Local Metric Expansion and Curvature

Irreducible local gravity first enters the metric at quadratic order in distance from a freely falling reference worldline. Constant and linear terms can be put into Minkowski form by choosing an orthonormal frame and freely falling coordinates. The quadratic coefficients are components of the Riemann tensor, so their effects scale as curvature times the square of the laboratory size.

Required background. We use the Riemann curvature tensor and the scope of local equivalence principles.

Helpful background. The calculation draws on Minkowski spacetime, tensor Taylor expansions, orthonormal frames, and geodesics and Jacobi fields.

Riemann normal expansion at one event

Choose Riemann normal coordinates $x^\mu$ centered at $p$, with an orthonormal coordinate basis there. Then

$$g_{\mu\nu}(p)=\eta_{\mu\nu}, \qquad \partial_\alpha g_{\mu\nu}(p)=0.$$

For the convention

$$[\nabla_\alpha,\nabla_\beta]V^\rho =R^\rho{}_{\sigma\alpha\beta}V^\sigma,$$

the leading expansion is

$$g_{\mu\nu}(x) =\eta_{\mu\nu} -\frac13R_{\mu\alpha\nu\beta}(p)x^\alpha x^\beta +O(L^3).$$

The absence of a linear term is the mathematical content of eliminating the connection at the central event. The quadratic coefficient is tensorial and cannot be eliminated if curvature is nonzero. Carroll relates the normal-coordinate expansion to the curvature symmetries and geodesic deviation Carroll 1997, Chapter 3, pp. 74–88, PDF.

Here $L$ measures coordinate distance in the chosen normal frame; the remainder is a Taylor statement with dimensionful coefficients. Cubic coefficients contain $\nabla R$, while quartic coefficients contain both $\nabla\nabla R$ and quadratic products of $R$. A controlled laboratory expansion therefore checks $\lVert R\rVert L^2$, $\lVert\nabla R\rVert L^3$, and, when needed, $(\lVert\nabla\nabla R\rVert+\lVert R\rVert^2)L^4$, with component bounds taken over the region used. A vanishing first derivative of curvature at the central event does not eliminate its second derivatives.

Small metric corrections and an accurate quadratic tidal model are different requirements. For the latter, the omitted terms must be small compared with the retained $\lVert R\rVert L^2$ term, not merely small compared with unity. If that retained term vanishes in a particular direction, use an absolute error bound and examine the next nonzero order.

Fermi normal coordinates along a geodesic

Riemann normal coordinates privilege one event. To describe a freely falling laboratory over time, take a timelike geodesic $z(T)$, parallel transport an orthonormal tetrad $e_{(a)}{}^\mu$ along it, and shoot spacelike geodesics orthogonal to the worldline. Their tetrad components define Fermi coordinates $(T,X^i)$ in a normal tube before those geodesics intersect.

Using tetrad components of curvature evaluated on $z(T)$, the metric through quadratic order is

$$\begin{aligned} g_{00}&=-1-R_{0i0j}X^iX^j+O(L^3),\\ g_{0i}&=-\frac23R_{0jik}X^jX^k+O(L^3),\\ g_{ij}&=\delta_{ij}-\frac13R_{ikjl}X^kX^l+O(L^3). \end{aligned}$$

Manasse and Misner derive these coefficients for a parallel-transported frame on Manasse and Misner 1963, p. 743, Eqs. (65)–(66). Their curvature convention has the opposite overall sign, $R^{\rm MM}_{abcd}=-R^{\rm site}_{abcd}$, so every quadratic curvature term changes sign on translation. The displayed equations are then fixed by the commutator convention above and checked below.

The construction is local in space and time. It fails as a single chart once orthogonal geodesics form caustics or the reference worldline leaves the region where the tetrad and curvature expansion are controlled.

Checking the time-time coefficient

For a slowly moving nearby free particle, use $T$ as parameter and keep only terms linear in $X^i$. The spatial geodesic equation becomes

$$\frac{d^2X^i}{dT^2}\simeq-\Gamma^i{}_{00}.$$

From $g_{00}=-1-R_{0k0l}X^kX^l$,

$$\Gamma^i{}_{00} =-\frac12\delta^{ij}\partial_jg_{00} =R^i{}_{0j0}X^j,$$

and therefore

$$\frac{d^2X^i}{dT^2} =-R^i{}_{0j0}X^j+O(L^2,vL).$$

This is the geodesic-deviation acceleration relative to the central observer. It independently fixes the sign of the $g_{00}$ term.

In the weak, stationary Newtonian limit $g_{00}=-(1+2\Phi)$ and

$$R_{0i0j}\simeq\partial_i\partial_j\Phi.$$

The same equation becomes $d^2X^i/dT^2=-\partial_i\partial_j\Phi\,X^j$, the difference between gravitational accelerations at neighboring points. The limit checks both sign and dimensions: $R$ has dimensions $L^{-2}$, so $RX^2$ is a dimensionless metric correction.

Acceleration is a different order

If the central worldline is accelerated rather than geodesic, Fermi-type coordinates acquire linear inertial terms. For a nonrotating observer in flat spacetime, $g_{00}$ begins as $-(1+a_iX^i)^2$. Such a term reflects the observer’s proper acceleration and can exist with $R_{\mu\nu\rho\sigma}=0$. Curvature remains identified by the quadratic tidal coefficients after acceleration and rotation have been calibrated.

This order counting is operationally useful. An accelerometer measures the linear inertial contribution along the reference worldline; a network of separated clocks or free probes measures the quadratic curvature contribution. Treating both as merely “gravity” obscures which effect a coordinate choice can remove.

Exercises

A freely falling laboratory has size $L$ in a region with curvature scale $\mathcal R$, meaning typical orthonormal components satisfy $|R_{abcd}|\sim\mathcal R^{-2}$. Estimate the leading fractional metric correction and state the locality condition.

Solution

Every quadratic term contains $R_{abcd}X^cX^d$, so its typical magnitude is $L^2/\mathcal R^2$. The leading locality condition is $L/\mathcal R\ll1$. If curvature varies on a separate scale $\mathcal L$, one must also require a small cubic correction, schematically $|\nabla R|L^3\sim L^3/(\mathcal R^2\mathcal L)\ll1$. To approximate the tidal correction itself by the quadratic term, require $L/\mathcal L\ll1$; smallness relative to the background Minkowski metric alone is insufficient.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv.
• Manasse, F. K., and Charles W. Misner. “Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry.” Journal of Mathematical Physics 4 (1963): 735–745. DOI.