Non-Affine Geodesic Parametrization

A geodesic is spacetime's straight line — the worldline a freely falling body traces when nothing but gravity acts. Its defining property, inherited from parallel transport, is that the curve carries its own tangent vector forward without ever turning it: the tangent is transported into a copy of itself. Write that condition with a well-chosen label along the curve and it collapses to the clean geodesic equation $u^\mu\nabla_\mu u^\nu=0$ . But "into a copy of itself" hides a choice. The geometric content is only that the transported tangent stay parallel to the original — pointed the same way. It never demanded that the two be equal in length. Relax that last requirement and the tangent may be carried into a multiple of itself, a scalar $f$ times $u^\nu$ , and the equation grows a right-hand side: the non-affine, or generalized, geodesic equation $u^\mu\nabla_\mu u^\nu=f(\lambda)\,u^\nu$ . That extra term is not a force and not a bending but a bookkeeping artifact of the label. A curve is a set of points; parametrize the same worldline two ways and it obeys two differently-shaped equations even though its image in the manifold is identical. The acceleration $f\,u^\nu$ is always tangent to the curve, never transverse — it can speed up or slow the advance of the parameter but can never push the particle off its geodesic. This is the precise sense in which "geodesic" is parametrization-invariant while "affinely parametrized geodesic" is not. Because the source is purely longitudinal it can always be integrated away; yet the non-affine form is unavoidable exactly where nature hands you a non-affine label — coordinate time on an orbit, or the Killing parameter on a horizon, where the constant non-affinity is the surface gravity.

Key takeaways

A geodesic is a curve whose tangent is parallel-transported along itself. With a general parameter $\lambda$ this reads $u^\mu\nabla_\mu u^\nu=f(\lambda)\,u^\nu$ — the right side is not zero but proportional to the tangent.
The non-affinity $f(\lambda)=\ddot\tau/\dot\tau$ measures how fast the label $\lambda$ stretches relative to an affine standard $\tau$ ; it vanishes exactly for affine parameters, which are unique up to $\tau\to a\tau+b$ .
The acceleration $f\,u^\nu$ is strictly longitudinal — parallel to the curve — so it changes only the parametrization speed, never the path; the image in the manifold is untouched.
You can always remove it: solving $\ddot\tau/\dot\tau=f$ integrates $\lambda$ back to an affine parameter, so affine and non-affine descriptions are physically equivalent.
The norm $g_{\mu\nu}u^\mu u^\nu$ is no longer conserved but obeys $u^\mu\nabla_\mu(g\,uu)=2f\,(g\,uu)$ — it rescales yet cannot change sign, so timelike stays timelike and null stays null.
Key idea: parallel transport of the tangent into a multiple of itself, $u^\mu\nabla_\mu u^\nu=f\,u^\nu$ , defines a geodesic; the acceleration term is always parallel to the curve, so it changes only the parametrization speed, never the path. Setting $f=0$ defines affine parameters, unique up to $\tau\to a\tau+b$ .

Where this lives today

The non-affine term is where geometry meets thermodynamics. Its most active modern arena is horizon surface gravity: the constant $\kappa$ from reading the horizon-generating Killing vector as a non-affine geodesic is the same $\kappa$ that fixes the Hawking temperature, and $\tau\propto e^{\kappa V}$ is the redshift that stretches vacuum modes into thermal radiation. Because a solar-mass hole's $T_H\approx6\times10^{-8}\ \mathrm{K}$ sits hopelessly below the cosmic microwave background, this cannot be measured astrophysically — which is why the last decade turned to analogue gravity. In 2016 and again with higher precision in 2019, Jeff Steinhauer's group at the Technion created sonic horizons in a flowing Bose-Einstein condensate, where the analogue surface gravity is set by the velocity gradient at the sonic point, and measured a thermal spectrum of phonons whose temperature tracks that analogue $\kappa$ — a laboratory echo of the non-affine relation derived here. The same machinery is routine in numerical relativity: ray-tracing photons around merging black holes and normalizing photon momenta require tracking $f$ whenever a coordinate time or Killing parameter is used instead of an affine one.

The physical picture

Imagine driving along a perfectly straight desert road. "Straight" is a property of the road, not of how you drive on it. You can floor the accelerator, brake, or coast — the speedometer swings wildly — yet you never leave the road. The non-affine term $f\,u^\nu$ is exactly this pedal action: a purely longitudinal contribution, always pointing along the road, never sideways. An affine parameter is cruise control at a fixed rate, where there is no longitudinal source term at all and the tangent vector you carry keeps a constant length. Geometrically, parallel transport carries the tangent vector along the curve. For any geodesic the transported tangent stays proportional to the actual tangent — they point the same way. If equal, the parameter is affine; if they differ by a scalar factor that varies along the curve, the parameter is non-affine, but the curve is still the very same geodesic. The acceleration term merely rescales the tangent's length, encoding a changing "clock rate." The decisive fact — the one the first figure makes visible — is that this rescaling has no transverse component: a transverse piece would turn the tangent, contradicting the geodesic condition.

Figure

The acceleration term f uᵘ points along the geodesic, never across it. A non-affine parameter speeds up or slows travel so the tangent u stretches, but the transverse part stays exactly zero — the path is untouched. In symbols aᵛ = f uᵛ, parallel to uᵛ.

The freedom to reparametrize is governed by a single first-order equation, $\ddot\tau/\dot\tau=f$ . Because $f$ couples only to the longitudinal direction, this one scalar equation removes the entire acceleration term — which is why the affine and non-affine descriptions are physically equivalent. The place this machinery bites hardest is where no affine label is free: on a black-hole horizon the natural label is the Killing parameter, non-affine with a constant $f=\kappa$ . The affine parameter then races ahead exponentially, $\tau\propto e^{\kappa V}$ , and that single constant $\kappa$ is the surface gravity that sets the Hawking temperature — plotted, with real numbers, in the second figure.

Figure

On a Killing horizon the affine parameter τ grows exponentially with the Killing parameter V as τ ∝ e^{κV}. The constant non-affinity κ is the surface gravity: equal steps in V multiply τ by the same factor. For a solar-mass Schwarzschild hole κ ≈ 1.5×10¹³ m/s² and the Hawking temperature is T_H = κ/2π ≈ 6.2×10⁻⁸ K.

The core equations

The whole subject is the difference between two ways of writing the same curve. The tangent is parallel-transported into a multiple of itself; with a general parameter $\lambda$ that multiple is a scalar function $f(\lambda)$ , so the equation of the curve is inhomogeneous.

u^\mu\nabla_\mu u^\nu = f(\lambda)\,u^\nu \qquad (1)

where: $u^\mu=\dfrac{\mathrm{d}x^\mu}{\mathrm{d}\lambda}$ — tangent vector in the parametrization $\lambda$ ; $\nabla_\mu$ — covariant derivative of the Levi-Civita connection; $f(\lambda)$ — scalar non-affinity function, vanishing iff $\lambda$ is affine; $\lambda$ — an arbitrary smooth label along the curve. The right side is proportional to $u^\nu$ , hence longitudinal — it rescales the tangent, never turns it. Written out in coordinates, the covariant directional derivative unpacks into a second derivative plus the connection term. The non-affine source then appears as a first-derivative term on the right — precisely the extra term textbook orbit equations carry when parametrized by anything but proper time.

\frac{\mathrm{d}^2 x^\nu}{\mathrm{d}\lambda^2} + \Gamma^\nu{}_{\alpha\beta}\frac{\mathrm{d}x^\alpha}{\mathrm{d}\lambda}\frac{\mathrm{d}x^\beta}{\mathrm{d}\lambda} = f(\lambda)\,\frac{\mathrm{d}x^\nu}{\mathrm{d}\lambda} \qquad (2)

where: $\Gamma^\nu{}_{\alpha\beta}$ — Christoffel symbols of the Levi-Civita connection; $x^\nu(\lambda)$ — coordinate functions of the curve. A non-zero right side is the fingerprint of a non-affine label; $f=0$ returns the standard geodesic equation. The non-affinity is not free-floating — it is fixed by how the chosen label relates to any affine parameter $\tau$ . It is the logarithmic rate at which the reparametrization Jacobian $\dot\tau=\mathrm{d}\tau/\mathrm{d}\lambda$ changes.

f(\lambda) = \frac{\mathrm{d}^2\tau/\mathrm{d}\lambda^2}{\mathrm{d}\tau/\mathrm{d}\lambda} = \frac{\mathrm{d}}{\mathrm{d}\lambda}\ln\!\left(\frac{\mathrm{d}\tau}{\mathrm{d}\lambda}\right) \qquad (3)

where: $\tau$ — any affine parameter; $\lambda$ — the general (possibly non-affine) parameter. Linear relabellings $\tau=a\lambda+b$ give $\ddot\tau=0$ , hence $f=0$ : the residual $\tau\to a\tau+b$ freedom. Because the tangent's length is no longer protected, the norm evolves. Metric compatibility $\nabla g=0$ lets the metric slide through the derivative, leaving a clean first-order law for the squared norm driven by the same $f$ .

u^\mu\nabla_\mu\big(g_{\nu\rho}u^\nu u^\rho\big) = 2f\,\big(g_{\nu\rho}u^\nu u^\rho\big) \qquad (4)

where: $g_{\nu\rho}u^\nu u^\rho$ — squared norm of the tangent, negative/positive for timelike/spacelike and $0$ for null. It rescales by $\exp\!\big(2\!\int f\,\mathrm{d}\lambda\big)$ but never crosses zero, so the causal character is preserved. The marquee application is a Killing horizon. Its generating Killing vector obeys a non-affine geodesic equation with a constant coefficient — the surface gravity — so the affine parameter along a generator grows exponentially in the Killing parameter.

\xi^\mu\nabla_\mu\xi^\nu = \kappa\,\xi^\nu,\qquad \tau\propto e^{\kappa V} \qquad (5)

where: $\xi^\mu$ — Killing vector generating the horizon, non-affinely parametrized by $V$ ; $\kappa$ — surface gravity, the constant non-affinity on the horizon; $\tau$ — affine parameter along the generator. This exponential relation underlies the Hawking temperature $T_H=\kappa/2\pi$ in units $\hbar=c=k_B=1$ .

Derivation AReparametrizing an affine geodesic produces a longitudinal acceleration

The goal is to prove the article's central claim: that changing the parameter of an affinely parametrized geodesic can only add a term proportional to the tangent, and to compute that coefficient exactly. We start from an affine geodesic, switch to an arbitrary label, and watch the covariant derivative act on the scalar Jacobian relating the two tangents — the payoff being the identification $f=\ddot\tau/\dot\tau$ , which turns "non-affine" into a precise, computable function.

Step 1. Begin with a geodesic affinely parametrized by $\tau$ , so its tangent $v^\mu=\mathrm{d}x^\mu/\mathrm{d}\tau$ satisfies $v^\mu\nabla_\mu v^\nu=0$ . Why: this is the homogeneous form we will deform; every subsequent term is measured against this baseline of "no source."

Step 2. Introduce a new label $\lambda$ with $\tau=\tau(\lambda)$ , invertible and smooth. By the chain rule the new tangent is the old one scaled by the Jacobian, $u^\nu=\mathrm{d}x^\nu/\mathrm{d}\lambda=(\mathrm{d}\tau/\mathrm{d}\lambda)\,v^\nu\equiv\dot\tau\,v^\nu$ . Why: reparametrization cannot rotate the tangent — it can only stretch it by a scalar, and that scalar $\dot\tau$ is the entire difference between the two descriptions.

Step 3. Apply the directional derivative with the Leibniz rule, treating $\dot\tau$ as a scalar field along the curve: $u^\mu\nabla_\mu u^\nu=\dot\tau\,v^\mu\nabla_\mu(\dot\tau\,v^\nu)=\dot\tau\big[(v^\mu\nabla_\mu\dot\tau)\,v^\nu+\dot\tau\,v^\mu\nabla_\mu v^\nu\big]$ . Why: the covariant derivative of a scalar is its ordinary derivative, so the connection touches only the vector factor, splitting the expression into a rescaling piece and a geodesic piece.

Step 4. The second bracketed term vanishes because $v^\mu$ is affinely geodesic, leaving $u^\mu\nabla_\mu u^\nu=\dot\tau\,(v^\mu\partial_\mu\dot\tau)\,v^\nu$ . Why: this is where the assumption of Step 1 pays off — the only surviving contribution is the one carried by the changing Jacobian, and it sits along $v^\nu$ , hence along $u^\nu$ .

Step 5. Convert the affine derivative to $\lambda$ : since $v^\mu\partial_\mu=\mathrm{d}/\mathrm{d}\tau$ , we have $v^\mu\partial_\mu\dot\tau=\mathrm{d}\dot\tau/\mathrm{d}\tau=(1/\dot\tau)\,\mathrm{d}\dot\tau/\mathrm{d}\lambda=\ddot\tau/\dot\tau$ , with $\ddot\tau=\mathrm{d}^2\tau/\mathrm{d}\lambda^2$ . Why: the natural derivative is with respect to $\lambda$ , the label we use; this step produces the ratio that becomes $f$ .

Step 6. Assemble, replacing $v^\nu=u^\nu/\dot\tau$ : $u^\mu\nabla_\mu u^\nu=\dot\tau\cdot(\ddot\tau/\dot\tau)\cdot(1/\dot\tau)\,u^\nu=(\ddot\tau/\dot\tau)\,u^\nu$ . Hence $f=\ddot\tau/\dot\tau$ . Why: the factors of $\dot\tau$ collapse to leave a single scalar times $u^\nu$ — the manifestly longitudinal source, with its coefficient now known in closed form. Cross-check: setting $\tau=a\lambda+b$ gives $\ddot\tau=0$ , so $f=0$ and equation (1) reduces to the standard geodesic equation, consistent with Wald §3.3 and Carroll §3.3.

Go deeper · Derivation B · 1 minIntegrating the non-affinity away to recover an affine parameter

Derivation BIntegrating the non-affinity away to recover an affine parameter

The goal is to show constructively that the acceleration term can always be removed — that for any prescribed $f(\lambda)$ there exists an affine parameter $\tau(\lambda)$ , given by an explicit double integral. This is what makes affine and non-affine geodesics equivalent: the source carries no invariant information, only a choice of clock. We invert $f=\ddot\tau/\dot\tau$ as a first-order equation for $\dot\tau$ and integrate twice.

Step 1. Demand that $\tau$ be affine, i.e. that reparametrizing to $\tau$ kills the source. From equation (3) this is the linear ODE $\ddot\tau-f(\lambda)\,\dot\tau=0$ . Why: "affine" is defined by $f=0$ in the new label; expressed in the old label $\lambda$ this becomes a differential condition on $\tau(\lambda)$ that we can solve directly.

Step 2. Divide by $\dot\tau$ (nonzero for any regular, orientation-preserving parametrization) to get $\dfrac{\mathrm{d}}{\mathrm{d}\lambda}\ln\dot\tau=f(\lambda)$ . Why: recognizing the left side as a logarithmic derivative reduces a second-order equation to a first-order one for $\dot\tau$ , which integrates immediately.

Step 3. Integrate once: $\dot\tau(\lambda)=\dot\tau_0\,\exp\!\Big[\int_{\lambda_0}^{\lambda} f(\lambda')\,\mathrm{d}\lambda'\Big]$ . Why: this "velocity factor" relates the two clocks' tick rates and is strictly positive, so orientation is preserved and $\dot\tau$ never vanishes.

Step 4. Integrate again: $\tau(\lambda)=\tau_0+\dot\tau_0\displaystyle\int_{\lambda_0}^{\lambda}\exp\!\Big[\int_{\lambda_0}^{\lambda'} f(\lambda'')\,\mathrm{d}\lambda''\Big]\mathrm{d}\lambda'$ . Why: the two constants $\tau_0$ (origin) and $\dot\tau_0$ (scale) are the residual freedom $\tau\to a\tau+b$ , and nothing in the construction fixes them.

Step 5. Specialize to constant $f=\kappa$ : the inner integral is $\kappa(\lambda-\lambda_0)$ , giving $\tau-\tau_0=\dfrac{\dot\tau_0}{\kappa}\big(e^{\kappa(\lambda-\lambda_0)}-1\big)$ . Why: this exponential is precisely the horizon relation — a constant non-affinity forces the affine parameter to grow geometrically, the structure that defines surface gravity in the next derivation. Verification: differentiating the boxed $\tau(\lambda)$ twice reproduces $\ddot\tau=f\dot\tau$ identically, matching the MTW §8.5 treatment of affine parametrization.

Go deeper · Derivation C · 2 minSurface gravity as the constant non-affinity of a horizon generator

Derivation CSurface gravity as the constant non-affinity of a horizon generator

The goal is to show that the non-affine geodesic equation is not a curiosity but the origin of one of the most important numbers in black-hole physics. On a Killing horizon the generating Killing vector is itself a non-affinely parametrized geodesic with a constant coefficient — the surface gravity $\kappa$ . Using only the definition of a Killing horizon and Killing's equation, we arrive at $\xi^\mu\nabla_\mu\xi^\nu=\kappa\,\xi^\nu$ together with the exponential affine relation and the Hawking temperature.

Step 1. A Killing horizon is a null hypersurface $H$ whose null normal is a Killing vector $\xi^\mu$ ; on $H$ the field becomes null, $\xi_\mu\xi^\mu=0$ , and its integral curves rule the surface. Why: this is the setting for every stationary black hole — the horizon is generated by the flow of a symmetry, so the generators are orbits of $\xi$ rather than of an abstract affine label.

Step 2. The scalar $\xi_\mu\xi^\mu$ is constant (zero) along $H$ , so its gradient is normal to $H$ . On a null surface the normal is proportional to $\xi$ itself, which defines $\kappa$ through $\nabla^\nu(\xi_\mu\xi^\mu)=-2\kappa\,\xi^\nu$ . Why: there is no other direction available — the gradient must be parallel to $\xi$ , and the proportionality constant is the surface gravity.

Step 3. Expand the left side with Killing's equation $\nabla_\mu\xi_\nu=-\nabla_\nu\xi_\mu$ : $\nabla^\nu(\xi_\mu\xi^\mu)=2\xi_\mu\nabla^\nu\xi^\mu=-2\xi_\mu\nabla^\mu\xi^\nu$ . Matching to Step 2 gives $\xi^\mu\nabla_\mu\xi^\nu=\kappa\,\xi^\nu$ . Why: Killing antisymmetry is the hinge, converting the gradient of the norm into the directional derivative of $\xi$ along itself — the non-affine operator of equation (1).

Step 4. This is equation (1) with $f=\kappa$ constant, so the generators are geodesics and $V$ (the parameter along the flow of $\xi$ ) is non-affine. By Step 5 of the previous derivation the affine parameter satisfies $\tau\propto e^{\kappa V}$ . Why: the horizon hands you the Killing parameter, not an affine one, and the price of that natural choice is precisely a constant non-affinity — the affine clock runs away exponentially.

Step 5. The zeroth law of black-hole mechanics states $\kappa$ is constant over the horizon, and quantizing fields on this exponentially-stretched geometry yields the Hawking temperature $T_H=\kappa/2\pi$ (natural units). Why: a single non-affinity constant, born from a reparametrization, fixes the temperature of the hole — the deepest payoff of taking the non-affine term seriously. Verification: for Schwarzschild $\kappa=c^4/4GM$ ; a solar mass gives $\kappa\approx1.52\times10^{13}\ \mathrm{m/s^2}$ and $T_H\approx6.2\times10^{-8}\ \mathrm{K}$ , the standard values (Wald §12.5).

Limiting cases

The special cases sharpen what the non-affine term does and, more importantly, what it never does.

Case	Result	Meaning
Affine parameter	$f=0\ \Rightarrow\ u^\mu\nabla_\mu u^\nu=0$	the standard geodesic equation; the tangent is transported into an exact copy of itself, no rescaling
Linear reparametrization	$\tau'=a\tau+b,\ a,b=\mathrm{const}$	maps one affine parameter to another, $f$ stays zero; this two-parameter set is the full residual freedom
Constant non-affinity	$f=\kappa\ \Rightarrow\ \tau\propto e^{\kappa\lambda}$	the affine parameter grows exponentially; the structure of surface gravity on horizon generators
Null geodesics	$g_{\mu\nu}u^\mu u^\nu=0$	no proper time exists, so an affine parameter is fixed abstractly; reparametrization freedom is essential
Norm along the flow	$g_{\mu\nu}u^\mu u^\nu\to e^{2\int f}\,(g\,uu)$	the norm rescales but keeps its sign — timelike stays timelike, null stays null

Common trapThe non-affine term means the particle is being pushed off its geodesic

It does not. The term $f(\lambda)\,u^\nu$ never signals a departure from the geodesic — the worldline is the same geometric geodesic it always was. The "acceleration" is an artifact of clock rate and is always tangent to the curve. Contrast a genuine four-acceleration $a^\nu=u^\mu\nabla_\mu u^\nu$ for a forced (non-geodesic) particle: with proper time that acceleration is transverse, $a^\nu u_\nu=0$ , because it bends the worldline. The non-affine term is the opposite — strictly parallel, $a^\nu\parallel u^\nu$ . So an $a^\nu$ with any component orthogonal to $u^\nu$ means genuine acceleration; a purely parallel $a^\nu$ is a non-affine parameter in disguise.

Worked examples

1 · Surface gravity and Hawking temperature of a solar-mass black hole (fully numeric)

A Schwarzschild black hole of mass $M=M_\odot=1.989\times10^{30}\ \mathrm{kg}$ has a Killing horizon generated by $\xi^\mu$ obeying $\xi^\mu\nabla_\mu\xi^\nu=\kappa\,\xi^\nu$ . Find the surface gravity $\kappa$ , the Hawking temperature $T_H$ , and state how the affine parameter grows along a generator.

Step 1. For Schwarzschild the surface gravity in SI units is $\kappa=\dfrac{c^4}{4GM}$ . Insert the constants: $c^4=(2.998\times10^8)^4=8.078\times10^{33}\ \mathrm{m^4/s^4}$ and $4GM=4(6.674\times10^{-11})(1.989\times10^{30})=5.310\times10^{20}\ \mathrm{m^3/s^2}$ . Then $\kappa=8.078\times10^{33}/5.310\times10^{20}=\mathbf{1.52\times10^{13}\ m/s^2}$ — the constant non-affinity of the Killing parameter.

Step 2. The Hawking temperature follows from the exponential affine relation via $T_H=\dfrac{\hbar\kappa}{2\pi c\,k_B}=\dfrac{\hbar c^3}{8\pi G M k_B}$ . Numerically $\hbar c^3=(1.0546\times10^{-34})(2.694\times10^{25})=2.841\times10^{-9}$ and $8\pi GMk_B=(25.13)(6.674\times10^{-11})(1.989\times10^{30})(1.381\times10^{-23})=4.61\times10^{-2}$ , giving $T_H=2.841\times10^{-9}/4.61\times10^{-2}=\mathbf{6.2\times10^{-8}\ K}$ .

Step 3. Along a horizon generator the Killing parameter $V$ is non-affine with $f=\kappa$ , so the affine parameter obeys $\tau\propto e^{\kappa V}$ : equal steps in $V$ multiply $\tau$ by the fixed factor $e^{\kappa\Delta V}$ . This exponential separation of the two clocks is the geometric root of the Hawking temperature — the horizon redshift stretches modes exponentially, at the rate $\kappa$ .

2 · Building the affine parameter from a prescribed non-affinity (fully worked)

A geodesic is handed to you in a parameter $\lambda>0$ with non-affinity $f(\lambda)=2/\lambda$ . Construct the affine parameter $\tau(\lambda)$ explicitly, verify it removes the source, and identify the residual freedom.

Step 1. Use the first integral $\dot\tau=\dot\tau_0\exp\!\big[\int_{\lambda_0}^\lambda f\,\mathrm{d}\lambda'\big]$ . Here $\int_{\lambda_0}^\lambda (2/\lambda')\,\mathrm{d}\lambda'=2\ln(\lambda/\lambda_0)$ , so $\dot\tau=\dot\tau_0\,(\lambda/\lambda_0)^2$ — the affine tick rate grows quadratically in $\lambda$ .

Step 2. Integrate again: $\tau(\lambda)=\tau_0+\dfrac{\dot\tau_0}{\lambda_0^2}\displaystyle\int_{\lambda_0}^{\lambda}\lambda'^2\,\mathrm{d}\lambda'=\tau_0+\dfrac{\dot\tau_0}{3\lambda_0^2}\big(\lambda^3-\lambda_0^3\big)$ . Choosing the convenient constants $\lambda_0=1,\ \tau_0=0,\ \dot\tau_0=1$ gives the clean result $\boldsymbol{\tau=\lambda^3/3}$ (up to the discarded constant).

Step 3. Verify: with $\tau=\lambda^3/3$ we get $\dot\tau=\lambda^2$ and $\ddot\tau=2\lambda$ , so $\ddot\tau/\dot\tau=2\lambda/\lambda^2=2/\lambda=f$ — the prescribed non-affinity is exactly reproduced. In the label $\tau$ the geodesic equation is homogeneous, $v^\mu\nabla_\mu v^\nu=0$ . The residual freedom is the pair $(\tau_0,\dot\tau_0)$ : any $a\tau+b$ is equally affine, which is why we could set them to $0$ and $1$ without loss.

Physics of the portrait

The non-affine geodesic equation is a statement about parallel transport, and parallel transport has an author — the geometer who made precise what it means to carry a vector along a curve "without turning," the exact content of $u^\mu\nabla_\mu u^\nu\propto u^\nu$ .

Origins · 1 minTullio Levi-Civita and how this was discovered

Tullio Levi-Civita

1873–1941 · parallel transport and the geometry of geodesics

Tullio Levi-Civita, born in Padua in 1873, was the student of Gregorio Ricci-Curbastro and his collaborator on the absolute differential calculus — the tensor calculus — laid out in their 1900 memoir Méthodes de calcul différentiel absolu. That formalism became the language of general relativity; Einstein leaned on it heavily and corresponded warmly with Levi-Civita. His decisive gift came in 1917, in a paper introducing parallel transport (parallelismo) on a curved manifold. Before it, "the tangent does not change along the curve" had no invariant meaning, because comparing vectors at different points requires a rule for carrying one to the other. Levi-Civita supplied exactly that rule, and with it the geodesic acquired its clean definition: the curve whose tangent is parallel-transported into a multiple of itself. The affine/non-affine distinction lives entirely inside his construction — an affine parameter is the label for which the transported tangent equals the original, a non-affine one the label for which it equals a scalar multiple. Levi-Civita worked most of his career in Rome, was stripped of his professorship by the Fascist racial laws of 1938, and died there in 1941. The equation $u^\mu\nabla_\mu u^\nu=f\,u^\nu$ is, in the end, a footnote to his 1917 insight: parallel transport is the whole game, and $f$ merely records whether the transport preserves length or only direction.

History

1854

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

1869

— Christoffel introduces the connection symbols that define covariant differentiation

1900

— Ricci-Curbastro and Levi-Civita codify the absolute differential calculus

1917

— Levi-Civita defines parallel transport, giving geodesics their parametrization-free meaning

1918

— Weyl introduces the affine connection, the setting in which affine parameters are natural

1973

— Bardeen, Carter and Hawking identify the constant non-affinity of a Killing horizon as its surface gravity

Going further

References. Misner, Thorne & Wheeler, Gravitation (1973), §3.3, §8.5 · Wald, General Relativity (1984), §3.3 and §12.5 (surface gravity) · Carroll, Spacetime and Geometry (2004), §3.3 · Schutz, A First Course in General Relativity (2009), §6.4 · Hartle, Gravity (2003), §8.2. The key results can be checked directly: $u^\mu\nabla_\mu u^\nu=f\,u^\nu$ with $f=\ddot\tau/\dot\tau$ follows by direct reparametrization; the affine double integral solves $\ddot\tau=f\dot\tau$ , and the specific case $f=2/\lambda$ yields $\tau\propto\lambda^3$ ; the norm law $u^\mu\nabla_\mu(g\,uu)=2f\,(g\,uu)$ follows from $\nabla g=0$ ; and $\xi^\mu\nabla_\mu\xi^\nu=\kappa\,\xi^\nu$ with $\tau\propto e^{\kappa V}$ is the horizon generator relation. The Schwarzschild values $\kappa=c^4/4GM\approx1.5\times10^{13}\ \mathrm{m/s^2}$ and $T_H\approx6.2\times10^{-8}\ \mathrm{K}$ for a solar mass are the standard textbook figures.

Builds on. The Geodesic Equation · Affine Parameters · Parallel Transport · Covariant Derivative · The Affine Connection. Leads to. Geodesics as Extremal Paths · Geodesic Deviation Equation · Tidal Forces in General Relativity.

Ask the tutor: Ask me to work through why a coordinate-time or Killing-parameter description of a given geodesic picks up its non-affine term, and how to integrate it back to proper time.

Is a non-affinely parametrized geodesic still a geodesic?

Yes — completely. The image of the curve in the manifold is unchanged; only the equation describing it changes shape. A geodesic is a geometric object — a set of points whose tangent is parallel-transported along itself — and that property is independent of how you label the points. The non-affine term $f\,u^\nu$ appears solely because your parameter ticks unevenly relative to an affine one, and you can always verify the curve is a genuine geodesic by finding the affine reparametrization that removes the term. So "non-affine geodesic" is not a different kind of path; it is the same path viewed through a stretched clock.

What does the function f physically represent?

$f(\lambda)=\ddot\tau/\dot\tau$ is the logarithmic rate at which your parameter stretches relative to an affine standard $\tau$ — a "clock acceleration" with dimensions of inverse $\lambda$ . If $f=0$ the two clocks run at proportional rates. Crucially $f$ carries no invariant information about the spacetime: two observers using different parameters on the same geodesic assign different $f$ . The one exception is the horizon, where the Killing parameter forces a constant $f=\kappa$ equal to the surface gravity, a genuine geometric invariant.

Why can the acceleration term always be removed?

Because it is purely longitudinal, and one scalar equation can absorb one scalar's worth of freedom. The term lives entirely along $u^\nu$ , so eliminating it means solving a single first-order condition, $\ddot\tau/\dot\tau=f$ , for $\tau(\lambda)$ — no transverse remainder, no integrability obstruction, just an ODE that always has a solution when $\dot\tau\neq0$ . Removing a transverse acceleration (a real force) is impossible this way, because it bends the path — the longitudinal-only structure is exactly what guarantees affine parameters exist for every geodesic.

Is the norm of the tangent conserved along a non-affine geodesic?

Not in general. By metric compatibility $u^\mu\nabla_\mu(g_{\nu\rho}u^\nu u^\rho)=2f\,(g_{\nu\rho}u^\nu u^\rho)$ , so the squared norm evolves by $\exp\!\big(2\!\int f\,\mathrm{d}\lambda\big)$ . It stretches with the tangent but can never change sign, because that factor is strictly positive: timelike stays timelike, null stays null. Only for an affine parameter ( $f=0$ ) is the norm strictly constant — exactly why proper time and proper distance, with $g_{\mu\nu}u^\mu u^\nu=\pm1$ , are affine. Conservation of the tangent's length is the hallmark of affine parametrization.

Why do null geodesics force you to confront non-affine parameters?

Because light has no proper time. For a timelike geodesic the metric hands you an affine parameter for free — proper time $\tau$ , with $g_{\mu\nu}u^\mu u^\nu=-1$ . For a null geodesic every point is at zero interval from every other, so arc length is identically zero and cannot serve at all. An affine parameter must instead be built from parallel transport, fixed only up to $\lambda\to a\lambda+b$ . Any physically motivated label — the affine parameter rescaled by photon frequency, or a coordinate time — is generically non-affine, injecting an $f\neq0$ you must track to normalize the photon's momentum. This is why the non-affine form is unavoidable in ray-tracing, redshift, and lensing.

How is surface gravity a special case of this equation?

On a Killing horizon the generating Killing vector $\xi^\mu$ satisfies $\xi^\mu\nabla_\mu\xi^\nu=\kappa\,\xi^\nu$ — literally equation (1) with a constant coefficient $f=\kappa$ , so the generators are geodesics run in the non-affine Killing parameter $V$ . Because $f$ is constant, the affine parameter grows exponentially, $\tau\propto e^{\kappa V}$ , and this stretching of null generators produces the thermal spectrum of Hawking radiation with $T_H=\kappa/2\pi$ . Surface gravity is thus the one context where the "artificial" non-affine coefficient becomes a physical invariant — constant over the horizon by the zeroth law.