Math 2.0

Relativity and gravitational theory Alternative theories of gravity Theorem

Lovelock's theorem

Builds on Lorentzian manifoldEinstein tensorCosmological constantEinstein field equationsEinstein–Hilbert actionTensor field

Statement

Let (M,g)(M, g) be a four-dimensional Lorentzian manifold. Let EabE^{ab} be a symmetric tensor field. Suppose that, in every coordinate system, its components at each point depend only on gabg_{ab} and its first and second partial derivatives there. Suppose that EabE^{ab} is divergence-free, ∇aEab=0\nabla_a E^{ab} = 0, for every metric gg. Then there are constants α\alpha and λ\lambda such that Eab=α Gab+λ gab,E^{ab} = \alpha \, G^{ab} + \lambda \, g^{ab}, where GabG^{ab} is the Einstein tensor.

Remarks

The Euler–Lagrange expression of any diffeomorphism-invariant Lagrangian density of the metric is divergence-free. So, in four dimensions, every such Lagrangian theory of the metric alone whose field equations are of second order gives the Einstein field equations with a cosmological constant. In this sense the Einstein–Hilbert action is essentially the only choice. Lovelock proved the result in the early 1970s. The dimension four is necessary: in dimension D≥5D \ge 5 further divergence-free tensors exist, such as the one obtained from the Gauss–Bonnet term R2−4RabRab+RabcdRabcdR^2 - 4R_{ab}R^{ab} + R_{abcd}R^{abcd}.

Going deeper

For a student

The theorem is the standard map of how to modify general relativity. Any alternative theory must break one of its hypotheses. It may add new fields such as a scalar field, allow higher derivatives, or work in more than four dimensions. It may also give up locality or change the geometry, for example by allowing torsion. Scalar–tensor theories, f(R)f(R) gravity, Lovelock gravity in higher dimensions and massive gravity each break one of these assumptions.

Reading path

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Cosmological constant, concept, no evidence levelCosmological constantconceptEinstein field equations, concept, no evidence levelEinstein field equati…conceptEinstein tensor, concept, no evidence levelEinstein tensorconceptEinstein–Hilbert action, concept, no evidence levelEinstein–Hilbert acti…conceptLorentzian manifold, concept, no evidence levelLorentzian manifoldconceptTensor field, concept, no evidence levelTensor fieldconceptLovelock's theorem, statement, no evidence levelLovelock's theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cosmological constant concept
  2. Einstein field equations concept
  3. Einstein tensor concept
  4. Einstein–Hilbert action concept
  5. Lorentzian manifold concept
  6. Tensor field concept
  7. Lovelock's theorem statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Lovelock%27s_theorem
About this page
TagDW4CF8A42M
URIm2://DW4CF8A42M
Kindstatement, theorem
WritingW0 stub
MSC83C05, 83D05
StatusThis copy of the site has no build data. Run m2 status DW4CF8A42M.
AboutEinstein field equations, Einstein tensor
UsesLorentzian manifold, Einstein tensor, Cosmological constant, Einstein field equations, Einstein–Hilbert action, Tensor field
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Changed3 November 2026

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