The Geodesic Initial-Value Problem

A point $p$ and tangent vector $v\in T_p\mathcal M$ determine one unique affinely parametrized geodesic near the initial event, provided the metric is sufficiently regular. This is a local ordinary-differential-equation result. It neither guarantees that the curve exists for every affine parameter nor establishes geodesic completeness of the spacetime.

Required background. Affine and Variational Geodesics supplies affine geodesics, Jacobi fields, and their variational characterization; What the Equivalence Principles Do and Do Not Imply supplies local-inertial reasoning and the scope of universal free fall.

Helpful background. Events, Intervals, and Minkowski Geometry supplies causal types and timelike and null worldlines; Tensor Fields and Natural Operations supplies coordinate-independent tensor operations; Differential Equations for Relativity supplies local existence, uniqueness, and numerical initial-value preparation.

Geodesic data as a first-order system

Choose a chart around $p$ and an affine parameter $\lambda$ with $\lambda=0$ initially. Writing $v^\mu=dx^\mu/d\lambda$, the geodesic equation is equivalent to

$$\frac{dx^\mu}{d\lambda}=v^\mu, \qquad \frac{dv^\mu}{d\lambda} =-\Gamma^\mu{}_{\alpha\beta}(x)v^\alpha v^\beta,$$

with data

$$x^\mu(0)=x_p^\mu, \qquad v^\mu(0)=v_p^\mu.$$

This is an autonomous first-order system on the tangent bundle. If $g_{\mu\nu}$ is $C^2$, then the Christoffel symbols are $C^1$ and hence locally Lipschitz in the state variables. Standard ODE existence and uniqueness therefore provides a unique local solution; a smooth metric gives a smooth solution with smooth dependence on the initial data. Reall states the theorem and reduces it to precisely this coordinate system in Reall 2013, §4.3, pp. 45–46, PDF.

The coordinate proof is local, but the result is geometric. If the solution reaches the edge of one chart while remaining at an ordinary point of the manifold, transform the tangent and continue in an overlapping chart. Uniqueness makes the pieces agree.

Causal normalization and affine scale

Along a solution,

$$\frac{d}{d\lambda} \left(g_{\mu\nu}v^\mu v^\nu\right) =2v_\mu v^\alpha\nabla_\alpha v^\mu=0.$$

Thus the initial value

$$\kappa=g(v,v)$$

is conserved. Choosing $\kappa=-1$ for future-directed timelike data makes $\lambda$ proper time up to an additive constant. Null data have $\kappa=0$, which does not fix the magnitude of $v_p$: multiplying the initial tangent by a positive constant follows the same oriented image with a rescaled affine parameter.

Initial point plus initial direction therefore suffices only for an unparametrized geodesic. Point plus initial tangent fixes the affine scale as well. The zero tangent gives the constant geodesic, a legitimate but physically uninteresting solution.

Maximal solutions and completeness

Local solutions patch uniquely to a maximal geodesic $\gamma:I\to\mathcal M$ on an open interval $I$ containing zero. Here “maximal” means not extendible as a geodesic in the given manifold; it does not mean longest or greatest proper time. A spacetime is geodesically complete only if every maximal affine geodesic of the specified causal class has parameter domain $\mathbb R$.

Nothing in the local ODE theorem forces $I=\mathbb R$. A coordinate component can diverge because a chart fails while the curve remains extendible, so coordinate blow-up alone is not incompleteness. Conversely, curvature need not diverge when incompleteness occurs: Minkowski spacetime with one event removed is flat, yet a straight geodesic aimed at the missing event ends after finite affine parameter within that punctured manifold. Carroll separates the local exponential-map construction from global domain and completeness questions in Carroll 1997, Ch. 3, pp. 72–73, PDF.

Geodesic incompleteness, a curvature singularity, an inextendible spacetime, and a boundary in a chosen representation are distinct claims. Establishing any of them requires additional global analysis.

Analytic and numerical diagnostics

For an analytic candidate $x^\mu(\lambda)$, check the initial data, substitute into every component of the geodesic equation, and verify the constant $g(v,v)$. If a Killing field $\xi^\mu$ is available, $\xi_\mu v^\mu$ supplies another independent check for affine motion.

For numerical integration, evolve both $x^\mu$ and $v^\mu$ and monitor the normalization error, equation residual, and chart domain. Repeat with smaller step size or tighter tolerance; a stable-looking plot is not a convergence test. Near a coordinate boundary, change charts or use regular variables before interpreting a failure. These diagnostics test an integration, not the global completeness of the underlying spacetime.

Exercise

Solve the initial-value problem in Cartesian Minkowski coordinates and then explain what changes if the event reached at $\lambda=1$ is removed from the manifold.

Solution

All Christoffel symbols vanish, so $dv^\mu/d\lambda=0$ and

$$x^\mu(\lambda)=x_p^\mu+\lambda v_p^\mu.$$

The solution exists for every real $\lambda$ in full Minkowski spacetime and preserves $v^2$. If the event $x_p^\mu+v_p^\mu$ is deleted, the same formula defines a geodesic only for parameter values that avoid $\lambda=1$. The branch approaching that event cannot be continued through $\lambda=1$ inside the punctured manifold, despite zero curvature everywhere on the manifold. Local uniqueness remains true; completeness fails globally.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997), Ch. 3, printed pp. 54–97. arXiv.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013), § 4.3, printed pp. 45–46. PDF.