Timelike Geodesics as Ideal Free Fall

A material body follows a timelike geodesic only after a controlled idealization: its representative worldline must describe a small, nonspinning, effectively neutral test body with no significant self-force or applied force. Within that regime, proper-time stationarity gives covariant acceleration zero and explains why the body’s mass does not appear in its trajectory.

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; Presymplectic Potentials and Currents supplies disciplined endpoint and boundary variations.

Timelike geodesic equation

For a future-directed timelike worldline $x^\mu(\tau)$, proper time fixes the four-velocity normalization,

$$u^\mu=\frac{dx^\mu}{d\tau}, \qquad u^\mu u_\mu=-1.$$

Stationarity of $-m\int d\tau$ under fixed-endpoint variations gives

$$a^\mu\equiv u^\nu\nabla_\nu u^\mu=0.$$

In coordinates this is

$$\frac{d^2x^\mu}{d\tau^2} +\Gamma^\mu{}_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau}=0.$$

The two coordinate terms are not separately vectors; their sum is. Consequently a freely falling body can have nonzero coordinate acceleration even though its accelerometer reads zero. Reall derives the proper-time equation and then identifies it with parallel transport of the tangent in Reall 2013, §§3.3 and 4.3, pp. 36–46, PDF.

The factor $m$ cancels from the equation. This is universality for ideal test bodies, not a statement that mass, composition, or internal structure can never affect real motion.

“Free” refers to vanishing nongravitational proper force. An orbiting spacecraft can have a visibly curved coordinate track while its occupants float and $a^\mu=0$; a laboratory held at fixed height has a stationary coordinate position but a nonzero accelerometer reading supplied by the floor. The equivalence principle motivates this local identification of inertial and freely falling motion, but it does not by itself derive a unique gravitational field equation or remove the approximations behind the representative worldline. Derivations from stress-energy conservation require additional localization and small-body hypotheses.

Normalization and Newtonian sign checks

Metric compatibility provides an invariant consistency check:

$$\frac{d}{d\tau}(u^\mu u_\mu) =2u_\mu a^\mu=0.$$

Thus initial unit normalization is preserved. More generally, any four-force compatible with constant rest mass must satisfy $u_\mu f^\mu=0$.

For a slowly moving particle in a weak, nearly stationary field, take

$$g_{00}=-(1+2\Phi), \qquad |\Phi|\ll1, \qquad |\mathbf v|\ll1.$$

To leading order, $\Gamma^i{}_{00}=\partial^i\Phi$, and the spatial geodesic equation becomes

$$\frac{d^2x^i}{dt^2}=-\partial^i\Phi.$$

This recovers the Newtonian sign and shows that the connection term, rather than a new force proportional to $m$, carries the gravitational acceleration. It is also a useful check on the site’s $(-,+,+,+)$ convention.

Test-body hierarchy

Let $L$ be the body’s size and let $\mathcal R$ denote a local curvature radius, schematically $|R_{\alpha\beta\gamma\delta}|\sim\mathcal R^{-2}$ in a body-adapted orthonormal frame. The condition $L/\mathcal R\ll1$ makes the metric’s tidal correction small across the body. It does not by itself make curvature nearly constant: one must separately control curvature derivatives, for example $L|\nabla R|\ll|R|$ when a nonzero representative curvature scale is available. Near a curvature zero, compare the absolute neglected terms with the requested accuracy instead of dividing by $|R|$.

When the curvature and its derivatives share one variation scale $\mathcal R$, spin-curvature effects are suppressed by $|S|/(m\mathcal R)$ and ordinary quadrupole effects by $(L/\mathcal R)^2$. These are useful estimates under that extra scale assumption. In a background with sharp variation, the actual force terms and their derivatives must be estimated directly.

Applied electromagnetic, pressure, thrust, or contact forces must also be small compared with the acceleration scale relevant to the problem. Finally, the body must not appreciably change the background: in a system with external mass scale $M$, a common perturbative requirement is $m/M\ll1$. At the next order the body’s regular self-field and radiation reaction alter its motion. These qualifications are developed on Forces, Structure, and Limits of Geodesic Motion.

The center of an extended body is itself part of the approximation. With spin, different spin-supplementary conditions can select different representative curves inside the worldtube; no coordinate choice turns that issue into exact point-particle motion.

Stationary is not globally maximal

A timelike geodesic is a stationary curve of the proper-time functional. In a sufficiently small normal neighborhood it locally maximizes proper time between its endpoints, but conjugate points or multiple geodesics can destroy maximality on a longer segment. Carroll emphasizes both the affine equation and this local qualification in Carroll 1997, Ch. 3, pp. 68–73, PDF.

This distinction also separates equations of motion from global causal questions. The local geodesic equation neither proves that a solution extends for arbitrary proper time nor determines which of several geodesics connects distant events.

Exercise

Verify the weak-field limit above, including its sign.

Solution

Stationarity and slow motion make the $00$ connection term dominant. Since $g^{ij}=\delta^{ij}+O(\Phi)$ and $\partial_0g_{\mu\nu}=0$,

$$\Gamma^i{}_{00} =-\frac12g^{ij}\partial_jg_{00} =\partial^i\Phi+O(\Phi^2).$$

Also $dt/d\tau=1+O(\Phi,v^2)$. Dividing the spatial geodesic equation by $(dt/d\tau)^2$ therefore gives $d^2x^i/dt^2=-\partial^i\Phi$ at leading order.

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 and 4.3, printed pp. 36–46. PDF.