Second (Differential) Bianchi Identity

The Bianchi identity is load-bearing in the computations that produced modern gravitational-wave astronomy. In numerical relativity — which simulates colliding black holes to generate the waveform templates LIGO and Virgo match against their data — the ten Einstein equations split into six evolution equations and four constraints (the Hamiltonian and momentum constraints) that must hold on every spatial slice. The twice-contracted Bianchi identity is precisely the statement that if the constraints hold on an initial slice, the evolution preserves them: constraint violation propagates as a wave that, analytically, stays zero. Real simulations run in finite precision, so tiny violations creep in and can blow up; entire formulations (BSSN, generalized harmonic gauge) are engineered so that the Bianchi-guaranteed constraint propagation stays controlled rather than amplified — BSSN by making the evolution system strongly hyperbolic, while generalized harmonic gauge and the Z4c/CCZ4 variants add explicit constraint-damping terms. Without that structural identity there would be no stable long-term binary-black-hole evolutions and no template banks — and GW150914 in 2015 would have had nothing to be compared against. The same identity, in vacuum form $\nabla^\mu C_{\mu\nu\rho\sigma}=0$, lets the emitted curvature be decomposed into Weyl scalars and read off as the two polarizations of a gravitational wave at infinity.

Parallel-transport a vector around a small closed loop and it comes back rotated; the rotation, per unit enclosed area, is the curvature threading the loop. Now build a small three-dimensional cell — think of a tiny cube — out of six such faces. Transport the vector around the full boundary of that cube by traversing each face loop in turn. Because the cube is a closed solid, every internal edge is shared by exactly two faces and is walked once in each direction, so those contributions cancel in pairs. The total rotation around the boundary of the boundary must therefore be zero. Take that cancellation to the infinitesimal limit and it becomes a statement about how curvature changes from face to face — its covariant derivative — summed cyclically over the three edge-directions of the cell. That statement is the second Bianchi identity. Nothing about a specific spacetime enters; the identity is forced by the mere fact that curvature is built from a connection.

Read the same fact through fields and it becomes a kinship with electromagnetism. Curvature is the field strength of the connection, exactly as $F_{\mu\nu}$ is the field strength of the electromagnetic potential. The differential Bianchi identity says this field strength has no "source" in its antisymmetrized derivative — the gravitational analogue of $\nabla\cdot\mathbf{B}=0$ and $\partial_{[\mu}F_{\nu\rho]}=0$, the half of Maxwell's equations that follows from $F=\mathrm{d}A$ with no charges anywhere. And just as those source-free equations let the other half be sourced consistently by a conserved current, the Bianchi identity lets Einstein's equations be sourced consistently by a conserved stress-energy tensor. The whole payoff is one chain of contractions, shown below.

The identity is best thought of as an integrability condition: it is exactly what is required for a candidate curvature to be the curvature of some connection, and it keeps the second-order field equations from over-determining the ten metric components. The four relations packaged in $\nabla^\mu G_{\mu\nu}=0$ remove four would-be dynamical equations, leaving six — precisely matched to the four-function freedom of choosing coordinates. Without it there would be no consistent theory of gravity of Einstein's form at all.

The identity has one content and several faces. Its most economical form packs the whole cyclic structure into an antisymmetrization bracket: the covariant derivative of Riemann, totally antisymmetrized over the derivative index and the first curvature pair, vanishes. The square bracket denotes the normalized sum over all orderings with signs, and the antisymmetry of Riemann in that first pair is what lets three indices, not four, carry the cycle.

where: $R_{\mu\nu\rho\sigma}$ — Riemann curvature tensor, fully covariant; $\nabla_\lambda$ — Levi-Civita (metric-compatible, torsion-free) covariant derivative; $[\,\lambda\mu\nu\,]$ — normalized total antisymmetrization over the three enclosed indices, the two curvature indices $\mu,\nu$ plus the derivative index $\lambda$. Physical reading: the rate of change of curvature, wrung of all its symmetric part across three directions, has nothing left over. Because Riemann is antisymmetric in its first pair, the $3!=6$ signed terms in the bracket collapse into three, and the identity wears its more recognizable cyclic form. This is the shape that makes the "Jacobi identity" fingerprint visible: the derivative index and the first pair of curvature indices march around a cycle.

where: the three terms are the cyclic permutations of $\lambda,\mu,\nu$ with the second pair $\rho\sigma$ held fixed. Physical reading: sum the covariant gradient of curvature around a closed cycle of three directions and it cancels, exactly as the holonomies around the faces of a cell cancel on the shared edges. The same identity, expressed in Cartan's language of curvature two-forms, is a single clean line: the exterior covariant derivative of the curvature form vanishes. This form makes the parallel with gauge theory exact and is the version that connects the identity to characteristic classes and topology.

where: $\mathcal{R}^a{}_b=\tfrac12 R^a{}_{b\mu\nu}\,\mathrm{d}x^\mu\wedge\mathrm{d}x^\nu$ — curvature two-form; $\omega^a{}_b$ — connection one-form; $\mathrm{D}$ — exterior covariant derivative. Physical reading: curvature is a "closed" object under the covariant exterior derivative, the direct analogue of $\mathrm{d}F=0$ for the abelian field strength and $\mathrm{D}F=0$ in Yang–Mills theory. The physics begins at the first contraction. Tracing the identity over one index of the derivative-cycle with one of the fixed pair yields a relation between the divergence of Riemann and the antisymmetrized gradient of the Ricci tensor. It already looks Maxwell-like, and in vacuum it becomes precisely a wave equation for curvature.

where: $R_{\nu\sigma}=R^\mu{}_{\nu\mu\sigma}$ — Ricci tensor; $\nabla^\mu=g^{\mu\lambda}\nabla_\lambda$ — the divergence acting on the first index. Physical reading: the divergence of the full curvature is not free but is fixed by how the Ricci tensor varies — the "source" for the divergence of Riemann is the gradient of its own trace. A second trace closes the loop. Contracting equation (4) over the remaining free pair produces a relation between the divergence of the Ricci tensor and the gradient of the Ricci scalar — the twice-contracted Bianchi identity, and the linchpin of everything that follows.

where: $R=g^{\nu\sigma}R_{\nu\sigma}$ — Ricci scalar. Physical reading: the Ricci tensor is not divergence-free — its divergence equals half the gradient of the scalar curvature — which is exactly why $R_{\mu\nu}=\kappa T_{\mu\nu}$ cannot be the field equation, and why a corrected object is needed. That corrected object is the Einstein tensor, and the factor $\tfrac12$ in (5) is precisely what its trace-reversed combination is built to cancel. The result is the identity that gives general relativity its consistency.

where: $G_{\mu\nu}$ — Einstein tensor, the unique symmetric, divergence-free tensor built from the metric and its first two derivatives that is linear in the second derivatives. Physical reading: the geometric side of the field equations conserves itself automatically, so imposing $G_{\mu\nu}=\kappa T_{\mu\nu}$ makes $\nabla^\mu T_{\mu\nu}=0$ a corollary rather than an extra law. Finally, feeding the Ricci decomposition of Riemann back into the once-contracted identity isolates the divergence of the Weyl (conformal) tensor, with the Ricci and scalar pieces playing the role of sources. In four dimensions the coefficients are fixed.

where: $C_{\mu\nu\rho\sigma}$ — Weyl conformal tensor, the totally trace-free part of Riemann. Physical reading: the divergence of the free, tidal part of curvature is sourced by the local matter (through Ricci); in vacuum the right-hand side vanishes and $\nabla^\mu C_{\mu\nu\rho\sigma}=0$ governs how tidal curvature — gravitational radiation — propagates.

How the identity specializes or generalizes across the settings that matter most:

1 · Schur's theorem: isotropy forces constant curvature

A spacetime is an "Einstein space" if its Ricci tensor is proportional to the metric, $R_{\mu\nu}=f\,g_{\mu\nu}$ for some scalar function $f$ — the natural expression of local isotropy. Use the twice-contracted Bianchi identity to show that in dimension $n>2$ the function $f$ cannot vary from point to point, so the scalar curvature is constant. This is Schur's theorem, and it is the reason the maximally symmetric cosmological models have a genuinely constant curvature radius.

Step 1. Take the trace of $R_{\mu\nu}=f\,g_{\mu\nu}$ with $g^{\mu\nu}$. Since $g^{\mu\nu}g_{\mu\nu}=n$, this gives $R=n f$, so $f=R/n$ and the Einstein-space condition reads $R_{\mu\nu}=\tfrac{R}{n}g_{\mu\nu}$.

Step 2. Take the divergence of both sides with $\nabla^\mu$. On the right, metric compatibility lets $\nabla^\mu$ pass through $g_{\mu\nu}$, so $\nabla^\mu\!\left(\tfrac{R}{n}g_{\mu\nu}\right)=\tfrac1n\nabla_\nu R$. On the left, the twice-contracted Bianchi identity (5) gives $\nabla^\mu R_{\mu\nu}=\tfrac12\nabla_\nu R$. Equating: $\tfrac12\nabla_\nu R=\tfrac1n\nabla_\nu R$.

Step 3. Rearrange to $\left(\tfrac12-\tfrac1n\right)\nabla_\nu R=0$, i.e. $\dfrac{n-2}{2n}\,\nabla_\nu R=0$. For any dimension $n>2$ the prefactor is nonzero, so $\nabla_\nu R=0$ — the scalar curvature is constant. Feeding this back, $R_{\mu\nu}=\tfrac{R}{n}g_{\mu\nu}$ with $R$ constant, so the full Ricci tensor is covariantly constant too.

Step 4. Check the boundary case. In $n=2$ the prefactor $\tfrac{n-2}{2n}$ vanishes and the argument gives no constraint — consistent with the fact that every 2-surface is trivially an Einstein space with $R$ free to vary (e.g. a generic curved sheet). In our four-dimensional world $n=4$ gives $\tfrac{n-2}{2n}=\tfrac14\neq0$, so isotropy of the Ricci tensor at every point of a 4-spacetime rigidly implies constant $R$ — the mathematical backbone of the de Sitter and anti–de Sitter geometries.

2 · The Bianchi identity delivers the cosmological continuity equation

In a spatially flat Friedmann–Lemaître–Robertson–Walker universe with metric $\mathrm{d}s^2=-\mathrm{d}t^2+a(t)^2\,\delta_{ij}\mathrm{d}x^i\mathrm{d}x^j$ and a perfect fluid $T^\mu{}_\nu=\mathrm{diag}(-\rho,p,p,p)$, show that $\nabla^\mu G_{\mu\nu}=0$ forces $\dot\rho+3H(\rho+p)=0$ with $H=\dot a/a$. Then read off how matter and radiation dilute with expansion.

Step 1. Because $\nabla^\mu G_{\mu\nu}=0$ is an identity, the field equations $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ immediately give $\nabla^\mu T_{\mu\nu}=0$ with no further work — the conservation of the source is inherited from the geometry. We only have to evaluate the $\nu=0$ (time) component of $\nabla_\mu T^\mu{}_0=0$.

Step 2. Expand the covariant divergence: $\nabla_\mu T^\mu{}_0=\partial_\mu T^\mu{}_0+\Gamma^\mu{}_{\mu\lambda}T^\lambda{}_0-\Gamma^\lambda{}_{\mu0}T^\mu{}_\lambda$. For flat FLRW the needed Christoffels are $\Gamma^i{}_{j0}=H\,\delta^i{}_j$ and $\Gamma^\mu{}_{\mu 0}=\partial_0\ln\sqrt{-g}=3H$, since $\sqrt{-g}=a^3$.

Step 3. Insert $T^0{}_0=-\rho$, $T^i{}_j=p\,\delta^i{}_j$: the time derivative gives $-\dot\rho$, the trace term gives $3H(-\rho)$, and the last term gives $-3Hp$. Collecting, $\nabla_\mu T^\mu{}_0=-\dot\rho-3H\rho-3Hp=0$.

Step 4. Flip the sign to reach the standard form: $\dot\rho+3H(\rho+p)=0$ — the cosmological continuity equation, obtained here purely from the Bianchi identity plus the field equations, with no independent conservation postulate.

Step 5. Read off the dilution laws with an equation of state $p=w\rho$. Then $\dot\rho/\rho=-3(1+w)\dot a/a$ integrates to $\rho\propto a^{-3(1+w)}$. Pressureless matter ($w=0$) dilutes as $\rho\propto a^{-3}$ — pure volume dilution. Radiation ($w=\tfrac13$) dilutes as $\rho\propto a^{-4}$ — an extra factor of $a$ from the cosmological redshift of each quantum. A cosmological constant ($w=-1$) gives $\rho=\text{const}$ — vacuum energy that does not dilute at all. Every one of these familiar scalings is, at root, a consequence of $\nabla^\mu G_{\mu\nu}=0$.

The identity carries the name of a geometer who was not the first to find it, in a story typical of nineteenth-century differential geometry — where the same structural truth was written down repeatedly before its physical weight was understood.

References. Misner, Thorne & Wheeler, Gravitation, §§13.5, 15.3 (the boundary of a boundary) · Wald, General Relativity, §3.2 · Carroll, Spacetime and Geometry, §3.6 · Hartle, Gravity, §21.2 · Schutz, A First Course in General Relativity, §6.5 · Nakahara, Geometry, Topology and Physics, §7.4 (differential-forms version). Several internal consistencies are worth noting. The antisymmetrized double partial of the connection vanishes in normal coordinates, which is the heart of the proof of $\nabla_{[\lambda}R_{\mu\nu]\rho\sigma}=0$. In the double contraction to $\nabla^\mu G_{\mu\nu}=0$, the factor $\tfrac12$ tracks the metric trace $g^{\sigma\nu}g_{\sigma\nu}=n$, while the cancellation itself is dimension-independent. The Weyl-divergence relation (7) is trace-free under $g^{\nu\rho}$ in $n=4$, and it is this trace-freedom that fixes the coefficients $\tfrac12$ and $\tfrac16$ appearing there. Schur's theorem follows from the prefactor $\tfrac{n-2}{2n}\nabla_\nu R=0$, and the FLRW continuity equation $\dot\rho+3H(\rho+p)=0$ yields the dilution laws $\rho\propto a^{-3(1+w)}$. No numerical constants beyond standard tensor traces enter any of these results.

Prerequisites. Riemann Curvature Tensor · Curvature as the Commutator of Covariant Derivatives · Algebraic Symmetries of the Riemann Tensor · First (Algebraic) Bianchi Identity. Leads to. Contracted Bianchi Identities · Einstein Tensor · Weyl Conformal Tensor · Ricci Decomposition of the Riemann Tensor.

Ask the tutor: Ask me to walk through the double contraction index by index, or to show exactly how the four contracted Bianchi identities become the constraint equations of the initial-value problem.