Weyl-Type Nonmetricity in Gravity Theories

• Weyl-type nonmetricity is the pure-trace form in which the covariant derivative of the metric is proportional to the metric, effectively acting as a gauge field for local dilatations.
• It underpins various gravity theories—including metric-affine, Einstein-Weyl, and Weyl-gauged models—by influencing black hole charges and cosmological dynamics.
• The nonmetricity tensor’s decomposition into trace and traceless parts leads to distinct physical effects such as anomalous transport, modified gravitational interactions, and the emergence of the second clock effect.

Searching arXiv for recent and foundational papers on Weyl-type nonmetricity to support the article. Search query: Weyl nonmetricity metric-affine gravity arXiv Weyl-type nonmetricity is the special, purely trace form of metric noncompatibility in which the covariant derivative of the metric is proportional to the metric itself. In metric-affine language one may define the nonmetricity tensor either as $$Q_{\alpha\mu\nu}\equiv \tilde\nabla_{\alpha}g_{\mu\nu}$$ or, with the opposite sign convention, as $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$; in the Weyl sector this becomes $Q_{\alpha\mu\nu}=\frac{1}{n}g_{\mu\nu}Q_\alpha$, or equivalently in four dimensions $Q_{\alpha\mu\nu}=g_{\mu\nu}W_\alpha$, while in the alternative sign convention it is written $\nabla_\lambda g_{\mu\nu}=-2A_\lambda g_{\mu\nu}$ or $\nabla_\lambda g_{\mu\nu}=-2w_\lambda g_{\mu\nu}$ (Bahamonde et al., 2022). The associated one-form—denoted in the literature by $W_\mu$, $A_\mu$, $w_\mu$, $b_\mu$, or $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$0—acts as a gauge field for local dilatations, and its presence is the defining feature of Weyl geometry and of a large class of metric-affine, Einstein-Weyl, and Weyl-gauged gravity theories (Gomes et al., 2018).

1. Definition, conventions, and irreducible structure

In a general metric-affine geometry the nonmetricity tensor measures the failure of the affine connection to preserve the metric. A standard decomposition isolates its trace and traceless parts:

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$1

In four dimensions one also writes

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$2

with the second term traceless in the last two indices. In this sense, Weyl-type nonmetricity is the pure-trace sector of a more general post-Riemannian structure, whereas the traceless sector is associated with shear degrees of freedom (Bahamonde et al., 2022).

The broader irreducible decomposition used in metric-affine and gauge-theoretic formulations includes, besides the trace vector $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$3, a second trace $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$4, a totally symmetric traceless part, and a mixed-symmetry traceless three-tensor. One convenient $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$5-dimensional expression decomposes $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$6 into the trace vector $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$7, the second trace $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$8, and traceless tensors $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$9 and $Q_{\alpha\mu\nu}=\frac{1}{n}g_{\mu\nu}Q_\alpha$0 (Condeescu et al., 2024). Another formulation writes, for arbitrary nonmetricity under conformal symmetry,

$Q_{\alpha\mu\nu}=\frac{1}{n}g_{\mu\nu}Q_\alpha$1

where $Q_{\alpha\mu\nu}=\frac{1}{n}g_{\mu\nu}Q_\alpha$2 is totally traceless and symmetric in its last two indices, and $Q_{\alpha\mu\nu}=\frac{1}{n}g_{\mu\nu}Q_\alpha$3 is antisymmetric in those indices (Delhom et al., 2019). This emphasizes that the Weyl sector is only one distinguished corner of the full nonmetricity tensor.

Geometrically, pure Weyl nonmetricity changes the length of vectors under parallel transport while preserving angles and the light-cone structure; by contrast, a traceless shear part distorts angles as well (Bahamonde et al., 2022). That distinction is central in metric-affine gravity, where the trace, dilation, and shear sectors are sourced by different components of matter hypermomentum.

2. Weyl connection, curvature, and gauge symmetry

For torsionless Weyl geometry, the affine connection is the Levi-Civita connection plus a vectorial deformation. In one standard convention,

$Q_{\alpha\mu\nu}=\frac{1}{n}g_{\mu\nu}Q_\alpha$4

so that

$Q_{\alpha\mu\nu}=\frac{1}{n}g_{\mu\nu}Q_\alpha$5

The associated Ricci tensor and scalar are

$Q_{\alpha\mu\nu}=\frac{1}{n}g_{\mu\nu}Q_\alpha$6

$Q_{\alpha\mu\nu}=\frac{1}{n}g_{\mu\nu}Q_\alpha$7

Under a local scale transformation $Q_{\alpha\mu\nu}=\frac{1}{n}g_{\mu\nu}Q_\alpha$8, the Weyl vector transforms as $Q_{\alpha\mu\nu}=\frac{1}{n}g_{\mu\nu}Q_\alpha$9, preserving the nonmetricity condition. In this sense, the Weyl vector is the gauge field for local changes of length scale (Gomes et al., 2018).

Equivalent formulas appear in other sign conventions. For example, with

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

one finds

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

where $Q_{\alpha\mu\nu}=g_{\mu\nu}W_\alpha$2 is the symmetric-teleparallel nonmetricity scalar built from the disformation tensor (Xu et al., 2020). In Cartan language the same structure is encoded as

$Q_{\alpha\mu\nu}=g_{\mu\nu}W_\alpha$3

with $Q_{\alpha\mu\nu}=g_{\mu\nu}W_\alpha$4 under infinitesimal Weyl rescalings (Vazquez-Mozo, 3 Jun 2026).

Weyl gauge symmetry can also be presented as a symmetry of generalized Weyl spaces $Q_{\alpha\mu\nu}=g_{\mu\nu}W_\alpha$5. Under

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

the full affine connection is invariant, as are the full Riemann and Ricci tensors, while the scalar curvature rescales as $Q_{\alpha\mu\nu}=g_{\mu\nu}W_\alpha$7 (Quiros, 2022). A gauge-covariant derivative $Q_{\alpha\mu\nu}=g_{\mu\nu}W_\alpha$8 can then be defined for a tensor of conformal weight $Q_{\alpha\mu\nu}=g_{\mu\nu}W_\alpha$9 (Quiros, 2022). Closely related Weyl-covariant constructions arise in quadratic Weyl gravity, where the connection

$\nabla_\lambda g_{\mu\nu}=-2A_\lambda g_{\mu\nu}$0

satisfies $\nabla_\lambda g_{\mu\nu}=-2A_\lambda g_{\mu\nu}$1 and is invariant under Weyl gauge transformations (Condeescu et al., 2023).

3. Dynamical realizations in gravity theories

A minimal dynamical model for the Weyl vector in metric-affine gravity is obtained from the homothetic curvature

$\nabla_\lambda g_{\mu\nu}=-2A_\lambda g_{\mu\nu}$2

together with the action

$\nabla_\lambda g_{\mu\nu}=-2A_\lambda g_{\mu\nu}$3

Variation with respect to $\nabla_\lambda g_{\mu\nu}=-2A_\lambda g_{\mu\nu}$4 yields a Maxwell-type equation,

$\nabla_\lambda g_{\mu\nu}=-2A_\lambda g_{\mu\nu}$5

so the Weyl vector couples directly to the dilation current, namely the trace part of the hypermomentum density (Bahamonde et al., 2022). This is the most direct realization of the statement that Weyl-type nonmetricity sources and is sourced by dilational matter.

A different metric-affine construction promotes the curvature scalar to the Weyl scalar $\nabla_\lambda g_{\mu\nu}=-2A_\lambda g_{\mu\nu}$6 and considers

$\nabla_\lambda g_{\mu\nu}=-2A_\lambda g_{\mu\nu}$7

Variation with respect to the Weyl vector gives

$\nabla_\lambda g_{\mu\nu}=-2A_\lambda g_{\mu\nu}$8

equivalently $\nabla_\lambda g_{\mu\nu}=-2A_\lambda g_{\mu\nu}$9, with $\nabla_\lambda g_{\mu\nu}=-2w_\lambda g_{\mu\nu}$0. The paper identifies this constraint as the mechanism that removes higher-derivative terms and keeps the theory second order (Gomes et al., 2018).

Weyl gauge symmetry also underlies purely quadratic gravity models. In four dimensions the most general purely quadratic Lagrangian invariant under the Weyl group is

$\nabla_\lambda g_{\mu\nu}=-2w_\lambda g_{\mu\nu}$1

with $\nabla_\lambda g_{\mu\nu}=-2w_\lambda g_{\mu\nu}$2 the dilation curvature, $\nabla_\lambda g_{\mu\nu}=-2w_\lambda g_{\mu\nu}$3 the Weyl-tensor square, and $\nabla_\lambda g_{\mu\nu}=-2w_\lambda g_{\mu\nu}$4 the Gauss-Bonnet term (Condeescu et al., 2024). A parallel formulation states that the unique Weyl-invariant quadratic action may be written with $\nabla_\lambda g_{\mu\nu}=-2w_\lambda g_{\mu\nu}$5, $\nabla_\lambda g_{\mu\nu}=-2w_\lambda g_{\mu\nu}$6, $\nabla_\lambda g_{\mu\nu}=-2w_\lambda g_{\mu\nu}$7, and $\nabla_\lambda g_{\mu\nu}=-2w_\lambda g_{\mu\nu}$8, depending on the Weyl-covariant connection (Condeescu et al., 2023).

The relation of Weyl-type nonmetricity to general metric-affine gravity predates these constructions. In a Weyl-invariant extension of metric-affine gravity, the pure Weyl piece is

$\nabla_\lambda g_{\mu\nu}=-2w_\lambda g_{\mu\nu}$9

and under $W_\mu$0 with $W_\mu$1 held fixed one has $W_\mu$2; the Weyl-invariant action then depends on affine curvature, torsion invariants, and the trace of nonmetricity through combinations such as $W_\mu$3 and $W_\mu$4 (Vazirian et al., 2013).

Action principles for Einstein-Weyl systems provide another perspective. In three dimensions, a metric-affine $W_\mu$5 action augmented by Lagrange multipliers and gravitational Chern-Simons terms yields the Einstein-Weyl equations when the traceless part of nonmetricity is consistently set to zero, while the Weyl vector obeys a special case of the generalized monopole equation,

$W_\mu$6

or $W_\mu$7 in differential-form language (Klemm et al., 2020). This illustrates that Weyl-type nonmetricity can be either an isolated sector or one component of a larger nonmetric theory whose reduction selects Einstein-Weyl geometry.

4. Black holes, charges, and compact-object observables

In the simplest pure-Weyl sector of metric-affine gravity one may set torsion $W_\mu$8 and the traceless shear part $W_\mu$9, so that

$A_\mu$0

For a static, spherically symmetric ansatz in Schwarzschild coordinates,

$A_\mu$1

with

$A_\mu$2

where $A_\mu$3 is the “dilatation charge” squared, analogous to the electric charge in Reissner-Nordström (Bahamonde et al., 2022). The metric function therefore acquires an additional $A_\mu$4 term sourced by the Weyl charge.

When torsion and the traceless shear part are also active, the same paper gives

$A_\mu$5

The resulting family is described there as the broadest family of static and spherically symmetric black hole solutions with spin, dilation and shear charges in metric-affine gravity so far (Bahamonde et al., 2022). In this enlarged sector there are two independent $A_\mu$6 charges associated with nonmetricity: a dilation charge $A_\mu$7 from the trace sector and a shear charge $A_\mu$8 from the traceless sector.

The stated phenomenological implications are correspondingly direct. Extra $A_\mu$9 terms modify perihelion precession, light deflection, and time delay around compact objects, while precision solar-system tests and observations of black hole shadows place bounds on $w_\mu$0 and $w_\mu$1 (Bahamonde et al., 2022). The same source further notes that black-hole thermodynamics can be extended to include $w_\mu$2 and $w_\mu$3 terms in the first law, although a Hamiltonian or Euclidean action analysis is needed to compute the corresponding entropies (Bahamonde et al., 2022).

A broader gauge-theoretic interpretation of these charged solutions is that the Weyl vector behaves much like a $w_\mu$4 gauge field for dilations (Bahamonde et al., 2022). This suggests a close analogy between homothetic curvature and an Abelian field strength, but with the charge now measuring matter dilation current rather than electric charge.

5. Cosmology, matter couplings, and transport phenomena

Weyl-type nonmetricity enters cosmology prominently through Weyl-type $w_\mu$5 gravity. In this framework one adopts the semi-metric connection

$w_\mu$6

so that

$w_\mu$7

and considers an action containing $w_\mu$8, a Proca-type kinetic term and mass for the Weyl vector, and a Lagrange multiplier enforcing $w_\mu$9 (Xu et al., 2020). In this setting the matter energy-momentum tensor is not covariantly conserved:

$b_\mu$0

and the nonzero divergence generates both an energy source term in the continuity equation and an extra force in the Euler equation, so matter generally follows nongeodesic motion (Xu et al., 2020).

For flat FLRW cosmology with a timelike Weyl field, $b_\mu$1, one has $b_\mu$2 in the convention used there, and the field equations can be recast as effective Friedmann equations,

$b_\mu$3

where $b_\mu$4 and $b_\mu$5 collect the Weyl and $b_\mu$6 contributions (Xu et al., 2020). A later observational study for the same framework uses the ansatz

$b_\mu$7

with 32 cosmic chronometers, 1701 Pantheon$b_\mu$8 SNe Ia, and 6 BAO points, obtaining the best-fit values

$b_\mu$9

together with $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$00 and $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$01 (Bhagat et al., 2024). Another study of logarithmic and strong-coupling $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$02 models reports transition redshifts $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$03 and $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$04, present-day equation-of-state parameters $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$05, an age of the Universe $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$06 Gyr, and satisfaction of the Null, Weak and Dominant Energy Conditions with controlled violation of the Strong Energy Condition (Bhagat et al., 16 Jun 2025).

Weyl-type nonmetricity also affects local dynamics. In Weyl-type $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$07 gravity the geodesic-deviation equation becomes

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$08

where the second term arises from the extra force produced by matter-nonmetricity coupling (Yang et al., 2021). In the weak-field limit the generalized Poisson equation takes the form

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$09

with modified $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$10 and PPN parameter $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$11 (Yang et al., 2021). The same analysis gives corrections to tidal forces and a shifted Roche limit through gradients of the Weyl vector and the extra force (Yang et al., 2021).

Outside gravitation, Weyl-type nonmetricity has recently appeared in anomalous transport theory. In a four-dimensional metric-affine background with pure Weyl nonmetricity, the Weyl-invariant four-form

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$12

reduces in the pure Weyl case to

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$13

Using transgression methods, the equilibrium partition function yields the covariant axial current

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$14

or equivalently

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$15

which exhibits nonmetricity-mediated chiral separation effects driven by the Weyl magnetic field and the fluid vorticity (Vazquez-Mozo, 3 Jun 2026). This is a purely geometric transport effect: in the Riemannian limit $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$16 one recovers the usual anomaly-induced currents, whereas here the geometric Weyl one-form itself sources the anomalous response.

6. Conceptual issues, dualities, and the second clock effect

The most persistent conceptual issue attached to Weyl-type nonmetricity is the second clock effect. In generalized Weyl spaces with torsion-free connection and nonmetricity $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$17, parallel transport changes the length of a vector by a path-dependent amount. For a vector of conformal weight $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$18, one formulation gives

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$19

while for a massive particle with four-momentum $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$20 one has

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$21

Atomic transition frequencies then acquire path dependence, which is the second clock effect (Quiros, 2022). The same line of argument is developed in generalized Weyl spacetimes $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$22, where the master equation for length change under gauge-invariant parallel transport yields a path-dependent mass variation and, consequently, history-dependent clock rates and line splittings (Quiros, 2021).

Both analyses conclude that the effect disappears only in Weyl integrable geometry, where the gauge field is a pure gradient:

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$23

In that case the loop integrals reduce to boundary terms, and for closed spatial loops the second clock effect vanishes (Quiros, 2022). The phenomenological argument of the second paper is sharper: generalized Weyl spaces, including symmetric teleparallel subclasses, are claimed not to be phenomenologically viable on stellar or solar-system scales because of the second clock effect and perihelion-shift constraints, with only the integrable-Weyl case evading the effect (Quiros, 2021). This remains one of the principal controversies surrounding physical realizations of nonintegrable Weyl geometry.

A distinct response does not preserve standard Weyl nonmetricity but modifies the geometric ansatz. Schrödinger’s length-preserving nonmetricity imposes

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$24

so that vector lengths remain invariant under parallel transport despite $Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$25. In the “Weyl-Schrödinger” specialization the connection becomes

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$26

and the resulting cosmology yields effective dark-energy terms in the metric formulation, whereas the Palatini variation collapses back to general relativity (Ming et al., 2023). This is not a standard Weyl geometry in the strict sense; rather, it is an alternative nonmetric theory motivated by the desire to avoid length-changing parallel transport.

Another important structural issue is the relation between vectorial nonmetricity and vectorial torsion. In Weyl gauge gravity, pure vector nonmetricity

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$27

is in one-to-one correspondence with vector torsion

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$28

via a projective redefinition of the affine connection (Condeescu et al., 2024). The same equivalence appears in the three-form/tangent-space description of Weyl quadratic gravity, where the torsionful metric-compatible connection

$Q_{\lambda\mu\nu}\equiv -\nabla_\lambda g_{\mu\nu}$29

is related by a projective shift to the torsionless nonmetric Weyl connection (Condeescu et al., 2023). More generally, the trace part of torsion can alternatively be interpreted as the trace part of nonmetricity in Einstein manifolds with torsion and nonmetricity (Klemm et al., 2018). This suggests that some physical effects attributed to Weyl-type nonmetricity may admit an equivalent torsional description, at least in the purely vectorial sector.

Taken together, these developments place Weyl-type nonmetricity at the intersection of metric-affine geometry, gauge symmetry, modified gravity, and anomalous transport. Its pure-trace form is mathematically simple and physically expressive: it gauges local dilatations, generates homothetic curvature, supports black-hole and cosmological solutions, and induces geometric transport effects. At the same time, the second clock effect, the role of integrability, and the projective duality with torsion remain central in assessing which realizations are viable as descriptions of nature.