Variational Trinity: Geometric Gravity Formulations

• Variational Trinity defines three equivalent gravity formulations where curvature, torsion, and non-metricity yield Einstein’s equations through distinct action principles.
• It employs unique variational methods and boundary term treatments to ensure well-posed dynamics and consistent canonical (ADM) decompositions.
• The framework extends to modified theories such as f(R), f(T), and f(Q) gravity, offering insights for quantum gravity, numerical relativity, and gauge/gravity duality.

The Variational Trinity, often termed the Geometric Trinity of Gravity, refers to a set of three variational formulations of the gravitational field—curvature-based (General Relativity, GR), torsion-based (Teleparallel Equivalent of General Relativity, TEGR), and non-metricity-based (Symmetric Teleparallel Equivalent of General Relativity, STEGR)—that, under specific boundary conditions, yield dynamically equivalent field equations. These formulations are distinguished fundamentally by the primary geometric object responsible for mediating gravitational interactions—respectively, the Riemann curvature, spacetime torsion, or non-metricity of the affine connection. The Trinity plays a central role in theoretical gravitational physics, metric-affine geometry, and modern attempts to generalize and quantize gravitational dynamics (Capozziello et al., 2021, Erdmenger et al., 2023, Capozziello et al., 11 Mar 2025).

1. Formulations and Action Principles

The three faces of the Variational Trinity are encoded in action functionals that differ only up to total divergences (boundary terms), but correspond to different choices of geometric variables and connections:

• Curvature-Based (Einstein–Hilbert):

$$S_{\rm curv} = \frac{1}{2\kappa^2} \int d^4x\,\sqrt{-g}\;R[g] + S_{\rm m}[g, \Psi]$$

where $R[g]$ is the Ricci scalar of the Levi–Civita connection, $g_{\mu\nu}$ the metric, and $S_{\rm m}$ the matter action.

• Torsion-Based (TEGR):

$$S_{\rm tors} = \frac{1}{2\kappa^2} \int d^4x\,e\,T[e^A{}_{\mu},\omega^A{}_{B\mu}] + S_{\rm m}[e^A{}_{\mu},\Psi]$$

where $e = \det e^A{}_\mu$, $T$ the torsion scalar, and $\omega^A{}_{B\mu}$ a flat spin connection.

• Nonmetricity-Based (STEGR):

$$S_{\rm nonmet} = \frac{1}{2\kappa^2} \int d^4x\,\sqrt{-g}\;Q[g_{\mu\nu},\Gamma^\alpha{}_{\mu\nu}] + S_{\rm m}[g, \Psi]$$

with $Q$ the non-metricity scalar, and $R[g]$0 a symmetric, flat, non-metric connection (Capozziello et al., 2021, Capozziello et al., 11 Mar 2025).

Each of these action functionals leads, after variation with respect to the appropriate dynamical fields (metric, tetrad, or connection), to field equations that reduce algebraically to the Einstein field equations in vacuum.

2. Dynamical Equivalence and Boundary Terms

The dynamical equivalence of the three formulations arises from two essential identities, expressing the Ricci scalar in terms of the torsion or non-metricity scalars plus total derivatives:

$R[g]$1

$R[g]$2

Thus, the differences between the actions are pure boundary terms, which vanish (or are absorbed into surface fluxes) under suitable asymptotic or boundary conditions. As a consequence, the field equations sourced by metric/tetrad/connection variations are identical modulo redefinition of the superpotential (March et al., 2023).

A summary of the action functionals and their equivalence:

Formulation	Action functional	Primary dynamical variable
GR	$R[g]$3	$R[g]$4
TEGR	$R[g]$5	$R[g]$6
STEGR	$R[g]$7	$R[g]$8

The equivalence is manifest only if the Gibbons–Hawking–York (GHY) boundary terms are carefully managed. In the teleparallel and symmetric teleparallel cases, the necessary GHY term vanishes for the variational problem to be well-posed; retaining it would lead to an ill-defined variational problem. Metric–affine decompositions clarify that, after imposing the teleparallel (resp. symmetric teleparallel) conditions, all relevant surface terms are built into the bulk torsion or non-metricity Lagrangians themselves (Erdmenger et al., 2023).

3. Hamiltonian Structure and Constraint Algebras

Upon $R[g]$9 (Arnowitt–Deser–Misner, ADM) decomposition, all three formulations admit well-defined canonical structures with associated momenta, constraint algebras, and evolution equations:

• General Relativity: The ADM Hamiltonian,

$g_{\mu\nu}$0

with constraints forming the Dirac algebra.

• TEGR: The Hamiltonian,

$g_{\mu\nu}$1

where $g_{\mu\nu}$2 are local Lorentz Gauss constraints.

• STEGR: The Hamiltonian,

$g_{\mu\nu}$3

with $g_{\mu\nu}$4 enforcing the coincident gauge (Capozziello et al., 2021).

In each case, the final Dirac constraint algebra is isomorphic to ADM gravity, with only additional Gauss-type constraints enforcing Lorentz or affine invariance in the teleparallel or symmetric teleparallel cases. This structure underpins both canonical quantization attempts and the numerical implementation of each formulation.

4. Boundary Terms, Variational Principles, and Extensions

The treatment of boundary terms is central to the trinity. In the generalized context of metric-affine gravity, the distinction and transformation of GHY boundary terms in moving between curvature, torsion, and non-metricity representations are key to establishing well-posedness and dynamical equivalence. For higher-curvature actions or $g_{\mu\nu}$5-like extensions, an analogous prescription applies: the Einstein–Hilbert bulk and boundary terms are rewritten in "teleparallel" variables, and the teleparallel (or non-metric) limit is taken only after discarding the relevant metric-affine GHY term. This results in well-posedness without retaining a standalone GHY term in the (S)TEGR theories (Erdmenger et al., 2023).

For generalized actions,

Theory	Action (bulk + boundary)	Dynamical equivalence condition
$g_{\mu\nu}$6	$g_{\mu\nu}$7	—
$g_{\mu\nu}$8	$g_{\mu\nu}$9	not equivalent unless $S_{\rm m}$0 form
$S_{\rm m}$1	$S_{\rm m}$2	not equivalent unless $S_{\rm m}$3 form

Only when the Lagrangians are functions of $S_{\rm m}$4 or $S_{\rm m}$5 does the dynamical equivalence to $S_{\rm m}$6 gravity persist; otherwise, new propagating degrees of freedom and symmetry breakings can occur (Capozziello et al., 11 Mar 2025).

5. Physical and Mathematical Consequences

Despite the equality of field equations, the trinity distinguishes itself via the underlying geometric structures, physical interpretations (geodesics vs. autoparallels), and the natural coupling to matter and spin. In the non-relativistic limit, the geometric trinity (via $S_{\rm m}$7 expansion) contracts to a single, weaker structure—Maxwell gravitation—unifying the Newton–Cartan, teleparallel, and symmetric teleparallel Newtonian theories into a Maxwell-type theory, in sharp contrast to the purely relativistic situation where GR itself exhausts the common core of dynamical content (March et al., 2023).

Boundary terms acquire further significance in holographic contexts. For gauge/gravity duality scenarios (e.g., AdS/CFT), the bulk/boundary dictionary adapts accordingly: spin and hypermomentum fluxes are naturally encoded via the deformation one-form $S_{\rm m}$8 of the metric-affine decomposition, and non-metricity/torsion variables absorb previously separate boundary counterterms. Topological invariants, such as Gauss–Bonnet and Pontryagin terms, also admit teleparallel or non-metric analogues with potentially distinct physical signatures in dual CFTs (Erdmenger et al., 2023).

6. Extensions: f-Theories and Modified Gravity

The trinity framework straightforwardly generalizes to extended theories by promoting the fundamental scalar (curvature, torsion, or non-metricity) to arbitrary functions, yielding $S_{\rm m}$9, $$S_{\rm tors} = \frac{1}{2\kappa^2} \int d^4x\,e\,T[e^A{}_{\mu},\omega^A{}_{B\mu}] + S_{\rm m}[e^A{}_{\mu},\Psi]$$0, and $$S_{\rm tors} = \frac{1}{2\kappa^2} \int d^4x\,e\,T[e^A{}_{\mu},\omega^A{}_{B\mu}] + S_{\rm m}[e^A{}_{\mu},\Psi]$$1 gravity. However, equivalence between these extended theories does not persist in general. Only boundary-corrected forms—$$S_{\rm tors} = \frac{1}{2\kappa^2} \int d^4x\,e\,T[e^A{}_{\mu},\omega^A{}_{B\mu}] + S_{\rm m}[e^A{}_{\mu},\Psi]$$2, $$S_{\rm tors} = \frac{1}{2\kappa^2} \int d^4x\,e\,T[e^A{}_{\mu},\omega^A{}_{B\mu}] + S_{\rm m}[e^A{}_{\mu},\Psi]$$3—remain dynamically equivalent to $$S_{\rm tors} = \frac{1}{2\kappa^2} \int d^4x\,e\,T[e^A{}_{\mu},\omega^A{}_{B\mu}] + S_{\rm m}[e^A{}_{\mu},\Psi]$$4. In the absence of appropriate boundary term inclusion, $$S_{\rm tors} = \frac{1}{2\kappa^2} \int d^4x\,e\,T[e^A{}_{\mu},\omega^A{}_{B\mu}] + S_{\rm m}[e^A{}_{\mu},\Psi]$$5 or $$S_{\rm tors} = \frac{1}{2\kappa^2} \int d^4x\,e\,T[e^A{}_{\mu},\omega^A{}_{B\mu}] + S_{\rm m}[e^A{}_{\mu},\Psi]$$6 theories generically propagate extra degrees of freedom not present in $$S_{\rm tors} = \frac{1}{2\kappa^2} \int d^4x\,e\,T[e^A{}_{\mu},\omega^A{}_{B\mu}] + S_{\rm m}[e^A{}_{\mu},\Psi]$$7, and Lorentz invariance is typically broken. This leads to the so-called Extended Geometric Trinity: $$S_{\rm tors} = \frac{1}{2\kappa^2} \int d^4x\,e\,T[e^A{}_{\mu},\omega^A{}_{B\mu}] + S_{\rm m}[e^A{}_{\mu},\Psi]$$8 (Capozziello et al., 11 Mar 2025).

A schematic summary:

Theory	Variables	Geometry	Actions Equivalent When
$$S_{\rm tors} = \frac{1}{2\kappa^2} \int d^4x\,e\,T[e^A{}_{\mu},\omega^A{}_{B\mu}] + S_{\rm m}[e^A{}_{\mu},\Psi]$$9	$e = \det e^A{}_\mu$0	curvature	—
$e = \det e^A{}_\mu$1	$e = \det e^A{}_\mu$2	torsion	includes boundary term
$e = \det e^A{}_\mu$3	$e = \det e^A{}_\mu$4	non-metricity	includes boundary term

A plausible implication is that the space of physically viable metric-affine gravity theories is tightly constrained by considerations of boundary terms and underlying geometric structure.

7. Outlook and Applications

The Variational Trinity undergirds developments in canonical quantum gravity, geometric reformulations of gravitational gauge theories, analysis of topological invariants, and the construction of viable extensions to GR. In numerical relativity, formulations based on torsion or non-metricity can offer practical or conceptual advantages in discretization or constraint handling, owing to their distinct Hamiltonian formulations. In gauge/gravity duality, the trinity demands a reformulation of the boundary dictionary when torsion and non-metricity are active geometric players.

The equivalence of gravitational dynamics in the trinity is ultimately established only on-shell and is highly sensitive to the treatment of boundary terms; subleading differences may influence observable physics in extended or modified theories, indicating fertile ground for future research, particularly in cosmological and high-energy domains (Capozziello et al., 2021, Erdmenger et al., 2023, Capozziello et al., 11 Mar 2025, March et al., 2023).