Geodesic Frames and Local Weightlessness

A freely falling laboratory is weightless because its central worldline has zero proper acceleration and its nonrotating frame can be chosen locally inertial. That statement removes connection effects at an event—or along the central geodesic in a suitable frame—but it does not remove curvature. Extended experiments detect the remaining tidal relative acceleration.

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.

A freely falling tetrad

Let $\gamma(\tau)$ be a timelike geodesic with tangent $u^\mu$. Choose an orthonormal tetrad on one event of $\gamma$, with $e_{\hat0}{}^\mu=u^\mu$, and parallel transport every leg:

$$u^\nu\nabla_\nu e_{\hat a}{}^\mu=0, \qquad g_{\mu\nu}e_{\hat a}{}^\mu e_{\hat b}{}^\nu =\eta_{\hat a\hat b}.$$

Metric compatibility preserves orthonormality. The timelike leg remains tangent to the geodesic, while the spatial triad defines a nonrotating local frame. An ideal accelerometer on the central worldline reads

$$a^{\hat i}=e^{\hat i}{}_{\mu} u^\nu\nabla_\nu u^\mu=0.$$

This is the operational content of free-fall weightlessness. It is independent of whether the chosen coordinates assign a nonzero $d^2x^i/dt^2$ to the laboratory.

Normal coordinates and their reach

At any event $p$, Riemann normal coordinates can be chosen so that

$$g_{\mu\nu}(p)=\eta_{\mu\nu}, \qquad \partial_\rho g_{\mu\nu}(p)=0, \qquad \Gamma^\mu{}_{\nu\rho}(p)=0.$$

Reall constructs these coordinates from the exponential map and stresses their pointwise domain in Reall 2013, §4.4, pp. 47–48, PDF. Fermi normal coordinates adapt the construction to an entire timelike geodesic and its parallel tetrad: on the central curve the metric is Minkowskian and its first derivatives vanish, while away from it curvature-dependent terms appear.

The word “local” is essential. Normal coordinates do not generally make $\Gamma^\mu{}_{\nu\rho}$ vanish throughout an open neighborhood. Nor do they provide one inertial frame covering separated laboratories. The frame exists on a tube narrow enough that the relevant orthogonal geodesics are unique; caustics or intersections limit that tube.

Riemann normal coordinates are centered on one event; geodesic Fermi coordinates are adapted to a timelike worldline. An accelerated observer may still carry a Fermi–Walker-transported, nonrotating tetrad, but acceleration-dependent inertial terms then remain on the central line. A rest frame is therefore not automatically a freely falling or locally inertial frame.

Curvature left after free fall

In normal coordinates, the Riemann tensor at $p$ is built from second derivatives of the metric because the connection vanishes there. In a geodesic Fermi chart $(T,X^i)$, the coordinate metric therefore has the schematic form

$$g_{ab}(T,X) =\eta_{ab}+O(RX^2),$$

where $X$ is spatial distance from the reference worldline and $a,b$ label these coordinates. The correction refers to coordinate components away from the central curve. Components in an orthonormal tetrad are exactly $\eta_{\hat a\hat b}$ at every event where that tetrad is defined; curvature instead appears in how different frames are related. No coordinate transformation can set a nonzero tensor $R^\mu{}_{\nu\rho\sigma}(p)$ to zero.

For a one-parameter family of affinely parametrized geodesics, let $\xi^\mu$ connect neighboring worldlines and choose the family parameters so that $[u,\xi]=0$. With the site’s curvature convention,

$$\frac{D^2\xi^\mu}{d\tau^2} =R^\mu{}_{\alpha\beta\gamma} u^\alpha u^\beta\xi^\gamma.$$

Thus each freely falling accelerometer can read zero while their separation accelerates. Reall derives this sign from the commutator definition in Reall 2013, §6.5, pp. 60–62, PDF. If $|R|\sim\mathcal R^{-2}$ and the laboratory size is $L$, relative acceleration is of order $L/\mathcal R^2$ and the fractional metric correction is of order $(L/\mathcal R)^2$.

What a local experiment can conclude

A pointlike nongravitational experiment in this frame has the special-relativistic form at the central event, subject to the matter theory and minimal-coupling assumptions. An extended experiment can compare separated test masses, transport a gyroscope around a loop, or measure phase accumulated over finite area; such procedures are sensitive to curvature. Carroll distinguishes vanishing connection coefficients from invariant curvature and develops tetrad components in Carroll 1997, Chs. 2–3, pp. 50–51 and 87–92, PDF.

A supported laboratory is different: its central worldline has $a^\mu\ne0$, and an accelerometer measures the support force. A null curve has no rest frame and no proper-time clock, although normal coordinates at any event on it still exist. Finally, spin, finite size, external forces, and self-force can make a body’s chosen center nongeodesic; local coordinates cannot transform those tensorial effects away.

Exercise

In normal coordinates at $p$, does $\Gamma^\mu{}_{\nu\rho}(p)=0$ imply $R^\mu{}_{\nu\rho\sigma}(p)=0$?

Solution

No. At $p$ the quadratic connection terms in the curvature vanish, but derivatives of the connection need not:

$$R^\mu{}_{\nu\rho\sigma}(p) =\partial_\rho\Gamma^\mu{}_{\sigma\nu}(p) -\partial_\sigma\Gamma^\mu{}_{\rho\nu}(p).$$

Equivalently, these are combinations of second derivatives of $g_{\mu\nu}$. Normal coordinates remove the metric’s first derivatives at one event, not its curvature-controlled second derivatives. Neighboring geodesics can therefore have nonzero relative acceleration there.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997), Chs. 2–3, printed pp. 40–97. arXiv.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013), §§ 4.4 and 6.5, printed pp. 47–48 and 60–62. PDF.