Curvature of Spacetime

Curvature is the irreducible, coordinate-free content of gravity. The equivalence principle removes the field at a point but leaves the tidal effects; the metric can be flattened to first order but not to second; parallel transport around a loop returns a vector rotated. All three are the same object — the Riemann curvature tensor $R^{\rho }{}_{\sigma \mu \nu }$. This page builds it, reads off its physical meaning through geodesic deviation, and contracts it into the Ricci and Einstein tensors that source the field equations.

We use $c=1$, mostly-plus signature, and fix sign conventions below.

The Riemann tensor from the commutator of derivatives

On flat space, second partial derivatives commute. On a curved manifold, covariant derivatives do not — and the failure to commute is the curvature. Acting on a vector field,

$$[{\nabla }_{\mu },{\nabla }_{\nu }]V^{\rho }=R^{\rho }{}_{\sigma \mu \nu }{V}^{\sigma }-T^{\lambda }{}_{\mu \nu }{\nabla }_{\lambda }{V}^{\rho },$$

with $T^{\lambda }{}_{\mu \nu }=0$ for the Levi-Civita connection. This defines the Riemann curvature tensor, which works out to be built from the Christoffel symbols and their first derivatives:

$$\boxed{R^{\rho }{}_{\sigma \mu \nu }={\partial }_{\mu }{\Gamma }^{\rho }{}_{\nu \sigma }-{\partial }_{\nu }{\Gamma }^{\rho }{}_{\mu \sigma }+{\Gamma }^{\rho }{}_{\mu \lambda }{\Gamma }^{\lambda }{}_{\nu \sigma }-{\Gamma }^{\rho }{}_{\nu \lambda }{\Gamma }^{\lambda }{}_{\mu \sigma }.}$$

Since $\Gamma \sim \partial g$, we have $R\sim {\partial }^{2}g$: curvature is the second derivative of the metric — exactly the $20$ components that could not be removed by choice of local inertial frame. Crucially, although $\Gamma$ is not a tensor, the combination above is: $R^{\rho }{}_{\sigma \mu \nu }$ transforms tensorially, so "$R=0$" is a coordinate-free statement.

Sign convention. We follow MTW/Wald/Carroll: the definition above, with $R_{\rho \sigma \mu \nu }=g_{\rho \lambda }{R}^{\lambda }{}_{\sigma \mu \nu }$, the Ricci tensor as the $13$-contraction $R_{\mu \nu }=R^{\lambda }{}_{\mu \lambda \nu }$, and the Einstein equation carrying $+8\pi G$. With this choice the sphere has positive scalar curvature. (Weinberg flips the overall sign of $R$; always check a text's conventions.)

Flat ⟺ zero curvature

A central theorem: $R^{\rho }{}_{\sigma \mu \nu }=0$ everywhere on a region iff the metric can be brought to constant (Minkowski) form ${\eta }_{\mu \nu }$ by a coordinate change throughout that region. So curvature is the exact obstruction to being globally flat — the precise sense in which a genuine gravitational field (curvature $\ne 0$) cannot be transformed away, only localized-away point by point.

Symmetries and how many components

Lowering the first index, $R_{\rho \sigma \mu \nu }$ obeys:

• Antisymmetry in the first and last pairs: $R_{\rho \sigma \mu \nu }=-R_{\sigma \rho \mu \nu }=-R_{\rho \sigma \nu \mu }$;
• Pair symmetry: $R_{\rho \sigma \mu \nu }=R_{\mu \nu \rho \sigma }$;
• First (algebraic) Bianchi identity: $R_{\rho [\sigma \mu \nu ]}=0$, i.e. $R_{\rho \sigma \mu \nu }+R_{\rho \mu \nu \sigma }+R_{\rho \nu \sigma \mu }=0$;
• Second (differential) Bianchi identity: ${\nabla }_{[\lambda }{R}_{\rho \sigma ]\mu \nu }=0$.

In $n$ dimensions these leave $\frac{n^2(n^2-1)}{12}$ independent components: $1$ in 2D, $6$ in 3D, and $20$ in 4D — matching the count of unremovable second derivatives of the metric on the previous page. The differential Bianchi identity is the linchpin that makes the field equations consistent with conservation of energy–momentum.

Geodesic deviation: curvature as tidal force

The physical meaning of Riemann is tidal acceleration. Take a one-parameter family of nearby geodesics and let $S^{\mu }$ be the separation vector between two infinitesimally close ones, $u^{\mu }$ their tangent. Their relative acceleration obeys the geodesic deviation equation:

$$\frac{D^2{S}^{\mu }}{d{\tau }^2}=-R^{\mu }{}_{\alpha \nu \beta }{u}^{\alpha }{S}^{\nu }{u}^{\beta }.$$

Two freely falling particles, each feeling no force, nonetheless accelerate toward or away from each other — and the coefficient is the Riemann tensor. This is the exact, relativistic version of the tidal-force argument: a single freely falling frame removes the field, but the relative motion of neighboring free-fallers cannot be removed, and it measures curvature directly. In the Newtonian limit the equation reduces to ${\ddot{S}}^i=-({\partial }_i{\partial }_j\Phi )S^j$, so the components $R^i{}_{0j0}$ become the Newtonian tidal tensor ${\partial }_i{\partial }_j\Phi$. "Tidal force = curvature" is literally an equation.

Contractions: Ricci, scalar, Weyl, Einstein

The $20$-component Riemann tensor splits into pieces with distinct physical roles.

• Ricci tensor (a $(0,2)$, symmetric tensor, $10$ components): $$R_{\mu \nu }=R^{\lambda }{}_{\mu \lambda \nu }=g^{\alpha \beta }{R}_{\alpha \mu \beta \nu }.$$ It measures how the volume of a small ball of freely falling test particles changes: a nonzero $R_{\mu \nu }$ means the geodesic congruence focuses (or defocuses). This is the part of curvature directly sourced by local matter in the field equations.
• Ricci scalar (one function): $R=g^{\mu \nu }{R}_{\mu \nu }$. The single number summarizing curvature at a point; positive for a sphere, negative for a saddle.
• Weyl (conformal) tensor $C_{\rho \sigma \mu \nu }$ (the remaining $10$ components in 4D): the totally trace-free part of Riemann. It carries the tidal / gravitational-wave degrees of freedom that persist in vacuum, where $R_{\mu \nu }=0$ but $C\ne 0$ (the field of the Sun outside its surface, a passing gravitational wave). Schematically, $$R_{\rho \sigma \mu \nu }=\underset{\text{free field (vacuum tides, waves)}}{\underbrace{C_{\rho \sigma \mu \nu }}}+\underset{\text{local matter}}{\underbrace{(\text{Ricci terms})}}.$$
• Einstein tensor (symmetric, $10$ components): $$\boxed{G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }.}$$

The contracted Bianchi identity

Contracting the differential Bianchi identity twice yields the single most important consequence for dynamics:

$$\boxed{{\nabla }_{\mu }{G}^{\mu \nu }=0}\text{(identically, for any metric).}$$

The Einstein tensor is automatically divergence-free — a geometric identity, true of every spacetime, independent of any field equation. This is precisely what allows $G_{\mu \nu }$ to be set proportional to the conserved stress–energy tensor $T_{\mu \nu }$: the identity ${\nabla }_{\mu }{G}^{\mu \nu }=0$ forces ${\nabla }_{\mu }{T}^{\mu \nu }=0$, so local energy–momentum conservation is built into the geometry, not imposed by hand. It also singles out $G_{\mu \nu }$ (rather than $R_{\mu \nu }$, which is not divergence-free) as the correct left-hand side of the field equations.

Measures of curvature and singularities

Because $R_{\mu \nu }$ can vanish in vacuum, scalar invariants built from the full Riemann tensor are used to characterize a spacetime coordinate-independently — notably the Kretschmann scalar $K=R_{\rho \sigma \mu \nu }{R}^{\rho \sigma \mu \nu }$. Its divergence signals a genuine curvature singularity (e.g. $K\sim M^2/r^6\to \infty$ at the center of a black hole), as opposed to a mere coordinate breakdown (the event horizon, where $K$ is finite). Distinguishing physical singularities from coordinate artifacts is a recurring theme in the exact solutions.

Summary

• The Riemann tensor $R^{\rho }{}_{\sigma \mu \nu }$ is the commutator of covariant derivatives, $\sim {\partial }^{2}g$: the $20$ irreducible second derivatives of the metric, and the exact obstruction to flatness.
• Its symmetries (antisymmetry, pair symmetry, algebraic + differential Bianchi) cut $256$ components to $20$ in 4D.
• Geodesic deviation shows Riemann is the relativistic tidal tensor; the Newtonian limit gives $R^i{}_{0j0}\to {\partial }_i{\partial }_j\Phi$.
• Contractions give the Ricci tensor and scalar (matter-sourced, volume-focusing), the Weyl tensor (vacuum tides and waves), and the Einstein tensor $G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }$, which is divergence-free by the contracted Bianchi identity — guaranteeing energy–momentum conservation in the field equations.

Next

We now have the geometric left-hand side. The Einstein Field Equations set $G_{\mu \nu }$ proportional to the matter stress–energy tensor, derive the constant from the Newtonian limit, establish uniqueness, and read off the first solutions.