---
title: Curvature and the Riemann Tensor
module: Curved Spacetime
moduleNumber: 5
lessonNumber: 5
order: 505
summary: >
  Curvature is the failure of parallel transport to commute: carrying a vector
  around an infinitesimal loop returns it rotated, and the rotation per unit area
  is the Riemann tensor. Its symmetries cut the components to twenty in four
  dimensions. Geodesic deviation makes it the equation of tidal forces, and its
  contractions — the Ricci tensor, the Ricci scalar, and the divergence-free
  Einstein tensor — assemble the objects the field equation is built from.
topics: [Curved Spacetime]
draft: false
sources:
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §3 — Curvature"
  - book: Hartle
    ref: "Gravity, Ch. 21 — Curvature and the Einstein Equation"
---

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 $V^\rho$,

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

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,

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

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 $\d a^\mu$ and $\d
b^\nu$; it returns changed by

$$
\Delta V^\rho = -\,R^\rho{}_{\sigma\mu\nu}\,V^\sigma\,\d a^\mu\,\d b^\nu.
$$

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.

$$
% caption: Parallel transport around an infinitesimal loop returns a vector rotated
% by an amount proportional to the enclosed area times the Riemann tensor. The
% rotation per unit area is the operational measure of curvature.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % loop parallelogram
  \draw[black, thick] (0,0) -- (2.4,0.3) -- (2.9,2.1) -- (0.5,1.8) -- cycle;
  \node[black, anchor=north] at (1.2,0.05) {loop};
  % start vector
  \draw[->, acc, very thick] (0,0) -- (0.9,0.35);
  \node[acc, anchor=north] at (0.55,0.4) {start};
  % returned vector (rotated) at same corner, slightly turned
  \draw[->, black, very thick] (0,0) -- (0.75,0.6);
  \node[black, anchor=west] at (0.7,0.75) {returned};
  % area label
  \node[black, anchor=center] at (1.6,1.1) {area A};
  \node[black, anchor=west] at (3.1,1.1) {rotation = A over a squared};
\end{tikzpicture}
$$

## 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 $\binom{n}{2}$ values; pair symmetry makes the tensor a symmetric
matrix on that pair-index, giving $\tfrac{1}{2}m(m+1)$ with $m = \binom{n}{2}$; the
cyclic identity removes $\binom{n}{4}$ more. The count of independent components
is

$$
N(n) = \frac{n^2(n^2 - 1)}{12}.
$$

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,

$$
\nabla_{[\lambda} R_{\rho\sigma]\mu\nu} = 0,
$$

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

$$
% caption: The symmetries of the Riemann tensor reduce its components stage by
% stage. In four dimensions the naive 256 collapse to 20 independent functions;
% the antisymmetries, pair symmetry, and cyclic identity each remove a block.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \foreach \x/\lab/\val in {0/all indices/256, 1.9/antisymmetry/36, 3.8/pair symmetry/21, 5.7/cyclic/20} {
    \draw[black, thick] (\x,0) rectangle (\x+1.6,3.0);
  }
  % bars of shrinking height (log-ish visual)
  \fill (0,0) rectangle (1.6,3.0);
  \fill (1.9,0) rectangle (3.5,1.7);
  \fill (3.8,0) rectangle (5.4,1.05);
  \fill (5.7,0) rectangle (7.3,1.0);
  \node[black, anchor=south] at (0.8,3.0) {256};
  \node[black, anchor=south] at (2.7,1.7) {36};
  \node[black, anchor=south] at (4.6,1.05) {21};
  \node[black, anchor=south] at (6.5,1.0) {20};
  \node[black, anchor=north] at (0.8,-0.1) {raw};
  \node[black, anchor=north] at (2.7,-0.1) {antisym};
  \node[black, anchor=north] at (4.6,-0.1) {pair};
  \node[black, anchor=north] at (6.5,-0.1) {cyclic};
\end{tikzpicture}
$$

## 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**,

$$
\frac{D^2 \xi^\mu}{\d\tau^2} = -\,R^\mu{}_{\nu\rho\sigma}\,u^\nu \xi^\rho u^\sigma,
$$

where $D/\d\tau = u^\nu\nabla_\nu$ 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,

$$
\frac{\d^2 \xi^i}{\d t^2} = -\,\partial_i\partial_j\Phi\;\xi^j,
$$

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 $g_{00}$.

$$
% caption: Two nearby geodesics, initially parallel, converge or diverge because
% of curvature; the second derivative of their separation is the Riemann tensor
% contracted with the four-velocity. This relative acceleration is the tidal force,
% the frame-independent signature of gravity.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,-0.2) -- (0,3.6) node[above] {time};
  % two geodesics converging
  \draw[acc, very thick] (1.2,0) .. controls (1.3,1.3) and (1.8,2.4) .. (2.2,3.3);
  \draw[acc, very thick] (3.2,0) .. controls (3.0,1.3) and (2.7,2.4) .. (2.6,3.3);
  \node[acc, anchor=north] at (1.2,-0.05) {geodesic};
  \node[acc, anchor=north] at (3.2,-0.05) {geodesic};
  % separation vectors at increasing times
  \draw[<->, black] (1.25,0.65) -- (3.1,0.65);
  \draw[<->, black] (1.55,1.9) -- (2.85,1.9);
  \draw[<->, black] (2.25,3.15) -- (2.55,3.15);
  \node[black, anchor=south] at (2.2,0.65) {separation};
  \node[black, anchor=west] at (3.2,2.4) {shrinks: tidal};
\end{tikzpicture}
$$

## 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,

$$
R_{\mu\nu} = R^\lambda{}_{\mu\lambda\nu},
$$

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

$$
R = g^{\mu\nu} R_{\mu\nu},
$$

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,

$$
\nabla_\mu\Big(R^{\mu\nu} - \tfrac{1}{2}g^{\mu\nu}R\Big) = 0.
$$

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

$$
G_{\mu\nu} = R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu}R,
\qquad
\nabla_\mu G^{\mu\nu} = 0.
$$

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.

$$
% caption: The Riemann tensor contracts to the Ricci tensor and then the Ricci
% scalar; the combination that is automatically divergence-free is the Einstein
% tensor. The trace-free remainder is the Weyl tensor, carrying vacuum tidal and
% radiative curvature.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, thick] (0,1.0) rectangle (2.2,2.2);
  \node[black, anchor=center] at (1.1,1.6) {Riemann};
  \draw[->, black] (2.25,1.6) -- (3.15,1.6);
  \node[black, anchor=south] at (2.7,1.6) {trace};
  \draw[black, thick] (3.2,1.0) rectangle (5.2,2.2);
  \node[black, anchor=center] at (4.2,1.6) {Ricci tensor};
  \draw[->, black] (5.25,1.6) -- (6.15,1.6);
  \node[black, anchor=south] at (5.7,1.6) {trace};
  \draw[black, thick] (6.2,1.0) rectangle (8.0,2.2);
  \node[black, anchor=center] at (7.1,1.6) {Ricci scalar};
  % Einstein tensor below
  \draw[acc, thick, fill=acc!10] (3.2,-0.7) rectangle (5.2,0.5);
  \node[acc, anchor=center] at (4.2,-0.1) {Einstein tensor};
  \draw[->, black] (4.2,0.95) -- (4.2,0.55);
  \node[black, anchor=west] at (5.3,-0.1) {divergence free};
  % Weyl remainder
  \draw[black, dashed] (0,-0.7) rectangle (2.2,0.5);
  \node[black, anchor=center] at (1.1,-0.1) {Weyl (trace free)};
  \draw[->, black, dashed] (1.1,0.95) -- (1.1,0.55);
\end{tikzpicture}
$$

> **Worked example.** On the 2-sphere of radius $a$ the single independent Riemann
> component is $R_{\theta\phi\theta\phi} = a^2\sin^2\theta$, so
> $R^\theta{}_{\phi\theta\phi} = \sin^2\theta$. The Ricci tensor is
> $R_{\mu\nu} = a^{-2} g_{\mu\nu}$ (a maximally symmetric space), and the Ricci
> scalar is $R = 2/a^2$, positive and constant. A larger sphere curves less, and
> the flat plane is the $a \to \infty$ limit with all curvature vanishing. This
> single number is what the Riemann tensor reduces to in two dimensions, matching
> the component count $N(2) = 1$.

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.

[^carroll-riem]: **Carroll**, _Lecture Notes on General Relativity_, §3 — the Riemann tensor from commuting covariant derivatives, its symmetries and independent-component count, geodesic deviation, and the Ricci and Einstein tensors with the Bianchi identities.

[^hartle-riem]: **Hartle**, _Gravity_, Ch. 21 — curvature and tidal forces, the Riemann tensor and its relation to the Newtonian tidal tensor, and the contractions leading to the Einstein tensor.