--- title: Quadratic Non-Metricity Gravity url: https://www.emergentmind.com/topics/quadratic-non-metricity-gravity type: topic --- # Quadratic Non-Metricity Gravity Quadratic non-metricity gravity denotes a family of metric-affine and symmetric teleparallel theories in which the gravitational Lagrangian contains terms quadratic in the non-metricity tensor \(Q_{\alpha\mu\nu}\), its traces, or in the non-metricity scalar \(\mathcal Q\) or \(\mathbb Q\). In the symmetric teleparallel setting one imposes vanishing curvature and torsion, so non-metricity is the only nontrivial geometric field strength; in broader metric-affine formulations, quadratic non-metricity terms appear alongside torsion and curvature invariants. Across the recent literature, the subject spans five-parameter quadratic non-metricity actions, quadratic \(f(Q)\) models such as \(f(\mathcal Q)=\mathcal Q+\alpha \mathcal Q^2\), Hamiltonian classifications, exact black-hole solutions, cosmological hyperfluid solutions, compact-star structure, and gauge-theoretic formulations related to Weyl symmetry [1909.09045][2007.05064][2508.16679]. ## 1. Geometric framework and invariant content The defining object is the non-metricity tensor \(Q_{\alpha\mu\nu}\), but the surveyed literature uses two sign conventions: \[ Q_{\alpha\mu\nu}\equiv \nabla_\alpha g_{\mu\nu} \qquad\text{or}\qquad Q_{\alpha\mu\nu}\equiv -\nabla_\alpha g_{\mu\nu}. \] Both conventions occur explicitly in recent work, and the associated traces are written either as \(Q_\mu,\tilde Q_\mu\) or \(Q_\mu,\bar Q_\mu\). In four dimensions, the parity-even pure-non-metricity sector is generated by five independent quadratic scalars, \[ I_1=Q_{\alpha\beta\gamma}Q^{\alpha\beta\gamma},\quad I_2=Q_{\alpha\beta\gamma}Q^{\beta\alpha\gamma},\quad I_3=Q_\alpha Q^\alpha,\quad I_4=\bar Q_\alpha \bar Q^\alpha,\quad I_5=Q_\alpha \bar Q^\alpha, \] or equivalent notations such as \(A_1,\dots,A_5\) and \(X_1,\dots,X_5\) [2007.05064][2401.11591]. In symmetric teleparallel geometry one restricts to vanishing curvature and vanishing torsion. A standard gauge choice is the coincident gauge, \(\Gamma^\alpha{}_{\mu\nu}=0\), so that all non-metricity is carried by derivatives of the metric and, in particular, \(Q_{\alpha\mu\nu}=\partial_\alpha g_{\mu\nu}\) in that gauge. This realizes symmetric teleparallelism as a gauge fixing of the general flat-connection framework [1909.09045][2508.00958]. Several papers emphasize irreducible decompositions of \(Q_{\alpha\mu\nu}\). One formulation introduces the Weyl vector \(W_\mu\equiv Q_\mu/4\), a second vector \(\Lambda_\mu\), an antisymmetric piece \(\Omega_\lambda{}^{\mu\nu}\), and a traceless symmetric piece \(q_{\lambda\mu\nu}\). Another cosmological formulation parameterizes homogeneous and isotropic non-metricity by three functions \(A(t)\), \(B(t)\), and \(C(t)\). These decompositions are technically useful because the constraint structure, cosmological dynamics, and matter couplings depend on which irreducible pieces are excited [2401.11591][2404.19498]. A broader metric-affine extension supplements the five parity-even pure-non-metricity invariants with parity-odd pieces. In the pure non-metricity subsector of the complete quadratic metric-affine action, one retains \(I_1,\dots,I_5\) together with a parity-odd invariant \[ J_1\equiv \varepsilon^{\rho\sigma\mu\nu}Q_{\rho\sigma}{}^\lambda Q_{\mu\nu\lambda}. \] This enlarges the algebraic landscape beyond the parity-preserving symmetric teleparallel models [2404.19498]. ## 2. Action principles and principal model classes The most widely used pure quadratic non-metricity action in four-dimensional symmetric teleparallel gravity is a five-parameter Lagrangian, \[ S_Q=\frac12 M_{\rm pl}^2\int d^4x\,\sqrt{-g}\, \bigl[ c_1 Q_{\alpha\mu\nu}Q^{\alpha\mu\nu} +c_2 Q_{\alpha\mu\nu}Q^{\mu\nu\alpha} +c_3 Q_\mu Q^\mu +c_4 \bar Q_\mu \bar Q^\mu +c_5 Q_\mu \bar Q^\mu \bigr], \] or equivalent normalizations and sign conventions [2401.11591][2007.05064]. Within this family, the Symmetric Teleparallel Equivalent of General Relativity (STEGR) is singled out by a specific choice of coefficients. Because conventions differ, two tuples appear in the recent literature: \[ (c_1,c_2,c_3,c_4,c_5)=\left(\tfrac14,-\tfrac12,-\tfrac14,0,\tfrac14\right) \] and \[ (c_1,c_2,c_3,c_4,c_5)=\left(-\tfrac14,\tfrac12,\tfrac14,0,-\tfrac12\right). \] Both are stated explicitly in the surveyed works and correspond to the same STEGR branch under differing sign and normalization conventions [1909.09045][2007.05064]. A second major class is the quadratic \(f(Q)\) model. In four-dimensional compact-star studies one uses \[ f(\mathcal Q)=\mathcal Q+\alpha \mathcal Q^2, \] with matter minimally coupled in the coincident gauge and field equations of the form \[ f_Q\,G_{\mu\nu} +\tfrac12 g_{\mu\nu}(f_Q\mathcal Q-f) +2f_{QQ}P^\lambda{}_{\mu\nu}\partial_\lambda \mathcal Q =\kappa T_{\mu\nu}. \] In \((2+1)\) dimensions, a closely related model is written as \[ f(\mathbb Q)=\mathbb Q+\tfrac12\alpha \mathbb Q^2-2\Lambda, \] supplemented, in the charged case, by a Maxwell sector [2507.14591][2508.16679]. Quadratic non-metricity also appears as a subsector of more general quadratic metric-affine gravity. One line of work studies the most general parity-even eleven-parameter action built from torsion, non-metricity, and their mixing; another extends this to the full seventeen-parameter theory including parity-odd terms. In these formulations, the independent connection can be solved algebraically under non-degeneracy conditions, and the resulting metric equations can be rewritten as Einstein equations with source corrections depending on hypermomentum and its derivatives [2109.13293][2112.09154]. A distinct but related construction is Weyl quadratic gravity. There the same quadratic action can be written in three equivalent languages: a non-metric torsion-free Weyl geometry, a metric Weyl-gauge-covariant formulation, and a metric formulation with vectorial torsion. The common action contains \[ \alpha F_{\mu\nu}F^{\mu\nu} +\beta \widetilde R^2 +\gamma \widetilde R_{\mu\nu}\widetilde R^{\mu\nu} +\delta C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} +\varepsilon \mathcal G, \] with the Weyl gauge field \(w_\mu\) carrying either vectorial non-metricity or vectorial torsion, depending on the chosen geometric representation [2312.13384]. ## 3. Dynamical structure, constraints, and relation to general relativity At the linearized level around Minkowski space, the generic five-parameter quadratic non-metricity theory propagates more than the massless graviton. In the coincident gauge, where \(Q_{\alpha\mu\nu}=\partial_\alpha h_{\mu\nu}\), the quadratic theory generally contains the spin-2 field \(h_{\mu\nu}^{\rm TT}\), a spin-1 vector \(V_\mu\), and two scalar combinations. The conditions \[ c_2+c_5=0,\qquad 2c_3+c_2+c_4=0,\qquad c_1>0 \] remove the unwanted vector and extra scalar sectors and leave a ghost-free massless spin-2 mode. Under the STEGR tuning the linearized field equations reduce to the Fierz-Pauli form [1909.09045]. The nonlinear Hamiltonian structure sharpens this picture. The five-parameter theory space can be partitioned into nine primary sectors according to degeneracy conditions on the coefficients \(c_i\). Sector \(0\) is generic and has no primary constraints; sectors I-VIII arise when the Hessian becomes degenerate. Sector V, defined by \[ c_3=-c_1,\qquad c_4=-2c_1-c_2,\qquad c_5=2c_1, \] contains GR and is identified as the only promising family of consistent extensions. In that sector one expects two tensor degrees of freedom after including secondary constraints. By contrast, sectors with \(c_1=0\) remove all kinetic terms for \(h_{ij}\) and are regarded as doubtful, while sectors with very many primary constraints propagate at most one or zero degrees of freedom and are classified as unphysical [2007.05064]. A more refined Hamiltonian treatment in general teleparallel quadratic gravity confirms that, in the symmetric teleparallel case, the lapse and shift constraints are first-class and generate the standard secondary Hamiltonian and momentum constraints obeying the ADM algebra. In the irreducible basis, the lapse constraint depends only on the trace-vector piece and the second vector, whereas the shift constraints also receive contributions from the antisymmetric piece \(\Omega\) and the traceless symmetric part \(q\). A specific result is that the coupling \(k_8\) multiplying \(\Omega_\lambda{}^{\mu\nu}\Omega^\lambda{}_{\mu\nu}\) affects the shift constraint but not the lapse constraint [2401.11591]. The relation to general relativity depends strongly on the formulation. In the full quadratic metric-affine theories of Iosifidis and collaborators, the connection equation is linear in the distortion and admits a unique algebraic solution in terms of the hypermomentum \(\Delta_\lambda{}^{\mu\nu}\), provided certain coefficient matrices are invertible. When \(\Delta_\lambda{}^{\mu\nu}=0\), the distortion vanishes, the connection becomes Levi-Civita, and the theory reduces exactly to GR. This vacuum reduction is proved both in the parity-even eleven-parameter theory and in the complete seventeen-parameter theory including parity-odd terms [2109.13293][2112.09154]. A notable misconception addressed explicitly in recent work is the claim that higher-order non-metricity gravity admits no spherically symmetric solution. An exact four-dimensional spherically symmetric solution was constructed in a quadratic non-metricity model, and the authors state that this contradicts all such claims in the literature [2508.00958]. ## 4. Spherically symmetric configurations and black holes In four dimensions, an exact static spherically symmetric solution has been obtained for the model \[ S=\int d^4x\,\sqrt{-g}\,\frac1{2\kappa^2}\,[Q+\alpha Q^2-\Lambda]+S_{\rm em}, \] with line element \[ ds^2=-S(r)\,dt^2+\frac{dr^2}{S(r)}+r^2 d\Omega^2, \qquad Q(r)=-\frac{2[S(r)+rS'(r)]}{r^2}. \] The off-diagonal field equation imposes \[ r^2S''-2S=0, \] whose solution, together with the diagonal equations, yields \[ S(r)=1-\frac{2m}{r}-\frac{\Lambda}{3}r^2, \qquad \alpha=-\frac{1}{4\Lambda}. \] The explicit interpretation given in that work is that the \(\alpha Q^2\) term acts exactly like a cosmological constant, \(\Lambda_{\rm eff}=-1/(4\alpha)\). When the Maxwell field is turned on, the same off-diagonal equation remains independent of the charge sector, and no genuinely charged Reissner-Nordström-(A)dS-type solution exists; the charged solution collapses to the uncharged one [2508.00958]. In \((2+1)\) dimensions, the quadratic \(f(\mathbb Q)\) model produces a substantially richer solution space. For the static circularly symmetric ansatz \[ ds^2=-\mu(r)\,dt^2+\frac{dr^2}{\nu(r)}+r^2 d\phi^2, \qquad \mathbb Q=-\frac{\mu'(r)\nu(r)}{r\mu(r)}, \] the uncharged sector with \(\alpha=0\) reproduces BTZ, \[ \mu(r)=-\Lambda r^2-m,\qquad \nu_1=1, \] whereas the quadratic branch \(\alpha\neq0\) gives \[ \mu(r)= -\frac{1\pm\sqrt{1-12\alpha\Lambda}}{6\alpha}\,r^2-m, \qquad \Lambda_{\rm eff}\equiv \frac{1\pm\sqrt{1-12\alpha\Lambda}}{6\alpha}. \] The charged BTZ branch at \(\alpha=0\) has \[ \phi(r)=c_1+c_2\ln(r/r_0),\qquad \mu(r)=-\Lambda r^2-m-2c_2^2\ln(r/r_0), \] while the charged quadratic branch admits closed-form integrals and a special elementary branch at \(\alpha=1/(12\Lambda)\) [2508.16679]. Geometric invariants distinguish the singularity structure. In the uncharged \((2+1)\)-dimensional branch one finds \[ R=2\Lambda_{\rm eff},\qquad \mathbb Q=\Lambda_{\rm eff},\qquad K=2\Lambda_{\rm eff}^2. \] In the charged quadratic case with \(\alpha\neq1/(12\Lambda)\), curvature and non-metricity invariants scale as \(O(1/r^2)\) near the center, whereas in the special branch \(\alpha=1/(12\Lambda)\) all curvature and non-metricity invariants diverge only as \(O(1/r)\). The paper identifies this as a milder central singularity than in BTZ and attributes the weakening to the quadratic \(\mathbb Q\) term. The same analysis shows that \(\mu(r)\) can develop up to three real zeros, corresponding to inner, event, and cosmological-type horizons [2508.16679]. Black-hole thermodynamics has also been worked out in that lower-dimensional quadratic \(f(\mathbb Q)\) setting. With outer horizon \(r_+\) defined by \(\mu(r_+)=0\), the surface gravity and Hawking temperature are \[ \kappa=\mu'(r_+)/2,\qquad T_H=\mu'(r_+)/(4\pi), \] and the entropy takes the Wald form \[ S=\frac14 A\,f_{\mathbb Q}(r_+)=\frac12\pi r_+\,[1+\alpha \mathbb Q(r_+)]. \] Over wide parameter ranges one finds \(T_H>0\), \(S>0\), \(C>0\), and the first law \[ dM=T_H\,dS+\Phi_H\,dQ \] holds identically. The same work also reports that the quadratic \(\mathbb Q\) term shifts the location of photon spheres and can admit stable circular photon orbits that are absent in pure BTZ [2508.16679]. ## 5. Cosmology, hypermomentum, and effective fluids Quadratic non-metricity cosmology has been developed chiefly in the metric-affine setting, where non-metricity is sourced by hypermomentum. On a spatially flat FLRW background, \[ ds^2=-dt^2+a^2(t)\delta_{ij}dx^i dx^j, \] the most general homogeneous and isotropic non-metricity is parameterized by three functions \(A(t)\), \(B(t)\), and \(C(t)\). The modified Friedmann equations then receive algebraic and derivative corrections built from these functions and from the couplings \(a_1,\dots,a_5\). When torsion and mixed torsion-non-metricity terms are switched off, the pure \(Q\)-quadratic sector is obtained directly from the full quadratic metric-affine equations [2109.06167][2404.19498]. A particularly explicit case is the torsionless model with only the \(a_3 Q_\mu Q^\mu\) invariant and a pure dilation hyperfluid. With \[ \Delta_{\alpha\mu\nu}=D(t)\,\delta_{\mu\alpha}u_\nu, \] the connection equation gives \[ Q_\mu=\frac{\kappa}{a_3}D(t)u_\mu, \] while the conservation laws imply \[ \dot\rho+3H(1+w)\rho=0,\qquad \dot D+3HD=0. \] In the non-Riemannian-dominated regime, one finds \[ H^2=B\kappa D^2,\qquad D(t)=D_0\left(\frac{a_0}{a(t)}\right)^3, \] and the exact scale factor \[ a(t)=a_0\,[1+B_1D_0(t-t_0)]^{1/3}. \] This is precisely the stiff-matter law \(a\propto t^{1/3}\), and the associated non-metricity sector behaves as \(p=\rho\propto a^{-6}\) [2109.06167]. The complete quadratic metric-affine cosmology enlarges the effective-fluid dictionary considerably. In the pure-non-metricity sector, pure dilation hypermomentum yields \(A=B=0\) and \(C=D(t)/4\), again producing a stiff-fluid component. Pure shear hypermomentum gives \(A\), \(B\), and \(C\) as linear combinations of the shear variables, and in a one-parameter family the effective equation of state becomes \[ w_{\rm eff}=-\frac53-\frac23\frac{s_5}{s_3}. \] Special choices reproduce a cosmological constant \((w_{\rm eff}=-1)\), a dark-matter-like component \((w_{\rm eff}=0)\), a domain-wall fluid \((w=-2/3)\), or a string-gas fluid \((w=-1/3)\). The same work notes that, compared to \(f(R)\) or \(R^2\) cosmology, non-metricity allows one to geometrize a much richer class of effective cosmic fluids descending from matter hypermomentum [2404.19498]. This body of results also clarifies a structural point. In the metric-affine quadratic theories of this type, non-metricity does not act merely as a passive reformulation of GR once hypermomentum is present; rather, the dilation and shear parts of matter source distinct non-metric components, and these feed back into the Friedmann system as effective fluids. A plausible implication is that observational distinctions between quadratic non-metricity cosmologies and scalar-curvature modifications will hinge on the matter microstructure encoded by hypermomentum rather than on the background metric sector alone. ## 6. Compact objects, hydrostatic equilibrium, and observational bounds Quadratic \(f(\mathcal Q)\) gravity has also been applied to relativistic stars. In the coincident gauge, with \[ f(\mathcal Q)=\mathcal Q+\alpha \mathcal Q^2, \] a static spherically symmetric interior metric \[ ds^2=-e^{a(r)}c^2dt^2+e^{b(r)}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2) \] and a perfect-fluid source lead to modified Tolman-Oppenheimer-Volkoff equations. Defining \[ m(r)=4\pi\int_0^r \rho(\tilde r)\,\tilde r^2 d\tilde r, \] the hydrostatic balance law can be written as \[ \frac{dp}{dr} = -\frac{[\rho c^2+p][m(r)+4\pi r^3 p/f_Q]}{r[r-2Gm(r)/c^2]} +\Delta_Q(r), \] with \[ \Delta_Q(r)= -\frac{2\alpha}{\kappa}\,e^{-b(r)}\,P^r{}_{rr}\,\frac{d\mathcal Q}{dr}, \qquad f_Q=1+2\alpha\mathcal Q. \] The term \(\Delta_Q\) is the additional force generated by the quadratic correction [2507.14591]. Using a Krori-Barua ansatz, \[ a(r)=s_0\Bigl(\frac rR\Bigr)^2+s_1,\qquad b(r)=s_2\Bigl(\frac rR\Bigr)^2, \] the resulting density and pressure profiles admit closed forms. Near the center, \[ \rho(r)=\rho_c-\tfrac12\rho_2 r^2+\cdots,\qquad p_r(r)=p_c-\tfrac12 p_2 r^2+\cdots, \] with \[ \rho_c=\frac{3s_2}{\kappa R^2}+O(\alpha),\qquad p_c=\frac{2s_0-s_2}{\kappa R^2}+O(\alpha). \] The explicit trend reported is that negative \(\alpha\) softens the pressure-density curve, whereas positive \(\alpha\) stiffens it. In an approximate linear equation of state fit, \(\alpha<0\) gives \(v_r^2\approx0.25\,c^2\) and \(v_\perp^2\approx0.18\,c^2\), while \(\alpha>0\) gives \(v_r^2\approx0.47\,c^2\) and \(v_\perp^2\approx0.28\,c^2\) [2507.14591]. Several stability criteria were tested. The Zeldovich ratio remains subluminal, \[ \frac{p_c}{\rho_c c^2}\approx \begin{cases} 0.112,&\alpha<0,\\ 0.178,&\alpha>0, \end{cases} \] the radial sound speed satisfies \(0\le v_r^2\le1\), and for \(\alpha<0\) one has \(v_r^2(r)\le0.3\,c^2\). Relative to the perturbative-QCD conformal bound \(v_s^20:\quad M\approx2.17\,M_\odot,\ R\approx12.11\,\mathrm{km},\ C\approx0.51. \] All lie below the Buchdahl limit \(8/9\), while the core and surface densities exceed nuclear saturation: \[ \rho_{\rm core}\approx7.2\times10^{14}\,\mathrm{g/cm^3}, \qquad \rho_{\rm surf}\approx(3.9\mbox{--}4.5)\times10^{14}\,\mathrm{g/cm^3}. \] The same analysis states the observational bound \[ |\alpha|\lesssim 3\,\mathrm{km}^2. \] Within that model, the quadratic non-metricity term therefore modifies hydrostatic equilibrium without spoiling the basic compactness and causality requirements [2507.14591]. ## 7. Broader metric-affine context, dualities, and interpretive issues Quadratic non-metricity gravity sits inside a wider network of metric-affine theories in which the connection is independent of the metric and matter may carry hypermomentum. In the eleven-parameter and seventeen-parameter quadratic metric-affine actions, the connection equation is linear in the distortion \(N^\lambda{}_{\mu\nu}\). Under non-degeneracy assumptions, one solves algebraically for \(N^\lambda{}_{\mu\nu}\), and then obtains closed forms for torsion and non-metricity: \[ T_{\mu\nu}{}^\alpha=N^\alpha{}_{[\mu\nu]},\qquad Q_{\alpha\mu\nu}=2N_{(\mu\nu)\alpha}. \] The metric field equations can then be recast as \[ \widetilde G_{\mu\nu}=\kappa T_{\mu\nu}+(\text{terms quadratic in }\Delta)+(\text{derivatives of }\Delta), \] so that spin, dilation, and shear components of hypermomentum act as effective geometric sources [2109.13293][2112.09154]. A second generalization is the vector-tensor correspondence. In a vacuum quadratic metric-affine action that includes all parity-even quadratic torsion and non-metricity invariants plus kinetic terms for the field strengths of the trace vectors, the theory is on-shell equivalent, up to boundary terms, to Riemannian gravity coupled to three interacting Proca fields, \[ A_\mu:=Q_\mu,\qquad B_\mu:=q_\mu,\qquad C_\mu:=T_\mu. \] The equivalent action contains mass terms, bilinear couplings, and Maxwell-type kinetic terms for these three vectors. In special subcases with only one kinetic term, a single vector propagates while the others remain auxiliary [2111.14214]. Weyl quadratic gravity provides a different unifying perspective. In the torsion-free non-metric formulation one has \[ Q_{\mu\nu\rho}=-2w_\mu g_{\nu\rho}, \] whereas in the metric torsionful formulation the same Weyl gauge field \(w_\mu\) appears as vectorial torsion, \[ T^\rho{}_{\mu\nu}=2\delta^\rho_{[\mu}w_{\nu]}. \] The two connections are related by a projective transformation, \[ \Gamma'{}^\rho_{\mu\nu}=\Gamma^\rho_{\mu\nu}+\delta^\rho_\mu A_\nu, \qquad A_\nu=w_\nu, \] and the quadratic action built from \(F^2\), \(R^2\), \(R_{\mu\nu}^2\), \(C^2\), and \(\mathcal G\) is invariant under this map. The physical statement advanced in that work is that vectorial non-metricity and vectorial torsion are dual descriptions with unchanged phenomenology [2312.13384]. Two interpretive tensions recur in the literature. First, vacuum quadratic metric-affine gravity with vanishing hypermomentum can reduce exactly to GR, whereas symmetric teleparallel \(f(Q)\) models and constrained quadratic non-metricity models can support nontrivial modified solutions such as the lower-dimensional black holes and the stellar configurations described above. Second, some constructions make the quadratic non-metricity sector behave effectively as a cosmological constant or as auxiliary vectors, while others produce new horizon structures, effective fluids, or modified stellar balance. This suggests that the decisive distinction is not the presence of quadratic non-metricity alone, but the combination of geometric constraints, matter couplings, and whether the connection is treated as an auxiliary metric-affine variable or fixed within a symmetric teleparallel sector. Source: https://www.emergentmind.com/topics/quadratic-non-metricity-gravity