Symmetric Teleparallel Gravity

• Symmetric Teleparallel Gravity is a geometric theory where gravity is encoded solely in nonmetricity, offering an alternative to curvature or torsion-based models.
• It employs a generalized quadratic Lagrangian and modified parallel transport prescriptions to reproduce standard geodesics and consistent fermion dynamics.
• The framework yields both classic solutions like Schwarzschild–de Sitter and innovative configurations, broadening the scope of gravitational phenomenology.

Symmetric teleparallel gravity (STPG) is a geometric theory of gravitation formulated on a non-Riemannian spacetime in which the only nonvanishing field strength is the nonmetricity, while the curvature and torsion of the connection are identically zero. The connection is generally not metric-compatible and fails to preserve the metric under parallel transport, giving rise to a nonmetricity tensor. STPG provides a distinctive arena in which gravitational dynamics are encoded purely in nonmetricity, offering an alternative to curvature-based formulations of general relativity and to torsion-driven models. The framework supports a general quadratic Lagrangian in nonmetricity, exhibits consistent couplings to spinor fields, delivers solutions with both Einsteinian and genuinely non-Einsteinian character, and admits equivalent dynamics to general relativity for a specific choice of coupling parameters.

1. Geometric Structure and Nonmetricity

In STPG, a non-Riemannian spacetime is constructed where the metric $g_{\alpha\beta}$ coexists with a flat, torsion-free, but non-metric compatible connection $\omega^{\alpha\beta}$. The only nontrivial Cartan structure equation is that for nonmetricity: $$Q_{\alpha\beta} \equiv - D g_{\alpha\beta} = -d g_{\alpha\beta} + (\omega_{\alpha\beta} + \omega_{\beta\alpha})$$ In a particular gauge, most conveniently the coordinate gauge $e^{\alpha}=dx^{\alpha}$ with $\omega^{\alpha\beta}=0$, the nonmetricity tensor simplifies to: $Q_{\alpha\beta} = -d g_{\alpha\beta}$ Thus, for any metric choice, nonmetricity is computable directly as the negative of the exterior derivative of the metric components in this gauge. Upon transformation to an orthonormal frame via a vierbein $h^a_{\phantom{a}\alpha}$, $Q_{ab}$ is simply the Lorentz-transformed nonmetricity, preserving the geometric action of the original tensor. For specific metric forms—such as conformal metrics $$g = e^{2\Phi}\eta_{\alpha\beta} dx^{\alpha}\otimes dx^{\beta}$$—the nonmetricity takes a particularly transparent form, $$Q_{\alpha\beta} = -\eta_{\alpha\beta}e^{2\Phi}d\Phi$$.

2. Autoparallel Curves and Parallel Transport

The standard geodesic equation, as derived from metric compatibility, fails in STPG when adopting the gauge where $\omega^{\alpha\beta}=0$, as the autoparallel equation $DV = 0$ naively reduces to $dV=0$, implying that all curves are straight lines, which is physically unsatisfactory. To address this, a generalized prescription for parallel transport of a tangent vector in a non-Riemannian geometry is introduced: $$D V^{\alpha} = ( a Q^{\alpha}_{\phantom{\alpha}\beta} + b q^{\alpha}_{\phantom{\alpha}\beta} ) V^{\beta} + c Q V^{\alpha}$$ where $q^{\alpha}_{\phantom{\alpha}\beta}$ is the antisymmetric part of the nonmetricity contribution and $Q = Q^\mu_{\phantom{\mu}\mu}$. For the parameter choice $a = b = 1, c = 0$, this reduces the autoparallel equation to the Riemannian geodesic equation: $$\frac{d^2 x^{\alpha}}{d \tau^2} + g^{\alpha\delta} ( \partial_\gamma g_{\beta\delta} + \partial_\delta g_{\beta\gamma} - \partial_\beta g_{\gamma\delta} ) \frac{dx^\gamma}{d\tau}\frac{dx^\delta}{d\tau} = 0$$ ensuring consistency with the usual motion of test particles in general relativity.

3. Lagrangian Formulation and Field Equations

The STPG action is constructed from the most general parity-conserving quadratic Lagrangian in nonmetricity, supplemented by Lagrange multipliers that enforce vanishing torsion and curvature: $$L = \sum_{I=1}^5 k_I Q^{ab} \wedge *^{(I)}_{ab} + \Lambda *1 + (\text{Lagrange multipliers}) - K$$ Here, $Q^{ab}$ is decomposed into five irreducible components, $k_I$ are real coupling constants, $\Lambda$ is the cosmological constant, and $K$ is the matter Lagrangian. A redefinition using new constants $(c_1, ..., c_5)$ clarifies the structure, and the Lagrangian becomes: $$L = c_1 Q_{ab} \wedge *Q^{ab} + c_2 \alpha_a \wedge *\alpha^a + c_3 \beta_a \wedge *\beta^a + c_4 Q \wedge *Q + c_5 \beta_a \wedge \gamma^a + \Lambda *1 + (\text{constraints}) - K$$ For the special values $c_1 = -1/2$, $c_2 = 1$, $c_3 = -1/2$, $c_4 = 1$, $c_5 = 0$, this Lagrangian becomes (modulo boundary terms) the Einstein-Hilbert action, and field equations derived via independent variations with respect to the coframe $e^a$, connection $\omega^{ab}$, and Lagrange multipliers reproduce general relativity.

4. Exact Solutions: Conformal, Spherically Symmetric, Cosmological, and pp-Wave

Multiple classes of exact solutions demonstrate the broader solution space of STPG:

• Conformal solutions: Using metrics conformal to Minkowski, $$g = e^{2\Phi}\eta_{\alpha\beta} dx^\alpha \otimes dx^\beta$$, the field equations impose algebraic constraints on the coupling constants, often leaving the function $\Phi$ arbitrary except under additional conditions.
• Spherically symmetric static solutions: For metrics of the form $$g = -F(r) dt^2 + G(r) dr^2 + r^2 (d\theta^2 + \sin^2\theta d\phi^2)$$ and with $G(r) = 1/F(r)$, solutions can reduce to $F^2(r) = 1 - M/r + (\Lambda/(6c)) r^2$, yielding the Schwarzschild–de Sitter solution (and its variants) for certain parameters. Other non-Einsteinian, non-reciprocal metric function solutions are also possible.
• Cosmological solutions: Analyzing spatially flat Robertson–Walker metrics in isotropic coordinates, the modified Friedmann equations are derived. Under a generalized equation of state (including Chaplygin gas behavior), scale factor solutions such as $S = K e^{2nt} [1 - K_1 e^{-nt}]^{2m/n}$ arise, where $m,n$ are determined by the nonmetricity-sector couplings.
• pp-wave solutions: For a metric $g = -dt^2 + p(u) dx^2 + q(u) dy^2 + dz^2$ (with $u = z-t$), the vacuum field equations (for GR-compatible couplings) yield $p''/p + q''/q = 0$. For more general choices, solutions not corresponding to those of Einstein gravity are obtainable, such as $pq = \text{constant}$.

These solutions illustrate both the ability to recover standard GR physics and the existence of genuinely new gravitational configurations in the STPG framework.

5. Spinor Coupling and Dirac Field Dynamics

Spin-1/2 fields are coupled to STPG utilizing a Clifford algebra-valued exterior calculus formalism, with the Dirac matrices appearing naturally via the coframe fields. The covariant derivative for the spinor is constructed via the Kosmann lift: $$\mathcal{D}\Psi = d\Psi + \frac{1}{4}(\omega^{ab} + 2\sigma^{ab})\Psi$$ In the coordinate gauge with $\omega^{ab}=0$, the Dirac Lagrangian becomes: $$K = \frac{i}{2}\left( \bar{\Psi} \gamma \wedge \mathcal{D}\Psi - \mathcal{D}\bar{\Psi}\wedge \gamma \Psi \right) + im \bar{\Psi} \Psi *1$$ Variation yields the generalized Dirac equation in a non-Riemannian background: $$* \gamma \wedge d\Psi - \text{(nonmetricity terms)} + m \Psi *1 = 0$$ Nonmetricity $Q$ enters explicitly in the interaction with fermions, and even in the absence of torsion, gravitational effects are transferred into the spinor sector. This ensures that any consistent theory of gravitating fermions within STPG must account for nonmetricity-induced modifications to their dynamics.

6. Conceptual Consequences and Theoretical Implications

STPG, especially for Lagrangians constructed from the most general quadratic invariants in nonmetricity, provides a setting in which Einsteinian gravity is subsumed as a special case, while a broader variety of gravitational phenomena emerge for other parameter choices. The framework allows for a direct algorithmic computation of nonmetricity from the metric in a natural gauge, and a parallel transport prescription ensures that geodesic motion can be made to agree with conventional theory. The theory delivers a flexible platform for coupling with spinor fields, enabling explorations of quantum aspects (via spinor dynamics) in a purely nonmetricity-based gravitational background.

The inclusion of both Einsteinian (e.g., Schwarzschild–de Sitter, standard cosmologies) and non-Einsteinian (non-reciprocal spherically symmetric metrics, arbitrary conformal factors, modified pp-waves) exact solutions shows that STPG constitutes a structurally complete alternative to GR for both classical and quantum matter. The equivalence with the Einstein–Hilbert Lagrangian in a limit underlines the internal consistency of the framework, while its natural extension to general quadratic and parity-conserving models permits new avenues for phenomenological and conceptual developments at both the classical and quantum level.

Table: Classes of Exact Solutions in Symmetric Teleparallel Gravity

Solution Class	Metric Form / Ansatz	Einsteinian Limit
Conformal	$g = e^{2\Phi}\eta_{\alpha\beta} dx^\alpha dx^\beta$	Yes, for certain couplings
Spherically Symmetric	$g = -F(r) dt^2 + G(r) dr^2 + r^2\, d\Omega^2$	Yes, $F^2=1-M/r+\Lambda r^2$ for $c=11/2$
Cosmological	$$g = -dt^2 + S^2(t)\,d\vec{x}^2/(1+\frac{kr^2}{4})^2$$	Yes, under specific parameters
pp-Wave	$g = -dt^2 + p(u) dx^2 + q(u) dy^2 + dz^2$	Yes, for $p''/p+q''/q=0$

The richness and flexibility of the STPG framework, the clear separation of geometric invariants, and its precise mapping onto Einsteinian dynamics under suitable conditions make it a robust and viable alternative geometric model for the gravitational interaction, supporting both standard and novel solutions and matter couplings.