Lovelock Uniqueness and Its Hypotheses

Lovelock’s theorem does not say that general relativity is the only possible theory of gravity. It classifies a sharply delimited set of pure-metric field tensors. In four dimensions, locality, diffeomorphism naturality, symmetry, identically vanishing divergence, smooth dependence, and differential order at most two leave only the Einstein tensor and the metric. Changing the dimension, derivative order, field content, connection, or locality assumption leaves the theorem’s domain.

Required background. We use the Riemann curvature tensor for natural second-order tensors, Hilbert stress–energy for the symmetric source paired with them, and the scope of the equivalence principles to avoid turning local metric physics into a uniqueness premise.

Helpful background. Minkowski geometry supplies the fixed Lorentzian signature, natural tensor operations supply the language of equivariance, and presymplectic potentials clarify why divergence identities arise in variational theories.

The four-dimensional statement

Let $X$ be a smooth four-dimensional manifold and let $g$ range over nondegenerate pseudo-Riemannian metrics of one fixed signature. Suppose $H_{\mu\nu}[g]$ satisfies all of the following:

1. Pure metric and local. At $x$, $H_g(x)$ depends only on the metric in an arbitrarily small neighborhood of $x$, with no extra fields or fixed background tensors.
2. Diffeomorphism natural. For every local diffeomorphism $\varphi$, $H[\varphi^*g]=\varphi^*H[g]$.
3. Differential order at most two. The value at $x$ depends only on the two-jet $j_x^2g$: in coordinates, on $g_{\alpha\beta}$ and its first and second derivatives at $x$.
4. Smooth regularity. That dependence on the two-jet is smooth on the bundle of nondegenerate metrics of the chosen signature.
5. Symmetry. $H_{\mu\nu}=H_{\nu\mu}$ for every $g$.
6. Identical divergence freedom. With the Levi-Civita connection of $g$, $\nabla^\mu H_{\mu\nu}[g]=0$ for every metric, without using a field equation.

Then there are real constants $a$ and $b$, independent of position and of the chosen metric, such that

$$H_{\mu\nu}=aG_{\mu\nu}+b g_{\mu\nu}.$$

This is the four-dimensional specialization of the smooth natural-tensor classification in Navarro and Navarro 2011, Definition 1.1, printed p. 2, and Theorem 2.6, printed p. 9, Open PDF. Lovelock’s 1971 paper and its 1972 four-dimensional companion are listed below for historical provenance; the proof locators used on this page are the accessible modern treatment.

No action principle, boundary condition, causal condition, or weak-field limit is among these hypotheses. Accordingly, the conclusion classifies the allowed left-hand tensor under the listed assumptions; it does not fix $a$, $b$, or the matter coupling. Writing

$$aG_{\mu\nu}+b g_{\mu\nu}=\kappa T_{\mu\nu}, \qquad a\ne0,$$

and dividing by $a$, the Newtonian limit fixes the ratio $\kappa/a=8\pi G$, while $\Lambda=b/a$. An overall rescaling of both sides is conventional.

Regularity and polynomial dependence

Some summaries state a separate assumption that $H_{\mu\nu}$ be linear in the second derivatives of the metric. The smooth-natural version above is more precise about where that property enters. It begins with a smooth two-jet map, not a polynomial ansatz. Symmetry, naturality, and divergence freedom then imply that the map is polynomial in the normal-coordinate second-derivative data, with degree at most

$$\left\lfloor\frac{D-1}{2}\right\rfloor.$$

In $D=4$ this degree is at most one. Thus linearity in those normal-tensor variables is a conclusion of the Navarro–Navarro formulation, whereas other formulations may impose a linearity or polynomial qualification at the outset. Smoothness still matters: an arbitrary singular or discontinuous rule on metric jets is not covered.

Reall gives the compact four-dimensional corollary $H_{ab}=\alpha G_{ab}+\beta g_{ab}$ in Reall 2013, § 6.7, printed p. 65, Eq. (6.34), PDF. That lecture-note passage is a statement of the theorem, not a proof; the proof architecture used here comes from Navarro and Navarro.

Proof architecture

The proof has four structural steps.

Reduce jets to an orthogonal-group problem. In normal coordinates at a point, $g_{ab}$ is the fixed signature matrix and $g_{ab,c}=0$. The remaining second derivatives form a vector space $N$ of normal tensors with specific pair and cyclic symmetries. A second-order natural symmetric tensor is therefore equivalent to a smooth $O(p,q)$-equivariant map $f:N\to S^2$. This is developed in Navarro and Navarro 2011, Definitions 1.1–1.2 and Theorem 1.3, printed pp. 2–3, Open PDF.

Differentiate the natural map. The successive derivatives $d^kf$ are themselves equivariant tensors. Divergence freedom forces an additional cyclic index symmetry on them. Definitions 1.1–1.10 establish the natural-bundle, normal-tensor, derivative, and divergence-free framework; Theorem 1.11 then embeds each homogeneous degree-$k$ possibility into an orthogonal-invariant space $(\mathrm{Div}^k)^{O(p,q)}$ Navarro and Navarro 2011, printed pp. 2–6, Open PDF.

Count invariants. Orthogonal invariant theory and the cyclic symmetries show that this invariant space is one-dimensional for $k\le\lfloor(D-1)/2\rfloor$ and zero above that value. Vanishing sufficiently high derivatives makes $f$ polynomial rather than merely smooth. These are Navarro and Navarro 2011, Lemma 2.2 and Corollary 2.3, printed pp. 6–7, Open PDF.

Exhibit the basis. Curvature forms and the metric construct Lovelock tensors $L_k$. Each is natural, symmetric, divergence-free, and homogeneous of degree $k$. The invariant count shows that no others exist: $L_0,\ldots,L_{\lfloor(D-1)/2\rfloor}$ form a basis. In four dimensions only $L_0$ and $L_1$ remain; after lowering indices they are proportional to $g_{\mu\nu}$ and $G_{\mu\nu}$. The construction and basis theorem are Navarro and Navarro 2011, Definition 2.5 and Theorem 2.6, printed p. 9, Open PDF.

This is a proof sketch of the classification’s architecture, not a replacement for the invariant-theory argument.

Explicit ways to leave the theorem

Higher dimension. For $D\ge5$, the Gauss–Bonnet or second Lovelock tensor is generally nonzero:

$$\begin{aligned} H^{\rm GB}_{\mu\nu}=2\big(&RR_{\mu\nu}-2R_{\mu\alpha}R^\alpha{}_\nu -2R^{\alpha\beta}R_{\mu\alpha\nu\beta}\\ &+R_\mu{}^{\alpha\beta\gamma}R_{\nu\alpha\beta\gamma}\big) -\frac12g_{\mu\nu}\mathcal L_{\rm GB}, \end{aligned}$$

where $\mathcal L_{\rm GB}=R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}-4R_{\alpha\beta}R^{\alpha\beta}+R^2$. It satisfies the same second-order, natural, symmetric, divergence-free conditions but is not a combination of $G_{\mu\nu}$ and $g_{\mu\nu}$. In four dimensions its metric Euler–Lagrange tensor vanishes identically.

Higher differential order. Varying an $R^2$ action gives a divergence-free tensor containing

$$2RR_{\mu\nu}-\frac12g_{\mu\nu}R^2 +2(g_{\mu\nu}\Box-\nabla_\mu\nabla_\nu)R,$$

which generally has fourth derivatives of the metric. It violates the order-two hypothesis.

Extra fields. A scalar–tensor term $F(\phi)R$ produces metric terms such as

$$F(\phi)G_{\mu\nu} +(g_{\mu\nu}\Box-\nabla_\mu\nabla_\nu)F(\phi).$$

Their divergence closes together with the scalar equation. This is not a pure-metric natural tensor.

Different connection or locality. Independent torsion or nonmetricity supplies additional tensors not built from the Levi-Civita two-jet. A nonlocal term such as $R\Box^{-1}R$ depends on boundary conditions or a Green function rather than only on $j_x^2g$. Each is an explicit departure from a stated hypothesis, not a counterexample to the theorem.

Exercises

Use the degree bound to determine which Lovelock tensors can occur in $D=4$, $D=5$, and $D=6$. What does this show about Gauss–Bonnet dynamics?

Solution

The largest degree is $p=\lfloor(D-1)/2\rfloor$. For $D=4$, $p=1$, so the basis contains only $L_0$ and $L_1$, corresponding to the metric and Einstein tensors. For $D=5$, $p=2$, and for $D=6$, $p=2$, so $L_2$, the Gauss–Bonnet tensor, is also allowed. Hence its local metric field tensor is identically absent in four dimensions but can contribute nontrivially from five dimensions onward.

References

• Lovelock, David. “The Einstein Tensor and Its Generalizations.” Journal of Mathematical Physics 12, no. 3 (1971): 498–501. DOI.
• Lovelock, David. “The Four-Dimensionality of Space and the Einstein Tensor.” Journal of Mathematical Physics 13, no. 6 (1972): 874–876. DOI.
• Navarro, Alberto, and José Navarro. “Lovelock’s Theorem Revisited.” Journal of Geometry and Physics 61, no. 10 (2011): 1950–1956. DOI; Open PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013), § 6.7, printed p. 65. PDF.