Bianchi Identity and Stress-Energy Conservation

The contracted Bianchi identity makes the Einstein tensor divergence-free for every metric, independently of any field equation. Einstein’s equation therefore accepts only sources whose divergence vanishes. For dynamical matter, however, that vanishing is generally an on-shell consequence of the matter equations and diffeomorphism covariance—not a separate geometric proof of every matter equation and not, by itself, a global conserved energy.

Required background. We use the Riemann curvature tensor for the differential Bianchi identity, Hilbert stress–energy for the source response, and the scope of the equivalence principles to distinguish local covariance from a dynamical conservation proof.

Helpful background. Minkowski geometry supplies the flat limit, natural tensor operations clarify diffeomorphism covariance, and presymplectic potentials prepare the off-shell Noether identity.

From curvature to the Einstein tensor

For the Levi-Civita connection, the differential Bianchi identity is

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

Contracting one curvature index and then contracting once more with the metric gives

$$\nabla^\mu R_{\mu\nu}-\frac12\nabla_\nu R=0.$$

Metric compatibility then turns this into

$$\nabla^\mu G_{\mu\nu} =\nabla^\mu\left(R_{\mu\nu}-\frac12Rg_{\mu\nu}\right)=0.$$

This is an identity: it holds off shell, before imposing Einstein’s equation. Reall states the contracted identity and its curvature form in Reall 2013, § 6.6, printed p. 63, Eqs. (6.30)–(6.31), PDF.

Taking the divergence of

$$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G\,T_{\mu\nu}$$

therefore yields

$$8\pi G\,\nabla^\mu T_{\mu\nu}=0$$

provided $G$ and $\Lambda$ are constants. An externally prescribed tensor with nonzero divergence cannot source this equation: the coupled system would have no solution. This compatibility is why replacing $G_{\mu\nu}$ by $R_{\mu\nu}$ alone fails for general matter; the latter obeys $\nabla^\mu R_{\mu\nu}=\tfrac12\nabla_\nu R$, not zero. The standard argument is given in Carroll 1997, § 4, printed pp. 109–111, Eqs. (4.37)–(4.43), PDF.

Where the matter equation enters

For scalar matter obtained from a diffeomorphism-covariant action, the off-shell identity can be written exactly. Define

$$\mathcal E_A =\frac{1}{\sqrt{-g}}\frac{\delta S_m}{\delta\phi^A}, \qquad \delta S_m =\int d^Dx\,\sqrt{-g}\left( \mathcal E_A\,\delta\phi^A -\frac12T_{\mu\nu}\,\delta g^{\mu\nu} \right),$$

where repeated field labels are summed. Under an infinitesimal diffeomorphism generated by a compactly supported $\xi^\mu$,

$$\delta_\xi\phi^A=\xi^\rho\nabla_\rho\phi^A, \qquad \delta_\xi g^{\mu\nu}=-2\nabla^{(\mu}\xi^{\nu)}.$$

Integrating the metric term by parts and using the arbitrariness of $\xi^\nu$ gives

$$\nabla^\mu T_{\mu\nu} =\mathcal E_A\nabla_\nu\phi^A.$$

Thus $\nabla^\mu T_{\mu\nu}=0$ follows when the scalar equations $\mathcal E_A=0$ hold. Reall gives the one-scalar derivation, using the equivalent covariant-metric variation, in Reall 2013, § 10.3, printed pp. 120–121, Eqs. (10.35)–(10.40), PDF. For tensor or gauge fields, Lie derivatives also contain derivatives of $\xi^\mu$, so the general off-shell identity involves the formal adjoint of the Lie-derivative generator acting on all Euler–Lagrange expressions; it is not generally the simple algebraic contraction displayed for scalars. Background structures held fixed can instead supply a force density, in which case only the stress tensor of the complete dynamical system is conserved.

A minimally coupled real scalar makes the distinction explicit. For

$$T_{\mu\nu} =\nabla_\mu\phi\nabla_\nu\phi -\frac12g_{\mu\nu} \left(\nabla_\rho\phi\nabla^\rho\phi+m^2\phi^2\right),$$

direct differentiation, commutation of covariant derivatives on a scalar, and the Leibniz rule give

$$\nabla^\mu T_{\mu\nu} =\left(\Box\phi-m^2\phi\right)\nabla_\nu\phi.$$

The Klein–Gordon equation makes the divergence vanish. The converse need not hold where $\nabla_\nu\phi=0$, so conservation alone is not logically identical to the scalar equation. For example, a nonzero constant field $\phi=\phi_0$ with $m\ne0$ has the conserved tensor $T_{\mu\nu}=-\tfrac12m^2\phi_0^2g_{\mu\nu}$, but fails $\Box\phi-m^2\phi=0$. It cannot be accepted as a solution of the Einstein–scalar theory merely because its stress tensor is compatible with Bianchi conservation. In the coupled system, the geometric identity and the matter Noether identity fit together; neither should be described as an extra independent evolution equation.

A pressureless fluid gives a complementary example. With $T^{\mu\nu}=\rho u^\mu u^\nu$ and $u^\mu u_\mu=-1$,

$$\nabla_\mu T^{\mu\nu} =u^\nu\nabla_\mu(\rho u^\mu) +\rho u^\mu\nabla_\mu u^\nu.$$

Contracting with $u_\nu$ yields the continuity equation $\nabla_\mu(\rho u^\mu)=0$. Projecting orthogonally to $u^\nu$ then gives $u^\mu\nabla_\mu u^\nu=0$ wherever $\rho\ne0$: ideal dust elements follow geodesics. For a fluid with pressure, the orthogonal projection instead gives the relativistic Euler equation, so pressure gradients accelerate the flow. Conservation encodes matter motion only after the constitutive form of $T^{\mu\nu}$ has been supplied.

This locates a common overstatement. Einstein’s equation plus Bianchi consistency can imply combinations of matter equations when the chosen stress tensor is sufficiently nondegenerate, but it does not create a missing matter model. Distinct fields can have the same stress tensor on a configuration, and the four divergence equations cannot generally replace all Euler–Lagrange equations of a multiplet.

Local balance is not automatically a global charge

Covariant divergence includes connection terms and expresses local energy–momentum balance. It does not provide a coordinate-independent integral of “total energy at time $t$” on an arbitrary spacetime. Such an integral requires additional geometric and boundary structure.

For example, if $\xi^\mu$ is a Killing field and $T_{\mu\nu}$ is symmetric and conserved, then

$$J^\mu=T^{\mu\nu}\xi_\nu, \qquad \nabla_\mu J^\mu=0.$$

Only after choosing suitable hypersurfaces and controlling boundary flux does this current yield a conserved charge. Moreover, $T_{\mu\nu}$ describes matter; it is not a generally covariant local stress tensor for the gravitational field.

Integrating $\nabla_\mu J^\mu=0$ over a finite spacetime region and applying Stokes’ theorem equates the charge difference on two hypersurfaces to flux through the remaining boundary. A time-independent charge requires the side flux to vanish or be accounted for. On a generic spacetime without a suitable Killing field, stress–energy conservation still holds locally, but there is no canonical current $T^{\mu\nu}\xi_\nu$ or preferred slicing from which to build a unique global energy.

In the flat-spacetime limit, Cartesian connection coefficients vanish and the covariant equation becomes $\partial_\mu T^{\mu\nu}=0$. Its time and spatial components recover the familiar local energy and momentum balance laws. This limiting case checks the curved-spacetime interpretation without erasing the additional structure needed for global charges.

Exercises

Prove the Killing-current statement and identify exactly where symmetry of $T_{\mu\nu}$ is used.

Solution

Apply the product rule:

$$\nabla_\mu J^\mu =(\nabla_\mu T^{\mu\nu})\xi_\nu +T^{\mu\nu}\nabla_\mu\xi_\nu.$$

The first term vanishes by stress–energy conservation. Symmetry gives

$$T^{\mu\nu}\nabla_\mu\xi_\nu =\frac12T^{\mu\nu}(\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu),$$

which vanishes by Killing’s equation. Hence $\nabla_\mu J^\mu=0$. Without symmetric $T^{\mu\nu}$, its antisymmetric part could contract the antisymmetric derivative of $\xi_\nu$.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv record.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013), §§ 6.6–6.7, printed pp. 63–64, and § 10.3, printed pp. 120–121. PDF.