Geodesics as Extremal Paths

Two definitions of a geodesic live inside general relativity, and the drama of this article is that they turn out to be the same curve. The geometric definition calls a geodesic the straightest possible path — a curve that parallel-transports its own tangent, never turning relative to the local notion of "straight ahead." The variational definition, developed here, calls a geodesic a curve whose proper time (timelike case) or proper length (spacelike case) is stationary against small wiggles that hold the endpoints fixed. For the Levi-Civita connection these two pictures coincide exactly, and that coincidence is a load-bearing pillar of the geometric theory of gravity: it lets us replace "force" with "shape of spacetime." A freely falling body feels nothing and is pushed by nothing; it simply threads the worldline of greatest elapsed proper time, and the deflection we once called gravity is what that extremal worldline looks like from a frame that is itself accelerating. This is Einstein's reforging of inertia into geometry, and it ties gravitation to the same action principles that organize classical mechanics and field theory, so that every symmetry of the metric hands us a conservation law for free. One technical subtlety is worth flagging at the outset: the natural proper-time integral carries a square root that is singular for light rays and clumsy to differentiate, so in practice one trades it for a quadratic "energy" functional that yields identical curves, automatically supplies an affine parameter, and survives the null limit. And the word extremal should always be read as stationary — timelike geodesics are local maxima of proper time, spacelike ones are saddle points, and beyond a conjugate point even that local claim can fail.

Key takeaways

A geodesic between two fixed events is a curve of stationary proper time (timelike) or proper length (spacelike) — the relativistic upgrade of "shortest path on a surface."
Timelike geodesics locally maximize proper time: the free-falling clock reads the longest interval, the entire content of the twin-paradox resolution.
Varying the proper-time functional reproduces the geodesic equation ẍ^μ + Γ^μ_αβ ẋ^α ẋ^β = 0 with the Levi-Civita connection, so free fall is fixed entirely by the metric.
The square-root action is awkward; the quadratic energy Lagrangian L = g_μν ẋ^μ ẋ^ν gives the same geodesics, forces an affine parameter, and handles null curves where length vanishes.
Every coordinate the metric ignores gives a conserved momentum — energy and angular momentum are cyclic-coordinate momenta, one Killing vector each.
Key idea: a geodesic is a stationary point of the proper-time or proper-length functional; varying it yields the geodesic equation, and the quadratic energy Lagrangian both simplifies the algebra and selects an affine parameter.

The core insight

A timelike or spacelike geodesic between fixed events is a stationary point of the proper-time / proper-length functional; varying that functional reproduces the geodesic equation, and using the quadratic "energy" Lagrangian both simplifies the calculation and selects an affine parameter.

Where this lives today

The geodesic variational principle is the computational engine of modern gravitational astrophysics. When the Event Horizon Telescope reconstructs the shadow of the black hole in M87 or Sagittarius A*, the bright ring it images is drawn by null geodesics winding around the photon sphere at $r = 3M$ before escaping — ray-tracing those paths is a direct integration of the geodesic equation with conserved $E$ and $L$ . When LIGO and Virgo model a binary black-hole inspiral, the innermost stable circular orbit of worked example 2 marks the transition from adiabatic orbit to plunge, setting the frequency at which the chirp turns over. Every GPS receiver silently trusts the maximal-aging principle: satellite clocks, higher in Earth's potential and moving faster, tick at a rate the weak-field geodesic calculation predicts to nanoseconds per day, without which position fixes would drift kilometers within hours. And at the theoretical frontier, the focusing of geodesic congruences at conjugate points — Morse's conjugate points, made Lorentzian — remains the technical core of the singularity theorems and of work on cosmic censorship, where whether geodesics extend past a would-be singularity is the whole question.

The physical picture

Run a stopwatch along every conceivable worldline joining two events. Most belong to accelerating observers who must fire rockets and feel g-forces to stay on them; exactly one class costs nothing to follow — the free-fall worldlines an unpowered body traces on its own. The variational principle says these effortless curves are precisely the ones on which the accumulated proper time is stationary, and for a timelike curve in a small enough region, stationary means maximal. The free faller ages the most. Gravity supplies no push; it only bends the geometry so that "aging the most" and "moving inertially" describe the same track, and what a hovering observer calls gravitational force is just the curvature of that maximal-aging path seen from an accelerated frame. The first figure makes the claim quantitative with the twin paradox.

Figure

Between the same two reunion events the straight inertial worldline banks the most proper time — τ = 10.0 yr — while a traveler out-and-back at v = 0.8c ages only 6.0 yr and one at v = 0.917c only 4.0 yr. The geodesic maximizes proper time; every accelerated detour costs elapsed time

The same variational engine, fed a real metric, produces the orbits astronomers track. Take the Schwarzschild geometry outside a spherical mass. Its geodesic Lagrangian ignores the coordinates t and φ, so their conjugate momenta — the energy E and angular momentum L — are conserved by the cyclic-coordinate rule. Combine them with the normalization g_μν ẋ^μ ẋ^ν = −1 and the radial motion collapses to a one-dimensional problem: a particle rolling in an effective potential. That potential carries an extra −ML²/r³ term absent from Newton, and that one term reshapes the orbital landscape — bending bound ellipses into precessing rosettes and, closer in, opening an innermost stable circular orbit below which no circular motion survives. The second figure is the object a relativist reaches for: the effective potential itself.

Figure

The Schwarzschild effective potential V(r) = −M/r + L²/2r² − ML²/r³ for a massive particle with L = 4M (units G = c = 1). Its two turning points are circular orbits — an unstable one at r = 4M (the barrier peak) and a stable one at r = 12M (the well) — and as L drops to 2√3 M ≈ 3.46M the pair merges into the innermost stable circular orbit at r = 6M

The core equations

The starting functional is the proper time itself — the reading of an ideal clock carried along the worldline. It is reparametrization invariant, which is a virtue, but the square root inside it is exactly what makes it hard to vary and useless for light.

S[\,x\,] = \tau = \int_{\lambda_1}^{\lambda_2}\sqrt{-g_{\mu\nu}(x)\,\frac{\mathrm{d}x^\mu}{\mathrm{d}\lambda}\frac{\mathrm{d}x^\nu}{\mathrm{d}\lambda}}\;\mathrm{d}\lambda \qquad (1)

where: $\tau$ — proper time elapsed along the worldline; $\lambda$ — arbitrary parameter labeling the curve; $g_{\mu\nu}$ — spacetime metric, signature $(-+++)$ ; $x^\mu(\lambda)$ — the worldline being varied. This is the curve's length measured with the spacetime ruler-and-clock, and a geodesic makes it stationary. Stationarity of any such action under endpoint-pinning variations is enforced by the Euler-Lagrange equations — the workhorse of everything below, with the only choice left being which Lagrangian to feed them.

\frac{\mathrm{d}}{\mathrm{d}\lambda}\!\left(\frac{\partial L}{\partial \dot x^\mu}\right) - \frac{\partial L}{\partial x^\mu} = 0 \qquad (2)

where: $L$ — Lagrangian (integrand of the chosen functional); $\dot x^\mu$ — $\mathrm{d}x^\mu/\mathrm{d}\lambda$ . These are $\delta S = 0$ written out, one equation per coordinate. The practical choice is the quadratic energy Lagrangian: it drops the square root entirely, gives the same extremal curves as (1), and selects an affine parameter for free.

L_{E} = g_{\mu\nu}(x)\,\frac{\mathrm{d}x^\mu}{\mathrm{d}\lambda}\frac{\mathrm{d}x^\nu}{\mathrm{d}\lambda} \qquad (3)

where: $L_E$ — quadratic Lagrangian whose extremals are affinely parametrized geodesics. This is the relativistic analogue of $\tfrac12 m\dot x^2$ , and it stays finite even when the proper length is zero. Feeding either functional into Euler-Lagrange yields the geodesic equation — the equation of motion of every freely falling body and light ray, with all of gravity buried in the single symbol $\Gamma$ .

\frac{\mathrm{d}^2 x^\mu}{\mathrm{d}\tau^2} + \Gamma^\mu{}_{\alpha\beta}\,\frac{\mathrm{d}x^\alpha}{\mathrm{d}\tau}\frac{\mathrm{d}x^\beta}{\mathrm{d}\tau} = 0 \qquad (4)

where: $\Gamma^\mu{}_{\alpha\beta}$ — Christoffel symbols of the Levi-Civita connection; $\tau$ — proper time, an affine parameter for timelike geodesics. The four-acceleration is zero once curvature is accounted for — "straight ahead" in a curved spacetime. The connection is not free to choose: the variation manufactures the Christoffel symbols, built from first derivatives of the metric, which is why the extremal definition forces the unique torsion-free, metric-compatible connection.

\Gamma^\mu{}_{\alpha\beta} = \tfrac12\, g^{\mu\nu}\big(\partial_\alpha g_{\nu\beta} + \partial_\beta g_{\nu\alpha} - \partial_\nu g_{\alpha\beta}\big) \qquad (5)

where: $g^{\mu\nu}$ — inverse metric; $\partial_\alpha$ — partial derivative $\partial/\partial x^\alpha$ . The symbols are symmetric in $\alpha\beta$ — the torsion-free property that makes the variational and parallel-transport geodesics agree. Any coordinate the metric fails to depend on is cyclic, and its conjugate momentum is conserved — the entire mechanism behind conserved energy and angular momentum.

p_\mu \equiv \frac{\partial L_E}{\partial \dot x^\mu} = 2\,g_{\mu\nu}\dot x^\nu,\qquad \partial_\sigma g_{\mu\nu}=0 \;\Rightarrow\; \frac{\mathrm{d}p_\sigma}{\mathrm{d}\lambda}=0 \qquad (6)

where: $p_\mu$ — canonical momentum conjugate to $x^\mu$ ; a vanishing $\partial_\sigma g_{\mu\nu}$ signals a Killing vector $\xi^\mu = \delta^\mu_\sigma$ and the conserved quantity $\xi_\nu \dot x^\nu$ — symmetry $\to$ conservation. Alongside the cyclic laws sits a universal one: for an affine parameter the normalization of the tangent is constant along the whole geodesic, and its value labels the curve's causal type.

g_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}\lambda}\frac{\mathrm{d}x^\nu}{\mathrm{d}\lambda} = \varepsilon,\qquad \varepsilon = \begin{cases}-1 & \text{timelike}\\ 0 & \text{null}\\ +1 & \text{spacelike}\end{cases} \qquad (7)

where: $\varepsilon$ — the fixed norm of the tangent for affine $\lambda$ . A timelike geodesic keeps unit-speed proper time, a null one stays null, and neither can slip between classes. Finally, whether a geodesic is a genuine extremum rather than merely stationary is decided by the second variation, governed by curvature through the geodesic deviation (Jacobi) equation. Where neighboring geodesics reconverge — a conjugate point — the maximal-aging property can be lost.

\frac{\mathrm{D}^2 \xi^\mu}{\mathrm{d}\tau^2} = -R^\mu{}_{\alpha\nu\beta}\,u^\alpha \xi^\nu u^\beta \qquad (8)

where: $\xi^\mu$ — separation vector to a neighboring geodesic; $R^\mu{}_{\alpha\nu\beta}$ — Riemann curvature tensor; $\mathrm{D}/\mathrm{d}\tau$ — covariant derivative along the geodesic; $u^\mu$ — unit tangent. Physical reading: tidal curvature focuses or defocuses the family of nearby geodesics, and focusing is what creates conjugate points.

Derivation AExtremizing proper time reproduces the geodesic equation

The goal is to take the proper-time functional (1) at face value, vary it under fixed-endpoint perturbations, and show that the resulting Euler-Lagrange equation is exactly the geodesic equation (4) with the Christoffel connection (5) — while seeing why proper time is itself the affine parameter that makes the algebra collapse. The strategy is to parametrize by proper time from the start, so that the awkward square root evaluates to one along the path and kills the term that would otherwise obstruct identifying the connection.

Step 1. Write $L = \sqrt{-g_{\alpha\beta}\dot x^\alpha \dot x^\beta}$ with dots meaning $\mathrm{d}/\mathrm{d}\lambda$ for arbitrary $\lambda$ , so $\tau = \int L\,\mathrm{d}\lambda$ . The two Euler-Lagrange pieces are $\partial L/\partial \dot x^\mu = -g_{\mu\nu}\dot x^\nu / L$ and $\partial L/\partial x^\mu = -\tfrac{1}{2}(\partial_\mu g_{\alpha\beta})\dot x^\alpha \dot x^\beta / L$ . Why the $L$ appears in every denominator: the square root differentiates to $1/(2\sqrt{\cdot})$ , and this factor is exactly what a clever parameter choice will neutralize.

Step 2. Impose $\lambda = \tau$ . Along the physical curve $L = \sqrt{-g_{\alpha\beta}(\mathrm{d}x^\alpha/\mathrm{d}\tau)(\mathrm{d}x^\beta/\mathrm{d}\tau)} = 1$ , since $\mathrm{d}\tau^2 = -g_{\alpha\beta}\,\mathrm{d}x^\alpha \mathrm{d}x^\beta$ by definition, and with $L$ constant it slides out of the derivative and the denominators. Why this is legitimate: the already-derived Euler-Lagrange expression may be evaluated on any parametrization, and $\tau$ is the one that makes $L$ constant, simplifying without changing the curve.

Step 3. The Euler-Lagrange equation reduces to $\tfrac{\mathrm{d}}{\mathrm{d}\tau}(g_{\mu\nu}\dot x^\nu) - \tfrac{1}{2}(\partial_\mu g_{\alpha\beta})\dot x^\alpha \dot x^\beta = 0$ . Carry out the total derivative, using $\tfrac{\mathrm{d}}{\mathrm{d}\tau}g_{\mu\nu} = (\partial_\alpha g_{\mu\nu})\dot x^\alpha$ , to get $g_{\mu\nu}\ddot x^\nu + (\partial_\alpha g_{\mu\nu})\dot x^\alpha \dot x^\nu - \tfrac{1}{2}(\partial_\mu g_{\alpha\beta})\dot x^\alpha \dot x^\beta = 0$ . Why the middle term must be symmetrized: $\dot x^\alpha \dot x^\nu$ is symmetric in $\alpha\nu$ , so only the symmetric part of $\partial_\alpha g_{\mu\nu}$ survives, which is what assembles the three-derivative combination of the connection.

Step 4. Replace $(\partial_\alpha g_{\mu\nu})\dot x^\alpha \dot x^\nu$ by its symmetrized form $\tfrac12(\partial_\alpha g_{\mu\nu}+\partial_\nu g_{\mu\alpha})\dot x^\alpha \dot x^\nu$ and contract the whole equation with the inverse metric $g^{\rho\mu}$ . The bracket becomes $\tfrac12 g^{\rho\mu}(\partial_\alpha g_{\mu\nu}+\partial_\nu g_{\mu\alpha}-\partial_\mu g_{\alpha\nu})$ , which is exactly $\Gamma^\rho{}_{\alpha\nu}$ , leaving $\ddot x^\rho + \Gamma^\rho{}_{\alpha\nu}\dot x^\alpha \dot x^\nu = 0$ . Why the connection is forced to be Levi-Civita: the combination that drops out is manifestly symmetric in its lower indices and built from the metric alone, so the extremal principle can produce no other connection. *As a check: in flat-space Cartesian coordinates every $\Gamma = 0$ , so the equation reduces correctly to $\ddot x^\mu = 0$ . The proper-time normalization $L = 1$ used in Step 2 is consistent with the conserved norm (7).*

Go deeper · Derivation B · 1 minThe energy Lagrangian gives the same geodesics and fixes an affine parameter

Derivation BThe energy Lagrangian gives the same geodesics and fixes an affine parameter

The goal is to justify the workhorse of every practical calculation: that extremizing the quadratic $L_E = g_{\mu\nu}\dot x^\mu \dot x^\nu$ yields the identical image curves as the square-root functional, and — the crucial payoff — that its solutions come already affinely parametrized, whereas the arc-length functional leaves the parametrization free. The strategy is to write $L = \sqrt{-L_E}$ in the timelike case and track how the two Euler-Lagrange operators differ, isolating the single obstruction term between them.

Step 1. By the chain rule, $\partial L/\partial \dot x^\mu = \tfrac{1}{2L}\,\partial(-L_E)/\partial \dot x^\mu$ , and likewise for the $x^\mu$ derivative, so both Euler-Lagrange ingredients of $L$ are those of $-L_E$ divided by $2L$ . Why this helps: $L$ and $L_E$ share the same derivative structure up to the scalar $1/(2L)$ , so their equations can only differ through how that factor varies along the curve.

Step 2. Inserting into the operator for $L$ and clearing $\tfrac12$ gives $\tfrac{\mathrm{d}}{\mathrm{d}\lambda}\big[\partial(-L_E)/\partial \dot x^\mu\big] - \partial(-L_E)/\partial x^\mu = \tfrac{1}{L}\tfrac{\mathrm{d}L}{\mathrm{d}\lambda}\,\partial(-L_E)/\partial \dot x^\mu$ . The left side is exactly the Euler-Lagrange operator of $L_E$ ; the right side is the obstruction. Why it is the whole story: if it vanishes, the two functionals share every solution curve.

Step 3. The obstruction vanishes iff $\mathrm{d}L/\mathrm{d}\lambda = 0$ , i.e. $L_E = \text{const}$ . But the equations of motion of $L_E$ conserve $L_E$ automatically — it is the normalization (7), constant for any affine parameter — so every solution of the energy Lagrangian sits at $\mathrm{d}L/\mathrm{d}\lambda = 0$ and thus also solves the arc-length equations. Why $L_E$ is self-consistently affine: its own flow preserves $g_{\mu\nu}\dot x^\mu \dot x^\nu$ , so it never generates the obstruction.

Step 4. Conversely the arc-length functional is reparametrization invariant and can never single out a parameter — it accepts $\lambda \to f(\lambda)$ for any smooth $f$ , and only affine choices make $L_E$ constant. So both functionals draw the same geodesics, but only $L_E$ pins the parameter to affine and only $L_E$ stays finite on null curves where $L_E = 0$ but the square root is singular. Why this matters for photons: the arc-length action vanishes on a null curve and cannot be varied, so the energy Lagrangian is the only viable variational description of light.

Go deeper · Derivation C · 1 minThe Newtonian limit: maximal aging becomes Newton's second law

Derivation CThe Newtonian limit: maximal aging becomes Newton's second law

The goal is to show that "extremize proper time" degenerates, in a weak static field with slow motion, into the elementary statement that a particle accelerates down the gradient of a potential — recovering $\ddot{\vec x} = -\nabla\Phi$ and the metric-potential identification $g_{tt} \approx -(1+2\Phi)$ that anchors the Newtonian correspondence. The strategy is to expand the proper-time integrand to first order in the potential and second order in the velocity, then read off the effective Lagrangian.

Step 1. Adopt the weak-field metric $\mathrm{d}s^2 = -(1+2\Phi)\,\mathrm{d}t^2 + (1-2\Phi)\,\delta_{ij}\,\mathrm{d}x^i \mathrm{d}x^j$ with $\Phi \ll 1$ and units $c = 1$ , keeping only first order in $\Phi$ . Why this form: it is the linearized metric a slow, weak source produces, with the two factors fixed by demanding the correct Newtonian and light-bending limits.

Step 2. The proper time per unit coordinate time along a worldline of speed $v$ (with $v^2 = \delta_{ij}\dot x^i \dot x^j$ ) is $\mathrm{d}\tau/\mathrm{d}t = \sqrt{(1+2\Phi) - (1-2\Phi)v^2}$ ; expanding for $\Phi, v^2 \ll 1$ and dropping the cross term gives $\mathrm{d}\tau/\mathrm{d}t \approx 1 + \Phi - \tfrac12 v^2$ . Why the signs land this way: aging is boosted by sitting deeper in the well ( $+\Phi$ ) and reduced by moving fast ( $-\tfrac12 v^2$ ) — gravitational and velocity time dilation in one line.

Step 3. So $\tau \approx \int(1 + \Phi - \tfrac12 v^2)\,\mathrm{d}t$ , and maximizing $\tau$ (discarding the constant, flipping the sign) is equivalent to extremizing $\int(\tfrac12 v^2 - \Phi)\,\mathrm{d}t$ . Why this is already the answer: $\tfrac12 v^2 - \Phi$ is exactly the nonrelativistic Lagrangian — kinetic minus potential energy per unit mass — of a particle in the field $\Phi$ .

Step 4. Its Euler-Lagrange equation is $\tfrac{\mathrm{d}}{\mathrm{d}t}\dot x^i = -\partial_i \Phi$ , i.e. $\ddot{\vec x} = -\nabla\Phi$ — Newton's second law for motion in a gravitational potential. Why the identification $g_{tt} = -(1+2\Phi)$ is confirmed: only that coefficient reproduces the correct $+\Phi$ in the effective Lagrangian, so the demand of Newtonian agreement fixes the time-time metric component to first order. *As a numerical check, the expansion $\sqrt{(1+2\Phi)-(1-2\Phi)v^2} = 1 + \Phi - \tfrac12 v^2$ holds to second order; for example, with $\Phi = 10^{-6}$ and $v = 10^{-3}$ both sides agree to the relevant order.*

Limiting cases

The variational characterization behaves differently in the three causal classes and in the weak-field regime, and knowing which functional survives where is the practical heart of the subject.

Case	Result	Meaning
Timelike geodesic	$\delta\!\int\!\sqrt{-g_{\mu\nu}\,\mathrm{d}x^\mu \mathrm{d}x^\nu}=0,\ \ g_{\mu\nu}u^\mu u^\nu=-1$	Proper time stationary and locally maximal; worldline of a free massive particle. $u^\mu$ is the four-velocity normalized to $-1$ .
Spacelike geodesic	$\delta\!\int\!\sqrt{g_{\mu\nu}\,\mathrm{d}x^\mu \mathrm{d}x^\nu}=0,\ \ g_{\mu\nu}t^\mu t^\nu=+1$	Proper length stationary but a saddle, not an extremum; relevant to spatial geometry, e.g. shortest distance on a constant-time slice.
Null geodesic	$g_{\mu\nu}k^\mu k^\nu=0,\ \ k^\nu\nabla_\nu k^\mu=0$	Length and proper time vanish identically, so the arc-length functional is useless; defined by the energy functional with an affine parameter, or by parallel transport of $k^\mu$ .
Newtonian / weak field	$\tau\approx\int\!\big(1+\Phi-\tfrac12 v^2\big)\,\mathrm{d}t$	Maximizing proper time reduces to extremizing $\int(\tfrac12 v^2-\Phi)\,\mathrm{d}t$ , the action of a particle in Newtonian potential $\Phi$ ; $v$ is coordinate speed.
Past a conjugate point	second variation loses definiteness	Beyond the first conjugate point a timelike geodesic is only stationary, no longer a local maximum; the Morse index counts the lost directions.

Common trapGeodesics minimize distance, so timelike ones minimize proper time

They do not. In Lorentzian signature the Riemannian intuition inverts: a timelike geodesic locally maximizes proper time — the free-falling clock reads the longest interval, not the shortest — and a spacelike geodesic is a saddle point. The word to trust is stationary, never shortest. Two further traps hide here. First, the square-root functional degenerates for null curves, where the length is identically zero and cannot be varied, so light needs the energy functional and an affine parameter. Second, even the local-maximum property holds only up to the first conjugate point: over a long enough arc, neighboring geodesics refocus and a nearby non-geodesic curve can beat the geodesic's proper time — exactly the phenomenon Morse theory was built to quantify.

Worked examples

1 · Maximal aging and the twin paradox (fully numeric)

Two reunion events A and B are separated by 10 years of coordinate time in flat spacetime. Twin S stays inertial (a geodesic); twin T rockets out at constant speed and returns. Compute the proper time each twin ages for $v = 0.8c$ and for $v = 0.917c$ , and identify which worldline is the geodesic.

Step 1. Twin S sits on a straight timelike worldline — a geodesic of flat spacetime — so between A and B she ages the full coordinate interval: $\tau_S = \int\sqrt{1 - v^2}\,\mathrm{d}t$ with $v = 0$ , giving $\tau_S = \mathbf{10.0\ \text{yr}}$ , the candidate for the maximum.

Step 2. Twin T moves at speed $v$ on each leg, so his integrand is $\sqrt{1 - v^2}$ throughout. For $v = 0.8c$ : $\sqrt{1 - 0.64} = 0.6$ , so over 10 years of coordinate time $\tau_T = 10 \times 0.6 = \mathbf{6.0\ \text{yr}}$ — four years less than his sister.

Step 3. Push to $v = 0.917c$ : $\sqrt{1 - 0.841} = \sqrt{0.159} \approx 0.40$ , giving $\tau_T = 10 \times 0.40 = \mathbf{4.0\ \text{yr}}$ . The faster the detour, the sharper the loss — a bent worldline is "longer" in coordinate terms but shorter in proper time, the reverse of Euclidean intuition.

Step 4. There is no paradox because the twins are not symmetric: only S is on a geodesic, feeling no acceleration, while T must fire engines to turn around. The variational principle delivers the verdict — the geodesic (inertial) worldline S maximizes proper time at 10.0 yr, and every accelerated alternative banks strictly less (6.0 yr, then 4.0 yr).

2 · Circular orbits and the ISCO from conserved quantities (numeric)

In the Schwarzschild metric, use the cyclic coordinates to extract the conserved energy and angular momentum, reduce the radial motion to an effective potential, and locate the circular orbits for $L = 4M$ and the innermost stable circular orbit. Evaluate the ISCO radius for a $10\,M_\odot$ black hole.

Step 1. The geodesic Lagrangian $L_E = g_{\mu\nu}\dot x^\mu \dot x^\nu$ for Schwarzschild does not contain $t$ or $\phi$ , so equation (6) gives two conserved momenta: $E = (1 - 2M/r)\,\dot t$ and $L = r^2\dot\phi$ (per unit mass, equatorial plane, units $G = c = 1$ ). These are the energy and angular momentum handed over for free by the two Killing vectors $\partial_t$ and $\partial_\phi$ .

Step 2. Insert $E$ and $L$ into the normalization (7), $g_{\mu\nu}\dot x^\mu \dot x^\nu = -1$ , and rearrange into $\tfrac12\dot r^2 + V(r) = \tfrac12(E^2 - 1)$ with $V(r) = -\dfrac{M}{r} + \dfrac{L^2}{2r^2} - \dfrac{ML^2}{r^3}$ . The final term, absent from Newton, is the source of all relativistic orbital novelty.

Step 3. Circular orbits are turning points $V'(r) = 0$ , i.e. $Mr^2 - L^2 r + 3ML^2 = 0$ . For $L = 4M$ this is $r^2 - 16Mr + 48M^2 = 0$ , with roots $r = \mathbf{12M}$ (potential minimum, a stable orbit) and $r = \mathbf{4M}$ (barrier peak, an unstable orbit) — the two dots in the second figure.

Step 4. As $L$ shrinks the two roots merge when the discriminant $L^4 - 12M^2L^2$ vanishes, i.e. $L = 2\sqrt{3}\,M \approx 3.46M$ , at which point $r = L^2/2M = \mathbf{6M}$ : the innermost stable circular orbit, below which no stable circular geodesic exists and matter must plunge. For a $10\,M_\odot$ black hole, $M = 14.77\ \text{km}$ , so the ISCO lies at $r = 6M = 88.6\ \text{km}$ — the inner edge of an accretion disk, fixed entirely by the geodesic principle.

Physics of the portrait

The idea that a geodesic is an extremal path is as old as the calculus of variations, but the deep question — when is a stationary geodesic actually a maximum or minimum, and what spoils it — belongs to one mathematician above all.

Origins · 2 minMarston Morse and how this was discovered

Marston Morse

1892–1977 · American founder of Morse theory and the calculus of variations "in the large," which counts when a geodesic stops being a true extremum

Harold Marston Morse, born in Waterville, Maine, took his doctorate at Harvard in 1917 under George David Birkhoff and spent the heart of his career at the Institute for Advanced Study in Princeton alongside Einstein and von Neumann. His life's work answered a question the classical variational principle leaves open. Euler and Lagrange had shown how to find curves that make an integral stationary, and Jacobi had already noticed in the 1830s that a geodesic ceases to be a genuine minimum once it passes a conjugate point — a place where an infinitesimally neighboring geodesic refocuses onto it. Morse turned this scattered insight into a global theory. His 1934 monograph, The Calculus of Variations in the Large, introduced what we now call the Morse index: the number of independent directions in which the second variation of the length functional is negative. The Morse index theorem states that this index equals the number of conjugate points along the geodesic, counted with multiplicity — a bridge between the local geometry of focusing and the global count of how far from a minimum a path has drifted. For general relativity the payoff is direct: a timelike geodesic maximizes proper time only up to its first conjugate point, beyond which the second variation (governed by the geodesic deviation equation, hence by the Riemann tensor) loses definiteness and nearby worldlines can overtake the geodesic's aging. Extended to the Lorentzian setting, this machinery became the analytic backbone of the Penrose-Hawking singularity theorems, where the focusing of geodesic congruences at conjugate points forces incomplete geodesics — the mathematical fingerprint of a singularity. Morse's "in the large" viewpoint thus completes the extremal picture: it explains not just that geodesics are stationary, but exactly when their extremal character survives.

History

1744

— Euler derives the equation for curves that extremize an integral, founding the calculus of variations

1755

— Lagrange introduces the variation δ and the Euler–Lagrange equations, systematizing stationary-action problems

1854

— Riemann, in his habilitation lecture, defines geodesics on a curved manifold directly through the metric

1915

— Einstein makes free fall the extremization of proper time, geometrizing inertia as motion along timelike geodesics

1934

— Morse's Calculus of Variations in the Large ties a geodesic's extremal character to its conjugate points

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §13.4 · Wald, General Relativity (1984), §3.3 and §9.3 (conjugate points) · Carroll, Spacetime and Geometry (2004), §3.3 · Hartle, Gravity (2003), Ch. 5–6 and 8 · Schutz, A First Course in General Relativity (2009), §6.4 and §7.2 · Morse, The Calculus of Variations in the Large (1934), for the index theory of conjugate points. The key results in this article are: the geodesic equation obtained by varying $\int\sqrt{-g_{\mu\nu}\dot x^\mu \dot x^\nu}\,\mathrm{d}\lambda$ and setting $\lambda = \tau$ ; the Christoffel formula and the energy-Lagrangian equivalence including the affine-parameter argument; the weak-field reduction $g_{tt} \approx -(1+2\Phi)$ recovering $\ddot{\vec x} = -\nabla\Phi$ ; the twin-paradox arithmetic $\tau = 10\sqrt{1-v^2}$ giving 6.0 yr at $v = 0.8c$ and 4.0 yr at $v = 0.917c$ ; the Schwarzschild effective potential $V = -M/r + L^2/2r^2 - ML^2/r^3$ with circular orbits at $r = 4M$ and $r = 12M$ for $L = 4M$ , and the ISCO at $r = 6M$ for $L = 2\sqrt{3}\,M$ , giving $88.6$ km for a $10\,M_\odot$ hole. Standard observational benchmarks include Mercury's $43''$ /century perihelion advance and the $1.75''$ solar light deflection.

Builds on: The Geodesic Equation · Christoffel Symbols · The Levi-Civita Connection · Affine Parameters. Leads to: Killing Vectors · The Geodesic Deviation Equation · Tidal Forces in General Relativity · Parallel Transport.

Ask the tutor: Ask me to vary any metric's geodesic Lagrangian with you — Schwarzschild orbits, the twin paradox, a light ray past the Sun — and we will read off the conserved quantities and the effective potential together.

Why is a timelike geodesic a maximum of proper time rather than a minimum?

It comes down to the minus sign in the Lorentzian metric. In ordinary Riemannian geometry the metric is positive-definite, so wiggling a geodesic can only add length and geodesics minimize distance. In spacetime with signature $(-+++)$ , the proper-time integrand carries an overall minus under the square root, and adding spatial wiggling to a timelike worldline increases the spatial contribution, which decreases the proper time. So the straight (inertial) path banks the most proper time, and any deviation costs aging. The clean physical statement is the twin paradox: the twin who never accelerates — the one on the geodesic — is always the older one at reunion.

What exactly is an affine parameter, and why does the energy Lagrangian pick one out?

An affine parameter is one in which the geodesic equation takes its clean form $\ddot x^\mu + \Gamma^\mu{}_{\alpha\beta}\dot x^\alpha \dot x^\beta = 0$ with no extra term on the right. Reparametrizing to $\sigma = f(\lambda)$ adds a term proportional to $\dot x^\mu$ , spoiling that form unless $f$ is linear, $\sigma = a\lambda + b$ — the affine freedom. Proper time and the natural parameter of the energy functional are affine; a random parametrization is not. The energy Lagrangian selects one because its equations conserve $L_E = g_{\mu\nu}\dot x^\mu \dot x^\nu$ automatically, and constancy of that norm is exactly the condition that the parameter runs uniformly. The arc-length functional, being reparametrization invariant, prefers no parameter at all.

How do conservation laws fall out of the variational picture?

Through cyclic coordinates. If the metric components do not depend on some coordinate $x^\sigma$ — for instance $t$ in a static spacetime or $\phi$ in an axisymmetric one — then $\partial L_E/\partial x^\sigma = 0$ , and the Euler-Lagrange equation immediately says $\mathrm{d}(g_{\sigma\nu}\dot x^\nu)/\mathrm{d}\lambda = 0$ . That conserved momentum is the projection of the tangent onto a Killing vector $\xi^\mu = \delta^\mu_\sigma$ , so time-translation symmetry gives conserved energy and rotational symmetry gives conserved angular momentum. This is the geodesic version of Noether's theorem, and it is what reduces the Schwarzschild orbit problem from four coupled equations to a single one-dimensional effective-potential problem.

Why must the connection be the Levi-Civita one for the two definitions to agree?

There are two independent ways to call a curve a geodesic. The variational way extremizes proper time; the geometric way parallel-transports the tangent along itself. Varying the metric functional produces the Christoffel symbols, which are by construction symmetric in their lower indices (torsion-free) and built from the metric (metric-compatible) — that is, the unique Levi-Civita connection. The parallel-transport definition, by contrast, works with whatever connection you hand it. Only when that connection is Levi-Civita do the two notions of "geodesic" pick out the same curves. Give the manifold a connection with torsion, and the autoparallels (parallel-transport curves) and the extremal curves generally split apart.

Do photons really follow geodesics if their proper time is zero?

Yes, but not via the proper-time functional, which is identically zero along any null curve and offers nothing to extremize. The resolution is the energy Lagrangian $L_E = g_{\mu\nu}\dot x^\mu \dot x^\nu$ with an affine parameter: its Euler-Lagrange equation is the same geodesic equation but stays finite when $L_E = 0$ . Equivalently, a null geodesic is the curve whose tangent $k^\mu$ parallel-transports itself, $k^\nu\nabla_\nu k^\mu = 0$ , while staying null, $k_\mu k^\mu = 0$ . Both routes give light bending, lensing, and the photon sphere; the arc-length picture is simply the wrong tool here.

What is a conjugate point and why does it matter for extremization?

A conjugate point is a place where an infinitesimally neighboring geodesic, starting from the same initial event, refocuses back onto the original geodesic — think of two great circles leaving the north pole and reconverging at the south pole. Its existence is governed by the geodesic deviation equation and hence by spacetime curvature. It matters because a geodesic is a genuine local extremum of the proper-time functional only up to its first conjugate point; past that point the second variation loses definiteness, and a nearby curve can beat the geodesic (longer proper time for timelike, shorter for spacelike). Morse's index theorem counts how many negative directions the second variation has acquired by counting conjugate points, and this focusing behavior is the analytic seed of the Penrose–Hawking singularity theorems.