Math 2.0

Relativity and gravitational theory Definition

Einstein–Gauss–Bonnet gravity

Builds on Lovelock theory of gravityLorentzian manifoldScalar curvatureRicci curvatureRiemann curvature tensorChern–Gauss–Bonnet theoremEuler characteristic

Definition

Let (M,g)(M, g) be a Lorentzian manifold of dimension DD, with scalar curvature RR, Ricci tensor RabR_{ab} and Riemann tensor RabcdR_{abcd}. The Gauss–Bonnet term is 𝒢=R2−4RabRab+RabcdRabcd.\mathcal{G} = R^2 - 4 R_{ab}R^{ab} + R_{abcd}R^{abcd}. Let κ>0\kappa > 0, Λ\Lambda and α\alpha be real constants. Einstein–Gauss–Bonnet gravity is the Lovelock theory with action S=12κ∫(R−2Λ+α 𝒢)−g dDx+Sm,S = \frac{1}{2\kappa} \int \left( R - 2\Lambda + \alpha \, \mathcal{G} \right) \sqrt{-g}\, d^D x + S_m, where SmS_m is the action of matter.

Remarks

The field equations are Gab+Λgab+αHab=κTabG_{ab} + \Lambda g_{ab} + \alpha H_{ab} = \kappa T_{ab}, where GabG_{ab} is the Einstein tensor and Hab=2(RRab−2RacRcb−2RacbdRcd+RacdeRbcde)−12gab 𝒢.H_{ab} = 2\left(R R_{ab} - 2R_{ac}R^c{}_b - 2R_{acbd}R^{cd} + R_{acde}R_b{}^{cde}\right) - \frac12 g_{ab}\,\mathcal{G}. Example: for D≥5D \ge 5 and α≠0\alpha \ne 0 the tensor HabH_{ab} is not identically zero, and the theory differs from general relativity. Non-example: the action with αR2\alpha R^2 in place of α 𝒢\alpha\,\mathcal{G} is not of this form, and its field equations are of fourth order. For D=4D = 4, HabH_{ab} vanishes identically and the theory is general relativity; on a compact oriented Riemannian 44-manifold the Chern–Gauss–Bonnet theorem gives ∫𝒢 dV=32π2χ(M)\int \mathcal{G}\, dV = 32\pi^2 \chi(M), where χ\chi is the Euler characteristic. Boulware and Deser found its static spherically symmetric black hole solutions in 1985.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Curvature form, concept, no evidence levelCurvature formconceptEuler characteristic, concept, no evidence levelEuler characteristicconceptGaussian curvature, concept, no evidence levelGaussian curvatureconceptGeodesic curvature, concept, no evidence levelGeodesic curvatureconceptLevi-Civita connection, concept, no evidence levelLevi-Civita connectionconceptLorentzian manifold, concept, no evidence levelLorentzian manifoldconceptLovelock theory of gravity, concept, no evidence levelLovelock theory of gr…conceptParametrised surface, concept, no evidence levelParametrised surfaceconceptPfaffian, concept, no evidence levelPfaffianconceptRegular surface, concept, no evidence levelRegular surfaceconceptRicci curvature, concept, no evidence levelRicci curvatureconceptRiemann curvature tensor, concept, no evidence levelRiemann curvature ten…conceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptScalar curvature, concept, no evidence levelScalar curvatureconceptTangent plane and normal vector of a surface, concept, no evidence levelTangent plane and nor…conceptOrientable surface, concept, no evidence levelOrientable surfaceconceptGauss–Bonnet theorem, statement, no evidence levelGauss–Bonnet theoremstatementChern–Gauss–Bonnet theorem, statement, no evidence levelChern–Gauss–Bonnet th…statementEinstein–Gauss–Bonnet gravity, definition, no evidence levelEinstein–Gauss–Bonnet…definition
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. Curvature form concept
  2. Euler characteristic concept
  3. Gaussian curvature concept
  4. Geodesic curvature concept
  5. Levi-Civita connection concept
  6. Lorentzian manifold concept
  7. Lovelock theory of gravity concept
  8. Parametrised surface concept
  9. Pfaffian concept
  10. Regular surface concept
  11. Ricci curvature concept
  12. Riemann curvature tensor concept
  13. Riemannian manifold concept
  14. Scalar curvature concept
  15. Tangent plane and normal vector of a surface concept
  16. Orientable surface concept
  17. Gauss–Bonnet theorem statement
  18. Chern–Gauss–Bonnet theorem statement
  19. Einstein–Gauss–Bonnet gravity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_gravity
About this page
Tag46H9M20BK9
URIm2://46H9M20BK9
Kinddefinition
WritingW0 stub
MSC83D05, 83E15
DefinesEinstein–Gauss–Bonnet gravity
UsesLovelock theory of gravity, Lorentzian manifold, Scalar curvature, Ricci curvature, Riemann curvature tensor, Chern–Gauss–Bonnet theorem, Euler characteristic
Changed3 November 2026

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