---
title: Symmetric Teleparallel Theories
url: https://www.emergentmind.com/topics/symmetric-teleparallel-theories
type: topic
---

# Symmetric Teleparallel Theories

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 [2306.09289].

## 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
\[
L^\alpha{}_{\mu\nu}=\tfrac12 g^{\alpha\lambda}\left(-Q_{\mu\lambda\nu}-Q_{\nu\lambda\mu}+Q_{\lambda\mu\nu}\right)
\]
is the disformation. Two traces of the nonmetricity tensor are standard,
\[
Q_\alpha = Q_{\alpha\mu}{}^\mu,\qquad \tilde Q_\alpha = Q_{\mu\alpha}{}^\mu,
\]
and they enter essentially all quadratic models [2306.09289].

A standard nonmetricity scalar is
\[
Q=\tfrac14 Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}
-\tfrac12 Q_{\alpha\mu\nu}Q^{\mu\alpha\nu}
-\tfrac14 Q_\alpha Q^\alpha
+\tfrac12 \tilde Q_\alpha Q^\alpha,
\]
or equivalently \(Q=P^\alpha{}_{\mu\nu}Q_\alpha{}^{\mu\nu}\) in terms of the nonmetricity conjugate \(P^\alpha{}_{\mu\nu}\). Because the full connection is flat, the Levi–Civita curvature scalar obeys
\[
\mathring R=-Q-B,\qquad
B=\nabla_\alpha(Q^\alpha-\tilde Q^\alpha),
\]
so the action \(\int \sqrt{-g}\,Q\) differs from Einstein–Hilbert only by a boundary term. This defines STEGR and places GR, TEGR, and STEGR in the “geometrical trinity” [2306.09289].

Beyond STEGR, the most general parity-even quadratic scalar built from \(Q_{\alpha\mu\nu}\) is
\[
\mathcal Q =
c_1 Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}
+c_2 Q_{\alpha\mu\nu}Q^{\mu\alpha\nu}
+c_3 Q_\alpha Q^\alpha
+c_4 \tilde Q_\alpha \tilde Q^\alpha
+c_5 Q_\alpha \tilde Q^\alpha,
\]
which defines the “Newer GR” class. STEGR is recovered for
\[
c_1=2,\quad c_2=-1,\quad c_3=-\tfrac12,\quad c_4=0,\quad c_5=1
\]
in one convention, or equivalently for the corresponding rescaled convention used in other papers [2306.09289].

## 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
\[
\Gamma^\alpha{}_{\mu\nu}=0.
\]
This is the coincident gauge. In that gauge,
\[
Q_{\alpha\mu\nu}=\partial_\alpha g_{\mu\nu},
\]
and the action resembles Einstein’s historical “TT action,” i.e. the part of \(\mathring R\) quadratic in the connection [2306.09289].

The covariant formulation introduces four scalar Stückelberg fields \(\xi^a(x)\) and defines
\[
\Gamma^\alpha{}_{\mu\nu}
=
\frac{\partial x^\alpha}{\partial \xi^a}\,\partial_\mu\partial_\nu \xi^a.
\]
These \(\xi^a\) are scalars, not components of a spacetime vector. They define a holonomic tetrad
\[
\theta^a{}_\mu=\partial_\mu \xi^a,
\qquad
\Gamma^\alpha{}_{\mu\nu}=e^\alpha{}_a\,\partial_\mu \theta^a{}_\nu,
\]
with the defining property
\[
\nabla_\mu \theta^a{}_\nu =0.
\]
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 [2306.09289].

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
\[
\mathring\nabla_\mu T^\mu{}_\nu=0.
\]
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 [2306.09289].

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 [2109.01525].

## 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** | \(S\sim \int \sqrt{-g}\,Q\) | Equivalent to Einstein–Hilbert up to a boundary term |
| **Newer GR** | \(\mathcal Q=c_1A_1+c_2A_2+c_3A_3+c_4A_4+c_5A_5\) | Most general quadratic parity-even nonmetricity theory |
| **\(f(Q)\)** | \(S\sim \int \sqrt{-g}\,f(Q)\) | Second-order metric equations in FLRW |
| **Scalar–nonmetricity** | \(\mathcal L_g=A(\phi)Q-B(\phi)g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi-2V(\phi)\) | Nonminimal scalar coupling to \(Q\) |
| **\(f(Q,B)\)** | \(S\sim \int \sqrt{-g}\,f(Q,B)\) | Separates second-order \(Q\) and boundary-term \(B\) contributions |
| **TDiff class** | \(Q-\frac{a_5}{4}Q_\alpha Q^\alpha\) 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 [2412.11730].

\(f(Q)\) gravity is the most studied nonlinear extension. Its metric field equations can be written as
\[
f_Q \mathring G_{\mu\nu}
+\tfrac12(f-f_Q Q)g_{\mu\nu}
-f_{QQ}P_{\mu\nu}{}^\alpha \partial_\alpha Q
=
T_{\mu\nu},
\]
which is the symmetric-teleparallel analogue of the familiar field equations in \(f(T)\) gravity [2306.09289].

Scalar–nonmetricity and \(f(Q,B)\) theories enlarge the scalar sector. In scalar–nonmetricity models the scalar can be nonminimally coupled to \(Q\), while in \(f(Q,B)\) one treats the boundary term \(B\) as an independent argument. The latter includes \(f(R)\) as the special case \(f(Q,B)=f(Q-B)\) [1810.08220].

A recent symmetry-based construction isolates a ghost-free subclass with transverse diffeomorphism symmetry. The action
\[
S=\frac{M_{\rm Pl}^2}{2}\int d^4x\sqrt{-g}\left[Q-\frac{a_5}{4}Q_\alpha Q^\alpha\right]
\]
is equivalent to a scalar-tensor theory
\[
S=\int d^4x\sqrt{-g}\left[
\frac{M_{\rm Pl}^2}{2}Q
-\frac12(\partial\varphi)^2
-\frac{M_{\rm Pl}^2\lambda_0}{2}e^{\varphi/\sqrt{a_5}}
\right],
\]
so it propagates the two STEGR tensor modes plus one healthy scalar, together with a global degree of freedom \(\lambda_0\) [2406.19355].

## 4. Cosmology

Homogeneous and isotropic symmetric teleparallel cosmology is more general than the flat-FLRW coincident-gauge ansatz commonly used in the \(f(Q)\) literature. The fully covariant classification yields the Robertson–Walker metric
\[
g_{tt}=-N^2(t),\quad
g_{rr}=\frac{A^2(t)}{\chi^2},\quad
g_{\vartheta\vartheta}=A^2(t)r^2,\quad
g_{\varphi\varphi}=A^2(t)r^2\sin^2\vartheta,
\]
with \(\chi=\sqrt{1-kr^2}\), 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 \(K(t)\). The corresponding cosmological nonmetricity can be written as
\[
Q_{\rho\mu\nu}
=
2Q_1\,n_\rho n_\mu n_\nu
+
2Q_2\,n_\rho h_{\mu\nu}
+
2Q_3\,h_{\rho(\mu}n_{\nu)},
\]
and, generically, the connection scalar \(K(t)\) enters the background dynamics as an additional scalar quantity [2109.01525].

In the special spatially flat branch with \(K_2=K_3=0\), the nonmetricity scalar becomes
\[
Q=6H^2,
\]
and for \(f(Q)\) gravity the modified Friedmann equations reduce to
\[
12H^2 f_Q-f = 2\kappa^2 \rho,
\]
\[
-48H^2\dot H f_{QQ}-4(\dot H+3H^2)f_Q+f=2\kappa^2 p.
\]
In this branch the connection scalar \(K\) completely decouples from the metric equations, which explains why most \(f(Q)\) 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 [2109.01525].

The broader quadratic cosmology program has also been developed in terms of the five invariants
\[
A=Q_{\alpha\mu\nu}Q^{\alpha\mu\nu},\quad
B=Q_{\alpha\mu\nu}\tilde Q^{\alpha\mu\nu},\quad
C=Q_\alpha Q^\alpha,\quad
D=Q_\alpha \tilde Q^\alpha,\quad
E=\tilde Q_\alpha \tilde Q^\alpha,
\]
with action
\[
S=\int d^4x\,\sqrt{-g}\,[f(A,B,C,D,E,F)+\mathcal L_m].
\]
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 [1905.09019].

Bouncing cosmologies have been studied in \(f(Q)\) gravity for the ansatz
\[
a(t)=\left(\frac{b^2+t^2}{b^2+t_0^2}\right)^{1/4}.
\]
For the linear model \(f(Q)=\alpha Q\), one finds
\[
\rho = -\frac{3 \alpha\, t^2}{4 (b^2 + t^2)^2},
\qquad
\rho+p=\frac{\alpha(b-t)(b+t)}{(b^2+t^2)^2},
\]
so \(\rho\ge 0\) requires \(\alpha<0\), and the bounce is accompanied by NEC violation near \(t=0\). For the nonlinear model \(f(Q)=Q+mQ^n\), positivity of the density is achieved for \(n\ge 0.5\) and \(m\le 0\), 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 [2107.02936].

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

The perturbative spectrum of quadratic symmetric teleparallel theories has been derived around Minkowski space. Writing
\[
g_{ab}=\eta_{ab}+h_{ab},\qquad
\Gamma_{cab}=\partial_a\partial_b u_c,
\]
the nonmetricity becomes
\[
Q_{abc}
=
\partial_a\big(h_{bc}-2\partial_{(b}u_{c)}\big),
\]
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 \(A(\Box)\) and \(C(\Box)\) as
\[
\Pi_{abcd}(k)
=
\frac{P^{(2)}_{abcd}}{A(-k^2)\,k^2}
+
\frac{3A(-k^2)-C(-k^2)}{2A(-k^2)C(-k^2)k^2}\,P^{(0)}_{abcd}.
\]
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 [1710.05708].

For the five-parameter quadratic “newer general relativity” class, gravitational waves propagate on the Minkowski light cone. The principal polynomial is proportional to \([\eta^{-1}(k,k)]^{15}\), so nontrivial wave covectors satisfy
\[
\eta^{\mu\nu}k_\mu k_\nu=0.
\]
Using the Newman–Penrose formalism, the allowed polarization content depends on \(c_2,c_4,c_5\). In particular, the condition
\[
c_2+c_4+c_5=0,\qquad c_5\neq 0
\]
selects theories with only the two tensor polarizations of GR, and this subset is larger than the single STEGR point [1808.02894].

For \(f(Q)\) gravity on Minkowski background, the linearized equations reduce to \(\delta G_{\mu\nu}=0\) provided \(f(0)=0\) and \(f'(0)\neq 0\). Hence, at this order, \(f(Q)\) has the same wave speed and tensor polarizations as GR. By contrast, scalar–nonmetricity theories can support an additional scalar degree of freedom, and \(f(Q,B)\) generically introduces a massive scalar breathing mode through
\[
(\Box-m^2)R^{(1)}=0
\]
with
\[
m^2=\frac{f_Q(0,0)}{3f_{BB}(0,0)}.
\]
These distinctions are central to gravitational-wave phenomenology in the nonmetricity framework [1810.08220].

In the weak-field regime, symmetric teleparallel theories require a gauge-invariant treatment of the connection because \(\Gamma^\mu{}_{\nu\rho}\) is not a tensor. The post-Newtonian formalism has therefore been extended by introducing a gauge vector \(\mathbf C^\mu\) relating the coincident gauge and the distinguished PPN gauge. The gauge-invariant connection perturbations are
\[
\boldsymbol{\Gamma}^{(2)i}{}_{jk}=-\partial_j\partial_k \mathbf C^{(2)i},
\qquad
\boldsymbol{\Gamma}^{(3)0}{}_{jk}=-\partial_j\partial_k \mathbf C^{(3)0},
\]
and the associated second- and third-order nonmetricity terms can be expressed in terms of the standard PPN potentials \(\mathbf U,\mathbf V_i,\mathbf W_i,\boldsymbol\chi\). This yields a systematic route to extract PPN parameters such as \(\gamma\) and \(\alpha_1\) for concrete symmetric teleparallel models [2111.06255].

## 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 \(Q_{\alpha\mu\nu}\). A family of conformal scalars \(Q_c\) is obtained by demanding
\[
\bar Q_c=\Omega^{-2}Q_c
\]
under \(g_{\mu\nu}\to \Omega^2 g_{\mu\nu}\). For a quadratic ansatz, the coefficients satisfy
\[
\begin{pmatrix}
c_1\\ c_2\\ c_3\\ c_4\\ c_5
\end{pmatrix}
=
\begin{pmatrix}
-b_1-4b_3\\
-4b_1-b_2\\
b_3\\
b_2\\
2b_1
\end{pmatrix},
\]
and the conformally invariant action is
\[
S_c=\int d^4x\,\sqrt{-g}\,Q_c^2.
\]
This construction yields second-order field equations and leads naturally to scalar-tensor representations of \(f(Q)\) and \(f(Q,B)\) gravity. In particular, nonminimally coupled scalar–\(Q\) dark-energy models can be reinterpreted as \(f(Q,B)\) theories [1908.05741].

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 [2412.11730].

The status of free-fall in nonmetric geometry remains conceptually subtle. A novel approach to autoparallels in coincident gauge proposes
\[
D\mathcal T^\mu
=
(aQ^\mu{}_\nu+b\,q^\mu{}_\nu)\mathcal T^\nu+c\,Q\,\mathcal T^\mu,
\]
leading to a generalized autoparallel equation that reproduces the standard Riemannian geodesic equation for \(a=b=1\) and \(c=0\). For other choices of \((a,b,c)\), the resulting orbital dynamics differ and can mimic flat rotation curves without dark matter [1102.1878]. A later Weyl-covariant analysis argues that the second clock effect can be removed by the transport prescription
\[
Du^\mu = Q^\mu{}_\nu u^\nu,
\]
and introduces a Weyl-invariant proper time \(d\hat\tau=\varphi\,d\tau\). 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 [2202.10165].

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 \(\lambda_0\), shows that symmetry reduction from full Diff to TDiff can be implemented without reintroducing the ghost-like instabilities that plague generic teleparallel modifications [2406.19355].

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, \(f(Q)\), scalar–nonmetricity, \(f(Q,B)\), 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 [2306.09289].

Source: https://www.emergentmind.com/topics/symmetric-teleparallel-theories