Curvature and geodesic deviation

Parallel transport means changing a vector so its covariant derivative along a path vanishes. In flat geometry, transport around a sufficiently small contractible loop returns the vector unchanged. In curved geometry a small loop can produce a mismatch proportional to its oriented area. The Riemann tensor encodes this path dependence. A rotating coordinate basis can create nonzero connection coefficients even when this tensor vanishes, so transport around the loop gives a stronger diagnostic than the connection at one point.

Adopt $[\nabla_\mu,\nabla_\nu]V^\rho=R^\rho{}_{\sigma\mu\nu}V^\sigma$. This defines our sign convention; some references reverse it. The connection is Levi-Civita and the signature remains $(+,-,-,-)$. Antisymmetry in the final two indices follows immediately from exchanging the derivatives. A meaningful comparison with another book requires matching its tensor definition and index ordering before comparing individual signs.

Take a smooth family of nearby geodesics with tangent $u^\mu$ and infinitesimal separation $\xi^\mu$. Choose the family parameters so the tangent and separation vector fields commute. Geodesic deviation then relates the second covariant derivative of the separation to curvature. The separation must be small enough for the linear approximation; finite clouds acquire corrections involving curvature gradients and higher powers of their size.

In a slowly moving weak-field laboratory, this reduces to the Newtonian tidal equation $\ddot\xi^i=-\partial_i\partial_j\Phi\,\xi^j$. A spatially uniform gravitational acceleration has zero Hessian and creates no relative acceleration in this approximation. Around a spherical mass, radial separations stretch while transverse separations compress. An observer cannot remove all of these effects by accelerating the laboratory, because they compare more than one neighboring worldline.

Contract the first and third curvature indices to obtain $R_{\sigma\nu}=R^\rho{}_{\sigma\rho\nu}$, the Ricci tensor. Contract again with the inverse metric for the scalar R. In four dimensions the full Riemann tensor has information left after these contractions. Its trace-free Weyl part describes, among other things, tidal geometry in a vacuum region where the Ricci tensor vanishes when the cosmological constant is zero.

This explains why the exterior of an isolated spherical body can have $R_{\mu\nu}=0$ while neighboring falling bodies still separate tidally. Coordinate-independent scalar contractions can help identify genuine singular behavior, but a single scalar is not a complete test of every possible geometry. Operational measurements such as geodesic deviation retain information about the chosen observer and directions, while the underlying curvature tensor transforms consistently between frames.