Third-Order Lagrangian Density

• Third-order Lagrangian densities are functions built from cubic curvature invariants, involving six spacetime derivatives to underpin higher-curvature gravity theories.
• They classify curvature invariants into second- and third-order trace categories, enabling analytically tractable field equations in static, spherically symmetric spacetimes.
• Their applications include modeling conformal anomalies in six dimensions and deriving analytic black hole solutions, informing both theoretical and practical gravity research.

A third-order Lagrangian density is a function constructed from curvature invariants (and potentially their derivatives) of degree three in the curvature or equivalently, containing six spacetime derivatives when considering gravitational theories. Such Lagrangians are central to the paper of higher-curvature theories of gravity, the construction of conformal invariants in six dimensions, and in the systematic treatment of trace anomalies and black hole solutions in modified gravity. The structure and classification of these Lagrangians underpin both formal advances in geometric analysis and practical developments in gravitational physics.

1. Structure and Algebraic Classification of Third-Order Lagrangian Densities

Third-order Lagrangian densities for gravity are formed by linear combinations of independent scalar invariants of mass dimension six, which typically involve cubic contractions of the Riemann, Ricci, or Weyl tensors, as well as terms with derivatives acting on curvature tensors. Formally, the generic class can be expressed as: $\mathcal{L} = \sum_{i=1}^{10} A^i L_i$ where the $L_i$ include representative invariants:

• $R^{abcd}R_{cdef}R_{~~ab}^{ef}$
• $R R^{abcd}R_{abcd}$
• $R^3$
• $\nabla_a R \nabla^a R$
• other six-derivative terms

Two of the initial twelve such invariants are redundant (modulo total derivatives), so only ten are independent. The choice of coefficients $A^i$ determines crucial properties of the resulting gravitational theory.

A central question is the order of the traced field equations, as generic higher-curvature actions give rise to very high-order field equations. By requiring the trace of the field equations to be only third order (or lower), the allowed forms of the Lagrangian are highly constrained. Specifically, the coefficients must be chosen so that higher-derivative terms cancel in the trace. This is operationalized via the identity: $$T^a_{~a} = \frac{n-D}{2}\mathcal{L} + (\text{total derivative}),$$ with $n$ the number of derivatives (here $n=6$) and $D$ the spacetime dimension (Oliva et al., 2010).

The resulting classification organizes invariants into those whose field equation traces are of second order and those of third order:

• Second-order trace invariants: The six-dimensional Euler density $\mathcal{E}_6$ and two cubic Weyl invariants, $W_1$ and $W_2$.
• Third-order trace invariants: Denoted $\Sigma$ and $\Theta$, constructed from the same blocks, but such that the trace is generically third order (unless further constraints are imposed).

2. Dimension-Dependent Structure and Conformal Properties

The set of independent invariants and their properties crucially depend on spacetime dimension, due to the identities satisfied by the Riemann, Ricci, and Weyl tensors.

• $D\geq6$: The standard basis is $\{\mathcal{E}_6, W_1, W_2\}$, with $\Sigma$ and $\Theta$ as additional invariants giving third-order traces.
• $D=5$: A 'special' invariant arises: $$\mathcal{N}_6 \equiv \frac{D-2}{D-5}(4W_1 + 8W_2 - \mathcal{E}_6)$$ which does not vanish due to lower-dimensional degeneracies and still gives second-order trace for static, spherically symmetric spacetimes.
• $D=3$: The only relevant invariant (among those considered) giving a third-order trace is $C_{abc}C^{abc}$, the square of the (identically vanishing) Weyl tensor's three-dimensional analog—the Cotton tensor (itself a measure of nonconformal flatness).

This dimensional dependence reflects deep geometric properties and is central to constructing conformal anomalies in $D=6$, where the theory based on cubic Weyl invariants is conformally invariant.

3. Static, Spherically Symmetric Solutions

When applied to static, spherically symmetric spacetimes, third-order Lagrangian densities admit exact analytic solutions characterized by significant simplification of the field equations. The general ansatz is: $$ds_D^2 = N(r)\left[ -f(r) dt^2 + \frac{dr^2}{f(r)} + r^2 d\Sigma^2_{D-2,\gamma} \right]$$ where $d\Sigma^2_{D-2,\gamma}$ is the metric on the $(D-2)$-dimensional maximally symmetric space (with curvature index $\gamma$).

For Lagrangians containing only cubic Weyl invariants,

• In $D=6$ (conformal case), $N(r)$ can be gauged away and the solution becomes: $$ds_6^2 = -\left[a r^2 + b r + K - \frac{c(1+e r)^{5/2}}{r^{1/2}}\right] dt^2 + \left[dr^2/\text{(same)}\right] + r^2 d\Sigma^2_{4, \gamma}$$ where integration constants are related by algebraic constraints, and the solution typically features curvature singularities at finite radius.
• For $D\neq 6$, the traced field equations force

$f(r) = a r^2 + b r + \gamma$

with arbitrary $N(r)$.

• In $D=5$, the special invariant $\mathcal{N}_6$ yields the unique solution

$$ds^2 = -[c r^{2/3} + \gamma] dt^2 + \frac{dr^2}{c r^{2/3} + \gamma} + r^2 d\Sigma_3^2$$

which is generically not conformally flat.

These explicit solutions demonstrate that proper algebraic tuning in the construction of the Lagrangian enables the field equations to be highly tractable in symmetric settings, a property often absent in more general higher-order curvature theories (Oliva et al., 2010).

4. Applications: Trace Anomalies and Theoretical Implications

The classification of third-order Lagrangian densities sheds light on diverse physical and mathematical structures:

• In six dimensions, specific combinations of cubic invariants reproduce the conformal anomaly (type-B anomaly) of quantum field theory, as the trace of the field equations is proportional to the classical action density itself, modulo total derivatives.
• The relationship between the Lagrangian and conformal invariants informs the construction of local counterterms and provides the geometric content underlying holographic trace anomalies in AdS/CFT.
• The explicit construction of black hole solutions and their thermodynamics in these theories is directly enabled by the simplification of the traced field equations. Notably, in $D=5$ the cubic invariant $\mathcal{N}_6$ yields nonconformally flat black hole spacetimes with physical horizons and curvature singularities, offering models for higher-curvature corrections in black hole physics.

The rigorous cancellation of higher-derivative terms in the trace offers a route to building higher-order gravitational theories that partially avoid pathologies typically associated with higher-derivative gravity (such as Ostrogradsky instability), at least as far as the trace sector is concerned.

5. Algebraic Constraints and Explicit Classification

The process of ensuring that the trace of the field equations is of reduced order leads to a system of linear constraints on the coefficients $A^i$. The subset of invariants yielding second-order traces is spanned by $\mathcal{E}_6$, $W_1$, $W_2$, and for $D=5$, the special invariant $\mathcal{N}_6$. Invariants like $\Sigma$ and $\Theta$ are essential when constructing models with a genuinely third-order trace.

The classification can be encapsulated schematically as: $$\mathcal{L} = a\,\mathcal{E}_6 + b\,W_1 + c\,W_2 + d\,\Sigma + e\,\Theta$$ where the free parameters are fixed by the requirements of the desired trace order and invariance properties.

Consistency with the homogeneity argument assures that the combination $T^a_{~a} = \frac{n-D}{2}\mathcal{L}$ is valid up to a total divergence, which underlies the entire algebraic classification.

6. Physical Significance and Broader Impact

The third-order Lagrangian density, when constructed appropriately, serves as the foundational object in a wide range of gravitational theories, including Lovelock gravity, the conformal invariants in six dimensions, and as a laboratory for investigating classical trace anomalies. Its explicit classification enables the construction of higher-derivative models where the resistance to the propagation of unphysical degrees of freedom is built into the trace structure.

This framework facilitates the systematic paper of static, spherically symmetric solutions, enriches the mathematics of gravitational actions (particularly in connection with boundary terms and topological densities), and provides analytic tools for exploring quantum anomalies and holographic dualities. The explicit identification of the relevant invariants and the precise algebraic classification directly support ongoing investigations in higher-curvature gravity, black hole thermodynamics, and the geometric origins of trace anomalies (Oliva et al., 2010).