Geometric Trinity of Gravity Equivalence

• Geometric Trinity of Gravity is an equivalence framework where gravitational dynamics are represented by curvature (GR), torsion (TEGR), or non-metricity (STEGR), all yielding the same classical field equations.
• The equivalence is achieved via boundary-term identities that ensure the different affine structures only introduce total divergences in the respective gravitational actions.
• Nonlinear extensions and gauge choices in f(R), f(T), and f(Q-B) models reveal that while linear equivalence holds, modifications can break the symmetry and lead to distinct phenomenological implications.

The geometric trinity of gravity is the statement that Einsteinian gravitational dynamics admits three exactly equivalent representations built on distinct geometric carriers: curvature in General Relativity (GR), torsion in the Teleparallel Equivalent of General Relativity (TEGR), and non-metricity in the Symmetric Teleparallel Equivalent of General Relativity (STEGR). In the metric-affine framework, these theories use different fundamental variables and different affine structures, yet their actions differ only by boundary terms and therefore generate the same classical field equations under the standard assumptions on boundary conditions. In the convention adopted in "Extended Geometric Trinity of Gravity" (Capozziello et al., 11 Mar 2025), the key identities are $\mathring{R} = \hat T - \tilde B$ and $\mathring{R} = Q - B$, with $e=\det e^a{}_\mu=\sqrt{-g}$; other sign conventions also occur in the literature.

1. Metric-affine setting and the three geometric sectors

A general affine connection $\Gamma^\lambda{}_{\mu\nu}$ carries three independent tensors: curvature, torsion, and non-metricity. In the notation used across the trinity literature, these are

$$R^\rho{}_{\sigma\mu\nu}(\Gamma),\qquad T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$

Relative to the Levi–Civita connection $\mathring{\Gamma}^\lambda{}_{\mu\nu}$, the general connection decomposes as

$$\Gamma^\lambda{}_{\mu\nu}=\mathring{\Gamma}^\lambda{}_{\mu\nu}+K^\lambda{}_{\mu\nu}+L^\lambda{}_{\mu\nu},$$

where $K^\lambda{}_{\mu\nu}$ is the contortion and $L^\lambda{}_{\mu\nu}$ is the disformation (Capozziello et al., 11 Mar 2025).

The curvature formulation, namely GR, takes the metric $g_{\mu\nu}$ as the fundamental variable and fixes the connection to be Levi–Civita, so that torsion and non-metricity vanish while curvature is generically nonzero. The torsion formulation, namely TEGR, uses a tetrad $\mathring{R} = Q - B$0 together with an inertial Lorentz spin connection $\mathring{R} = Q - B$1; in the Weitzenböck gauge $\mathring{R} = Q - B$2, the connection becomes

$\mathring{R} = Q - B$3

so curvature vanishes, metric compatibility is preserved, and torsion is nonzero. The non-metricity formulation, namely STEGR, uses the metric and a flat, torsionless affine connection; in the coincident gauge one can set $\mathring{R} = Q - B$4 globally, so that all gravitational information is carried by non-metricity (Jimenez et al., 2019).

This tripartite classification is not merely taxonomic. It isolates three distinct ways of encoding the same massless spin-2 dynamics: through the Riemann tensor of a Levi–Civita connection, through the torsion of a flat metric-compatible connection, or through the non-metricity of a flat torsionless connection. A plausible implication is that the trinity is best understood as a structural equivalence inside metric-affine gravity rather than as a claim that curvature, torsion, and non-metricity are simultaneously present in the same GR sector.

2. Exact equivalence of GR, TEGR, and STEGR

In the curvature representation, the Einstein–Hilbert action is

$\mathring{R} = Q - B$5

In the teleparallel representation, the action is written in terms of the torsion scalar

$\mathring{R} = Q - B$6

with

$\mathring{R} = Q - B$7

and

$\mathring{R} = Q - B$8

In the symmetric teleparallel representation, the non-metricity scalar is

$\mathring{R} = Q - B$9

with $e=\det e^a{}_\mu=\sqrt{-g}$0 and $e=\det e^a{}_\mu=\sqrt{-g}$1, and the action is

$e=\det e^a{}_\mu=\sqrt{-g}$2

or equivalently with $e=\det e^a{}_\mu=\sqrt{-g}$3 once the sign convention is tracked consistently (Capozziello et al., 11 Mar 2025).

The equivalence is established by the boundary-term identities. For TEGR,

$e=\det e^a{}_\mu=\sqrt{-g}$4

and for STEGR,

$e=\det e^a{}_\mu=\sqrt{-g}$5

Thus the three Einstein-level actions differ only by total divergences (Capozziello et al., 11 Mar 2025).

These identities are not formal curiosities. They require specific geometric assumptions. TEGR needs a flat connection, metric compatibility, and the correct treatment of the inertial spin connection; STEGR needs a flat and torsionless connection, with the coincident gauge available globally. Under these assumptions, GR, TEGR, and STEGR produce the same classical gravitational dynamics even though they assign gravity to different geometric objects.

3. Nonlinear extensions and the Extended Geometric Trinity

The exact equivalence of the linear Einstein-level trinity does not survive arbitrary nonlinearization. In metric language, one may consider

$e=\det e^a{}_\mu=\sqrt{-g}$6

whose metric field equations are

$e=\det e^a{}_\mu=\sqrt{-g}$7

with trace

$e=\det e^a{}_\mu=\sqrt{-g}$8

The naïve teleparallel and symmetric teleparallel analogues $e=\det e^a{}_\mu=\sqrt{-g}$9 and $\Gamma^\lambda{}_{\mu\nu}$0 are not, in general, equivalent to $\Gamma^\lambda{}_{\mu\nu}$1, because the boundary terms that supported the linear equivalence are no longer external to the function (Capozziello et al., 11 Mar 2025).

The equivalence is restored only when the boundary combinations are retained inside the nonlinear function. The resulting theories are

$\Gamma^\lambda{}_{\mu\nu}$2

or, equivalently, the actions

$\Gamma^\lambda{}_{\mu\nu}$3

This is the Extended Geometric Trinity of Gravity (Capozziello et al., 11 Mar 2025).

The $\Gamma^\lambda{}_{\mu\nu}$4 analysis makes the same point from the symmetric teleparallel side. The field equations of $\Gamma^\lambda{}_{\mu\nu}$5 reduce exactly to the metric $\Gamma^\lambda{}_{\mu\nu}$6 equations when $\Gamma^\lambda{}_{\mu\nu}$7, since then $\Gamma^\lambda{}_{\mu\nu}$8, and the connection equation

$\Gamma^\lambda{}_{\mu\nu}$9

becomes trivial in the $$R^\rho{}_{\sigma\mu\nu}(\Gamma),\qquad T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$0 sector (Capozziello et al., 2023). In that framework, $$R^\rho{}_{\sigma\mu\nu}(\Gamma),\qquad T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$1 remains second order in the metric, whereas $$R^\rho{}_{\sigma\mu\nu}(\Gamma),\qquad T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$2 becomes fourth order; this is why $$R^\rho{}_{\sigma\mu\nu}(\Gamma),\qquad T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$3 aligns with $$R^\rho{}_{\sigma\mu\nu}(\Gamma),\qquad T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$4, while pure $$R^\rho{}_{\sigma\mu\nu}(\Gamma),\qquad T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$5 does not (Capozziello et al., 2023).

At the linearized level around Minkowski, $$R^\rho{}_{\sigma\mu\nu}(\Gamma),\qquad T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$6, $$R^\rho{}_{\sigma\mu\nu}(\Gamma),\qquad T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$7, and $$R^\rho{}_{\sigma\mu\nu}(\Gamma),\qquad T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$8 all carry the same $$R^\rho{}_{\sigma\mu\nu}(\Gamma),\qquad T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$9 tensor $\mathring{\Gamma}^\lambda{}_{\mu\nu}$0 $\mathring{\Gamma}^\lambda{}_{\mu\nu}$1 massive scalar content. By contrast, analytic $\mathring{\Gamma}^\lambda{}_{\mu\nu}$2 and $\mathring{\Gamma}^\lambda{}_{\mu\nu}$3 show only the two GR tensor modes at first perturbative order, so their inequivalence to $\mathring{\Gamma}^\lambda{}_{\mu\nu}$4 already appears in their perturbative sector (Capozziello et al., 11 Mar 2025).

4. Symmetries, gauges, and the status of the Equivalence Principle

The trinity is dynamically unified but not foundation-neutral. In TEGR, local Lorentz invariance is exact only when the inertial spin connection is treated correctly. Gauge-fixing $\mathring{\Gamma}^\lambda{}_{\mu\nu}$5 in all frames is harmless in pure TEGR but generically breaks local Lorentz invariance in $\mathring{\Gamma}^\lambda{}_{\mu\nu}$6 models, producing frame-dependent results unless the covariant formulation is restored by solving for the appropriate inertial spin connection (Capozziello et al., 11 Mar 2025).

In STEGR and $\mathring{\Gamma}^\lambda{}_{\mu\nu}$7 theories, the analogous issue concerns the affine connection. The coincident gauge $\mathring{\Gamma}^\lambda{}_{\mu\nu}$8 is always accessible in symmetric teleparallelism, but in modified $\mathring{\Gamma}^\lambda{}_{\mu\nu}$9 models the connection must still be treated consistently; otherwise the dynamics may become coordinate-dependent. The covariant $$\Gamma^\lambda{}_{\mu\nu}=\mathring{\Gamma}^\lambda{}_{\mu\nu}+K^\lambda{}_{\mu\nu}+L^\lambda{}_{\mu\nu},$$0 formulation with an independent flat, torsionless connection removes these spurious gauge artifacts (Capozziello et al., 11 Mar 2025).

This gauge-structural distinction extends to the Equivalence Principle. One line of analysis argues that in GR the Equivalence Principle is a foundational input, while TEGR and STEGR recover it without making it foundational. In that reading, GR ties causal and inertial structure through the Levi–Civita connection, whereas TEGR and STEGR treat inertial or affine structure more explicitly and recover the Equivalence Principle as an emergent property of the classical sector (Mancini et al., 11 Jan 2025). With minimally coupled matter, free fall still follows Levi–Civita geodesics in all three formulations, so empirical agreement is preserved at the level of standard test-body motion (Mancini et al., 11 Jan 2025).

A plausible implication is that the trinity separates two questions that are often conflated: whether the classical field equations are equivalent, and whether the underlying spacetime ontology is the same. The first answer is affirmative under the standard assumptions; the second remains interpretively contested.

5. Boundary terms, variational principle, conserved charges, and $$\Gamma^\lambda{}_{\mu\nu}=\mathring{\Gamma}^\lambda{}_{\mu\nu}+K^\lambda{}_{\mu\nu}+L^\lambda{}_{\mu\nu},$$1 structure

Boundary terms are not peripheral in the trinity; they are the mechanism of equivalence. In the metric-affine treatment of the Gibbons–Hawking–York problem, the boundary term that appears in the identities $$\Gamma^\lambda{}_{\mu\nu}=\mathring{\Gamma}^\lambda{}_{\mu\nu}+K^\lambda{}_{\mu\nu}+L^\lambda{}_{\mu\nu},$$2 and $$\Gamma^\lambda{}_{\mu\nu}=\mathring{\Gamma}^\lambda{}_{\mu\nu}+K^\lambda{}_{\mu\nu}+L^\lambda{}_{\mu\nu},$$3 is the difference between the GR GHY term and the corresponding metric-affine GHY term. For TEGR and STEGR, the consistent GHY-like boundary term must vanish, because their bulk actions contain only first derivatives through torsion or non-metricity, and adding an extra boundary counterterm would spoil the standard Dirichlet variational problem (Erdmenger et al., 2023).

The same boundary logic controls conserved charges. In a general metric-affine Noether analysis, the diffeomorphism current is exact on shell, $$\Gamma^\lambda{}_{\mu\nu}=\mathring{\Gamma}^\lambda{}_{\mu\nu}+K^\lambda{}_{\mu\nu}+L^\lambda{}_{\mu\nu},$$4, so the physical charge is quasilocal. In GR, the superpotential is the Komar/Iyer–Wald expression

$$\Gamma^\lambda{}_{\mu\nu}=\mathring{\Gamma}^\lambda{}_{\mu\nu}+K^\lambda{}_{\mu\nu}+L^\lambda{}_{\mu\nu},$$5

In TEGR,

$$\Gamma^\lambda{}_{\mu\nu}=\mathring{\Gamma}^\lambda{}_{\mu\nu}+K^\lambda{}_{\mu\nu}+L^\lambda{}_{\mu\nu},$$6

and in STEGR,

$$\Gamma^\lambda{}_{\mu\nu}=\mathring{\Gamma}^\lambda{}_{\mu\nu}+K^\lambda{}_{\mu\nu}+L^\lambda{}_{\mu\nu},$$7

These yield the same conserved charges once the boundary terms are treated correctly, and they support the view that gravitational energy–momentum and entropy are quasilocal and holographic rather than local bulk densities (Jiménez et al., 2021).

The trinity also admits parallel $$\Gamma^\lambda{}_{\mu\nu}=\mathring{\Gamma}^\lambda{}_{\mu\nu}+K^\lambda{}_{\mu\nu}+L^\lambda{}_{\mu\nu},$$8 formulations. A full $$\Gamma^\lambda{}_{\mu\nu}=\mathring{\Gamma}^\lambda{}_{\mu\nu}+K^\lambda{}_{\mu\nu}+L^\lambda{}_{\mu\nu},$$9 split has been derived for TEGR and STEGR alongside the standard ADM split of GR, producing general 3-tetrad and 3-metric evolution equations and recovering, respectively, the metric $K^\lambda{}_{\mu\nu}$0 formulation of GR, the tetrad $K^\lambda{}_{\mu\nu}$1 formulation of TEGR, and the metric $K^\lambda{}_{\mu\nu}$2 formulation of STEGR (Capozziello et al., 2021). This is especially relevant for Hamiltonian analysis, numerical relativity, and degree-of-freedom counting, since the same Einsteinian content is then realized in curvature, torsion, and non-metricity variables.

6. Generalizations: Gauss–Bonnet, unimodular, non-relativistic, and pre-geometric formulations

The trinity extends beyond the Einstein–Hilbert scalar in several directions. For curvature-only actions built from the curvature two-form, a generalized geometrical trinity can be formulated by replacing curvature with the quadratic $K^\lambda{}_{\mu\nu}$3 terms built from contortion and disformation, provided the teleparallel flatness constraint is imposed and the corresponding GHY term is removed (Erdmenger et al., 2023).

A particularly explicit higher-order case is the Gauss–Bonnet topological scalar. In metric teleparallel form, the teleparallel Gauss–Bonnet construction requires $K^\lambda{}_{\mu\nu}$4 independent torsion invariants versus $K^\lambda{}_{\mu\nu}$5 in the corresponding effective field theory basis; in symmetric teleparallel form it requires $K^\lambda{}_{\mu\nu}$6 independent non-metricity invariants versus $K^\lambda{}_{\mu\nu}$7; and in the most general teleparallel setting the Gauss–Bonnet combination uses $K^\lambda{}_{\mu\nu}$8 invariants out of an EFT basis of $K^\lambda{}_{\mu\nu}$9 (Bajardi et al., 2023). These counts show that the Gauss–Bonnet invariant selects a highly constrained subset of the torsion and non-metricity operator space. The same work also emphasizes that pseudo-invariant constructions such as $L^\lambda{}_{\mu\nu}$0 and $L^\lambda{}_{\mu\nu}$1 are structurally delicate (Bajardi et al., 2023).

A unimodular version of the trinity also exists. By employing a Weyl compensator formalism, one can construct a WTDiff-invariant teleparallel theory that is equivalent, at the nonlinear level, to unimodular gravity and hence to GR with an adjustable cosmological constant; together with the curvature and symmetric teleparallel representations, this yields a geometrical trinity of unimodular gravity (Nakayama, 2022).

The trinity furthermore has a non-relativistic counterpart. Taking the $L^\lambda{}_{\mu\nu}$2 limit of STEGR produces a flat, torsionless, non-metric affine theory dubbed STENC, and this theory is equivalent to Newton–Cartan gravity and its teleparallel equivalent, completing a non-relativistic trinity with curvature, torsion, and non-metricity nodes (Wolf et al., 2023). A related analysis identifies Maxwell gravitation as the dynamical common core of that non-relativistic trinity, while also arguing that no analogous distinct common core exists in the relativistic case beyond GR itself (March et al., 2023).

At a more foundational level, a pre-geometric origin for the trinity has been proposed in a gauge-theoretic Yang–Mills-like framework with a Higgs-like field. In that construction, spontaneous symmetry breaking yields the effective metric, tetrads, spin connection, and the gauge choices corresponding to Levi–Civita, Weitzenböck, and coincident connections, thereby reproducing the GR, TEGR, and STEGR actions in the broken phase (Capozziello et al., 16 Jun 2026).

7. Cosmology, quantum-sensitive tests, and conceptual debate

The cosmological literature shows both the power and the limits of trinity equivalence. In minisuperspace, the Extended Geometric Trinity is recovered by including the divergence terms inside the nonlinear functional, so that $L^\lambda{}_{\mu\nu}$3, $L^\lambda{}_{\mu\nu}$4, and $L^\lambda{}_{\mu\nu}$5 become equivalent at both classical and quantum minisuperspace level; by contrast, pure $L^\lambda{}_{\mu\nu}$6 and pure $L^\lambda{}_{\mu\nu}$7 remain inequivalent to $L^\lambda{}_{\mu\nu}$8 (Battista et al., 28 Feb 2026). This reinforces the central structural lesson of the extended trinity: equivalence survives nonlinearization only when the boundary combinations are preserved.

Quantum-sensitive probes have also begun to test how far the equivalence may extend beyond classical field equations. In the study of false-vacuum decay, the Euclidean on-shell action difference between bounce and false-vacuum configurations was shown to be identical in GR, TEGR, and STEGR, so that the semiclassical tunneling exponent satisfies

$L^\lambda{}_{\mu\nu}$9

under minimal scalar coupling, $g_{\mu\nu}$0-symmetric bounce conditions, and the appropriate asymptotics (Branchina et al., 27 Jun 2026). This is a nontrivial example in which boundary-term equivalence survives a semiclassical observable.

At the same time, the trinity remains conceptually disputed. A strong critical position argues that, so long as TEGR and STEGR remain exactly equivalent to GR, they add flat parallel-transport structures that are unobservable and therefore amount to formal rephrasings rather than genuinely distinct physical theories; on that view, there is "no geometric trinity of gravity" in a substantive physical sense (Golovnev, 2024). Other work, however, stresses that even when bulk dynamics coincide, the different roles assigned to curvature, torsion, and non-metricity alter the formulation of the Equivalence Principle, the treatment of inertial structure, the variational problem, and the organization of conserved charges (Mancini et al., 11 Jan 2025).

The most balanced reading is therefore twofold. First, the Geometric Trinity of Gravity is a precise and mathematically nontrivial equivalence statement about how Einsteinian dynamics can be encoded in curvature, torsion, or non-metricity. Second, once one moves to modified gravities, matter couplings, boundary-sensitive observables, gauge fixing, or foundational interpretation, the three formulations need not remain interchangeable. This is exactly why the subject has become a productive interface between metric-affine geometry, modified gravity, cosmology, and the conceptual analysis of spacetime (Capozziello et al., 11 Mar 2025).