Gauss–Bonnet and Topological Curvature Terms

The Gauss–Bonnet density is quadratic in curvature, yet a constant coefficient makes no local contribution to the bulk metric equation in four dimensions. This is not a cancellation that holds for every curvature-squared action. It is the critical-dimensional Euler theorem: under precise smoothness, orientation, boundary, dimension, and coupling hypotheses, the integral is a topological invariant. A boundary, a field-dependent coefficient, a dimension above four, or an additional field changes the conclusion in a controlled and identifiable way.

Required background. Gravitational Actions and Boundary Terms supplies the distinction between a bulk Euler–Lagrange equation and a boundary variation; Curvature supplies the Riemann symmetries and contractions used in the proof.

Helpful background. Einstein Equations supplies the Bianchi identity; Formulations of General Relativity supplies hypersurface and boundary data; Quantum Gravity provides the separate EFT and ultraviolet context for higher-curvature terms.

The four-dimensional Euler statement

Define

$$\mathcal G =R_{abcd}R^{abcd}-4R_{ab}R^{ab}+R^2.$$

Assume first that $(M,g)$ is a smooth, compact, oriented, closed four-dimensional Riemannian manifold. With the site’s Riemann-tensor sign—the same sign appearing in all three squared terms—the Chern–Gauss–Bonnet theorem states

$$\frac{1}{32\pi^2}\int_M d^4x\sqrt{g}\,\mathcal G =\chi(M).$$

The right side is the Euler characteristic and does not vary under a smooth metric deformation. The Lovelock-density construction and its dimensional dependence are given in Padmanabhan and Kothawala 2013, § 2.2, equations (12)–(26), while the critical-dimensional Euler argument is made explicit in Padmanabhan and Kothawala 2013, § 2.5, equations (49)–(54).

For a Lorentzian gravitational region, the local variational identity below is the operative statement. Global use of the Euler formula requires the corresponding topology and a mathematically appropriate continuation or pseudo-Riemannian formulation; it should not be inferred merely because the spacetime has four coordinates.

Why the bulk field equation vanishes

Vary the density before using the dimension. Up to a boundary divergence,

$$\delta\!\left(\sqrt{-g}\,\mathcal G\right) =\sqrt{-g}\,H_{ab}\,\delta g^{ab}+\partial_a\Theta^a,$$

where

$$\begin{aligned} H_{ab}={}&2\left( RR_{ab}-2R_{ac}R_b{}^c-2R^{cd}R_{acbd} +R_a{}^{cde}R_{bcde} \right)\\ &-\frac12g_{ab}\mathcal G. \end{aligned}$$

The proof is clearest in generalized-delta notation. The $m$th Lovelock density contains $m$ Riemann tensors contracted with a totally antisymmetric Kronecker delta. Its metric equation contains a delta antisymmetrized over $2m+1$ indices. For Gauss–Bonnet, $m=2$; in $D=4$ that antisymmetrization has five slots but only four independent directions, so it vanishes identically:

$$H_{ab}\equiv0\qquad(D=4).$$

This is an algebraic identity for every smooth four-dimensional metric, not an equation imposed on special solutions. It proves that

$$S_{\rm GB}=\alpha\int_M d^4x\sqrt{-g}\,\mathcal G$$

with constant $\alpha$ has no local bulk metric equation under these conditions. The classification of divergence-free, second-order metric concomitants underlying the result is established in Lovelock 1971, pp. 498–501, equations (1.1)–(3.5).

Boundaries are part of the theorem

If $\partial M$ is nonempty, the divergence $\partial_a\Theta^a$ need not integrate to zero. The Euler characteristic is then represented by a bulk term plus a boundary contribution built from intrinsic and extrinsic curvature. Likewise, a well-posed Dirichlet variation of a Lovelock action requires its boundary completion, often called the Myers term for Gauss–Bonnet. The boundary-sensitive construction is reviewed in Padmanabhan and Kothawala 2013, § 2.6, equations (55)–(68).

Consequently, “the term vanishes in four dimensions” is inaccurate. The density is generally nonzero, its integral records topology, and its boundary variation can matter. What vanishes is the constant-coupling local bulk Euler–Lagrange tensor after the correct boundary problem has been specified.

How the conclusion is evaded

Three changes invalidate a hypothesis rather than contradicting the theorem.

Dimension above four. When $D\gt4$, antisymmetrization over five indices no longer vanishes, and $H_{ab}$ supplies a nontrivial second-order Lovelock equation. In the Einstein-normalized action $S=(16\pi G_D)^{-1}\int\sqrt{-g}(R+\alpha_{\rm GB}\mathcal G)$, the relative coupling $\alpha_{\rm GB}$ has dimensions of length squared. It differs from the coefficient $\alpha$ of the standalone integral above: $\alpha=\alpha_{\rm GB}/(16\pi G_D)$. The Einstein expansion is controlled only when $\lvert\alpha_{\rm GB}\rvert\mathcal R\ll1$ for the relevant curvature scale $\mathcal R$ and the selected solution branch approaches GR.

Field-dependent coefficient. For

$$S\supset\int d^4x\sqrt{-g}\,F(\phi)\mathcal G,$$

integrations by parts differentiate $F$. The metric equation then contains curvature contracted with $\nabla_a\nabla_bF$, while the scalar equation contains $F_{,\phi}\mathcal G$. A dynamical scalar coupling is therefore not a constant coefficient and can support scalarized or hairy solutions.

Singular dimensional prescription. Multiplying the $D$-dimensional coupling by $1/(D-4)$ and then sending $D\to4$ is not ordinary four-dimensional Lovelock gravity. The rescaled field equation retains a Weyl-dependent piece whose limit is not an intrinsic conserved four-dimensional tensor, as shown in Gürses, Şişman, and Tekin 2020, § II, equations (6)–(20). Dropping that piece violates the Bianchi identity. A consistent nontrivial two-degree-of-freedom construction can instead break temporal diffeomorphisms; that different theory and the nonuniqueness of the limit are shown in Aoki, Gorji, and Mukohyama 2020, §§ IV–VI, equations (4.1)–(6.7).

These are discriminating counterexamples to an overbroad slogan, not failures of the Euler theorem. Empirical bounds on their couplings remain outside this proof in Tests of Gravity, and proposed microscopic origins remain separate in Quantum Gravity.

An independent normalization check

For a unit round four-sphere,

$$R_{abcd}R^{abcd}=24, \qquad R_{ab}R^{ab}=36, \qquad R=12.$$

Hence $\mathcal G=24$. Since $\operatorname{Vol}(S^4)=8\pi^2/3$,

$$\int_{S^4}d^4x\sqrt g\,\mathcal G =24\frac{8\pi^2}{3}=64\pi^2 =32\pi^2\chi(S^4),$$

with $\chi(S^4)=2$. This verifies the normalization without using the desired field-equation conclusion. In contrast, evaluating $H_{ab}$ from the displayed constant-curvature tensors gives zero component by component only in $D=4$.

Common pitfalls

Confuse zero variation with zero density. The Euler density on $S^4$ is positive and its integral is nonzero even though its four-dimensional bulk metric variation vanishes.

Omit the boundary hypothesis. A bounded region needs the boundary contribution or boundary data that make the variational statement valid.

Call a singular dimensional limit “4D Gauss–Bonnet.” Additional fields, broken symmetry, or retained higher-dimensional data define a different theory and must be named explicitly.

Exercises

Verify the Euler normalization for a round four-sphere of radius $L$, and then identify exactly which hypothesis fails for $F(\phi)\mathcal G$ and for the rescaled $D\to4$ prescription.

Solution

The invariants scale as $L^{-4}$ while the volume scales as $L^4$, so the integral remains $64\pi^2$ and gives $\chi=2$. A nonconstant $F(\phi)$ violates the constant-coupling hypothesis because integrations by parts act on $F$. The rescaled dimensional prescription changes the theory before taking the limit; its Weyl-dependent contribution is neither fixed by intrinsic four-dimensional data nor removable while preserving the Bianchi identity.

References

• Aoki, Katsuki, Mohammad Ali Gorji, and Shinji Mukohyama. 2020. “A Consistent Theory of $D\to4$ Einstein–Gauss–Bonnet Gravity.” Physics Letters B 810: 135843. DOI.
• Gürses, Metin, Tahsin Çağrı Şişman, and Bayram Tekin. 2020. “Is There a Novel Einstein–Gauss–Bonnet Theory in Four Dimensions?” European Physical Journal C 80: 647. DOI.
• Lovelock, David. 1971. “The Einstein Tensor and Its Generalizations.” Journal of Mathematical Physics 12: 498–501. DOI.
• Padmanabhan, T., and Dawood Kothawala. 2013. “Lanczos–Lovelock Models of Gravity.” Physics Reports 531: 115–171. DOI.