Teleparallel Equivalent of General Relativity (TEGR)

• TEGR is a reformulation of gravity using torsion on a flat, metric-compatible manifold, reproducing Einstein's field equations.
• The TEGR action is derived from the torsion scalar via a tetrad field, differing only by a total divergence from the Einstein–Hilbert action.
• TEGR offers a framework for localizing gravitational energy and clarifying gauge structures through conserved currents and inertial spin connections.

The Teleparallel Equivalent of General Relativity (TEGR) provides an exact reformulation of Einstein’s general relativity in terms of torsion rather than curvature. In this approach, gravitation is encoded in the torsion of a globally flat, metric-compatible Weitzenböck connection constructed from a tetrad (vierbein) field, while the metric structure is recovered as a derived object. TEGR reproduces all the classical field equations and experimental predictions of GR, yet it offers new perspectives for the localization of gravitational energy, the definition of conserved quantities, and the gauge structure of gravity. This makes TEGR a foundational pillar of the so-called "geometric trinity" alongside conventional GR (curvature-based) and the symmetric teleparallel framework (non-metricity-based).

1. Geometric Structure and Fundamental Variables

TEGR is formulated on a four-dimensional, time-orientable, globally parallelizable manifold $M$ equipped with an orthonormal tetrad field $e^a{}_\mu(x)$, where $a = 0,1,2,3$ is a Lorentz index and $\mu = 0,1,2,3$ a spacetime index. The spacetime metric is constructed as

$g_{\mu\nu} = \eta_{ab}\,e^a{}_\mu\,e^b{}_\nu,$

with $\eta_{ab} = \mathrm{diag}(-1, +1, +1, +1)$. The fundamental connection is the Weitzenböck connection,

$$\Gamma^\lambda{}_{\mu\nu} = e_a{}^\lambda\,\partial_\nu e^a{}_\mu,$$

which is metric-compatible, curvature-free ($R^\rho{}_{\sigma\mu\nu}(\Gamma) = 0$), but possesses nontrivial torsion: $$T^\rho{}_{\mu\nu} = \Gamma^\rho{}_{\nu\mu} - \Gamma^\rho{}_{\mu\nu} = e_a{}^\rho (\partial_\mu e^a{}_\nu - \partial_\nu e^a{}_\mu).$$ The difference between $\Gamma^\lambda{}_{\mu\nu}$ and the Levi-Civita connection defines the contortion tensor. These objects collectively encode all gravitational degrees of freedom; the six tetrad gauge freedoms correspond to local Lorentz rotations, while ten correspond to the metric field (Mancini et al., 11 Jan 2025).

2. Action Principle and Field Equations

The TEGR action is constructed from the torsion scalar $e^a{}_\mu(x)$0: $e^a{}_\mu(x)$1 where the superpotential $e^a{}_\mu(x)$2 is given by

$e^a{}_\mu(x)$3

with $e^a{}_\mu(x)$4 the contortion. The Lagrangian density reads

$e^a{}_\mu(x)$5

$e^a{}_\mu(x)$6, $e^a{}_\mu(x)$7. This Lagrangian differs from the Einstein–Hilbert Lagrangian by a total divergence: $e^a{}_\mu(x)$8

where $e^a{}_\mu(x)$9 is the Ricci scalar of the Levi–Civita connection and $a = 0,1,2,3$0 (Bahamonde et al., 2015). As a result, the field equations obtained by varying the action with respect to $a = 0,1,2,3$1,

$a = 0,1,2,3$2

are algebraically equivalent to Einstein’s equations in standard GR (Maluf, 2013, Mancini et al., 11 Jan 2025).

3. Hamiltonian Formulation and Constraint Structure

The Hamiltonian analysis begins with canonical momenta conjugate to $a = 0,1,2,3$3. In ADM-like $a = 0,1,2,3$4 decompositions, the Hamiltonian is expressed in terms of lapse, shift, spatial triad variables, and their canonical conjugates (Pati et al., 2022, Okolow, 2011).

Primary constraints arise due to the absence of time derivatives for $a = 0,1,2,3$5, reflecting diffeomorphism and internal Lorentz gauge invariance. Additional first-class constraints enforce the spatial constraints and local Lorentz invariance. The total Hamiltonian takes the form: $a = 0,1,2,3$6 with $a = 0,1,2,3$7 and $a = 0,1,2,3$8 corresponding to diffeomorphism and Lorentz-gauge generators, respectively. Their Poisson brackets close to yield the Poincaré algebra: $a = 0,1,2,3$9 guaranteeing consistency with the relativistic symmetry structure (Maluf et al., 2023, Neto et al., 2014).

4. Local Lorentz Invariance, Spin Connection, and Gauge Structure

A central innovation in modern TEGR formulations is the introduction of the inertial (flat) spin connection $\mu = 0,1,2,3$0. Under local Lorentz rotations $\mu = 0,1,2,3$1,

$\mu = 0,1,2,3$2

The TEGR Lagrangian, being a total divergence in the $\mu = 0,1,2,3$3 dependence, is invariant under local Lorentz transformations provided the appropriate inertial connection is used (Krssak et al., 2018, Nashed et al., 2016, Emtsova et al., 2019). This removes the "good/bad tetrad" ambiguity: any tetrad is allowed if accompanied by its compatible spin connection determined by the "turning off gravity" principle, which requires the torsion to vanish in the absence of gravitation (Emtsova et al., 23 May 2025, Emtsova et al., 2023).

In the principal bundle formalism, the correct choice of absolute structures and treatment of the non-dynamical connection determines whether the full diffeomorphism group or a subgroup is realized as the gauge group (Brezina et al., 31 Jul 2025).

5. Conserved Currents, Superpotentials, and Gravitational Energy-Momentum

TEGR provides a Noether-current formalism for conserved charges, based on diffeomorphism invariance: $\mu = 0,1,2,3$4 with the antisymmetric superpotential

$\mu = 0,1,2,3$5

Integrating over spatial boundaries yields the energy-momentum and angular momentum as surface charges, e.g.,

$\mu = 0,1,2,3$6

where $\mu = 0,1,2,3$7 (Maluf et al., 2023, Emtsova et al., 2019). These formal expressions admit variational and geometric interpretations and are valid in full generality, including for gravitational radiation and black hole spacetimes.

6. Gravitational Energy Localization, Gauge Dependence, and Physical Implications

A notable distinction of TEGR is the localization of gravitational energy-momentum. TEGR definitions yield locally conserved, reference-frame-dependent energy-momentum tensors not available in purely metric GR. In gravitational wave spacetimes, particles can exchange kinetic energy with the field locally, as shown via the inertial acceleration tensor of the tetrad and corresponding work-energy relations. This provides an explicit, frame-adapted, and observer-dependent localization of gravitational energy (Maluf et al., 2023, Emtsova et al., 2023).

The calculation of Bondi mass loss, the energy of cosmological horizons, and black hole irreducible mass underlines the utility and physical consistency of TEGR charges. However, since both energy and angular momentum depend on the observer’s frame, their values are covariant under coordinate changes but not invariant under all local Lorentz transformations (reflecting the inertial content carried by different tetrad choices). The requirement of appropriate gauge fixing, often implemented by the vanishing of certain skew-symmetric tensors or by using covariant inertial spin connections, ensures that conserved charges are physically meaningful and invariant under the residual gauge (Nashed et al., 2016).

7. Philosophical, Foundational, and Experimental Perspectives

TEGR demonstrates that all gravitational effects and classical tests of GR can be equivalently encoded in a globally flat but torsionful geometry, decoupling the notion of gravity from curvature. This underdetermines the metrical, affine, and teleparallel descriptions at the empirical level. The equivalence principle, fundamental to GR, appears as an emergent property in TEGR, rather than a postulate (Mulder et al., 24 Apr 2025, Mancini et al., 11 Jan 2025).

Operationalization and visualization of torsion are feasible (parallelogram closure, spin-torsion couplings, gradiometer measurements), and torsion poses no new barriers to physical intuition compared to curvature (Mulder et al., 24 Apr 2025). At the quantum or high-energy level, differences may potentially arise—e.g., in effective field theory corrections, modified gravity extensions ($\mu = 0,1,2,3$8 vs $\mu = 0,1,2,3$9), or violations of the equivalence principle detectable in future precision experiments (Mylova et al., 2022, Mancini et al., 11 Jan 2025).

Table: Geometric Structures in GR vs TEGR

Structure	General Relativity (GR)	Teleparallel Equivalent (TEGR)
Fundamental field	Metric $g_{\mu\nu} = \eta_{ab}\,e^a{}_\mu\,e^b{}_\nu,$0	Tetrad $g_{\mu\nu} = \eta_{ab}\,e^a{}_\mu\,e^b{}_\nu,$1
Connection	Levi–Civita ($g_{\mu\nu} = \eta_{ab}\,e^a{}_\mu\,e^b{}_\nu,$2)	Weitzenböck ($g_{\mu\nu} = \eta_{ab}\,e^a{}_\mu\,e^b{}_\nu,$3)
Field strength	Curvature ($g_{\mu\nu} = \eta_{ab}\,e^a{}_\mu\,e^b{}_\nu,$4)	Torsion ($g_{\mu\nu} = \eta_{ab}\,e^a{}_\mu\,e^b{}_\nu,$5)
Standard Lagrangian	$g_{\mu\nu} = \eta_{ab}\,e^a{}_\mu\,e^b{}_\nu,$6	$g_{\mu\nu} = \eta_{ab}\,e^a{}_\mu\,e^b{}_\nu,$7 with $g_{\mu\nu} = \eta_{ab}\,e^a{}_\mu\,e^b{}_\nu,$8
Energy-momentum	Nonlocal, pseudo-tensor, frame-dependent	Local, tensorial under coordinate but not local Lorentz
Gauge structure	Diffeomorphisms + local Lorentz	Diffeomorphisms + local Lorentz (with inertial connection)

• (Maluf, 2013) The teleparallel equivalent of general relativity (Maluf).
• (Maluf et al., 2023) Tetrad Fields, Reference Frames, and the Gravitational Energy-Momentum in the Teleparallel Equivalent of General Relativity.
• (Krssak et al., 2018) Teleparallel Theories of Gravity: Illuminating a Fully Invariant Approach.
• (Pati et al., 2022) Hamilton's equations in the covariant teleparallel equivalent of general relativity.
• (Emtsova et al., 2019) Conserved Currents and Superpotentials in Teleparallel Equivalent of General Relativity.
• (Mulder et al., 24 Apr 2025) Is spacetime curved? Assessing the underdetermination of general relativity and teleparallel gravity.
• (Mancini et al., 11 Jan 2025) Equivalent Gravities and Equivalence Principle: Foundations and experimental implications.
• (Emtsova et al., 23 May 2025) The Noether formalism for constructing conserved quantities in teleparallel equivalents of general relativity.
• (Ferraro et al., 2016) Hamiltonian formulation of teleparallel gravity.
• (Nashed, 2011) Gravitational radiation fields in teleparallel equivalent of general relativity and their energies.
• (Neto et al., 2014) The angular momentum of plane-fronted gravitational waves in the teleparallel equivalent of general relativity.

---

In summary, TEGR offers a manifestly gauge-invariant, dynamically equivalent, and conceptually distinct formulation of gravitation that localizes gravitational energy-momentum, brings gauge-theoretic methods into gravitational dynamics, and provides a robust foundation for both classical and quantum investigations of gravity. It establishes that gravity can be interpreted purely as a spacetime torsional gauge theory, challenging the unique status of curvature as the geometric origin of gravitational effects (Mulder et al., 24 Apr 2025, Maluf et al., 2023, Krssak et al., 2018).