Symmetric Teleparallel Theories

• Symmetric Teleparallel Theories are metric–affine models characterized by a flat, torsion-free connection where nonmetricity encodes gravitational interactions.
• They comprise distinct classes such as STEGR, f(Q), and scalar–nonmetricity models, each yielding unique cosmological dynamics and gravitational wave signatures.
• Their covariant formulations and ghost-free constructions provide practical frameworks to explore strong-field regimes, modified gravity, and nontrivial cosmological solutions.

Symmetric teleparallel theories are metric–affine theories in which the independent affine connection is flat and torsion-free but not metric compatible, so gravitation is encoded in nonmetricity rather than curvature or torsion. In this sense they provide a third, geometrically distinct formulation of gravity alongside curvature-based general relativity and torsion-based teleparallel gravity. The central objects are a metric $g_{\mu\nu}$, an affine connection $\Gamma^\alpha{}_{\mu\nu}$, the nonmetricity tensor $Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}$, and actions built from quadratic or nonlinear invariants of $Q_{\alpha\mu\nu}$. The resulting theory space includes the symmetric teleparallel equivalent of general relativity (STEGR), the five-parameter quadratic “Newer General Relativity” class, $f(Q)$ models, scalar–nonmetricity theories, $f(Q,B)$ extensions, conformal constructions, and recently identified ghost-free subclasses with transverse-diffeomorphism symmetry (Blixt et al., 2023).

1. Geometric definition and basic invariants

In a general metric–affine geometry, curvature, torsion, and nonmetricity are defined by

$$R^\alpha{}_{\beta\mu\nu},\qquad T^\alpha{}_{\mu\nu}=2\Gamma^\alpha{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$

Symmetric teleparallel geometry imposes

$$R^\alpha{}_{\beta\mu\nu}=0,\qquad T^\alpha{}_{\mu\nu}=0,\qquad Q_{\alpha\mu\nu}\neq 0,$$

so the connection is flat and symmetric, but non-metric. The connection can be decomposed as

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

where $\mathring{\Gamma}^\alpha{}_{\mu\nu}$ is the Levi–Civita connection and

$\Gamma^\alpha{}_{\mu\nu}$0

is the disformation. Two traces of the nonmetricity tensor are standard,

$\Gamma^\alpha{}_{\mu\nu}$1

and they enter essentially all quadratic models (Blixt et al., 2023).

A standard nonmetricity scalar is

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

or equivalently $\Gamma^\alpha{}_{\mu\nu}$3 in terms of the nonmetricity conjugate $\Gamma^\alpha{}_{\mu\nu}$4. Because the full connection is flat, the Levi–Civita curvature scalar obeys

$\Gamma^\alpha{}_{\mu\nu}$5

so the action $\Gamma^\alpha{}_{\mu\nu}$6 differs from Einstein–Hilbert only by a boundary term. This defines STEGR and places GR, TEGR, and STEGR in the “geometrical trinity” (Blixt et al., 2023).

Beyond STEGR, the most general parity-even quadratic scalar built from $\Gamma^\alpha{}_{\mu\nu}$7 is

$\Gamma^\alpha{}_{\mu\nu}$8

which defines the “Newer GR” class. STEGR is recovered for

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

in one convention, or equivalently for the corresponding rescaled convention used in other papers (Blixt et al., 2023).

2. Coincident gauge, covariantization, and the connection sector

Because a flat, torsion-free connection is pure gauge, one can always locally choose coordinates such that

$Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}$0

This is the coincident gauge. In that gauge,

$Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}$1

and the action resembles Einstein’s historical “TT action,” i.e. the part of $Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}$2 quadratic in the connection (Blixt et al., 2023).

The covariant formulation introduces four scalar Stückelberg fields $Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}$3 and defines

$Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}$4

These $Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}$5 are scalars, not components of a spacetime vector. They define a holonomic tetrad

$Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}$6

with the defining property

$Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}$7

This “fundamental covariantly conserved teleparallel tetrad” provides a Weitzenböck-like description of symmetric teleparallel geometry: the tetrad is holonomic, the connection is flat and torsionless, and the metric is independent rather than orthonormal with respect to the tetrad (Blixt et al., 2023).

A central result of the covariant analysis is that the connection equation is not independent. In vacuum it is equivalent to the Levi–Civita divergence of the metric equation; with matter it is equivalent to the covariant conservation law

$Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}$8

Accordingly, the only physical difference between the covariantized theory and a coincident-gauge theory with an arbitrary source placed on the right-hand side is the requirement that the energy–momentum tensor be covariantly conserved (Blixt et al., 2023).

The coincident gauge is, however, a coordinate condition rather than an invariant statement about a solution. In homogeneous and isotropic cosmology, standard FLRW form of the metric and coincident gauge need not be simultaneously realizable. The fully covariant analysis of homogeneous and isotropic symmetric teleparallel geometry shows that imposing both conditions at once restricts one to a very special spatially flat branch; generically the coordinates in which the metric is FLRW do not coincide with the coordinates in which the connection vanishes (Hohmann, 2021).

3. Principal classes of symmetric teleparallel theories

The theory space is organized by the choice of scalar invariant or function thereof. A concise classification is:

Class	Defining structure	Characteristic feature
STEGR	$Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}$9	Equivalent to Einstein–Hilbert up to a boundary term
Newer GR	$Q_{\alpha\mu\nu}$0	Most general quadratic parity-even nonmetricity theory
$Q_{\alpha\mu\nu}$1	$Q_{\alpha\mu\nu}$2	Second-order metric equations in FLRW
Scalar–nonmetricity	$Q_{\alpha\mu\nu}$3	Nonminimal scalar coupling to $Q_{\alpha\mu\nu}$4
$Q_{\alpha\mu\nu}$5	$Q_{\alpha\mu\nu}$6	Separates second-order $Q_{\alpha\mu\nu}$7 and boundary-term $Q_{\alpha\mu\nu}$8 contributions
TDiff class	$Q_{\alpha\mu\nu}$9 and generalizations	Ghost-free, equivalent to GR plus a scalar and a global mode

The “Newer GR” class is the natural quadratic arena for strong-field and wave analyses. In spherical symmetry, recent work derives the most general stationary, spherically symmetric symmetric teleparallel connection starting from the coincident gauge and uses it to construct vacuum solutions in two one-parameter PPN-compatible subclasses of Newer GR (Hohmann et al., 2024).

$f(Q)$0 gravity is the most studied nonlinear extension. Its metric field equations can be written as

$f(Q)$1

which is the symmetric-teleparallel analogue of the familiar field equations in $f(Q)$2 gravity (Blixt et al., 2023).

Scalar–nonmetricity and $f(Q)$3 theories enlarge the scalar sector. In scalar–nonmetricity models the scalar can be nonminimally coupled to $f(Q)$4, while in $f(Q)$5 one treats the boundary term $f(Q)$6 as an independent argument. The latter includes $f(Q)$7 as the special case $f(Q)$8 (Soudi et al., 2018).

A recent symmetry-based construction isolates a ghost-free subclass with transverse diffeomorphism symmetry. The action

$f(Q)$9

is equivalent to a scalar-tensor theory

$f(Q,B)$0

so it propagates the two STEGR tensor modes plus one healthy scalar, together with a global degree of freedom $f(Q,B)$1 (Bello-Morales et al., 2024).

4. Cosmology

Homogeneous and isotropic symmetric teleparallel cosmology is more general than the flat-FLRW coincident-gauge ansatz commonly used in the $f(Q,B)$2 literature. The fully covariant classification yields the Robertson–Walker metric

$f(Q,B)$3

with $f(Q,B)$4, together with the most general homogeneous and isotropic flat, torsionless connection. After imposing the teleparallel conditions, the connection is characterized by one extra scalar function $f(Q,B)$5. The corresponding cosmological nonmetricity can be written as

$f(Q,B)$6

and, generically, the connection scalar $f(Q,B)$7 enters the background dynamics as an additional scalar quantity (Hohmann, 2021).

In the special spatially flat branch with $f(Q,B)$8, the nonmetricity scalar becomes

$f(Q,B)$9

and for $$R^\alpha{}_{\beta\mu\nu},\qquad T^\alpha{}_{\mu\nu}=2\Gamma^\alpha{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$0 gravity the modified Friedmann equations reduce to

$$R^\alpha{}_{\beta\mu\nu},\qquad T^\alpha{}_{\mu\nu}=2\Gamma^\alpha{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$1

$$R^\alpha{}_{\beta\mu\nu},\qquad T^\alpha{}_{\mu\nu}=2\Gamma^\alpha{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$2

In this branch the connection scalar $$R^\alpha{}_{\beta\mu\nu},\qquad T^\alpha{}_{\mu\nu}=2\Gamma^\alpha{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$3 completely decouples from the metric equations, which explains why most $$R^\alpha{}_{\beta\mu\nu},\qquad T^\alpha{}_{\mu\nu}=2\Gamma^\alpha{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$4 cosmology has been formulated in flat FLRW coincident gauge. The covariant analysis shows that these results are consistent, but only within one special sector of the full homogeneous and isotropic theory (Hohmann, 2021).

The broader quadratic cosmology program has also been developed in terms of the five invariants

$$R^\alpha{}_{\beta\mu\nu},\qquad T^\alpha{}_{\mu\nu}=2\Gamma^\alpha{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$5

with action

$$R^\alpha{}_{\beta\mu\nu},\qquad T^\alpha{}_{\mu\nu}=2\Gamma^\alpha{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$6

On a spatially flat FLRW minisuperspace, the Noether Symmetry Approach yields a classification of the models admitting point symmetries and provides exact solutions, including power-law and de Sitter–like expansions in a conformally invariant quadratic subclass (Dialektopoulos et al., 2019).

Bouncing cosmologies have been studied in $$R^\alpha{}_{\beta\mu\nu},\qquad T^\alpha{}_{\mu\nu}=2\Gamma^\alpha{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$7 gravity for the ansatz

$$R^\alpha{}_{\beta\mu\nu},\qquad T^\alpha{}_{\mu\nu}=2\Gamma^\alpha{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$8

For the linear model $$R^\alpha{}_{\beta\mu\nu},\qquad T^\alpha{}_{\mu\nu}=2\Gamma^\alpha{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=\nabla_\alpha g_{\mu\nu}.$$9, one finds

$$R^\alpha{}_{\beta\mu\nu}=0,\qquad T^\alpha{}_{\mu\nu}=0,\qquad Q_{\alpha\mu\nu}\neq 0,$$0

so $$R^\alpha{}_{\beta\mu\nu}=0,\qquad T^\alpha{}_{\mu\nu}=0,\qquad Q_{\alpha\mu\nu}\neq 0,$$1 requires $$R^\alpha{}_{\beta\mu\nu}=0,\qquad T^\alpha{}_{\mu\nu}=0,\qquad Q_{\alpha\mu\nu}\neq 0,$$2, and the bounce is accompanied by NEC violation near $$R^\alpha{}_{\beta\mu\nu}=0,\qquad T^\alpha{}_{\mu\nu}=0,\qquad Q_{\alpha\mu\nu}\neq 0,$$3. For the nonlinear model $$R^\alpha{}_{\beta\mu\nu}=0,\qquad T^\alpha{}_{\mu\nu}=0,\qquad Q_{\alpha\mu\nu}\neq 0,$$4, positivity of the density is achieved for $$R^\alpha{}_{\beta\mu\nu}=0,\qquad T^\alpha{}_{\mu\nu}=0,\qquad Q_{\alpha\mu\nu}\neq 0,$$5 and $$R^\alpha{}_{\beta\mu\nu}=0,\qquad T^\alpha{}_{\mu\nu}=0,\qquad Q_{\alpha\mu\nu}\neq 0,$$6, again with NEC and SEC violation near the bounce. In both cases the effective equation of state enters a phantom regime around the bounce, while homogeneous perturbations in the Hubble parameter decay away from the bounce (Mandal et al., 2021).

5. Linearized regime, post-Newtonian structure, and gravitational waves

The perturbative spectrum of quadratic symmetric teleparallel theories has been derived around Minkowski space. Writing

$$R^\alpha{}_{\beta\mu\nu}=0,\qquad T^\alpha{}_{\mu\nu}=0,\qquad Q_{\alpha\mu\nu}\neq 0,$$7

the nonmetricity becomes

$$R^\alpha{}_{\beta\mu\nu}=0,\qquad T^\alpha{}_{\mu\nu}=0,\qquad Q_{\alpha\mu\nu}\neq 0,$$8

and the exact propagator of the most general quadratic, parity-even, generally covariant, infinite-derivative theory can be written in terms of two form factors $$R^\alpha{}_{\beta\mu\nu}=0,\qquad T^\alpha{}_{\mu\nu}=0,\qquad Q_{\alpha\mu\nu}\neq 0,$$9 and $$\Gamma^\alpha{}_{\mu\nu}=\mathring{\Gamma}^\alpha{}_{\mu\nu}+L^\alpha{}_{\mu\nu},$$0 as

$$\Gamma^\alpha{}_{\mu\nu}=\mathring{\Gamma}^\alpha{}_{\mu\nu}+L^\alpha{}_{\mu\nu},$$1

This makes explicit the spin-2 and spin-0 content of the theory and underlies the construction of ghost-free non-local models with entire form factors (Conroy et al., 2017).

For the five-parameter quadratic “newer general relativity” class, gravitational waves propagate on the Minkowski light cone. The principal polynomial is proportional to $$\Gamma^\alpha{}_{\mu\nu}=\mathring{\Gamma}^\alpha{}_{\mu\nu}+L^\alpha{}_{\mu\nu},$$2, so nontrivial wave covectors satisfy

$$\Gamma^\alpha{}_{\mu\nu}=\mathring{\Gamma}^\alpha{}_{\mu\nu}+L^\alpha{}_{\mu\nu},$$3

Using the Newman–Penrose formalism, the allowed polarization content depends on $$\Gamma^\alpha{}_{\mu\nu}=\mathring{\Gamma}^\alpha{}_{\mu\nu}+L^\alpha{}_{\mu\nu},$$4. In particular, the condition

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

selects theories with only the two tensor polarizations of GR, and this subset is larger than the single STEGR point (Hohmann et al., 2018).

For $$\Gamma^\alpha{}_{\mu\nu}=\mathring{\Gamma}^\alpha{}_{\mu\nu}+L^\alpha{}_{\mu\nu},$$6 gravity on Minkowski background, the linearized equations reduce to $$\Gamma^\alpha{}_{\mu\nu}=\mathring{\Gamma}^\alpha{}_{\mu\nu}+L^\alpha{}_{\mu\nu},$$7 provided $$\Gamma^\alpha{}_{\mu\nu}=\mathring{\Gamma}^\alpha{}_{\mu\nu}+L^\alpha{}_{\mu\nu},$$8 and $$\Gamma^\alpha{}_{\mu\nu}=\mathring{\Gamma}^\alpha{}_{\mu\nu}+L^\alpha{}_{\mu\nu},$$9. Hence, at this order, $\mathring{\Gamma}^\alpha{}_{\mu\nu}$0 has the same wave speed and tensor polarizations as GR. By contrast, scalar–nonmetricity theories can support an additional scalar degree of freedom, and $\mathring{\Gamma}^\alpha{}_{\mu\nu}$1 generically introduces a massive scalar breathing mode through

$\mathring{\Gamma}^\alpha{}_{\mu\nu}$2

with

$\mathring{\Gamma}^\alpha{}_{\mu\nu}$3

These distinctions are central to gravitational-wave phenomenology in the nonmetricity framework (Soudi et al., 2018).

In the weak-field regime, symmetric teleparallel theories require a gauge-invariant treatment of the connection because $\mathring{\Gamma}^\alpha{}_{\mu\nu}$4 is not a tensor. The post-Newtonian formalism has therefore been extended by introducing a gauge vector $\mathring{\Gamma}^\alpha{}_{\mu\nu}$5 relating the coincident gauge and the distinguished PPN gauge. The gauge-invariant connection perturbations are

$\mathring{\Gamma}^\alpha{}_{\mu\nu}$6

and the associated second- and third-order nonmetricity terms can be expressed in terms of the standard PPN potentials $\mathring{\Gamma}^\alpha{}_{\mu\nu}$7. This yields a systematic route to extract PPN parameters such as $\mathring{\Gamma}^\alpha{}_{\mu\nu}$8 and $\mathring{\Gamma}^\alpha{}_{\mu\nu}$9 for concrete symmetric teleparallel models (Hohmann, 2021).

6. Conformal, strong-field, and conceptual developments

Conformal model building in symmetric teleparallel geometry exploits the fact that the metric is the fundamental variable and the field equations remain second order when the Lagrangian is algebraic in $\Gamma^\alpha{}_{\mu\nu}$00. A family of conformal scalars $\Gamma^\alpha{}_{\mu\nu}$01 is obtained by demanding

$\Gamma^\alpha{}_{\mu\nu}$02

under $\Gamma^\alpha{}_{\mu\nu}$03. For a quadratic ansatz, the coefficients satisfy

$\Gamma^\alpha{}_{\mu\nu}$04

and the conformally invariant action is

$\Gamma^\alpha{}_{\mu\nu}$05

This construction yields second-order field equations and leads naturally to scalar-tensor representations of $\Gamma^\alpha{}_{\mu\nu}$06 and $\Gamma^\alpha{}_{\mu\nu}$07 gravity. In particular, nonminimally coupled scalar–$\Gamma^\alpha{}_{\mu\nu}$08 dark-energy models can be reinterpreted as $\Gamma^\alpha{}_{\mu\nu}$09 theories (Gakis et al., 2019).

Strong-field spherical solutions in Newer GR show that symmetric teleparallel theory space admits vacuum geometries with no GR counterpart. Using the most general stationary, spherically symmetric flat, torsionless connection, two families of vacuum solutions have been found. One family is asymptotically flat and has zero Komar mass, while the other is non-asymptotically flat and has divergent Komar mass. Both are horizonless, exhibit naked singularities, and do not admit bound or stable circular orbits; neither realizes a traversable wormhole in the explicit examples studied (Hohmann et al., 2024).

The status of free-fall in nonmetric geometry remains conceptually subtle. A novel approach to autoparallels in coincident gauge proposes

$\Gamma^\alpha{}_{\mu\nu}$10

leading to a generalized autoparallel equation that reproduces the standard Riemannian geodesic equation for $\Gamma^\alpha{}_{\mu\nu}$11 and $\Gamma^\alpha{}_{\mu\nu}$12. For other choices of $\Gamma^\alpha{}_{\mu\nu}$13, the resulting orbital dynamics differ and can mimic flat rotation curves without dark matter (Adak et al., 2011). A later Weyl-covariant analysis argues that the second clock effect can be removed by the transport prescription

$\Gamma^\alpha{}_{\mu\nu}$14

and introduces a Weyl-invariant proper time $\Gamma^\alpha{}_{\mu\nu}$15. In that framework, the physically relevant worldlines of spinless test bodies are identified with “meta-geodesics” derived from the Weyl-invariant point-particle action rather than with the raw autoparallels of the symmetric teleparallel connection (Pala et al., 2022).

A major recent development is the identification of a class of ghost-free symmetric teleparallel theories based on transverse diffeomorphisms. Their non-linear equivalence to K-essence- and Horndeski-type scalar-tensor theories, together with the presence of a global integration constant $\Gamma^\alpha{}_{\mu\nu}$16, shows that symmetry reduction from full Diff to TDiff can be implemented without reintroducing the ghost-like instabilities that plague generic teleparallel modifications (Bello-Morales et al., 2024).

Symmetric teleparallel theories therefore occupy a mathematically broad and conceptually nontrivial sector of metric–affine gravity. STEGR preserves exact equivalence with GR, while Newer GR, $\Gamma^\alpha{}_{\mu\nu}$17, scalar–nonmetricity, $\Gamma^\alpha{}_{\mu\nu}$18, conformal, and TDiff-symmetric models realize distinct deformations with different cosmological sectors, perturbative spectra, and strong-field solutions. The central structural issues are the role of the independent but pure-gauge connection, the interplay between coincident gauge and covariance, the interpretation of matter couplings and conservation laws, and the control of extra scalar sectors so that modified nonmetricity dynamics remain free of ghosts and compatible with observation (Blixt et al., 2023).