Trinity of Gravity: Geometric Equivalence

• Trinity of Gravity is defined as three classically equivalent formulations of Einsteinian gravity that encode gravitation via curvature, torsion, or non-metricity.
• Each formulation employs distinct gauge choices and geometric variables, with differences confined to boundary terms in their action formulations.
• The framework facilitates extensions to modified gravity theories, cosmological applications, and quantum treatments while clarifying conserved charges and energy-momentum definitions.

The Trinity of Gravity, more commonly termed the Geometric Trinity of Gravity, denotes three classically equivalent formulations of Einsteinian gravitation in which the gravitational interaction is encoded, respectively, in curvature, torsion, or non-metricity. In its standard form, the trinity comprises General Relativity (GR), the Teleparallel Equivalent of General Relativity (TEGR), and the Symmetric Teleparallel Equivalent of General Relativity (STEGR). These formulations employ different geometric variables and distinct gauge choices for the affine connection, yet their actions differ only by boundary terms and they yield the same classical field equations and physical solutions in the Einstein-like sector (Jimenez et al., 2019).

1. The three geometric formulations

In metric-affine language, spacetime is endowed with a metric $g_{\mu\nu}$ and an affine connection $\Gamma^\alpha_{\mu\nu}$. The general connection admits the decomposition

$$\Gamma^\alpha_{\mu\nu} = \left\{^{\alpha}_{\mu\nu}\right\} + K^\alpha_{\mu\nu} + L^\alpha_{\mu\nu},$$

where $\left\{^{\alpha}_{\mu\nu}\right\}$ is the Levi-Civita connection, $K^\alpha_{\mu\nu}$ is the contortion, and $L^\alpha_{\mu\nu}$ is the disformation. The associated geometric tensors are the curvature $R^\alpha_{\ \beta\mu\nu}$, the torsion $T^\alpha_{\ \mu\nu} = 2\Gamma^\alpha_{[\mu\nu]}$, and the non-metricity $Q_{\alpha\mu\nu} = \nabla_\alpha g_{\mu\nu}$ (Jimenez et al., 2019).

The trinity is distinguished by which of these structures is taken as nontrivial:

Formulation	Nonzero structure	Vanishing structures
GR	Curvature	Torsion, non-metricity
TEGR	Torsion	Curvature, non-metricity
STEGR	Non-metricity	Curvature, torsion

In GR, gravitation is described by the spacetime curvature of the Levi-Civita connection, which is torsion-free and metric-compatible. In TEGR, the connection is flat and metric-compatible but torsionful; this is the Weitzenböck framework. In STEGR, the connection is flat and torsion-free but not metric-compatible; in the coincident gauge one has $\Gamma^\lambda_{\mu\nu} = 0$, and gravity is entirely encoded in the non-metricity tensor (Branchina et al., 27 Jun 2026).

The corresponding Einstein-like actions are

$\Gamma^\alpha_{\mu\nu}$0

$\Gamma^\alpha_{\mu\nu}$1

$\Gamma^\alpha_{\mu\nu}$2

with $\Gamma^\alpha_{\mu\nu}$3 the torsion scalar and $\Gamma^\alpha_{\mu\nu}$4 the non-metricity scalar in the specific combinations that reproduce Einstein dynamics (Jimenez et al., 2019).

2. Dynamical equivalence and boundary-term identities

The classical equivalence of the three formulations is encoded in identities relating the Ricci scalar to torsion and non-metricity scalars plus total divergences. In the teleparallel case,

$\Gamma^\alpha_{\mu\nu}$5

while in the symmetric teleparallel case

$\Gamma^\alpha_{\mu\nu}$6

Equivalent sign conventions also appear in the literature as

$\Gamma^\alpha_{\mu\nu}$7

These relations imply that the Einstein-Hilbert action and the TEGR or STEGR actions differ only by boundary terms, so that the resulting field equations coincide in the linear Einsteinian sector (Capozziello et al., 11 Mar 2025).

This equivalence is exact for the Einstein-Hilbert, TEGR, and STEGR actions, but it is generally lost for nonlinear extensions unless the boundary terms are retained inside the argument of the nonlinear function. Thus $\Gamma^\alpha_{\mu\nu}$8 and $\Gamma^\alpha_{\mu\nu}$9 are not dynamically equivalent to $$\Gamma^\alpha_{\mu\nu} = \left\{^{\alpha}_{\mu\nu}\right\} + K^\alpha_{\mu\nu} + L^\alpha_{\mu\nu},$$0, whereas the combinations $$\Gamma^\alpha_{\mu\nu} = \left\{^{\alpha}_{\mu\nu}\right\} + K^\alpha_{\mu\nu} + L^\alpha_{\mu\nu},$$1, $$\Gamma^\alpha_{\mu\nu} = \left\{^{\alpha}_{\mu\nu}\right\} + K^\alpha_{\mu\nu} + L^\alpha_{\mu\nu},$$2, or equivalently $$\Gamma^\alpha_{\mu\nu} = \left\{^{\alpha}_{\mu\nu}\right\} + K^\alpha_{\mu\nu} + L^\alpha_{\mu\nu},$$3, restore equivalence. This defines the Extended Geometric Trinity of Gravity, consisting of $$\Gamma^\alpha_{\mu\nu} = \left\{^{\alpha}_{\mu\nu}\right\} + K^\alpha_{\mu\nu} + L^\alpha_{\mu\nu},$$4, $$\Gamma^\alpha_{\mu\nu} = \left\{^{\alpha}_{\mu\nu}\right\} + K^\alpha_{\mu\nu} + L^\alpha_{\mu\nu},$$5, and $$\Gamma^\alpha_{\mu\nu} = \left\{^{\alpha}_{\mu\nu}\right\} + K^\alpha_{\mu\nu} + L^\alpha_{\mu\nu},$$6 (Capozziello et al., 11 Mar 2025).

The same logic extends beyond the Einstein-Hilbert action. It has been shown that the equivalence between curvature, teleparallel, and symmetric teleparallel descriptions can be generalized to any action built out of the curvature two-form, provided the appropriate boundary contribution is accounted for. This broader construction has been termed the generalized geometrical trinity of gravity (Erdmenger et al., 2023).

A related higher-order result concerns the Gauss–Bonnet topological scalar. In metric-teleparallel, symmetric teleparallel, and general teleparallel formalisms, the full expressions can be written explicitly in terms of torsion, non-metricity, and Levi-Civita covariant derivatives. The counting of invariant terms shows that the Gauss–Bonnet combination excludes some terms that would otherwise be allowed in a generic effective field theory expansion, indicating that the preservation of General Relativity symmetries at higher order is highly nontrivial in teleparallel formulations (Bajardi et al., 2023).

3. Gauge choices, connection fixing, and pre-geometric origin

A central technical feature of the trinity is that the same gravitational dynamics can be represented through different gauge choices for the connection. In the broken phase of a pre-geometric gauge theory, the emergent metric and connection can be arranged so as to reproduce each member of the trinity by a suitable gauge fixing. In particular, the Levi-Civita gauge yields the curvature formulation, the Weitzenböck gauge imposes $$\Gamma^\alpha_{\mu\nu} = \left\{^{\alpha}_{\mu\nu}\right\} + K^\alpha_{\mu\nu} + L^\alpha_{\mu\nu},$$7 and produces the torsion formulation, and the coincident gauge gives the non-metricity formulation through

$$\Gamma^\alpha_{\mu\nu} = \left\{^{\alpha}_{\mu\nu}\right\} + K^\alpha_{\mu\nu} + L^\alpha_{\mu\nu},$$8

This construction realizes GR, TEGR, and STEGR as distinct emergent phases of the same pre-geometric gauge-theoretic structure (Capozziello et al., 16 Jun 2026).

In that framework, spacetime initially has no metric structure. The fundamental fields are a gauge field $$\Gamma^\alpha_{\mu\nu} = \left\{^{\alpha}_{\mu\nu}\right\} + K^\alpha_{\mu\nu} + L^\alpha_{\mu\nu},$$9 and a Higgs-like field $\left\{^{\alpha}_{\mu\nu}\right\}$0, with spontaneous symmetry breaking generating the tetrad and spin connection according to

$\left\{^{\alpha}_{\mu\nu}\right\}$1

and the emergent metric

$\left\{^{\alpha}_{\mu\nu}\right\}$2

A prototypical pre-geometric Lagrangian is

$\left\{^{\alpha}_{\mu\nu}\right\}$3

which after spontaneous symmetry breaking reduces to the Einstein-Hilbert action plus cosmological constant (Capozziello et al., 16 Jun 2026).

This perspective suggests that the trinity is not merely a set of algebraic rewritings, but a structured set of low-energy realizations of a more fundamental metric-affine gauge theory. A plausible implication is that the familiar distinction between curvature, torsion, and non-metricity can be interpreted as an emergent choice of geometric language rather than a distinction at the pre-geometric level.

4. Conserved charges, boundary structure, and variational principles

The geometrical trinity has been used to reformulate the long-standing problem of gravitational energy, momentum, and entropy in a covariant metric-affine language. In a generic metric-affine theory, the Noether current $\left\{^{\alpha}_{\mu\nu}\right\}$4 satisfies

$\left\{^{\alpha}_{\mu\nu}\right\}$5

and on shell it becomes exact, $\left\{^{\alpha}_{\mu\nu}\right\}$6, so that the associated charge reduces to a surface integral. The corresponding conserved quantity takes the form

$\left\{^{\alpha}_{\mu\nu}\right\}$7

This establishes a holographic characterization of gravitational entropy and energy-momentum across the trinity (Jiménez et al., 2021).

Within this program, the canonical resolution of the gravitational energy-momentum problem identifies an inertial frame in which the bulk gravitational energy-momentum current vanishes, while the physically meaningful charge is encoded entirely in a boundary integral. In symmetric teleparallel gravity, the surface charge can be written as

$\left\{^{\alpha}_{\mu\nu}\right\}$8

with $\left\{^{\alpha}_{\mu\nu}\right\}$9 the gravitational excitation associated with non-metricity. Closely related results show that a single canonical excitation tensor unifies classical pseudotensor constructions due to Einstein, von Freud, Landau-Lifshitz, Bergmann-Thomson, Papapetrou, Weinberg, and Møller in a background-independent framework (Gomes et al., 2022).

The boundary structure is equally important for the variational problem. In the metric formulation of GR, a Gibbons-Hawking-York term must be added for a well-posed variation with boundary. By contrast, in TEGR and STEGR the corresponding GHY term must vanish for consistency. More precisely, the boundary term $K^\alpha_{\mu\nu}$0 that relates GR to TEGR or STEGR equals the difference between the GHY term of GR and that of metric-affine gravity. This result is one of the mechanisms underlying the generalized geometrical trinity for actions built from the curvature two-form (Erdmenger et al., 2023).

These results have direct implications for black-hole thermodynamics and gravitational radiation. In the trinity framework, black-hole entropy is realized as a Noether charge, and quasilocal energy is defined by surface integrals in a canonical frame. The Schwarzschild mass is recovered quasilocally, while in true vacuum gravitational waves carry no local energy according to the canonical current; the physical content is encoded in boundary data and matter interaction (Gomes et al., 2022).

5. Extensions, applications, and domain-specific realizations

The trinity has been extended to several adjacent settings. One extension concerns unimodular gravity. By employing the Weyl compensator formalism, a Weyl transverse diffeomorphism invariant teleparallel theory can be constructed whose low-energy dynamics has a single spin two graviton without a scalar degree of freedom and is equivalent to unimodular gravity, as well as Einstein’s general relativity with an adjustable cosmological constant, at the nonlinear level. Combined with the corresponding symmetric teleparallel construction, this yields a geometrical trinity of unimodular gravity (Nakayama, 2022).

Another extension is non-relativistic. By taking the non-relativistic limit of STEGR, one obtains a non-metric theory equivalent to Newton–Cartan theory and its teleparallel equivalent, thereby completing a non-relativistic geometric trinity. The three nodes are Newton–Cartan theory, Teleparallel Newton–Cartan, and Symmetric Teleparallel Newton–Cartan, characterized respectively by curvature, torsion, and non-metricity. In this regime, the dynamical common core of the non-relativistic trinity is Maxwell gravitation, whereas in the relativistic case no analogous distinct common core exists beyond GR itself (Wolf et al., 2023, March et al., 2023).

Cosmological model building has provided a large testing ground. In scalar-field dark-energy models with Yukawa-like non-minimal coupling $K^\alpha_{\mu\nu}$1, where $K^\alpha_{\mu\nu}$2, the coupling has the same mathematical form in GR, TG, and STG. Minimal coupling can admit a dust-like attractor with $K^\alpha_{\mu\nu}$3 in quasiquintessence models, while non-minimal coupling removes this behavior and drives the system toward $K^\alpha_{\mu\nu}$4 across the three gravity scenarios (Carloni et al., 2024). In a distinct analysis of a dilaton-inspired scalar field, late-time accelerated expansion is found in GR and teleparallel gravity, but no attractor points are found in the symmetric teleparallel case for the chosen parameters when a non-coincident gauge is used (Carloni et al., 27 Apr 2025).

At the minisuperspace level, Noether symmetry methods have been used to study the extended trinity in classical and quantum cosmology. In this setting, equivalence among the different formulations is restored by including the divergence terms that relate $K^\alpha_{\mu\nu}$5, $K^\alpha_{\mu\nu}$6, and $K^\alpha_{\mu\nu}$7. Exact cosmological solutions and Wheeler–DeWitt equations have been derived for $K^\alpha_{\mu\nu}$8, $K^\alpha_{\mu\nu}$9, and $L^\alpha_{\mu\nu}$0 models, with explicit demonstration that the boundary-augmented theories reproduce the richer solution space of extended GR (Battista et al., 28 Feb 2026).

6. Semiclassical robustness and foundational controversies

Although the trinity is classically equivalent, an important question is whether this equivalence survives beyond classical dynamics. A nontrivial semiclassical test is vacuum decay. In the analysis of false-vacuum tunneling mediated by an $L^\alpha_{\mu\nu}$1-symmetric bounce, the decay rate

$L^\alpha_{\mu\nu}$2

has been computed in GR, TEGR, and STEGR. Because the Lagrangians differ only by boundary terms and these boundary contributions cancel between the bounce and false-vacuum configurations, the tunneling exponents coincide: $L^\alpha_{\mu\nu}$3 This provides a concrete example in which equivalence extends beyond classical field equations to a semiclassical quantum observable (Branchina et al., 27 Jun 2026).

The trinity is also the subject of an active foundational dispute. One line of work emphasizes that GR, TEGR, and STEGR are dynamically equivalent but differ in how they relate to the Equivalence Principle (EP). In that view, the EP is fundamental in the construction of GR, whereas TEGR and STEGR recover it without assuming it as a foundational principle; this suggests that the EP may be emergent rather than fundamental (Mancini et al., 11 Jan 2025).

A contrary critique argues that there is no genuine “trinity” in a foundational sense, because once the metric is taken as physical, the Levi-Civita connection is uniquely fixed and alternative metric-affine structures amount, in the GR-equivalent sector, to unobservable auxiliary constructions. On that reading, teleparallel and symmetric teleparallel equivalents are mathematically valid but physically redundant descriptions unless the dynamics is genuinely modified (Golovnev, 2024).

These positions do not negate the formal equivalence results; rather, they identify a distinction between dynamical equivalence and foundational interpretation. The former is established at the level of the Einsteinian field equations and several nontrivial applications. The latter remains contested, especially when questions of the Equivalence Principle, gauge redundancy, observable structure, and theory underdetermination are placed in the foreground.

7. Significance

The Trinity of Gravity has become a unifying framework for metric-affine gravitation. It organizes GR, TEGR, and STEGR as three representations of the same classical spin-2 dynamics; clarifies how boundary terms, conserved charges, and gauge choices govern their equivalence; and provides a systematic platform for extensions such as $L^\alpha_{\mu\nu}$4, $L^\alpha_{\mu\nu}$5, and $L^\alpha_{\mu\nu}$6, unimodular constructions, higher-order topological invariants, and non-relativistic limits (Jimenez et al., 2019).

Its significance is both technical and conceptual. Technically, the trinity enables reformulations with first-derivative actions, alternate Hamiltonian and $L^\alpha_{\mu\nu}$7 decompositions, and covariant treatments of quasilocal energy and entropy. Conceptually, it sharpens the distinction between physical dynamics and geometric representation. Whether curvature, torsion, and non-metricity should be regarded as equally fundamental structures, emergent descriptions, or merely alternative calculational languages remains an open interpretive question. What is firmly established is that, in the Einstein-equivalent sector, these three geometric languages encode the same classical gravitational dynamics, and in at least some semiclassical settings they continue to do so (Capozziello et al., 2021, Branchina et al., 27 Jun 2026).