First-Order Formalism in Gravitational Theories

Updated 16 December 2025

First-Order Formalism for Gravitational Theories is a framework where the metric and connection are treated as independent variables, streamlining complex gravitational dynamics.
It enables methods like Palatini, teleparallel, and supersymmetric fake superpotential approaches that reduce higher-order derivatives to first-order systems.
The formalism provides a robust Hamiltonian structure, addresses ghost-related challenges, and facilitates analytic solutions in extended and modified gravity models.

A first-order formalism in gravitational theories refers to formulations in which the metric and (generally) the connection are treated as independent variables, and the Lagrangian involves at most first derivatives of these fields. This approach stands in contrast to second-order (metric) formalisms, where the connection is assumed to be Levi-Civita and all field equations are derived solely from metric variations, often resulting in higher-derivative field equations for extended gravitational actions. The first-order formalism encompasses Palatini gravity, Einstein–Cartan theory, teleparallel gravity, multisymplectic and Hamiltonian methods, as well as supersymmetric and fake-superpotential techniques for specific models and backgrounds.

1. Foundations and Motivations

The adoption of first-order methods in gravity is motivated both by structural features of field theory and practical considerations:

In the Palatini formalism, one starts from an action of the form $S[g, \Gamma] = \int \sqrt{-g} \, \mathcal{L}(g, \Gamma)$ , where $g_{\mu\nu}$ is the metric and $\Gamma^\lambda_{\mu\nu}$ an a priori independent (torsionless) affine connection. This approach is particularly well-suited for extended gravities built from higher powers of the curvature, but it is also central to teleparallel, affine, and Cartan formulations (Koivisto et al., 2013, Alvarez et al., 2017).
In teleparallel gravity, gravity is described in terms of a vielbein $e^a{}_\mu$ and a (typically curvatureless) Weitzenböck connection, with the torsion tensor $T^a{}_{\mu\nu}$ encoding gravitational degrees of freedom (Dupuis et al., 2019, Peshkov et al., 2022).
Hamiltonian and multisymplectic approaches emerge naturally when the Poincaré–Cartan form of a Lagrangian—or its suitable regular extensions—can be projected onto first-jet spaces, enabling canonical phase-space structures even for originally second-order Lagrangians (María et al., 2015, Gaset et al., 2017).
In supersymmetric, domain-wall, or holographic flow settings, first-order "fake superpotential" formalisms systematically reduce the order of differential equations for BPS or BPS-like solutions (Ferrara et al., 2012, Menezes, 2014, Korovin, 2013).

The independence of connection and metric generally persists only for actions with at least quadratic (or more complicated) dependence on the curvature; for the Einstein–Hilbert term, the Palatini variation restricts $\Gamma$ to be the Levi–Civita connection up to the presence of torsion in the Einstein–Cartan theory.

2. First-Order Variational Schemes: Palatini, Metricity, and Teleparallelism

The prototypical first-order action is the Palatini variational principle. For a generalized Lagrangian density $f(R(\Gamma),\ldots)$ , the action reads

$S[g, \Gamma] = \int d^4x \, \sqrt{-g} \, f(R(\Gamma), R(g), Q(\Gamma,\ldots), \ldots) + S_\text{m}[g, \Psi]$

where $R(\Gamma)$ is the Ricci scalar of $\Gamma$ , $Q$ curvature invariants, and matter couples only to $g$ (Koivisto et al., 2013, Alvarez et al., 2017).

Field equations follow by independent variations:

$\delta g_{\mu\nu}$ yields a generalized Einstein equation coupling $R_{\mu\nu}(\Gamma)$ , $R_{\mu\nu}(g)$ , and higher invariants to the stress-energy tensor.
$\delta\Gamma$ , for actions of the $f(R(\Gamma))$ type, typically enforces that $\Gamma$ is the Levi–Civita connection of a conformally related metric $\hat{g}_{\mu\nu} = f'(R) g_{\mu\nu}$ in pure Palatini $f(R)$ , or more general algebraic conditions in genuinely hybrid or quadratic formalisms (Koivisto et al., 2013, Alvarez et al., 2017, Alvarez et al., 2017).

A powerful alternative is to enforce metric compatibility ( $\nabla_\gamma g_{\alpha\beta}=0$ ) as a constraint with a Lagrange multiplier $k^{\alpha\beta\gamma}$ , resulting in a first-order-plus-constraint formalism that is strictly equivalent to the second-order metric theory and encodes all higher-derivative terms via derivatives of the Lagrange multiplier (Benisty et al., 2018).

In teleparallel gravity, the first-order formalism is realized using the vielbein $e^a{}_\mu$ (the tetrad) and a connection with vanishing curvature but nontrivial torsion. Teleparallel equivalents of GR (TEGR) and their $f(T)$ extensions have field equations of second-order in derivatives of the tetrad, but a first-order hyperbolic reduction (in the pure-tetrad formalism) allows a system closely resembling the structure of Maxwell's equations, suitable for numerical relativity (Peshkov et al., 2022, Dupuis et al., 2019).

3. First-Order Reduction and Hamiltonian Structure

The existence of a first-order Hamiltonian formalism for variational gravitational problems is intimately tied to the projectability of the Poincaré–Cartan (P–C) form to lower jet bundles and a regularity condition—namely, affine dependence in highest derivatives and nondegeneracy of the associated fiber Hessian (María et al., 2015, Gaset et al., 2017):

For a second-order Lagrangian $L$ affine in $y^{a}_{ij}$ (the second derivatives), the P–C form $\Omega_L$ descends to $J^1E$ , and the Hamilton–Cartan equations can be written as a first-order system for the variables $(y^a, y^a_{i}; p^i_{a})$ with

$\frac{\partial p^i_a}{\partial x^i} + \frac{\partial H}{\partial y^a} = 0, \qquad \frac{\partial y^a}{\partial x^i} - \frac{\partial H}{\partial p^i_a} = 0$

where $H$ is the Hamiltonian constructed from $L$ (María et al., 2015).

The canonical Einstein–Hilbert Lagrangian is second order but affine in $g_{ij,kl}$ , and the projectibility and regularity conditions are satisfied, enabling a first-order Hamilton–Dirac formalism entirely on gauge-invariant (metric) phase space. The associated symplectic structure is naturally induced on the solution space (María et al., 2015, Gaset et al., 2017).
For generalized BF theories, as well as the Einstein–Hilbert case, an explicit first-order equivalent Lagrangian can be constructed whenever a global solution exists for the exactness of the coefficient $1$-form $w'_L$ built from the coefficients of the affine terms (María et al., 2015).

4. Exemplars: Teleparallel, Quadratic, and Conformal Gravity

Teleparallel Gravity and Dual Loop Quantum Gravity

Teleparallel gravity admits a first-order formalism in which the dynamical content resides in the torsion, rather than curvature. The Weitzenböck connection (curvature free) yields a quadratic torsion scalar as the Lagrangian density. The Einstein–Cartan action can be recast as a first-order teleparallel action up to a boundary term: $S_\mathrm{TP}^{(1)}[e, \Delta ; \bullet \omega] = \frac{1}{2\kappa} \int_M \epsilon_{IJKL} e^I \wedge e^J \wedge \left( D_{\bullet\omega} \Delta + \Delta \wedge \Delta \right)$ Variation yields flatness for the total connection and recovers the teleparallel Einstein equations. Discretization of the first-order action leads to the so-called "dual loop gravity" variables, with complementary phase spaces to standard LQG; the two approaches are linked by integration-by-parts in the presymplectic potential (Dupuis et al., 2019).

Quadratic Gravity

Quadratic curvature invariants in a first-order (Palatini) formalism define a class of unitary, power-counting renormalizable candidate theories for gravity. In four dimensions, the most general such action is a linear combination of 12 independent curvature-squared operators. In first-order settings, the metric and (torsionless) connection are varied independently, leading to field equations inequivalent to their second-order (metric) counterparts (Alvarez et al., 2017, Alvarez et al., 2017):

The propagator structure is crucial: due to the absence of higher-derivative kinetic terms for the metric (no $1/p^4$ poles), physical quanta propagate with $1/p^2$ fall-off; no Ostrogradsky ghosts appear (Alvarez et al., 2017).
Weyl invariance is present in the UV, but upon coupling to matter or quantum corrections, an Einstein–Hilbert term is dynamically generated in the IR (Alvarez et al., 2017, Alvarez et al., 2017).

First-Order Approaches to Conformal and Holographic Gravities

Gauge-inspired first-order formalisms unify conformal and scale-invariant gravities. In $SU(2,2)$ gauge-Higgs models, the gravitational field equations are first-order in the connection and Higgs field, and spontaneous symmetry breaking recovers the conformal extension of GR as well as fourth-order Weyl gravity as special limits (Zlosnik et al., 2016).

First-order reduction is also central in constructing holographic duals of defect and boundary CFTs. Here, the domain-wall or defect equations are cast as first-order systems using a fake superpotential or similar technique, allowing for closed-form analytic solutions and full characterization of operator spectra and correlation functions (Korovin, 2013, Ferrara et al., 2012).

5. First-Order Flow Equations: Superpotential and BPS-like Methods

A distinct application is the recasting of field equations in backgrounds with enhanced symmetry (domain walls, braneworlds, multi-centered black holes, or curved RG flows) as first-order ODE or PDE systems:

In thick-brane models in modified teleparallel gravity, the field equations for the warp factor $A(y)$ and scalar field $\phi(y)$ , generally second order, can be systematically reduced to first-order (BPS-like) systems by postulating a superpotential $W(\phi)$ such that $A' = -\frac{1}{3} W(\phi)$ , with a corresponding first-order equation for $\phi$ (Menezes, 2014, Moreira et al., 2021).
For multi-centered black holes, dyonic solutions, or holographic interfaces, a "fake superpotential" controls the gradient flow, yielding first-order equations for the scalar moduli and associated charges, again reducing the complexity of the solution space (Ferrara et al., 2012, Korovin, 2013).
Such reductions make underlying integrability, BPS structure, or duality properties manifest and enable explicit analytic solutions not available in the original second-order formulation.

6. Viability, Ghosts, and Physical Implications

From a dynamical and quantum perspective, first-order gravitational theories display critical distinctions from their second-order analogues:

In pure metric $f(R)$ , a dynamical scalar mode (the "scalaron") emerges, whereas in Palatini $f(R)$ , the extra mode is non-dynamical or algebraic. Only pure Palatini $f(R)$ , hybrid $f(X)\equiv f[R(\Gamma)-R(g)]$ , and certain restricted classes avoid ghosts and tachyons around flat background (Koivisto et al., 2013).
In truly quadratic (or higher-derivative) gravity, the notorious spin-2 ghost of metric $R^2 + R_{\mu\nu}^2$ theory is absent in the Palatini formalism thanks to the first-order propagator structure (Alvarez et al., 2017, Alvarez et al., 2017).
Constraints can be implemented to deform or enforce metric compatibility, torsion-freeness, or (in alternative settings) Weyl or Carroll/Galilei limits, resulting in geometrically rich and physically distinct solution spaces (Benisty et al., 2018, Guerrieri et al., 2020, Guerrieri et al., 2021).

Specific first-order formulations also enable systematic Hamiltonian or multisymplectic (covariant) decompositions and readily yield conserved charges, symplectic structures, and Noether–Wald identities, directly relevant to black hole thermodynamics and AdS/CFT correspondences (Nam et al., 2016, Gaset et al., 2017).

Table: Broad Taxonomy of First-Order Gravitational Formalisms

Formalism	Key Independent Variables	Characteristic Features
Palatini / Metric-affine	$(g_{\mu\nu}, \Gamma^\lambda_{\mu\nu})$	Second-order EL eqns in $g$ , geometric interpretation; quadratic extensions inequivalent to metric case (Koivisto et al., 2013)
Teleparallel / Pure tetrad $f(T)$	$e^a{}_\mu$ , (Weitzenböck conn.)	Torsion as geometry, Maxwell-like evolution, loop/dual-loop duality (Peshkov et al., 2022, Dupuis et al., 2019)
Supersymmetric / BPS flow	Metric, scalar, gauge, auxiliary	First-order "fake superpotential" ODE/PDE, flows, attractors (Ferrara et al., 2012, Menezes, 2014, Korovin, 2013)
Multisymplectic Hamiltonian	Jet/phase space variables	Projectible Poincaré–Cartan form, geometric constraint algorithm (María et al., 2015, Gaset et al., 2017)
Gauge-invariant conformal	Gauge, Higgs, frame, connection	First-order polynomial action, emergent metric/scale, spontaneous symmetry reduction (Zlosnik et al., 2016)

7. Outlook: Extensions and Open Problems

First-order formalisms have become fundamental in the theoretical development of gravitational physics, enabling ghost-free UV completions in quadratic gravity, concrete realization of geometric dualities (teleparallel vs. metric), and the construction of analytic solutions in settings with nontrivial topological, cosmological, or quantum properties.

Several open directions and challenges persist:

Full classification of ghost- and tachyon-free extensions in the Palatini and hybrid schemes, especially beyond linear order and in the presence of matter (Koivisto et al., 2013, Alvarez et al., 2017).
Quantization and non-perturbative dynamics, particularly for first-order fields (e.g., contorsion) in loop/dual-loop quantum gravity (Dupuis et al., 2019).
Complete geometric understanding and global analysis of first-order multisymplectic structures, especially in nontrivial topologies and with topological matter couplings (Gaset et al., 2017).
Application of first-order reductions in numerically challenging regimes, e.g., dynamical collapse in $f(R)$ gravity, benefiting from the hyperbolic structure and boundary regularity analyses (LeFloch et al., 7 Dec 2025).
Extension to non- and ultra-relativistic limits (Galilei/Carroll) and their geometric realization (Guerrieri et al., 2020, Guerrieri et al., 2021).

The first-order formalism thus provides a rigorous, unified, and versatile framework that underlies both classical and quantum gravitational theories, accommodating advanced field-theoretic structures, and enabling new physical insights and analytic techniques across General Relativity and its modifications.