Affine and Variational Geodesics

A geodesic can be defined by parallel transport of its tangent or, for a Levi–Civita connection, by stationarity of an appropriate metric action. The two definitions agree when the curve is regularly parametrized and the variational problem is chosen correctly. Affine and nonaffine parameters must be separated, and null curves require an energy or auxiliary-field action because their metric length vanishes identically.

Required background. Affine Connections and Covariant Derivatives supplies the general connection and covariant-derivative framework; The Levi-Civita Connection supplies the metric-compatible torsion-free specialization, and Parallel Transport Along Curves supplies covariant differentiation along a curve. The Riemann Curvature Tensor fixes the convention used when geodesic variations are linearized later in the chapter.

Helpful background. Calculus and Linear Algebra for Relativity supplies the integration-by-parts calculation behind the Euler–Lagrange equation. Presymplectic Potentials and Currents is an advanced continuation from boundary terms to presymplectic currents; the finite-dimensional geodesic variation is derived here and does not assume that formalism.

Affine geodesics

Let $x^\mu(\lambda)$ be a smooth regular curve with tangent $u^\mu=dx^\mu/d\lambda$. It is an affinely parametrized geodesic when its tangent is parallel along itself:

$$\nabla_u u=0, \qquad \frac{d^2x^\rho}{d\lambda^2} +\Gamma^\rho{}_{\mu\nu} \frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}=0.$$

If $\lambda\mapsto a\lambda+b$ with constant $a\ne0$, the equation retains this form. A general monotone reparametrization $s=s(\lambda)$ instead produces

$$\nabla_vv=f(s)v, \qquad v=\frac{dx}{ds},$$

where $f(s)=(d^2\lambda/ds^2)/(d\lambda/ds)$. This follows by writing $v=(d\lambda/ds)u$ and applying the product rule. It describes the same unparametrized path but not an affine parameter. Solving $d\log\lvert d\lambda/ds\rvert/ds=f(s)$ removes $f$ locally. Along an affine Levi–Civita geodesic, $d[g(u,u)]/d\lambda=2g(\nabla_u u,u)=0$, so its timelike, null, or spacelike causal character is constant.

Variation of the energy action

For fixed endpoints, consider

$$E[x]=\frac12\int_{\lambda_0}^{\lambda_1} g_{\mu\nu}(x)\dot x^\mu\dot x^\nu\,d\lambda.$$

Its Euler–Lagrange equation is

$$\frac{d}{d\lambda}(g_{\rho\nu}\dot x^\nu) -\frac12\partial_\rho g_{\mu\nu}\dot x^\mu\dot x^\nu=0.$$

Raise the index and insert the Levi–Civita formula to obtain the affine geodesic equation. Thus energy-critical curves use a parameter whose speed squared is constant. The invariant first-variation form is even clearer. If $V$ is the variation field and vanishes at both endpoints,

$$\delta E =-\int_{\lambda_0}^{\lambda_1}g(V,\nabla_u u)\,d\lambda,$$

after metric compatibility and integration by parts. Since $V$ is arbitrary in the interior, stationarity gives $\nabla_u u=0$. The equivalence, including the role of parameter choice, is developed in Calegari 2022, §§6.1–6.3, pp. 47–51.

Length, proper time, and the null caveat

For a non-null regular curve one may vary

$$L[x]=\int \sqrt{\lvert g(\dot x,\dot x)\rvert}\,d\lambda.$$

This functional is reparametrization invariant. Its Euler–Lagrange equation determines the geodesic path but initially permits $\nabla_vv=fv$; choosing constant speed makes the parameter affine. In a convex normal neighborhood, a timelike geodesic maximizes proper time between its endpoints among future-directed causal curves in that neighborhood. This differs from the local length-minimizing property of short Riemannian geodesics. Spacelike Lorentzian geodesics have no corresponding general minimum principle: in spacetime dimension at least three, even a straight spacelike segment in Minkowski space is a saddle of length under nearby spacelike variations.

For a null curve, $g(\dot x,\dot x)=0$. Length is zero on the set of null curves and the square-root integrand is not differentiable at zero under arbitrary variations. Thus the usual length variation cannot derive a null geodesic equation. One may vary the energy action over unrestricted curves and then select solutions with null initial tangent; their norm stays zero. Restricting the energy action to null curves before varying would again make it identically zero. Alternatively use an auxiliary positive function, the einbein $e$:

$$S[x,e]=\frac12\int \left(e^{-1}g(\dot x,\dot x)-e m^2\right)d\lambda.$$

Varying $e$ gives $g(\dot x,\dot x)=-e^2m^2$, while varying $x$ gives $\nabla_{\dot x}\dot x=(\dot e/e)\dot x$. At $m=0$ these are nullity and the nonaffine geodesic equation; choosing constant $e$ gives an affine parameter. The simpler energy-action method, supplemented by a timelike or null normalization, is explained in Tong 2019, §1.2.6, eqs. (1.32)–(1.34), pp. 28–29, PDF.

Checks and limits

In Minkowski Cartesian coordinates $\Gamma^\rho{}_{\mu\nu}=0$, hence $x^\mu=A^\mu\lambda+B^\mu$. Under $s=\lambda^2$ on a positive interval the same straight line satisfies a nonaffine equation with acceleration parallel to its tangent; the path has not become curved. This explicitly separates affine structure from image geometry.

On a round sphere, great circles satisfy the constrained ambient equation $\ddot X=-\lvert\dot X\rvert^2X/a^2$, whose acceleration is purely normal. Its tangential covariant acceleration vanishes, so great circles are intrinsic geodesics. These checks reproduce the flat and curved cases without appealing solely to coordinate formulas.

The endpoint conditions in the variational statement deserve care. Fixed endpoints remove the boundary term $[g(V,u)]_{\lambda_0}^{\lambda_1}$ after integration by parts. If endpoints are allowed to move on specified submanifolds, that term instead yields transversality conditions: the geodesic tangent must meet the endpoint submanifold orthogonally in the Riemannian problem. Changing the admissible variations changes the theorem, even though the interior Euler–Lagrange equation is still geodesic.

Energy and length also differ away from critical constant-speed curves. Cauchy–Schwarz relates them in the positive-definite case, but energy depends on parametrization while length does not. It is therefore inaccurate to say that the two functionals are identical. Their stationary unparametrized paths agree after choosing constant speed; energy then fixes an affine representative. In Lorentzian signature the causal class must remain fixed for the square-root functional, and no positive-definite inequality can be imported unchanged.

Finally, the coordinate equation is invariant even though its two displayed terms are not separately vectors. Under a coordinate change, the inhomogeneous transformation of $\Gamma^\rho{}_{\mu\nu}$ cancels the corresponding second-derivative term in $d^2x^\rho/d\lambda^2$. This is why setting Christoffel symbols to zero at one point does not make geodesicity a coordinate-dependent notion.

The geometric construction here does not identify geodesics with physical free fall; that interpretation requires the equivalence principle and matter dynamics. Nor does local stationarity imply global maximality, minimality, uniqueness, or completeness.

Exercises

1. Changing parameter. Show that $x^\mu(s)=A^\mu\sqrt{s}+B^\mu$ for $s\gt0$ and $A\ne0$ traces a straight geodesic but is not affinely parametrized.

Solution

In Cartesian Minkowski coordinates, $v^\mu=A^\mu/(2\sqrt{s})$ and $dv^\mu/ds=-A^\mu/(4s^{3/2})=-(1/2s)v^\mu$. Thus $\nabla_vv=-(1/2s)v$, not zero. The acceleration is parallel to the tangent, so a change back to $\lambda=\sqrt{s}$ removes it and gives $d^2x^\mu/d\lambda^2=0$.

2. A spacelike saddle. In $ds^2=-dt^2+dx^2+dy^2$, compare the segment $(0,s,0)$, $0\le s\le L$, with the endpoint-fixed curves $(\epsilon f(s),s,0)$ and $(0,s,\epsilon f(s))$, where $f(0)=f(L)=0$ and $f$ is smooth and nonconstant.

Solution

For sufficiently small $\epsilon$, both curves remain spacelike. Their lengths are

$$\begin{aligned} L_{\mathrm{time}}(\epsilon) &=L-\frac{\epsilon^2}{2}\int_0^L(f')^2ds+O(\epsilon^4),\\ L_{\mathrm{space}}(\epsilon) &=L+\frac{\epsilon^2}{2}\int_0^L(f')^2ds+O(\epsilon^4). \end{aligned}$$

The time excursion shortens the curve, while the spatial excursion lengthens it. Since both variations can be arbitrarily small and keep the endpoints and causal character fixed, the original straight spacelike geodesic is a stationary saddle, not a local minimum or maximum.

References

• Danny Calegari, Introduction to Differential Geometry, preliminary version, May 26, 2022, University of Chicago. PDF.
• David Tong, General Relativity, University of Cambridge Part III Mathematical Tripos lecture notes, 2019. PDF.