First-Order Palatini Formulation

Updated 7 June 2026

The first-order (Palatini) formulation is a geometric variational framework where the coframe and affine connection are treated as independent variables, yielding on-shell metric compatibility and vanishing torsion.
It underpins a range of modified gravity theories and is foundational in canonical approaches to general relativity and Loop Quantum Gravity.
Its independent variation recovers Einstein’s equations in vacuum and supports Hamiltonian analysis via Ashtekar–Barbero variables, impacting cosmology and nonlocal gravity models.

The first-order (Palatini) formulation is a geometric variational framework in which the fundamental fields are the coframe (or tetrad/vierbein) and the affine connection, treated as independent variables. Unlike the metric (second-order) formulation, where the connection is determined uniquely as the Levi–Civita connection of the metric, the Palatini approach allows for independent variation of the affine structure, leading to field equations that impose metric compatibility and vanishing torsion as on-shell conditions rather than as off-shell identities. This formalism underlies a diverse family of modified gravity theories and has been foundational in canonical treatments of general relativity and the formulation of Loop Quantum Gravity.

1. Fundamental Objects and Action Principles

In four-dimensional spacetime, the first-order (Palatini) formulation employs:

The coframe (vierbein) $e^I{}_\mu$ , a basis of 1-forms, mapping spacetime indices $\mu,\,\nu,\ldots$ to an internal Lorentz frame $I,\,J,\ldots$ with Minkowski signature.
The connection 1-form $\omega_\mu{}^{IJ}$ , an independent $\mathfrak{so}(1,3)$ -valued gauge field.

The Palatini action, including a cosmological constant $\Lambda$ , is: $S_{\rm P}[e,\omega] = \frac{1}{2\kappa} \int_\mathcal{M} \epsilon_{IJKL}\, e^I \wedge e^J \wedge \left( F^{KL}[\omega] - \tfrac{\Lambda}{6}\, e^K \wedge e^L \right)$ where

$F^{IJ}[\omega] = d\omega^{IJ} + \omega^I{}_K \wedge \omega^{KJ}$

is the curvature 2-form of the connection, and $\epsilon_{IJKL}$ is the antisymmetric symbol. The spacetime metric arises as

$g_{\mu\nu} = \eta_{IJ}\, e^I{}_\mu\, e^J{}_\nu.$

This generalizes in affine or metric-affine gravity (Gaset et al., 2018), scalar-tensor extensions (0708.1163), and higher-curvature/modified actions (Júnior et al., 2024, Harko et al., 2010, Briscese et al., 2015).

2. Independent Variation and Field Equations

Variation of the action with respect to $\mu,\,\nu,\ldots$ 0 and $\mu,\,\nu,\ldots$ 1 yields:

Connection Variation (Palatini equation):

$\mu,\,\nu,\ldots$ 2

imposing the vanishing of the torsion 2-form $\mu,\,\nu,\ldots$ 3. This uniquely recovers $\mu,\,\nu,\ldots$ 4, the Levi–Civita spin connection (Yoon, 2017, Yoon, 2018).

Coframe Variation:

$\mu,\,\nu,\ldots$ 5

reduces, using $\mu,\,\nu,\ldots$ 6, to Einstein’s equations in the vacuum.

For more general actions in the coframe and connection (such as non-minimal, higher-curvature, or matter-coupled theories), the field equations generalize substantially and may yield algebraic, non-dynamical, or additional dynamical degrees of freedom (Olmo et al., 2011, Júnior et al., 2024, Harko et al., 2010, Briscese et al., 2015, Yoon, 2018).

3. Hamiltonian Structure and Constraint Analysis

After 3+1 decomposition, canonical analysis identifies the physical phase space as:

Canonical pairs: $\mu,\,\nu,\ldots$ 7, with $\mu,\,\nu,\ldots$ 8 the densitized triad and $\mu,\,\nu,\ldots$ 9 related to the extrinsic curvature.
First-class constraints:
- Gauss (internal Lorentz/gauge invariance): $I,\,J,\ldots$ 0
- Vector (spatial diffeomorphism): $I,\,J,\ldots$ 1
- Hamiltonian (scalar): $I,\,J,\ldots$ 2

The symplectic structure is encoded by: $I,\,J,\ldots$ 3 where $I,\,J,\ldots$ 4 are the Ashtekar–Barbero variables for real Barbero–Immirzi parameter $I,\,J,\ldots$ 5 (Yoon, 2017).

Constraint algebra closes and encodes gauge and diffeomorphism invariance. Second-class constraints arise in the presence of nondegeneracy or auxiliary conditions (Ciaglia et al., 2023).

4. Equivalence to Metric-Based Theories and Modifications

In vacuum and with appropriate compatibility conditions (vanishing torsion and metric compatibility $I,\,J,\ldots$ 6), the on-shell Palatini action reduces to the Einstein–Hilbert action: $I,\,J,\ldots$ 7 with dynamical content identical to classical general relativity (Yoon, 2017, Yoon, 2018, Gaset et al., 2018, Ciaglia et al., 2023).

For modified actions, such as

$I,\,J,\ldots$ 8 gravity, the Palatini variation yields scalar-tensor theories with a non-dynamical scalar degree of freedom, mapped to Brans–Dicke theories with $I,\,J,\ldots$ 9 (Olmo et al., 2011).
Higher-order, nonlocal, or non-minimal matter couplings, the Palatini approach leads to field equations that can be second-order (unlike the fourth-order metric case), may avoid introducing extra propagating degrees of freedom, and can alter classical solutions or perturbative spectra (Harko et al., 2010, Briscese et al., 2015, Júnior et al., 2024, Lykkas, 2022).
In scalar-tensor models, correct Palatini variation enforces metric compatibility via Lagrange multipliers; omitting this can result in physically distinct, typically strongly-coupled sectors (0708.1163).

5. Applications: Modified Gravity, Matter Couplings, and Phenomenology

The first-order Palatini formalism generalizes naturally to incorporate non-minimal matter couplings $\omega_\mu{}^{IJ}$ 0, leading to algebraic connections and auxiliary fields, non-geodesic motions, and potential energy-momentum non-conservation:

Field equations become algebraic in $\omega_\mu{}^{IJ}$ 1, with the connection being Levi–Civita of a conformally related “auxiliary” metric depending on derivatives of the gravitational action (Júnior et al., 2024, Harko et al., 2010).
Equations of motion for matter include an extra force orthogonal to four-velocity in non-conserving scenarios (Harko et al., 2010).
In cosmology and the weak-field limit, the Palatini approach modifies the effective Newton constant and cosmic expansion without introducing new propagating scalars, in contrast to metric formulations (Júnior et al., 2024, Giovannini, 2019, Lykkas, 2022).

For inflation and cosmological model building, higher-order Palatini models (e.g., $\omega_\mu{}^{IJ}$ 2) produce flattened Einstein-frame potentials without extra dynamical degrees of freedom, affecting observable tensors-to-scalar ratios (Giovannini, 2019, Lykkas, 2022, Gialamas et al., 2020). In the context of axion cosmology, non-minimal couplings and first-order variations modify instanton and wormhole solutions (Cheong et al., 2022).

6. Covariant Phase Space, Symplectic Geometry, and Gauge Content

The multisymplectic and covariant phase space formulations treat the Palatini theory as a gauge field theory with constraints and degeneracies in its multisymplectic form:

The presymplectic two-form on solution space arises naturally from the Schwinger–Weiss principle and encodes both the local gauge symmetries and the diffeomorphism invariance (Ciaglia et al., 2023, Gaset et al., 2018).
Gauge directions (local Lorentz/diffeomorphisms) generate degeneracies, and the reduced phase space inherits a canonical symplectic structure; the Peierls/DeWitt bracket is realized as a covariant Poisson bracket on the reduced phase space (Ciaglia et al., 2023).
Conservation laws and surface charges (e.g., ADM mass, angular momentum) admit rigorous construction and agree with the metric formulation when using Killing vector fields (Barnich et al., 2020).

In two dimensions, symplectic (Faddeev–Jackiw) and Dirac constraint analyses coincide on the gauge structure and highlight subtleties in gauge fixing within the first-order framework (McKeon, 2016).

7. Generalizations, Nonrelativistic and Nonlocal Extensions, and Continuum Mechanics

The Palatini approach extends to:

Nonrelativistic expansions: explicit first-order Galilean-covariant actions clarify the emergence of torsion and Newton–Cartan structures in the $\omega_\mu{}^{IJ}$ 3 limit (Hansen et al., 2020).
Nonlocal gravity and infinite-derivative actions: Palatini variations produce coupled, generally non-dynamical field equations that retain the Einstein branch as a solution, often failing to resolve curvature singularities (Briscese et al., 2015).
Elasticity and continuum mechanics: the coframe–connection Palatini approach provides a natural variational framework for Cosserat (micropolar) media with independent translations and rotations, exposing the underlying geometric structures and balance laws (Steinberg, 18 Mar 2026).