Curvature and the Riemann Tensor

The equivalence principle removed gravity locally but left a residue no frame could erase: the tidal field, the relative acceleration of neighbouring freely falling particles. That residue is curvature. This lesson defines curvature as the non-commuting of covariant derivatives, packages it in the Riemann tensor, reduces its independent components using its symmetries, connects it to observable tidal forces through the equation of geodesic deviation, and forms the contractions — Ricci tensor, Ricci scalar, Einstein tensor — that source the field equation of the next lesson.

Curvature from non-commuting derivatives

On a flat manifold the components of a constant vector are the same everywhere and parallel transport is path-independent. On a curved manifold it is not, and the cleanest measure is the commutator of two covariant derivatives. Acting on a vector $V^{\rho }$,

For the torsion-free Levi-Civita connection the last term vanishes, and the commutator is a pure algebraic (non-derivative) operation on $V$: the result at a point depends only on $V$ there, not on its derivatives. The coefficient is the Riemann curvature tensor, built from the Christoffel symbols and their first derivatives,

That $R$ is a genuine tensor, despite being assembled from the non-tensorial $\Gamma$'s, follows because the commutator of tensor operations is a tensor. Its vanishing is the invariant criterion for flatness: $R^{\rho }{}_{\sigma \mu \nu }=0$ everywhere if and only if coordinates exist in which $g_{\mu \nu }={\eta }_{\mu \nu }$ throughout. This settles the ambiguity of the Rindler metric — nonconstant metric, yet every Riemann component zero, hence flat — against the sphere, whose Riemann tensor is nonzero.

Curvature as loop holonomy

The geometric meaning is parallel transport around a closed loop. Carry a vector around an infinitesimal parallelogram spanned by displacements $\mathbf{d}{a}^{\mu }$ and $\mathbf{d}{b}^{\nu }$; it returns changed by

The change is first order in the enclosed area and vanishes as the loop shrinks, consistent with local flatness, while its rate per unit area is the Riemann tensor. For the 2-sphere of radius $a$, transporting a vector around a loop enclosing area $A$ rotates it by $A/a^2$, so the curvature scale is $1/a^2$; a larger sphere is flatter, and the plane ($a\to \infty$) has zero curvature. Holonomy is the operational definition — a gyroscope carried around a loop in a curved spacetime returns pointing in a measurably different direction, the basis of the geodetic-precession test.

Symmetries and component counting

The Riemann tensor with all indices lowered, $R_{\rho \sigma \mu \nu }=g_{\rho \lambda }{R}^{\lambda }{}_{\sigma \mu \nu }$, obeys a set of algebraic identities that drastically cut its independent components.

• Antisymmetry in the last pair: $R_{\rho \sigma \mu \nu }=-R_{\rho \sigma \nu \mu }$.
• Antisymmetry in the first pair: $R_{\rho \sigma \mu \nu }=-R_{\sigma \rho \mu \nu }$.
• Pair symmetry: $R_{\rho \sigma \mu \nu }=R_{\mu \nu \rho \sigma }$.
• First Bianchi identity (cyclic): $R_{\rho [\sigma \mu \nu ]}=0$, i.e. $R_{\rho \sigma \mu \nu }+R_{\rho \mu \nu \sigma }+R_{\rho \nu \sigma \mu }=0$.

The first two make $R$ antisymmetric within each of the two index pairs, so each pair ranges over $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ values; pair symmetry makes the tensor a symmetric matrix on that pair-index, giving $\frac{1}{2}m(m+1)$ with $m=\left(\genfrac{}{}{0ex}{}{n}{2}\right)$; the cyclic identity removes $\left(\genfrac{}{}{0ex}{}{n}{4}\right)$ more. The count of independent components is

In $n=2$ this is $1$ (a single number, the Gaussian curvature); in $n=3$ it is $6$; in $n=4$ it is $20$. Twenty functions carry the full curvature of spacetime. There is also a second Bianchi identity, a differential one,

which is not a counting statement but the source of the conservation law that makes the Einstein equation consistent, as the contraction below shows.

Geodesic deviation and tidal forces

Curvature becomes physical through the relative motion of nearby geodesics. Take a one-parameter family of geodesics and let ${\xi }^{\mu }$ be the separation vector joining a point on one to the corresponding point on its neighbour. The separation does not stay constant; its second derivative along the geodesics is the equation of geodesic deviation,

where $D/\mathbf{d}\tau =u^{\nu }{\nabla }_{\nu }$ is the covariant derivative along the geodesic and $u^{\mu }$ is the four-velocity. Two freely falling particles at rest relative to one another do not stay at rest if the Riemann tensor is nonzero: they accelerate toward or away from each other. This relative acceleration is the tidal force, and it is coordinate-independent — unlike the acceleration of a single particle, which a change of frame can null. The residue the equivalence principle could not remove is $R^{\mu }{}_{\nu \rho \sigma }$ itself.

The correspondence with Newtonian tides is exact in the weak-field limit. The Newtonian tidal equation derived from the equivalence-principle lesson,

matches the geodesic-deviation equation with $R^i{}_{0j0}\to {\partial }_i{\partial }_j\Phi /c^2$ in the appropriate limit. The Newtonian tidal tensor ${\partial }_i{\partial }_j\Phi$ is the slow-field shadow of the Riemann tensor, precisely as the potential $\Phi$ was the shadow of $g_{00}$.

Ricci tensor, scalar, and the Einstein tensor

The field equation does not use the full twenty-component Riemann tensor as its source term; it uses its contractions. The Ricci tensor is the trace on the first and third indices,

symmetric ($R_{\mu \nu }=R_{\nu \mu }$) with ten independent components in four dimensions. Its further trace is the Ricci scalar (scalar curvature),

a single function measuring average curvature at a point. In two dimensions $R=2/a^2$ on the sphere, twice the Gaussian curvature. Contracting the second Bianchi identity twice yields the key algebraic fact,

The symmetric, divergence-free tensor in parentheses is the Einstein tensor,

Its vanishing divergence is not imposed but follows identically from the Bianchi identity, and it supplies just the property a source term for a conserved stress–energy tensor requires. The Ricci tensor and scalar discard part of the Riemann tensor; the remainder, the trace-free part, is the Weyl tensor, which carries the tidal and radiative curvature that survives in vacuum where $R_{\mu \nu }=0$. Gravitational waves and the field outside a star live in the Weyl tensor, while the Ricci part is tied directly to local matter by the field equation.

The Einstein tensor is symmetric, built from the metric and its first two derivatives, and divergence-free by construction — the exact profile required of the geometric side of a field equation whose source is the conserved stress–energy tensor. Assembling that equation, matching its weak-field limit to Newtonian gravity, and adding the cosmological constant is the work of the final lesson.