Affine Parameters

A geodesic is the curved-spacetime successor to the straight line: the one curve that carries its own tangent vector forward by parallel transport, bending as little as the geometry allows. But a curve is a set of points, and a parametrization is the extra choice of how fast a label runs along it. That second choice is not physically neutral. Written one way, the geodesic obeys the clean autoparallel law $u^\nu\nabla_\nu u^\mu=0$ ; written another way — the very same points, relabelled — it grows a spurious right-hand side $f(\lambda)\,u^\mu$ that looks like acceleration but bends nothing. The parametrizations for which the clean form holds are the affine parameters, and they are not a single function but a whole family, any two members related by $\lambda\mapsto a\lambda+b$ . This distinction is the quiet hinge on which most of geometric general relativity turns. It reconciles the two definitions of a geodesic — "parallel-transport your tangent" and "extremize arc length" — that otherwise seem to give different equations. It explains why proper time is the natural clock of a falling body but is useless for a photon, whose worldline has zero length between any two of its points. And it is the fine print behind every conservation law you will ever use in a black-hole spacetime: the energy $-\xi_\mu p^\mu$ and angular momentum $\eta_\mu p^\mu$ that reduce orbits to one-dimensional potential problems are constant only when the tangent is affinely parametrized. Choose a bad parameter and none of these statements is false — each simply arrives dressed in correction terms proportional to $f$ that hide the geometry. Affine parametrization is the gauge choice that makes the geometry visible.

Key takeaways

An affine parameter is a privileged labelling of a geodesic for which the tangent parallel-transports into itself with no leftover acceleration: $u^\nu\nabla_\nu u^\mu=0$ .
Any admissible parameter is related to an affine one by an affine transformation $\tilde\lambda=a\lambda+b$ ( $a\neq0$ ); this two-constant freedom is the residual gauge of the geodesic equation.
A generic reparametrization leaves the path untouched but adds a source $f(\lambda)\,u^\mu$ , stretching or shrinking the tangent — same road, different odometer.
Timelike and spacelike geodesics come with a built-in affine parameter, arc length ( $\tau$ or $s$ ); null geodesics have no arc length, so their affine parameter must be supplied intrinsically by parallel transport.
Affine parametrization is exactly what makes the norm $g_{\mu\nu}u^\mu u^\nu$ constant and what makes the Killing charges $\xi_\mu u^\mu$ conserved — both fail by a term $\propto f$ otherwise.
Key idea: A geodesic satisfies $u^\nu\nabla_\nu u^\mu=f(\lambda)\,u^\mu$ ; the parameters that kill the right-hand side ( $f=0$ ) are the affine ones, unique up to $\lambda\to a\lambda+b$ , and only they make the tangent's length constant and the conservation laws hold without corrections.

The core insight

A geodesic satisfies $u^\nu\nabla_\nu u^\mu=f(\lambda)\,u^\mu$ ; the parametrizations that kill the right-hand side ( $f=0$ ) are the affine parameters, unique up to $\lambda\to a\lambda+b$ , and affine parametrization is precisely what makes the tangent's magnitude constant and the Killing conservation laws hold without corrections.

Where this lives today

Every image of a black hole you have seen is a picture drawn by affine parameters. The Event Horizon Telescope's reconstructions of M87 and Sagittarius A, and every synthetic frame of Interstellar's Gargantua, are made by numerically integrating null geodesics backward from a virtual camera — and the integrator uses the autoparallel equation precisely because it is the form free of spurious acceleration terms, with the conserved $E$ and $L$ of Worked Example 2 serving as the constants that keep the integration honest. The same machinery runs GRMHD simulations of accretion flows, strong-lensing reconstructions of distant galaxies, and the ray-tracing pipelines that turn a spacetime metric into a sky. In gravitational-wave astronomy, the geodesic-deviation equation that models how a passing wave stretches a detector's arms presupposes a common affine parameter on neighbouring worldlines; in cosmology, the "affine distance" along a light ray is the natural variable behind redshift–distance relations and the integrated Sachs–Wolfe effect. And at the foundational end, relativistic geodesy and the timing models of pulsar-timing arrays lean on the fact that a freely falling clock ticks proper time — the affine parameter of its own timelike geodesic — as the equivalence principle demands. Wherever light or matter falls freely and someone needs a number out of it, an affine parameter is doing the quiet work.

The physical picture

Imagine driving along a fixed mountain road. The road itself — every twist and rise fixed by the terrain — is the geodesic, a purely geometric object. Your trip odometer is the parameter: it labels where you are, but nothing about the road forces it to advance at any particular rate. An affine parameter is the "cruise-control" labelling in which your velocity vector keeps a constant length as measured by the geometry — no phantom pushes. Flooring and releasing the accelerator gives a perfectly legal but non-affine labelling: the road is identical, but the tangent vector you carry stretches and squeezes, and the equation of motion sprouts a term $f(\lambda)\,u^\mu$ that is pure bookkeeping, pointing straight along the road and bending it not at all.

Figure

A single geodesic, two labellings. Affine (top): the parameter advances so the tangent vector keeps constant length, u·∇u = 0. Non-affine (bottom): the ticks bunch and the tangent stretches, u·∇u = f u — the road is identical, only the odometer differs

The cleanest place to see the phantom acceleration is flat space. On a plane in Cartesian coordinates the straight line $x^\mu(\lambda)=a^\mu\lambda+b^\mu$ has $\ddot x^\mu=0$ — constant velocity — precisely because $\lambda$ enters linearly. Replace $\lambda$ by $\sigma$ with $\lambda=\sigma^3$ and the identical line becomes $x^\mu=a^\mu\sigma^3$ , whose $\ddot x^\mu=6a^\mu\sigma\neq0$ . Nothing bent; the "acceleration" is entirely an artifact of running the label as $\sigma^3$ instead of $\lambda$ . That artifact is the term $f\,u^\mu$ , and $f=2/\sigma$ here. The second figure tracks its fingerprint — the growth of the tangent's own length — with real numbers.

Figure

Flat-space straight line x = σ³ carried with the non-affine label σ: the tangent norm g(u,u) = 9σ⁴ grows without bound (reaching 144 at σ = 2), obeying d/dσ g(u,u) = 2f·g(u,u) with f = 2/σ; the affine label λ = σ³ instead pins the norm flat at 1

For light there is no ruler to fall back on. Every pair of points on a null geodesic is separated by zero interval, so proper time and proper distance both collapse to nothing along a light ray. The affine parameter is then the only natural clock a photon carries, and it is not borrowed from any metre-stick — it is handed to the ray by the geometry itself, through the demand that the photon's four-momentum parallel-transport. This is why affine parametrization, far from being a technicality, is the physical carrier of frequency and redshift along a beam of light: the ratio of the tangent's contraction against observers at emission and reception is the redshift.

The core equations

A geodesic is defined by parallel-transporting its own tangent; the covariant derivative of $u^\mu$ along the curve can therefore only point back along $u^\mu$ . Expanding the covariant derivative into partials and the connection turns this statement into a second-order ODE for the coordinates. When the tangential source is set to zero, we get the autoparallel form.

u^\nu\nabla_\nu u^\mu = 0 \qquad (1)

where: $u^\mu=\mathrm{d}x^\mu/\mathrm{d}\lambda$ — tangent vector to the curve; $\nabla_\nu$ — covariant derivative of the chosen (metric-compatible, torsion-free) connection; $\lambda$ — the affine parameter labelling points along the geodesic. Read it as: "the tangent does not change at all as it is carried along the curve" — the sharpest possible statement of moving as straight as the geometry permits. Drop the requirement that the source vanish and you get the most general equation a geodesic can satisfy in an arbitrary parametrization. Because parallel transport cannot rotate the tangent off the curve, the right-hand side must be purely tangential — a scalar times $u^\mu$ .

u^\nu\nabla_\nu u^\mu = f(\lambda)\,u^\mu \qquad (2)

where: $f(\lambda)$ — scalar function measuring the non-affineness, vanishing if and only if $\lambda$ is affine; $u^\mu$ — tangent vector, here of generally non-constant length. Read it as: "the same path, but the odometer is running unevenly"; $f$ has no transverse component because any such piece would bend the curve and it would no longer be a geodesic. Two affine parameters describe the identical geodesic with the identical clean equation, so they can differ only by a transformation that preserves the linear structure of the parameter axis. Solving $\lambda''=f\lambda'$ with $f=0$ (Derivation 1) delivers exactly this residual freedom.

\tilde\lambda = a\lambda + b, \qquad a\neq 0 \qquad (3)

where: $a$ — nonzero rescaling constant that sets the unit of the parameter; $b$ — additive constant that sets the origin. Read it as the geodesic's gauge group: fixing "affine" still leaves a two-parameter family, the direct analogue of choosing the zero and the tick-size of a clock. Metric compatibility lets the metric slide through the directional derivative, so the rate of change of the tangent's squared length is governed entirely by the source $f$ . This single identity is the engine behind the whole classification of geodesics.

\frac{\mathrm{d}}{\mathrm{d}\lambda}\big(g_{\mu\nu}u^\mu u^\nu\big) = 2f(\lambda)\,g_{\mu\nu}u^\mu u^\nu \qquad (4)

where: $g_{\mu\nu}$ — metric tensor with $\nabla_\rho g_{\mu\nu}=0$ ; $g_{\mu\nu}u^\mu u^\nu$ — squared norm of the tangent. Read it as: "the norm changes only through the non-affineness"; set $f=0$ and the norm is a constant of the motion — the tangent neither stretches nor shrinks. With an affine parameter the norm is fixed once and for all, and its sign classifies the geodesic permanently. For non-null curves the value can be normalized to unity, which simultaneously fixes the scale constant $a$ .

g_{\mu\nu}u^\mu u^\nu = \varepsilon, \qquad \varepsilon\in\{-1,\,0,\,+1\} \qquad (5)

where: $\varepsilon=-1$ timelike (with $\lambda=\tau$ , proper time), $\varepsilon=0$ null, $\varepsilon=+1$ spacelike (with $\lambda=s$ , proper distance), in signature $-+++$ . Read it as the birth certificate of a geodesic: whether a worldline is that of matter, light, or a spacelike section is decided by this one conserved number. Contracting an affine tangent with a Killing vector produces a quantity that is conserved along the geodesic — the machinery that turns symmetries into constants of motion. The proof (Derivation 3) needs the affine equation to kill one of the two Leibniz terms.

\frac{\mathrm{d}}{\mathrm{d}\lambda}\big(\xi_\mu u^\mu\big) = 0 \qquad (6)

where: $\xi^\mu$ — Killing vector field obeying $\nabla_{(\mu}\xi_{\nu)}=0$ ; $\xi_\mu u^\mu$ — the conserved charge, e.g. energy $E=-\xi_{(t)}{}_\mu p^\mu$ or angular momentum $L=\xi_{(\phi)}{}_\mu p^\mu$ . Read it as: "each spacetime symmetry hands a geodesic one conserved number" — provided, and only provided, the parameter is affine, or an extra $f\,\xi_\mu u^\mu$ would spoil it.

Derivation AFrom an arbitrary parameter to an affine one

The goal is to prove the central structural claim of the whole topic: that every geodesic, however badly parametrized, can be relabelled to the clean autoparallel form, and that the freedom left over after doing so is exactly the affine group $\lambda\mapsto a\lambda+b$ . We begin from the general geodesic equation carrying a source $f(\sigma)$ , change the parameter to a new label $\lambda(\sigma)$ , and demand that the source cancel; the condition on $\lambda$ is a simple linear ODE whose solution both defines the affine parameter and exposes its two-constant ambiguity.

Step 1. Write the geodesic with a generic parameter $\sigma$ , so the coordinates obey $\frac{\mathrm{d}^2x^\mu}{\mathrm{d}\sigma^2}+\Gamma^\mu_{\nu\rho}\frac{\mathrm{d}x^\nu}{\mathrm{d}\sigma}\frac{\mathrm{d}x^\rho}{\mathrm{d}\sigma}=f(\sigma)\frac{\mathrm{d}x^\mu}{\mathrm{d}\sigma}$ , with $f$ encoding the residual tangential acceleration. Why: this is the most general form Eq. (2) takes in coordinates — the bracket on the left is the covariant acceleration, the right-hand side its purely tangential remainder.

Step 2. Introduce a new parameter $\lambda(\sigma)$ and use the chain rule, $\frac{\mathrm{d}x^\mu}{\mathrm{d}\sigma}=\lambda'\frac{\mathrm{d}x^\mu}{\mathrm{d}\lambda}$ , where a prime denotes $\mathrm{d}/\mathrm{d}\sigma$ . Why: reparametrizing is nothing but a change of the independent variable, and the tangent vector rescales by the Jacobian $\lambda'$ of that change.

Step 3. Differentiate once more and substitute, collecting terms into $(\lambda')^2\big[\frac{\mathrm{d}^2x^\mu}{\mathrm{d}\lambda^2}+\Gamma^\mu_{\nu\rho}\frac{\mathrm{d}x^\nu}{\mathrm{d}\lambda}\frac{\mathrm{d}x^\rho}{\mathrm{d}\lambda}\big]+\lambda''\frac{\mathrm{d}x^\mu}{\mathrm{d}\lambda}=f\,\lambda'\frac{\mathrm{d}x^\mu}{\mathrm{d}\lambda}$ . Why: the second derivative produces two pieces — the covariant-acceleration bracket scaled by $(\lambda')^2$ , and a leftover $\lambda''\,u^\mu_\lambda$ from differentiating the Jacobian — and the bracket is precisely the affine form we are hunting for.

Step 4. Demand that the bracket vanish; that forces the surviving terms to balance, giving the condition $\lambda''=f\,\lambda'$ . Why: killing the bracket is the definition of " $\lambda$ is affine," so the leftover $\lambda''\,u^\mu_\lambda$ must cancel the source $f\lambda'\,u^\mu_\lambda$ — one scalar equation for one unknown function $\lambda(\sigma)$ .

Step 5. Integrate the linear ODE: $\ln\lambda'=\int f\,\mathrm{d}\sigma+\text{const}$ , hence $\lambda(\sigma)=a\!\int e^{\int f\,\mathrm{d}\sigma}\,\mathrm{d}\sigma+b$ , with integration constants $a\neq0$ and $b$ . Why: this proves existence — an affine parameter always exists for any geodesic — and when $f=0$ already it collapses to $\lambda=a\sigma+b$ , so any two affine parameters differ by exactly an affine transformation, establishing Eq. (3). *As a check: setting $f=0$ in the solution collapses it to $\lambda=a\sigma+b$ , confirming the affine-transformation freedom. For the flat-space example $x=\sigma^3$ one has $f=2/\sigma$ , giving $\lambda\propto\int e^{2\ln\sigma}\mathrm{d}\sigma=\int\sigma^2\mathrm{d}\sigma\propto\sigma^3$ — the cube, as expected.*

Go deeper · Derivation B · 2 minProper time as the canonical affine parameter for timelike geodesics

Derivation BProper time as the canonical affine parameter for timelike geodesics

The goal is to show that the variational definition of a timelike geodesic — the extremal of proper time — automatically delivers the autoparallel equation with proper time itself as the parameter, so that "extremize arc length" and "parallel-transport the tangent" agree, and agree specifically in the affine gauge. We work from the quadratic (energy) action, take its Euler–Lagrange equation, and read off both the connection and the normalization that fixes the scale, exhibiting proper time as an affine parameter with $g_{\mu\nu}u^\mu u^\nu=-1$ .

Step 1. Adopt the quadratic action $S=\tfrac12\!\int g_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}\tau}\frac{\mathrm{d}x^\nu}{\mathrm{d}\tau}\,\mathrm{d}\tau$ rather than the square-root arc-length action. Why: the square-root form is reparametrization-invariant and its Euler–Lagrange equation carries an $f\,u^\mu$ term for a generic parameter; the quadratic form is legitimate only when the parameter is proportional to arc length, so choosing it is already a bet that $\tau$ is affine — a bet Step 5 confirms.

Step 2. Vary $x^\mu$ to obtain the Euler–Lagrange equation $\frac{\mathrm{d}}{\mathrm{d}\tau}\big(g_{\mu\nu}\dot x^\nu\big)-\tfrac12\partial_\mu g_{\nu\rho}\dot x^\nu\dot x^\rho=0$ , with dots denoting $\mathrm{d}/\mathrm{d}\tau$ . Why: this is the standard first variation of a Lagrangian quadratic in velocities; the first term is the momentum's rate of change, the second the "force" from the coordinate dependence of the metric.

Step 3. Expand $\frac{\mathrm{d}}{\mathrm{d}\tau}(g_{\mu\nu}\dot x^\nu)=g_{\mu\nu}\ddot x^\nu+\partial_\rho g_{\mu\nu}\dot x^\rho\dot x^\nu$ , symmetrize the derivative term, and contract with the inverse metric to get $\ddot x^\mu+\tfrac12 g^{\mu\sigma}(\partial_\nu g_{\sigma\rho}+\partial_\rho g_{\sigma\nu}-\partial_\sigma g_{\nu\rho})\dot x^\nu\dot x^\rho=0$ . Why: the symmetric combination of three metric derivatives is exactly twice the Christoffel symbol, so the bracket is engineered to reassemble the connection.

Step 4. Identify the connection to obtain $\ddot x^\mu+\Gamma^\mu_{\nu\rho}\dot x^\nu\dot x^\rho=0$ . Why: this is Eq. (1) written in coordinates with $\tau$ as the parameter and with no right-hand side, so proper time is affine — the two definitions of a geodesic have merged.

Step 5. Check the norm: since the Lagrangian $L=\tfrac12 g_{\mu\nu}\dot x^\mu\dot x^\nu$ has no explicit $\tau$ -dependence, its value is conserved along the solution, and normalizing proper time by $\mathrm{d}\tau^2=-g_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu$ fixes that constant to $g_{\mu\nu}\dot x^\mu\dot x^\nu=-1$ . Why: conservation of $L$ is Eq. (4) with $f=0$ , and the value $-1$ is what pins the scale constant $a$ , leaving only the origin $b$ free for a timelike geodesic. *Note: the square-root action gives the same path but requires the affine choice to simplify to the connection form, and the conserved norm $-1$ is the signature- $-+++$ definition of proper time.*

Go deeper · Derivation C · 2 minAffine parametrization conserves the tangent norm and the Killing charges

Derivation CAffine parametrization conserves the tangent norm and the Killing charges

The goal is to demonstrate the two payoffs that make affine parametrization worth insisting on: that the tangent's squared length is conserved, and that every Killing symmetry yields a conserved charge — and to expose, in each case, the single term that would spoil the result if the parameter were non-affine. Both proofs are one line of Leibniz plus the geodesic equation, and both hinge on the same factor $u^\nu\nabla_\nu u^\mu$ being zero rather than $f\,u^\mu$ .

Step 1. Differentiate the norm along the curve using metric compatibility: $u^\rho\nabla_\rho(g_{\mu\nu}u^\mu u^\nu)=2g_{\mu\nu}u^\nu(u^\rho\nabla_\rho u^\mu)$ . Why: because $\nabla_\rho g_{\mu\nu}=0$ the metric passes through the derivative untouched, so all the dynamics sits in the covariant acceleration $u^\rho\nabla_\rho u^\mu$ .

Step 2. Insert the general geodesic equation $u^\rho\nabla_\rho u^\mu=f\,u^\mu$ to get $2f\,g_{\mu\nu}u^\mu u^\nu$ , which is Eq. (4); with an affine parameter $f=0$ and the norm is exactly conserved. Why: this is where the classification $\varepsilon\in\{-1,0,+1\}$ becomes permanent — a timelike geodesic can never become null under affine evolution, because its norm cannot budge.

Step 3. Now form the Killing charge and differentiate: $u^\nu\nabla_\nu(\xi_\mu u^\mu)=(u^\nu\nabla_\nu\xi_\mu)u^\mu+\xi_\mu(u^\nu\nabla_\nu u^\mu)$ . Why: the Leibniz rule splits the charge's evolution into a piece that tests the symmetry of the field $\xi$ and a piece that tests the geodesic equation — and the second piece is exactly the one that vanishes only in the affine gauge.

Step 4. Drop the second term because the parameter is affine ( $u^\nu\nabla_\nu u^\mu=0$ ), leaving $u^\nu u^\mu\nabla_\nu\xi_\mu$ , then symmetrize: $u^\nu u^\mu\nabla_{(\nu}\xi_{\mu)}=0$ by Killing's equation. Why: the contraction $u^\nu u^\mu$ is symmetric in its indices, so it annihilates the antisymmetric part of $\nabla_\nu\xi_\mu$ and sees only the symmetric part, which Killing's equation sets to zero — the charge is conserved.

Step 5. Trace the failure for a non-affine parameter: the dropped term is instead $\xi_\mu(f\,u^\mu)=f\,(\xi_\mu u^\mu)$ , so $\frac{\mathrm{d}}{\mathrm{d}\sigma}(\xi_\mu u^\mu)=f\,\xi_\mu u^\mu\neq0$ in general. Why: this makes the stakes concrete — with the wrong parameter, "energy" and "angular momentum" drift at a rate set entirely by the non-affineness $f$ , even though the underlying symmetry is untouched. *Both identities are the norm identity Eq. (4) and the conserved-quantity theorem, each failing by a term $\propto f$ off the affine gauge.*

Limiting cases

How the affine character of the standard parameters manifests across the three types of geodesic, and what a non-affine choice costs:

Case	Result	Meaning
Timelike geodesic	$g_{\mu\nu}u^\mu u^\nu=-1,\ \lambda=\tau$	Proper time is automatically affine; the normalization $-1$ (signature $-+++$ ) fixes the scale $a$ , leaving only the origin $b$ — a falling body's own clock.
Spacelike geodesic	$g_{\mu\nu}u^\mu u^\nu=+1,\ \lambda=s$	Proper distance is affine; identical structure to the timelike case with the sign flipped — the parameter of a spatial geodesic in a fixed slice.
Null geodesic	$g_{\mu\nu}u^\mu u^\nu=0$	Arc length is identically zero and useless as a label; the affine parameter is defined intrinsically by $u^\nu\nabla_\nu u^\mu=0$ and is fixed only up to $\lambda\to a\lambda+b$ — both constants remain free.
Non-affine parameter	$u^\nu\nabla_\nu u^\mu=f\,u^\mu$	Same path, but the norm drifts as $\frac{\mathrm{d}}{\mathrm{d}\lambda}(g_{\mu\nu}u^\mu u^\nu)=2f\,g_{\mu\nu}u^\mu u^\nu$ , and every Killing charge drifts as $f\,\xi_\mu u^\mu$ .
Flat space, $x^\mu=a^\mu\sigma^3$	$f=2/\sigma$ , affine $\lambda\propto\sigma^3$	A perfectly straight line made to look accelerated by a bad label; the phantom $\ddot x^\mu=6a^\mu\sigma$ is pure parametrization, removed by $\lambda=\sigma^3$ .

Common trapWriting Γ in the geodesic equation already assumes an affine parameter

The equation $\ddot x^\mu+\Gamma^\mu_{\nu\rho}\dot x^\nu\dot x^\rho=0$ is not the definition of a geodesic for an arbitrary parameter — it is the definition only once you have chosen an affine one. With a generic parameter the right-hand side is $f\,\dot x^\mu\neq0$ , and forgetting this is the most common way to get wrong answers in a numerical integration: you write the clean equation, feed it a coordinate time that is not affine, and watch your "conserved" energy slowly drift. Two further traps hide here. First, for a null geodesic proper time is identically zero and can never be the affine parameter — you must supply one through parallel transport of the photon momentum, not borrow it from a ruler. Second, the affine condition is a statement about the tangent's length, not about "constant speed" in any coordinate sense; coordinate speed is frame-dependent and says nothing about whether $\lambda$ is affine.

Worked examples

1 · Curing a phantom acceleration in flat spacetime (fully worked)

A particle moves along the $x$ -axis of flat Minkowski space on the perfectly straight worldline $x(\sigma)=\sigma^3$ (with $t,y,z$ held fixed for the spatial illustration; Christoffel symbols all zero in Cartesian coordinates). The naïve acceleration is nonzero. Find the source $f(\sigma)$ , verify the norm-growth law numerically, and construct the affine parameter.

Step 1. Compute the tangent and its coordinate derivative: $u=\mathrm{d}x/\mathrm{d}\sigma=3\sigma^2$ and $\mathrm{d}u/\mathrm{d}\sigma=6\sigma$ . With zero connection the geodesic equation reads $\mathrm{d}u/\mathrm{d}\sigma=f\,u$ , i.e. $6\sigma=f\,(3\sigma^2)$ , so $\boxed{f=2/\sigma}$ . The line is straight, yet $f\neq0$ — the label $\sigma$ is non-affine.

Step 2. Track the tangent norm (Euclidean along $x$ ): $g(u,u)=u^2=9\sigma^4$ . Check Eq. (4): $\frac{\mathrm{d}}{\mathrm{d}\sigma}(9\sigma^4)=36\sigma^3$ , while $2f\,g(u,u)=2\cdot\frac{2}{\sigma}\cdot9\sigma^4=36\sigma^3$ . They agree identically. Numerically the norm runs $9$ at $\sigma=1$ , $\mathbf{144}$ at $\sigma=2$ , and $729$ at $\sigma=3$ — the tangent stretches steeply even though nothing physical accelerates.

Step 3. Build the affine parameter from Derivation 1: $\lambda\propto\int e^{\int f\,\mathrm{d}\sigma}\mathrm{d}\sigma=\int e^{2\ln\sigma}\mathrm{d}\sigma=\int\sigma^2\,\mathrm{d}\sigma=\tfrac13\sigma^3+\text{const}$ , so $\lambda=\sigma^3$ up to the affine freedom. In this label $x=\lambda$ , hence $\mathrm{d}x/\mathrm{d}\lambda=1$ (constant) and $\mathrm{d}^2x/\mathrm{d}\lambda^2=0$ . The phantom acceleration is gone and the norm is pinned at $g(u,u)=1$ . The road never changed; only the odometer did.

2 · Conserved charges and the photon sphere in Schwarzschild (numbers)

In Schwarzschild spacetime (geometric units $G=c=1$ ) a photon moves on a null geodesic. Using the two Killing vectors, define its conserved energy and angular momentum, find the radius of the unstable circular photon orbit, and evaluate the critical impact parameter — the observable that fixes a black-hole shadow.

Step 1. The metric $\mathrm{d}s^2=-(1-2M/r)\mathrm{d}t^2+(1-2M/r)^{-1}\mathrm{d}r^2+r^2\mathrm{d}\Omega^2$ has a timelike Killing vector $\xi_{(t)}=\partial_t$ and a rotational one $\xi_{(\phi)}=\partial_\phi$ . By Eq. (6), and only because the photon momentum $p^\mu=\mathrm{d}x^\mu/\mathrm{d}\lambda$ is affinely parametrized, the charges $E=-\xi_{(t)\mu}p^\mu=(1-2M/r)\,\dot t$ and $L=\xi_{(\phi)\mu}p^\mu=r^2\dot\phi$ are constant along the ray (dots are $\mathrm{d}/\mathrm{d}\lambda$ ). A non-affine label would let both drift by $f E$ and $f L$ .

Step 2. Impose the null norm $g_{\mu\nu}p^\mu p^\nu=0$ (Eq. 5, $\varepsilon=0$ ) in the equatorial plane and solve for the radial motion: $\dot r^2=E^2-\frac{L^2}{r^2}\big(1-\frac{2M}{r}\big)$ . The effective potential $V(r)=\frac{L^2}{r^2}(1-2M/r)$ has a single maximum. Setting $V'(r)=0$ gives $-2r^{-3}+6Mr^{-4}=0$ , i.e. $\boxed{r_{\rm ph}=3M}$ — the photon sphere, an unstable circular light orbit.

Step 3. A circular orbit needs $\dot r=0$ with $r=r_{\rm ph}$ , so $E^2=V(3M)=\frac{L^2}{9M^2}(1-\tfrac{2}{3})=\frac{L^2}{27M^2}$ . The critical impact parameter is therefore $b_c=L/E=\sqrt{27}\,M=\mathbf{3\sqrt3\,M\approx5.196\,M}$ . Any photon aimed with $b<b_c$ is captured; $b>b_c$ escapes; $b=b_c$ spirals onto the photon sphere.

Step 4. Put in a real black hole: for Sagittarius A*, $M\approx4.15\times10^6\,M_\odot$ , so $GM/c^2\approx6.13\times10^6\ \mathrm{km}$ and $b_c=5.196\times6.13\times10^6\approx\mathbf{3.19\times10^7\ km}$ . Projected to Earth this sets the angular radius of the shadow the Event Horizon Telescope images — a number that exists only because $E$ and $L$ are conserved, which is to say only because the null geodesic is affinely parametrized.

Physics of the portrait

The affine parameter is the fingerprint of a deeper idea: that a manifold can carry a rule for parallel transport — a connection — that is logically prior to, and independent of, any notion of length. That separation of "straightness" from "distance" was Hermann Weyl's, and it is why a photon, which has no proper length to call its own, still has a perfectly good affine clock.

Origins · 1 minHermann Weyl and how this was discovered

Hermann Weyl

1885–1955 · founder of the affine connection and the geometry of parallel transport

Weyl was the mathematician who first prised the geodesic apart from the ruler. In Raum–Zeit–Materie (1918) and the papers surrounding it, he introduced the affine connection (der affine Zusammenhang) as an independent geometric structure — a rule specifying how to parallel-transport a vector — and defined a geodesic, or "autoparallel," as a curve that transports its own tangent, with no reference to the metric at all. This was a genuine conceptual break: Riemann and Levi-Civita had derived the connection from the metric, but Weyl saw that the connection could stand on its own, and that the natural class of parameters along an autoparallel was fixed only up to a linear transformation — the affine parameters. The same 1918 work launched his ill-fated but hugely influential gauge theory, an attempt to unify gravity and electromagnetism by letting lengths as well as directions change under transport; Einstein's objection (that atomic clocks would then remember their histories) sank the physical theory but seeded the modern notion of gauge invariance. Weyl's separation of affine from metric structure is exactly what lets general relativity treat null geodesics on the same footing as timelike ones: the metric supplies proper time when it can, but when it cannot — for light — the affine parameter that Weyl isolated is still there, handed to the ray by the connection alone. His reach ran from the Weyl tensor and the Weyl equation for massless fermions to the representation theory of Lie groups; the affine parameter is one of his quietest and most durable legacies.

History

1854

— Riemann's habilitation lecture founds the intrinsic geometry of curved manifolds

1869

— Christoffel introduces the symbols Γ that make covariant differentiation possible

1900

— Ricci and Levi-Civita systematize the absolute differential (tensor) calculus

1917

— Levi-Civita defines parallel transport, giving the connection an intrinsic meaning

1918

— Weyl introduces the affine connection and isolates the affine parameter of an autoparallel

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, §10.4 and §13.2 (parallel transport and geodesics) · Wald, General Relativity, §3.1–3.3 (geodesics and the affine parameter) · Carroll, Spacetime and Geometry, §3.1, §3.3, §3.8 (geodesics, Killing vectors) · Hartle, Gravity, §8.1–8.3 (geodesics and conserved quantities) · Schutz, A First Course in General Relativity, §6.4. Key results to keep in mind: the reparametrization ODE is $\lambda''=f\lambda'$ , with the flat-space example $x=\sigma^3\Rightarrow f=2/\sigma\Rightarrow\lambda\propto\sigma^3$ ; the norm-evolution identity Eq. (4) follows from metric compatibility $\nabla g=0$ , with the numerical instance $g(u,u)=9\sigma^4$ ( $=144$ at $\sigma=2$ ); the timelike Euler–Lagrange derivation confirms $g_{\mu\nu}\dot x^\mu\dot x^\nu=-1$ in signature $-+++$ ; the Schwarzschild photon sphere is $r_{\rm ph}=3M$ with critical impact parameter $b_c=3\sqrt3\,M\approx5.196\,M$ . Norm values $\{-1,0,+1\}$ are signature-dependent ( $-+++$ used here); the Sgr A* number uses $M\approx4.15\times10^6\,M_\odot$ .

Prerequisites. Parallel Transport · The Covariant Derivative · The Geodesic Equation · Christoffel Symbols. Leads to. Geodesics as Extremal Paths · The Geodesic Deviation Equation · Killing Vectors · Tidal Forces in General Relativity.

Ask the tutor: Curious how a photon can carry an affine parameter with no proper time to call its own, or how conserved energy quietly breaks when you pick the wrong label — ask, and we will trace it through a real geodesic together.

What is the difference between a geodesic and an affinely-parametrized geodesic?

A geodesic is a curve — a set of points in spacetime that is "as straight as possible," meaning it parallel-transports its own tangent direction. An affinely-parametrized geodesic adds a choice: a way of labelling those points so that the tangent vector keeps a constant length as it is carried along, giving the clean equation $u^\nu\nabla_\nu u^\mu=0$ . The same geodesic can be parametrized non-affinely, in which case the equation gains a term $f(\lambda)u^\mu$ that points along the curve and changes nothing geometric — it only signals that the label is running unevenly. So "geodesic" is about the path; "affine" is about the odometer.

Why does affine parametrization matter if it changes no physics?

Because it is what makes the physics readable. Nothing observable depends on the choice — just as nothing depends on which inertial frame you use — but the formulas change dramatically. In the affine gauge the tangent norm is conserved (so a geodesic's timelike/null/spacelike character is fixed once and for all), the Killing charges energy and angular momentum are constant, and the geodesic-deviation equation takes its clean curvature form. Use a non-affine parameter and every one of these acquires a correction term proportional to $f$ : your "conserved" quantities drift, your classification wobbles, and a numerical integrator quietly loses accuracy. Affine parametrization buys simplicity and reliability for the price of a gauge choice.

How can a photon have a parameter if it experiences no proper time?

This is exactly the situation Weyl's separation of connection from metric was built for. Along a null geodesic every pair of points is at zero interval, so proper time and proper distance both vanish and cannot label the ray. But the connection still knows how to parallel-transport the photon's four-momentum, and demanding that transport ( $p^\nu\nabla_\nu p^\mu=0$ ) defines a parameter directly — the affine parameter — with no metric length involved. It is fixed only up to $\lambda\to a\lambda+b$ , since there is no normalization $\pm1$ to pin the scale. Physically this affine parameter is the carrier of the photon's frequency: comparing the tangent against observers at emission and reception gives the redshift.

If f u only points along the curve, why not just keep the non-affine form?

You can — the path is a genuine geodesic either way — but you pay for it everywhere downstream. The term $f\,u^\mu$ contributes nothing to the geometry, yet it infiltrates every derived quantity: the norm $g_{\mu\nu}u^\mu u^\nu$ is no longer constant, the Killing charges drift at rate $f\,\xi_\mu u^\mu$ , and the tidy statement $\ddot x^\mu+\Gamma^\mu_{\nu\rho}\dot x^\nu\dot x^\rho=0$ becomes $=f\dot x^\mu$ . In a numerical ray-trace this shows up as spuriously non-conserved energy that flags integration error you cannot cleanly diagnose. Since removing $f$ costs only a one-line reparametrization (Derivation 1), the affine form is almost always worth adopting immediately.

Is proper time the only affine parameter for a timelike geodesic?

No — it is the canonical one, but any affine transform of it is equally affine. If $\tau$ is proper time then $\tilde\lambda=a\tau+b$ is also an affine parameter for the same worldline; the geodesic equation stays clean under it. What is special about proper time is that its normalization $g_{\mu\nu}u^\mu u^\nu=-1$ fixes the scale constant $a=1$ , so proper time is the affine parameter whose tangent is a unit four-velocity — the one a physical clock reads. Choosing a different affine parameter rescales the four-velocity's length away from unity but leaves every geometric statement intact. For null geodesics there is no such normalization, so both constants genuinely remain free.

Does any of this change if the connection has torsion or is non-metric?

The core statement survives, but two conveniences can fail. The autoparallel equation $u^\nu\nabla_\nu u^\mu=0$ and the affine reparametrization freedom $\lambda\to a\lambda+b$ depend only on having a connection, so they hold for any affine connection — torsion and all — since torsion is antisymmetric and drops out of the symmetric contraction $u^\nu u^\rho$ in the geodesic equation. What needs metric compatibility $\nabla g=0$ is the norm-evolution law Eq. (4): without it the metric no longer slides through the derivative, and even an affine parameter need not conserve $g_{\mu\nu}u^\mu u^\nu$ . Weyl's original non-metric geometry is precisely the case where lengths change under transport, which is what Einstein objected to. In standard general relativity the connection is the metric-compatible torsion-free Levi-Civita connection, so all the clean results hold.