# Contracted Bianchi identity and energy–momentum conservation
The contracted Bianchi identity is the differential identity of [Riemannian geometry](https://www.edgepedia.org/riemannian-geometry) stating that the covariant divergence of the Ricci tensor equals half the gradient of the Ricci scalar, \( \nabla_{\beta} R^{\alpha\beta} = \tfrac{1}{2} g^{\alpha\beta} \nabla_{\beta} R \). In general relativity it is the reason the [Einstein tensor](https://www.edgepedia.org/einstein-tensor) is automatically divergence-free, and therefore the reason the [Einstein field equations](https://www.edgepedia.org/einstein-field-equations) enforce local conservation of energy and momentum, \( \nabla_{\nu} T^{\mu\nu} = 0 \), rather than leaving it as a separate postulate.[1](https://www.roe.ac.uk/japwww/teaching/gr/GR_notes_2021)
| Key fact | Detail |
|---|---|
| Identity | \( \nabla_{\beta} R^{\alpha\beta} = \tfrac{1}{2} g^{\alpha\beta} \nabla_{\beta} R \), the twice-contracted form of the second Bianchi identity[1](https://www.roe.ac.uk/japwww/teaching/gr/GR_notes_2021) |
| Consequence | The Einstein tensor \( G^{\mu\nu} = R^{\mu\nu} - \tfrac{1}{2} g^{\mu\nu} R \) satisfies \( \nabla_{\mu} G^{\mu\nu} = 0 \) identically[1](https://www.roe.ac.uk/japwww/teaching/gr/GR_notes_2021) |
| Field equations | \( G^{\mu\nu} = -\tfrac{8\pi G}{c^{4}} T^{\mu\nu} \) in the Edinburgh sign convention; the coupling constant is fixed by the Newtonian limit[1](https://www.roe.ac.uk/japwww/teaching/gr/GR_notes_2021) |
| Conservation | \( \nabla_{\nu} T^{\mu\nu} = 0 \) follows from the field equations plus the identity; it is not an independent assumption[7](https://nbow.io/en/products/ai-powered-research/treatise-of-physics/chapter/05-general-relativity-cosmology/03-einstein-field-equations.html) |
| Equation count | The identity turns the four \(tt\) and \(ti\) equations into initial-value constraints, leaving 6 of 10 equations as evolution equations[2](https://physics.umd.edu/grt/taj/675d/675cnotes.html) |
| Variational origin | Diffeomorphism invariance of the Einstein–Hilbert action plus the matter equations of motion implies \( \nabla_{\mu} T^{\mu\nu} = 0 \) (Noether's second theorem)[6](https://home.iiserb.ac.in/~suvankar/downloads/Reall.pdf) |
| Scope | The twice-contracted identity holds on any smooth manifold with a metric, independent of any field equation[8](https://arxiv.org/abs/1901.00813) |
## The identity itself
The second Bianchi identity of the Riemann tensor, \( R^{a}{}_{b[cd;e]} = 0 \), arises from the fact that mixed partial derivatives commute; it can be proved simply by evaluating in local inertial coordinates at a point. Contracting on two index pairs yields the contracted form \( R^{a}{}_{b;a} = \tfrac{1}{2} R_{,b} \), or in fully covariant notation \( \nabla_{\beta} R^{\alpha\beta} = \tfrac{1}{2} \nabla^{\alpha} R \).[2](https://physics.umd.edu/grt/taj/675d/675cnotes.html) The Edinburgh lecture notes state the same result as \( \nabla_{\beta} R^{\alpha\beta} = g^{\alpha\beta} \nabla_{\beta} R / 2 \).[1](https://www.roe.ac.uk/japwww/teaching/gr/GR_notes_2021)
**Geometric, not dynamical.** The twice-contracted second Bianchi identity is a differential curvature identity that holds on any smooth manifold equipped with a metric; it does not require the Einstein equations to hold. When a Lorentzian metric does solve Einstein's equations with a matter stress-energy tensor as source, the identity then implies the physical laws of energy and momentum conservation for that matter.[8](https://arxiv.org/abs/1901.00813)
## Why the naive field equation fails
In building the field equations, the stress-energy tensor \( T^{\mu\nu} \) replaces mass density \( \rho \) as the source, so a second-rank tensor involving derivatives of the metric is needed on the geometric side. The obvious candidate, making the Ricci tensor proportional to the stress-energy tensor, \( R^{\alpha\beta} = \text{constant} \cdot T^{\alpha\beta} \), fails on consistency grounds: the contracted Bianchi identity gives \( \nabla_{\beta} R^{\alpha\beta} = \tfrac{1}{2} \nabla^{\alpha} R \neq 0 \) in general, while conservation of the stress-energy tensor requires \( \nabla_{\beta} T^{\alpha\beta} = 0 \).[1](https://www.roe.ac.uk/japwww/teaching/gr/GR_notes_2021)
The UMD notes show the same obstruction in a slightly different way. Taking the covariant divergence of a proposed relation \( R_{mn} = 4\pi[(1-x)T_{mn} + x T g_{mn}] \) and using both stress-tensor conservation and \( R^{mn}{}_{;n} = \tfrac{1}{2} R^{,n} \), the equation is consistent only for \( x = -1 \), which forces the coefficient combination \( 2T_{mn} - T g_{mn} \) and hence the Einstein tensor \( G_{mn} = R_{mn} - \tfrac{1}{2} R g_{mn} = 8\pi T_{mn} \).[2](https://physics.umd.edu/grt/taj/675d/675cnotes.html)
## The Einstein tensor and conservation
With the identity in hand, one can construct a tensor whose covariant divergence is always zero: the Einstein tensor, \( G^{\mu\nu} \equiv R^{\mu\nu} - \tfrac{1}{2} g^{\mu\nu} R \). The simplest consistent field equation is then \( G^{\mu\nu} = a T^{\mu\nu} \), and fixing the constant \( a \) by the [Newtonian limit](https://www.edgepedia.org/newtonian-limit) gives \( G^{\mu\nu} = -\tfrac{8\pi G}{c^{4}} T^{\mu\nu} \) in the Edinburgh convention (many texts write \( G_{\mu\nu} = 8\pi G T_{\mu\nu} \) with \( c = 1 \) and the opposite metric signature).[1](https://www.roe.ac.uk/japwww/teaching/gr/GR_notes_2021) • [2](https://physics.umd.edu/grt/taj/675d/675cnotes.html)
Taking the covariant divergence of both sides of the field equations, the left side vanishes identically by the Bianchi identity, so the right side must vanish too: \( \nabla_{\mu} T^{\mu\nu} = 0 \). Hence general relativity does not require an additional assumption to guarantee local conservation of energy and momentum; rather, this conservation follows as the integrability condition of its field equations, just as the Maxwell equations have charge conservation as theirs.[7](https://nbow.io/en/products/ai-powered-research/treatise-of-physics/chapter/05-general-relativity-cosmology/03-einstein-field-equations.html)
**What the conservation law says.** The conservation law \( \nabla_{\nu} T^{\mu\nu} = 0 \) is local, not a statement about a globally conserved quantity. Written out, it reads \( \partial T^{\mu\nu}/\partial x^{\nu} + \Gamma^{\mu}{}_{\alpha\nu} T^{\alpha\nu} + \Gamma^{\nu}{}_{\alpha\nu} T^{\alpha\mu} = 0 \), so the gravitational field, through the [Christoffel symbols](https://www.edgepedia.org/christoffel-symbols), exchanges energy and momentum with matter. A globally conserved quantity requires a divergence-free current, which can be built from the stress tensor when a Killing vector \( X_b \) exists: \( j^{a} = T^{ab} X_b \) satisfies \( j^{a}{}_{;a} = 0 \), the first term vanishing by conservation and the second by Killing's equation.[1](https://www.roe.ac.uk/japwww/teaching/gr/GR_notes_2021) • [2](https://physics.umd.edu/grt/taj/675d/675cnotes.html)
Applied to a perfect fluid, the \( u_a \) component of \( T^{ab}{}_{;b} = 0 \) gives the first law of thermodynamics, \( dE + p\,dV = 0 \), while the spatial components give the relativistic Euler equation, \( (\rho + p) u^{a} u_{i;a} = -p_{,i} \).[2](https://physics.umd.edu/grt/taj/675d/675cnotes.html)
## Constraint structure of the field equations
The identity also fixes the character of the Einstein equations as a system. The \( tt \) and \( ti \) components contain no second time derivatives of the metric, so they form four initial-value constraint equations; this can be deduced directly from the Bianchi identity and stress-tensor conservation. If the constraints hold at one time, the Bianchi identity implies they hold at all times, so of the ten Einstein equations only \( 10 - 4 = 6 \) are independent evolution equations. This non-uniqueness at the level of tensor components reflects the four arbitrary coordinate functions available in a generally covariant theory.[2](https://physics.umd.edu/grt/taj/675d/675cnotes.html)
## Variational viewpoint: Noether's second theorem
The same conservation law can be derived from the action. [Diffeomorphism invariance](https://www.edgepedia.org/diffeomorphism-invariance) of the [Einstein–Hilbert action](https://www.edgepedia.org/einstein-hilbert-action) implies the contracted Bianchi identity, and diffeomorphism invariance plus the equations of motion for the matter fields implies \( \nabla_{\mu} T^{\mu\nu} = 0 \).[6](https://home.iiserb.ac.in/~suvankar/downloads/Reall.pdf) Under Noether's second theorem, the symmetry group involves four arbitrary functions \( \xi^{\mu} \), and the result is not four conserved charges but rather four differential identities.[7](https://nbow.io/en/products/ai-powered-research/treatise-of-physics/chapter/05-general-relativity-cosmology/03-einstein-field-equations.html) Concretely, an infinitesimal diffeomorphism generated by a compactly supported vector field \( X \) produces a metric variation \( \delta_X g = \mathcal{L}_X g \) that must leave the total action stationary whenever the matter Euler–Lagrange equations hold, forcing \( \nabla^{\mu} T_{\mu\nu} = 0 \).[3](https://jetbundle.github.io/physics/field-theory/chapter-4/) David Tong's notes make the same point from the path-integral perspective: the Bianchi identity is a result of diffeomorphism invariance, and restricted to flat space the conservation law reduces to the ordinary Noether-current form \( \partial_{\mu} T^{\mu\nu} = 0 \).[4](https://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S4.html)
## Beyond Einstein gravity
The mechanism survives in broad classes of modified gravity. Tomi Koivisto, a cosmologist working on modified gravity models, proved explicitly that the covariant divergence of the energy-momentum tensor vanishes in theories whose action depends nonlinearly on the curvature scalar and its couplings with a scalar field, in both the metric and Palatini (first-order) formalisms; this generalized Bianchi identity can also be deduced directly from the covariance of the extended gravitational action.[5](https://arxiv.org/abs/gr-qc/0505128) In unimodular gravity, by contrast, the theory has its own conservation equation obtained from the [Bianchi identities](https://www.edgepedia.org/bianchi-identities) rather than the standard \( \nabla_{\mu} T^{\mu\nu} = 0 \).[9](https://arxiv.org/abs/2112.06644)
## References
1. [Relativity & Gravitation lecture notes, Institute for Astronomy, University of Edinburgh (2021)](https://www.roe.ac.uk/japwww/teaching/gr/GR_notes_2021)
2. [Physics 675 lecture notes, T. Jacobson, University of Maryland (Fall 2006)](https://physics.umd.edu/grt/taj/675d/675cnotes.html)
3. [Classical field dynamics, JETBUNDLE field theory notes](https://jetbundle.github.io/physics/field-theory/chapter-4/)
4. [The Einstein Equations, General Relativity, David Tong, University of Cambridge](https://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S4.html)
5. [T. Koivisto (2005). Covariant conservation of energy momentum in modified gravities. arXiv:gr-qc/0505128](https://arxiv.org/abs/gr-qc/0505128)
6. [H. S. Reall. General Relativity lecture notes, University of Cambridge (2012)](https://home.iiserb.ac.in/~suvankar/downloads/Reall.pdf)
7. [The Einstein Field Equations, Treatise of Physics](https://nbow.io/en/products/ai-powered-research/treatise-of-physics/chapter/05-general-relativity-cosmology/03-einstein-field-equations.html)
8. [Weak second Bianchi identity for static, spherically symmetric spacetimes with timelike singularities. arXiv:1901.00813](https://arxiv.org/abs/1901.00813)
9. [Nonconservative unimodular gravity: a viable cosmological scenario. arXiv:2112.06644](https://arxiv.org/abs/2112.06644)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › General relativity and curved spacetime › Foundations and field equations › Einstein field equations › Bianchi identities and energy–momentum conservation*
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