary derivatives in (5) may be swapped for covariant ones with any torsion-free connection and the answer is unchanged, because the Christoffel contributions cancel by the symmetry $\Gamma^\lambda_{\mu\nu}=\Gamma^\lambda_{\nu\mu}$ . This is the algebraic proof that $\mathcal{L}_X$ needs no connection.

\mathcal{L}_X T = X^\lambda \nabla_\lambda T - (\nabla_\lambda X^{\mu})\,T^{\cdots\lambda\cdots} + (\nabla_{\nu} X^{\lambda})\,T^{\cdots}{}_{\cdots\lambda\cdots} \qquad (6)

where: $\nabla$ — any symmetric (torsion-free) connection; the result is independent of which one, and applied to the metric it yields Killing's equation in a line. A Killing vector is defined by demanding that dragging the metric change nothing: $\mathcal{L}_\xi g=0$ . Feeding $g$ through (6) and using metric compatibility $\nabla g=0$ turns this into the vanishing of the symmetrized covariant derivative of $\xi$ .

\mathcal{L}_\xi g_{\mu\nu} = \nabla_\mu \xi_\nu + \nabla_\nu \xi_\mu = 2\,\nabla_{(\mu}\xi_{\nu)} = 0 \qquad (7)

where: $\xi_\mu=g_{\mu\nu}\xi^\nu$ — the covariant components of the Killing vector; $g_{\mu\nu}$ — the metric; $\nabla_{(\mu}\xi_{\nu)}$ — the symmetrized derivative. It is an overdetermined first-order system whose solutions form the finite-dimensional Lie algebra of infinitesimal isometries. For differential forms there is a slicker route: Cartan's "magic" formula expresses $\mathcal{L}_X$ on any $p$ -form through the exterior derivative and the interior product, with no coordinates at all.

\mathcal{L}_X \alpha = \mathrm{d}\,(\iota_X \alpha) + \iota_X\,(\mathrm{d}\alpha) \qquad (8)

where: $\alpha$ — a differential $p$ -form; $\mathrm{d}$ — the exterior derivative; $\iota_X$ — the interior product (contraction of $X$ into the first slot). It packages "differentiate then contract" and its reverse into the flow derivative. The physical harvest is a conservation law: contract a Killing vector with the tangent to a geodesic and the result is constant along the curve — every symmetry becomes a first integral of the motion.

Q = \xi_\mu\, \frac{\mathrm{d}x^\mu}{\mathrm{d}\tau} = \xi_\mu u^\mu = \text{const along a geodesic} \qquad (9)

where: $u^\mu=\mathrm{d}x^\mu/\mathrm{d}\tau$ — the geodesic tangent (four-velocity); $\tau$ — an affine parameter (proper time for timelike geodesics); $Q$ — the conserved charge tied to $\xi$ .

Derivation AThe Lie bracket as the failure of two flows to close

The goal here is to give the bracket $[X,Y]$ its geometric meaning as the obstruction to commuting flows, and thereby to explain why commuting vector fields — and only they — can serve as a coordinate grid. We track a single point through a small loop: flow along $X$ for parameter $\epsilon$ , then $Y$ , then back along $X$ , then back along $Y$ , and show it lands a distance $\epsilon^2[X,Y]$ from its start, so the bracket is exactly the order- $\epsilon^2$ non-closure of the flow rectangle.

Step 1. Expand each flow to second order from its defining equation $\dot x^\mu = X^\mu(x)$ , giving $\phi^X_\epsilon(x)^\mu = x^\mu + \epsilon X^\mu(x) + \tfrac12\epsilon^2 X^\nu\partial_\nu X^\mu + O(\epsilon^3)$ , and likewise for $Y$ . Why: the second-order term, where each field is re-evaluated at the already-displaced point, is what carries the bracket.

Step 2. Compose the four maps in order $\phi^Y_{-\epsilon}\circ\phi^X_{-\epsilon}\circ\phi^Y_{\epsilon}\circ\phi^X_{\epsilon}$ , at each stage evaluating the next field at the point reached so far and keeping terms through $O(\epsilon^2)$ . Why: evaluating $Y$ at the $X$ -displaced point rather than at $p$ is what breaks the symmetry between the two orders.

Step 3. Collect the surviving terms. Every first-order displacement cancels between the forward and backward legs, and the $\tfrac12\epsilon^2 X^\nu\partial_\nu X^\mu$ self-terms cancel too, leaving $\phi^Y_{-\epsilon}\circ\phi^X_{-\epsilon}\circ\phi^Y_{\epsilon}\circ\phi^X_{\epsilon}(p) = p + \epsilon^2\big(X^\nu\partial_\nu Y^\mu - Y^\nu\partial_\nu X^\mu\big) + O(\epsilon^3)$ . Why: the mixed term $X^\nu\partial_\nu Y^\mu - Y^\nu\partial_\nu X^\mu$ is exactly the bracket components $[X,Y]^\mu$ , so the gap is $\epsilon^2[X,Y]_p$ .

Step 4. Read off the meaning: the loop closes to order $\epsilon^2$ if and only if $[X,Y]=0$ , in which case the two flows commute and $X,Y$ can be completed to a commuting coordinate basis $\partial_a,\partial_b$ . Why: this is the geometric content of $\mathcal{L}_X Y=[X,Y]$ — the Lie derivative of $Y$ along $X$ is the rate at which the $X$ -flow drags $Y$ out of alignment, which is the same non-closure seen infinitesimally. *As a check, the explicit example $X=\partial_x,\ Y=x\partial_y$ gives $[X,Y]=\partial_y$ , the gap drawn in the first figure.*

Go deeper · Derivation B · 1 minFrom metric invariance to Killing's equation

Derivation BFrom metric invariance to Killing's equation

The aim is to convert the geometric statement "the flow of $\xi$ is an isometry", written compactly as $\mathcal{L}_\xi g=0$ , into the working equation $\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu=0$ that one solves to find symmetries. The bridge is the covariant form of the Lie derivative plus the metric compatibility of the Levi-Civita connection, which collapse three terms into a clean symmetrized derivative and expose Killing's equation as the vanishing of the shear-and-expansion part of $\nabla\xi$ .

Step 1. Apply the covariant Lie-derivative formula (6) to the $(0,2)$ tensor $g$ , giving $\mathcal{L}_\xi g_{\mu\nu} = \xi^\lambda\nabla_\lambda g_{\mu\nu} + g_{\lambda\nu}\nabla_\mu\xi^\lambda + g_{\mu\lambda}\nabla_\nu\xi^\lambda$ , one correction term for each lower index of $g$ . Why: using $\nabla$ rather than $\partial$ is legal because the Lie derivative is connection-independent, and it lets metric compatibility do the next step's work.

Step 2. Kill the transport term. For the Levi-Civita connection $\nabla_\lambda g_{\mu\nu}=0$ identically, so the first term drops and only the two index-correction terms remain. Why: metric compatibility says the metric is covariantly constant, which is what makes the $\xi^\lambda\nabla_\lambda g$ piece vanish.

Step 3. Lower the free index with $g$ inside the covariant derivative — again permitted by compatibility — to write $g_{\lambda\nu}\nabla_\mu\xi^\lambda=\nabla_\mu\xi_\nu$ and likewise for the other term, yielding $\mathcal{L}_\xi g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu$ . Why: this is the symmetrized covariant derivative $2\nabla_{(\mu}\xi_{\nu)}$ , whose vanishing is Killing's equation.

Step 4. Demand isometry, $\mathcal{L}_\xi g=0$ , to obtain $\nabla_{(\mu}\xi_{\nu)}=0$ . Why: the symmetric part of $\nabla\xi$ is the strain of the flow (its expansion and shear); setting it to zero says the flow moves the geometry rigidly, preserving every length and angle — the infinitesimal statement of an isometry. *Note that the antisymmetric part $\nabla_{[\mu}\xi_{\nu]}$ is unconstrained, consistent with a Killing flow carrying rotation but no strain.*

Go deeper · Derivation C · 1 minWhy a Killing vector conserves a charge along geodesics

Derivation CWhy a Killing vector conserves a charge along geodesics

The target is the single most useful consequence of Killing symmetry: that the contraction of a Killing vector with the tangent to any geodesic is constant along the curve, so each symmetry supplies one first integral of the motion. The proof is a two-line collaboration between the geodesic equation, which kills the derivative of the tangent, and Killing's equation, which kills the symmetric derivative of $\xi$ — together forcing the total derivative of $\xi_\mu u^\mu$ to vanish.

Step 1. Differentiate the scalar $\xi_\mu u^\mu$ along the curve using the tangent as the directional operator: $\dfrac{\mathrm{d}}{\mathrm{d}\tau}(\xi_\mu u^\mu) = u^\nu\nabla_\nu(\xi_\mu u^\mu)$ . Why: along a curve with tangent $u$ , the ordinary $\tau$ -derivative of any scalar equals $u^\nu\nabla_\nu$ acting on it, which lets covariant machinery take over.

Step 2. Apply the Leibniz rule to split the derivative into a piece acting on $\xi$ and a piece acting on $u$ : $u^\nu\nabla_\nu(\xi_\mu u^\mu) = u^\nu u^\mu\nabla_\nu\xi_\mu + \xi_\mu\,u^\nu\nabla_\nu u^\mu$ . Why: isolating the two factors lets each defining equation strike its own term.

Step 3. Use the geodesic equation $u^\nu\nabla_\nu u^\mu=0$ to eliminate the second term outright. Why: a geodesic parallel-transports its own tangent, so the derivative of $u$ along the motion is zero — this is what "straightest possible path" means covariantly.

Step 4. Use Killing's equation on the first term: $u^\nu u^\mu\nabla_\nu\xi_\mu = u^\nu u^\mu\nabla_{(\nu}\xi_{\mu)} = 0$ , since the symmetric product $u^\nu u^\mu$ projects onto the symmetric part of $\nabla_\nu\xi_\mu$ , which vanishes. Hence $\mathrm{d}(\xi_\mu u^\mu)/\mathrm{d}\tau=0$ . Why: only the symmetric part of $\nabla\xi$ survives contraction with $u^\nu u^\mu$ , and that is exactly what Killing's equation zeroes — so the charge is conserved. *Specializing to Schwarzschild's $\partial_t$ and $\partial_\phi$ returns the conserved energy $E$ and angular momentum $L$ of orbit theory.*

Limiting cases

The Lie derivative simplifies sharply in special cases, and the Killing structure constrains the geometry it lives on:

Case	Result	Meaning
Scalar field	$\mathcal{L}_X f = X^\mu\partial_\mu f = X(f)$	Just the directional derivative — no connection, no bracket subtlety; it agrees with $\nabla_X f$ .
Vector field	$\mathcal{L}_X Y = [X,Y]$	The Lie bracket; antisymmetric and Jacobi-satisfying, making vector fields a Lie algebra.
Commuting fields	$[X,Y]=0 \iff \phi^X_s\circ\phi^Y_t=\phi^Y_t\circ\phi^X_s$	Flows commute exactly when the bracket vanishes; such fields complete to a coordinate basis.
Maximally symmetric space	$\#\,\text{Killing vectors}=\tfrac12 n(n+1)$	In $n$ dimensions: $n$ translations plus $\tfrac12 n(n-1)$ rotations; $10$ for 4D Minkowski (Poincaré).
Conformal Killing limit	$\mathcal{L}_\xi g_{\mu\nu}=2\,\omega\,g_{\mu\nu}$	Metric preserved only up to scale, $\omega=\tfrac1n\nabla_\lambda\xi^\lambda$ ; true Killing needs $\omega=0$ .

Common trapThe Lie derivative is not a pointwise directional derivative of the tensor

The most common trap is to treat $\mathcal{L}_X T$ like $\nabla_X T$ — as an object that depends on $X$ only at the point where you evaluate it. It does not. Because the coordinate formula contains $\partial_\lambda X^\mu$ , the Lie derivative depends on $X$ throughout a neighbourhood, and it is not $C^\infty$ -linear in $X$ : in general $\mathcal{L}_{fX}T\neq f\,\mathcal{L}_X T$ . For a vector argument the failure is explicit, $\mathcal{L}_{fX}Y=[fX,Y]=f[X,Y]-(Yf)X$ , and the stray $-(Yf)X$ is precisely the non-tensorial piece. A second trap: replacing $\partial_\mu$ by $\nabla_\mu$ in the formulas does not mean the connection is doing any work — the Christoffel terms cancel by torsion-freeness, so use $\nabla$ only for algebraic convenience, never as an ingredient.

Worked examples

1 · The three Killing vectors of the round 2-sphere (fully worked)

Take the unit-radius- $a$ sphere with metric $\mathrm{d}s^2=a^2(\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2)$ . Verify that $\xi=\partial_\phi$ solves Killing's equation, and count how many independent Killing vectors the sphere admits.

Step 1. Write the candidate in components. With coordinates $(\theta,\phi)$ , $\xi^\mu=(0,1)$ and the metric $g_{\theta\theta}=a^2,\ g_{\phi\phi}=a^2\sin^2\theta$ give the lowered field $\xi_\mu=g_{\mu\nu}\xi^\nu=(0,\ a^2\sin^2\theta)$ . The nonzero Christoffel symbols are $\Gamma^\theta_{\phi\phi}=-\sin\theta\cos\theta$ and $\Gamma^\phi_{\theta\phi}=\Gamma^\phi_{\phi\theta}=\cot\theta$ .

Step 2. Test the three independent components of $\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu-2\Gamma^\lambda_{\mu\nu}\xi_\lambda$ . The $(\theta\theta)$ component is $2\partial_\theta\xi_\theta-2\Gamma^\lambda_{\theta\theta}\xi_\lambda=\mathbf{0}$ (since $\xi_\theta=0,\ \Gamma^\phi_{\theta\theta}=0$ ), and $(\phi\phi)$ is $2\partial_\phi\xi_\phi-2\Gamma^\theta_{\phi\phi}\xi_\theta=\mathbf{0}$ (since $\xi_\theta=0$ ).

Step 3. The mixed component is the real test: $(\theta\phi)$ gives $\partial_\theta\xi_\phi+\partial_\phi\xi_\theta-2\Gamma^\lambda_{\theta\phi}\xi_\lambda=\partial_\theta(a^2\sin^2\theta)-2\Gamma^\phi_{\theta\phi}\xi_\phi=2a^2\sin\theta\cos\theta-2\cot\theta\cdot a^2\sin^2\theta=2a^2\sin\theta\cos\theta-2a^2\sin\theta\cos\theta=\mathbf{0}$ . All three components vanish, so $\partial_\phi$ is a Killing vector — the manifest one from the metric's $\phi$ -independence.

Step 4. Count the rest. As a maximally symmetric 2-space the sphere admits $\tfrac12 n(n+1)=\tfrac12\cdot 2\cdot 3=\mathbf{3}$ Killing vectors: the generators of $SO(3)$ , rotations about the $x$ , $y$ , and $z$ axes. Only $\partial_\phi$ is aligned with a coordinate; the other two are "hidden" rotations, and there are no translational Killing vectors because the sphere is curved. Result: $\partial_\phi$ is Killing, and $S^2$ has exactly $3$ Killing vectors.

2 · Conserved charges and the photon capture radius in Schwarzschild

In Schwarzschild spacetime $\mathrm{d}s^2=-(1-2GM/r)\,\mathrm{d}t^2+(1-2GM/r)^{-1}\mathrm{d}r^2+r^2(\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2)$ , use the Killing vectors to build the conserved quantities and find the critical impact parameter below which light is captured.

Step 1. The metric is independent of $t$ and $\phi$ , so $\xi_{(t)}=\partial_t$ and $\xi_{(\phi)}=\partial_\phi$ are Killing. Their conserved charges for a geodesic of tangent $u^\mu=\dot x^\mu$ are $E=-\xi_{(t)\mu}u^\mu=(1-2GM/r)\,\dot t$ and $L=\xi_{(\phi)\mu}u^\mu=r^2\sin^2\theta\,\dot\phi$ . Working in the equatorial plane $\theta=\pi/2$ , these are the conserved energy and angular momentum per unit mass.

Step 2. For a photon the null condition $g_{\mu\nu}u^\mu u^\nu=0$ combines with the two constants to give $\dot r^2=E^2-\dfrac{L^2}{r^2}\Big(1-\dfrac{2GM}{r}\Big)$ . The Killing constants have removed the $t$ and $\phi$ equations entirely, leaving a one-dimensional radial problem with effective potential $V(r)=\dfrac{L^2}{r^2}\Big(1-\dfrac{2GM}{r}\Big)$ .

Step 3. Capture versus escape is decided by the peak of $V(r)$ . Setting $V'(r)=0$ gives the photon sphere $r_{\rm ph}=3GM/c^2$ , where the maximum equals $E^2$ when the impact parameter $b\equiv L/E$ reaches $b_{\rm crit}=r_{\rm ph}/\sqrt{1-2GM/(c^2 r_{\rm ph})}$ .

Step 4. Plug in $r_{\rm ph}=3GM/c^2$ : the factor $1-2GM/(c^2 r_{\rm ph})=1-\tfrac23=\tfrac13$ , so $b_{\rm crit}=\dfrac{3GM/c^2}{\sqrt{1/3}}=3\sqrt{3}\,\dfrac{GM}{c^2}\approx \mathbf{5.196}\,\dfrac{GM}{c^2}$ . Light with $b<b_{\rm crit}$ spirals in and is captured; $b>b_{\rm crit}$ escapes. For Sgr A* ( $GM/c^2\approx 6.0\times10^9\ \mathrm{m}$ ) this is $b_{\rm crit}\approx 3.1\times10^{10}\ \mathrm{m}$ — the shadow radius the Event Horizon Telescope images, set by exactly this Killing-derived number. Result: $b_{\rm crit}=3\sqrt{3}\,GM/c^2$ , straight from the two Killing vectors.

Physics of the portrait

The Lie derivative and its bracket are the working end of an idea Sophus Lie spent his life building — that continuous symmetry is itself a mathematical object, generated by vector fields and organized by their commutators.

Origins · 1 minSophus Lie and how this was discovered

Sophus Lie

1842–1899 · founder of continuous transformation groups and the geometry of flows

Marius Sophus Lie was a Norwegian mathematician who set out to do for differential equations what Galois had done for polynomials: understand them through their symmetries. Working alongside Felix Klein around 1870, he conceived of continuous groups of transformations — now called Lie groups — captured infinitesimally by the vector fields generating their flows, with the commutator of two generators encoding the group's structure. That commutator is the Lie bracket, the flow-derivative it controls is the Lie derivative, and the algebra of generators is the Lie algebra; his three-volume Theorie der Transformationsgruppen (1888–1893, with Friedrich Engel) founded the whole subject. The physical half of this article rests on a contemporary, Wilhelm Killing, who in the early 1890s wrote down the equation for the vector fields whose flows preserve a metric — the isometry generators now bearing his name. When Emmy Noether tied continuous symmetry to conservation in 1918, Lie's flows and Killing's equation became the language in which "why is energy conserved?" finally had an answer.

History

1842

— Sophus Lie is born in Nordfjordeid, Norway

1870

— Lie, working with Felix Klein, begins the theory of continuous transformation groups

1888

— Lie and Engel publish the first volume of Theorie der Transformationsgruppen

1892

— Wilhelm Killing writes the equation for the vector fields generating isometries

1918

— Emmy Noether links continuous symmetries to conserved quantities, giving Killing vectors their dynamical meaning

Going further

Texts. Nakahara, Geometry, Topology and Physics, 2nd ed., Ch. 5 (Lie derivatives, flows, isometries) · Stone & Goldbart, Mathematics for Physics (CUP 2009), differential-geometry chapters · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed. · Wald, General Relativity (1984), App. C and the symmetries chapter · Weinberg, Gravitation and Cosmology, Ch. 13 (Killing vectors and maximally symmetric spaces) · Riley, Hobson & Bence, Mathematical Methods for Physics and Engineering, 3rd ed. Key results to keep in mind: the coordinate formula (5) remains valid when $\partial$ is replaced by $\nabla$ for any torsion-free connection; Killing's equation $\nabla_{(\mu}\xi_{\nu)}=0$ follows from $\mathcal{L}_\xi g=0$ by metric compatibility; the maximal Killing-vector count is $\tfrac12 n(n+1)$ — $10$ for 4D Minkowski, $3$ for $S^2$ , $6$ for 3-space; and the Schwarzschild conserved quantities $E$ , $L$ and the photon capture radius $b_{\rm crit}=3\sqrt3\,GM/c^2$ are both worked in full above.

Prerequisites. Manifolds and Tangent Spaces · Covariant and Contravariant Tensors · The Metric Tensor · The Covariant Derivative · Christoffel Symbols and Affine Connections. Leads to. Parallel Transport and Geodesics · Exterior Calculus and the Exterior Derivative · Differential Forms.

Ask the tutor: Ask me to solve Killing's equation for a metric of your choice, or to trace how a single Killing vector turns into a conserved energy, momentum, or shadow radius.

What is the difference between the Lie derivative and the covariant derivative?

Both differentiate tensor fields, but they need different inputs. The covariant derivative $\nabla_X T$ asks how $T$ changes relative to parallel transport, which requires a connection — an extra rule for carrying a tensor between neighbouring tangent spaces. The Lie derivative $\mathcal{L}_X T$ asks how $T$ changes as the flow of $X$ drags it, using only the flow itself, so it needs no connection and no metric. A practical tell: $\nabla_X T$ at a point depends on $X$ only at that point, while $\mathcal{L}_X T$ depends on $X$ throughout a neighbourhood, because it differentiates $X$ too.

Why is the Lie derivative not tensorial in X?

A tensorial (pointwise) dependence would mean $\mathcal{L}_{fX}T=f\,\mathcal{L}_X T$ for every function $f$ — the value at a point set by $X$ at that point alone. But the coordinate formula contains derivatives $\partial_\lambda X^\mu$ , so multiplying $X$ by $f$ also differentiates $f$ . The cleanest instance is $\mathcal{L}_{fX}Y=[fX,Y]=f[X,Y]-(Yf)X$ : the extra $-(Yf)X$ is the non-tensorial term. This is exactly why the Lie derivative is not a "directional derivative of $T$ " in the way $\nabla_X T$ is — it measures a flow effect, which is inherently non-local in $X$ .

How do I actually find the Killing vectors of a given metric?

Two routes. The quick one: scan the metric for coordinates it does not depend on. If $g_{\mu\nu}$ is independent of $x^a$ , then $\partial_a$ is immediately a Killing vector — this catches $\partial_t$ and $\partial_\phi$ in Schwarzschild. The complete one: write $\xi_\mu$ with unknown component functions and solve $\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu=0$ as coupled first-order PDEs. This also finds the "hidden" symmetries, like the two non- $\phi$ rotations of the sphere, and its integrability conditions cap the count at $\tfrac12 n(n+1)$ .

What is a conformal Killing vector, and how does it differ?

A conformal Killing vector preserves the metric only up to a local rescaling: $\mathcal{L}_\xi g_{\mu\nu}=2\omega\,g_{\mu\nu}$ with $\omega=\tfrac1n\nabla_\lambda\xi^\lambda$ , rather than the strict $\mathcal{L}_\xi g=0$ of a true Killing vector. Its flow preserves angles and the light-cone structure but not lengths. These generate the conformal symmetries central to massless field theory, 2D CFT, and null geodesics — a photon's path depends only on the conformal class of the metric, so conformal Killing vectors give conserved quantities for light even when full isometries are absent. A true Killing vector is the case $\omega=0$ .

Why does each Killing vector produce a conserved quantity?

Because a Killing vector is a symmetry of the geometry, and symmetry means something is not being pushed by the dynamics. Concretely, take the scalar $\xi_\mu u^\mu$ along a geodesic and differentiate: one term dies by the geodesic equation (the tangent is parallel-transported), and the other dies by Killing's equation (the symmetric $u^\mu u^\nu$ contracts against a derivative whose symmetric part vanishes). The result is $\mathrm{d}(\xi_\mu u^\mu)/\mathrm{d}\tau=0$ . This is the geometric face of Noether's theorem: the isometry generated by $\xi$ is a continuous symmetry, and its Noether charge is precisely $\xi_\mu u^\mu$ .

What is Cartan's magic formula good for?

Cartan's formula $\mathcal{L}_X\alpha=\mathrm{d}(\iota_X\alpha)+\iota_X(\mathrm{d}\alpha)$ computes the Lie derivative of any differential form using only the exterior derivative and interior product — no coordinates, no index gymnastics. It is the fastest route to transport theorems: on a volume form it gives $\mathcal{L}_u(\mathrm{d}V)=(\nabla\cdot u)\,\mathrm{d}V$ , the divergence as the fractional rate of volume change under the flow. On a closed form ( $\mathrm{d}\alpha=0$ ) it shows the Lie derivative is exact — the mechanism behind conserved fluxes and Helmholtz's frozen-in vorticity. It is the workhorse of geometric mechanics and the Reynolds transport theorem.