Free Fall, Proper Time, and Geodesic Motion

Free fall turns a geometric curve into a physical prediction only after three questions are answered: which action represents the probe, which parameter labels its worldline, and which neglected effects are small enough to call the body “ideal.” This chapter develops that chain for massive and massless probes, then separates local existence and local weightlessness from the global and finite-size claims they do not imply.

The central result is simple but conditional. A structureless, force-free test body follows a geodesic of the Levi-Civita connection. A massive body’s proper time can be an affine parameter; a null curve has no proper time and retains a constant affine-rescaling freedom. Curvature remains measurable through relative motion even when every pointlike accelerometer in a freely falling laboratory reads zero.

Helpful background. Events, Intervals, and Minkowski Geometry reviews causal classification and proper time; Tensor Fields and Natural Operations supplies coordinate-independent contractions and variations; Affine and Variational Geodesics supplies the geometric geodesic and Jacobi-field framework; What the Equivalence Principles Do and Do Not Imply fixes the scope of local inertial reasoning; Presymplectic Potentials and Currents provides the boundary-term discipline useful in action variations.

Worldlines, actions, and parameters

For a timelike curve, the elapsed proper time is

$$d\tau^2=-g_{\mu\nu}dx^\mu dx^\nu,$$

and stationarity of $-m\int d\tau$ gives

$$u^\nu\nabla_\nu u^\mu=0, \qquad u^2=-1.$$

The square-root action is invariant under arbitrary orientation-preserving relabelings. A quadratic action gives the same unparametrized geodesics only after an affine gauge has been chosen. An einbein action keeps reparametrization invariance manifest, supplies the mass-shell constraint, and remains meaningful for $m=0$.

For a null tangent $k^\mu$, proper time vanishes. The affine equation is $k^\nu\nabla_\nu k^\mu=0$; an arbitrary parameter instead gives a right-hand side $f k^\mu$. Two affine parameters differ by $\lambda'=a\lambda+b$, not by an arbitrary function. Reall develops proper-time extremization, affine geodesics, and normal coordinates in Reall 2013, §§3.3 and 4.3–4.4, pp. 36–48, PDF, while Carroll gives a complementary variational and geometric treatment in Carroll 1997, Ch. 3, pp. 68–73, PDF.

Two routes through the chapter

For a first graduate course, follow the sequence below from action to limitations. It derives each equation before interpreting it, and it keeps the massive/null distinction visible from the beginning. This route is the shortest preparation for geometric optics, curvature, tidal gravity, and the Einstein equation.

For a calculation-focused route, begin with the action, move directly to the initial-value problem, and then compare the timelike and null parameter choices. Finish with the approximation hierarchy before treating an orbit as a physical body. Readers interested mainly in laboratory interpretation can instead pair timelike free fall with geodesic frames, then use the force-and-structure page to diagnose departures from weightlessness.

Chapter contents

The six pages form one cumulative argument:

1. The Test-Particle Action compares the proper-length, quadratic, and einbein descriptions, including the massless limit.
2. Timelike Geodesics as Ideal Free Fall derives $a^\mu=0$, checks the Newtonian sign, and states the test-body hierarchy.
3. Null Geodesics and Affine Parameters explains why null proper time cannot be a parameter and constructs an affine one locally.
4. The Geodesic Initial-Value Problem reduces the equation to a first-order system and separates local existence from completeness.
5. Geodesic Frames and Local Weightlessness relates parallel tetrads and normal coordinates to zero accelerometer readings while retaining curvature.
6. Forces, Structure, and Limits of Geodesic Motion classifies applied force, spin, multipole, self-force, and radiation-reaction corrections.

After the sequence, a reader should be able to select the correct worldline action, identify affine versus nonaffine equations, formulate geodesic initial data, perform normalization and weak-field checks, and state quantitatively why a real body’s motion may be approximated by a geodesic.

Local statements and physical boundaries

The geodesic equation is a local differential equation. Smooth initial data determine a unique local solution in a chart, and overlapping charts extend it while the curve remains on the manifold. Geodesic completeness asks whether every maximal affine solution extends to all real parameter values; no local existence theorem answers that global question.

Likewise, local inertial coordinates set the connection to zero at an event, but curvature depends on connection derivatives and survives. A laboratory of size $L$ in curvature radius $\mathcal R$ sees metric corrections of order $(L/\mathcal R)^2$ and relative accelerations of order $L/\mathcal R^2$. Pointlike weightlessness and extended tidal response are therefore compatible.

This chapter does not catalog orbits in named spacetimes, prove global completeness theorems, or calculate self-force and finite-size dynamics. Those subjects require additional geometry, field equations, boundary conditions, and approximation schemes. Here they appear only far enough to prevent the monopole geodesic from being applied outside its domain.

Review the chapter

A curve has a null tangent at one event, and coordinates have been chosen so that every Christoffel symbol vanishes there. Can one conclude that the curve is an affinely parametrized null geodesic and that gravity is absent at that event?

Solution

Neither conclusion follows. Nullness at one event is only $k^2=0$ there; the curve must satisfy $k^\nu\nabla_\nu k^\mu=0$ to be affinely geodesic, or an equation proportional to $k^\mu$ to be a null pregeodesic. In the chosen coordinates the affine equation at that event reduces to $d^2x^\mu/d\lambda^2=0$, but the curve’s second derivative is extra information, not fixed by $\Gamma=0$.

Moreover, vanishing Christoffel symbols are pointwise and coordinate dependent. The Riemann tensor contains derivatives of the connection and can be nonzero, producing geodesic deviation in an extended experiment. The correct conclusions are local free-fall simplicity and zero proper acceleration for an actual geodesic—not vanishing curvature or absence of gravity in every operational sense.

Onward connection

Light and Geometric Optics turns the chapter’s null-geodesic kinematics into a controlled wave approximation. There the phase gradient supplies the null tangent, Maxwell’s equations fix amplitude and polarization transport, and caustics expose the boundary between a regular wave field and a nonuniform ray description.

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), §§ 3.3, 4.3–4.4, and 5.1, printed pp. 36–50. PDF.