Metric–Affine Dynamics and Matter Couplings

Metric–affine gravity varies the metric and affine connection independently, so matter can source geometry through both stress–energy and hypermomentum. The latter splits into spin, dilation, and shear currents. Whether those currents merely generate auxiliary torsion and nonmetricity or excite propagating connection modes depends on the gravitational action. Projective symmetry provides a sharp consistency test: in an Einstein–Hilbert Palatini sector it removes one vectorial connection component and requires compatible matter coupling.

Required background. Frames, Tetrads, and Spinors supplies Lorentz and general frame currents; Differential Forms and Integration supplies curvature and torsion; and Gravitational Actions and Boundary Terms supplies independent connection variation.

Helpful background. Einstein Equations fixes the Palatini GR limit, while Formulations of General Relativity compares metric, tetrad, and connection variables.

Independent metric and connection equations

A general metric–affine action has the form

$$S[g,\Gamma,\Psi] =\frac{1}{2\kappa}\int d^4x\,\sqrt{-g}\, F(g,R(\Gamma),T,Q) +S_{\mathrm m}[g,\Gamma,\Psi], \qquad \kappa=8\pi G.$$

The two matter responses are

$$T_{\mu\nu} =-\frac{2}{\sqrt{-g}} \frac{\delta S_{\mathrm m}}{\delta g^{\mu\nu}}, \qquad \Delta_\lambda{}^{\mu\nu} =-\frac{2}{\sqrt{-g}} \frac{\delta S_{\mathrm m}} {\delta\Gamma^\lambda{}_{\mu\nu}}.$$

Here the middle lower connection index is the derivative index, as in $\nabla_\mu V^\rho=\partial_\mu V^\rho+\Gamma^\rho{}_{\mu\sigma}V^\sigma$. This ordering matters in every projective trace below. The metric equation is sourced by $T_{\mu\nu}$; the connection equation is sourced by $\Delta_\lambda{}^{\mu\nu}$.

Lower the last hypermomentum index and expose the general-linear representation carried by its first and last slots:

$$\Delta^\mu{}_{\lambda\nu} =g_{\nu\rho}\Delta_\lambda{}^{\mu\rho} =\tau^\mu{}_{\lambda\nu} +\frac14 g_{\lambda\nu}D^\mu +\Sigma^\mu{}_{\lambda\nu},$$

where

$$\tau^\mu{}_{\lambda\nu} =\Delta^\mu{}_{[\lambda\nu]}, \qquad D^\mu=\Delta_\lambda{}^{\mu\lambda}, \qquad \Sigma^\mu{}_{\lambda\nu} =\Delta^\mu{}_{(\lambda\nu)} -\frac14g_{\lambda\nu}D^\mu.$$

Thus spin is the antisymmetric current, dilation is the trace, and shear is the symmetric trace-free current. These names classify sources; they do not by themselves prove a one-to-one response. The connection kinetic operator can mix torsion and nonmetricity irreducible pieces.

Which matter carries hypermomentum?

A minimally coupled scalar has $\nabla_\mu\phi=\partial_\mu\phi$ and therefore no direct affine current. A Yang–Mills field strength must be constructed with the exterior derivative, not an affine covariant curl, to preserve its internal gauge symmetry; it likewise has no minimal hypermomentum. Fermions require a spin connection, and their standard minimal coupling sources the antisymmetric spin sector, reducing in the simplest parity-even case to an axial torsion current.

Dilation and shear require matter or nonminimal operators that are sensitive to the trace or symmetric part of the independent connection. Examples include idealized media with microstructure and allowed currents coupled to the nonmetricity traces. In a controlled action these couplings must be stated explicitly; calling all matter “minimally coupled” does not determine them. The systematic scalar, fermion, and gauge-field construction in Rigouzzo and Zell 2023, §§2.1–2.3, pp. 5–10, PDF shows both the fermion sensitivity and why gauge-boson field strengths cannot directly contain the affine connection.

If $F$ is linear in curvature and at most quadratic without derivatives in $T$ and $Q$, the connection equation is often algebraic. Solving it then turns hypermomentum into local matter contact interactions, as in Einstein–Cartan theory. Curvature-squared terms can instead provide derivatives—and hence kinetic terms—for pieces of $\Gamma$. Propagation is therefore an action-level conclusion, not a consequence of the word “independent.”

Projective symmetry and its trace condition

In the site index order a projective transformation is

$$\Gamma^\lambda{}_{\mu\nu} \longmapsto \Gamma^\lambda{}_{\mu\nu} +\delta^\lambda_\nu A_\mu.$$

Direct substitution gives

$$\delta T^\lambda{}_{\mu\nu} =\delta^\lambda_\nu A_\mu-\delta^\lambda_\mu A_\nu, \qquad \delta Q_{\alpha\mu\nu}=2A_\alpha g_{\mu\nu},$$

and

$$\delta R^\lambda{}_{\rho\mu\nu} =2\delta^\lambda_\rho\partial_{[\mu}A_{\nu]}.$$

The Ricci tensor changes only by an antisymmetric curl, so $g^{\mu\nu}R_{\mu\nu}(\Gamma)$ is invariant. For the Einstein–Hilbert Palatini action, define the connection Euler tensor $P_\lambda{}^{\mu\nu}$ so that its equation is

$$P_\lambda{}^{\mu\nu}=\kappa\Delta_\lambda{}^{\mu\nu}.$$

Projective invariance is precisely the off-shell Noether identity

$$P_\nu{}^{\mu\nu}\equiv0.$$

Taking the same trace of the field equation requires

$$\boxed{\Delta_\nu{}^{\mu\nu}=D^\mu=0}.$$

Equivalently, varying the matter action under the projective shift gives $\delta S_{\mathrm m}=-(1/2)\int\sqrt{-g}\,D^\mu A_\mu$. Thus a projectively invariant matter action has zero dilation current. If a chosen matter coupling has $D^\mu\ne0$ while the gravitational sector remains projectively invariant, the trace equation is inconsistent rather than dynamical. One must restrict the coupling, add a justified symmetry-breaking term that determines the projective vector, or impose an appropriate constrained geometry. The Palatini identity and matter obstruction are developed in Iosifidis 2019, ch. 2, §2.1, pp. 56–66, Eqs. (2.1)–(2.55), PDF.

Independent projective-mode check

Start from Minkowski metric and zero connection, then perform a projective shift:

$$\Gamma^\lambda{}_{\mu\nu}=\delta^\lambda_\nu A_\mu.$$

It produces nonzero torsion and nonmetricity exactly as above, while

$$R^\lambda{}_{\rho\mu\nu} =2\delta^\lambda_\rho\partial_{[\mu}A_{\nu]}, \qquad R_{\rho\nu}=2\partial_{[\rho}A_{\nu]}, \qquad R=0.$$

The arbitrary $A_\mu$ is therefore invisible to the Einstein–Hilbert scalar even when its curl is nonzero. This verifies both the scalar invariance and the missing trace equation. A term such as $Q_\mu Q^\mu$ is not projectively invariant and can make a combination involving $A_\mu$ auxiliary; curvature-squared terms can make it propagate. Neither change is an innocent gauge fixing, so its constraint and kinetic signs must be recalculated.

Results and open questions through 2026-08-19

The evidence cutoff for this active theory-space assessment is 2026-08-19. Conclusions differ because “metric–affine gravity” covers distinct actions, geometric constraints, matter couplings, and background regimes.

Formulation or calculation	Evidence and supported result	Assumptions or regime	Limitation or negative evidence
Einstein–Hilbert Palatini sector	Iosifidis 2019, ch. 2, §2.1, pp. 56–66, Eqs. (2.1)–(2.55), PDF derives the traceless Palatini tensor, projective vector, and hypermomentum condition.	Curvature-linear gravitational action; site index order is the transpose of the source convention.	The result does not classify generic quadratic curvature actions; non-projective dilation matter requires a changed theory.
Algebraic connection with Standard Model fields	Rigouzzo and Zell 2023, §§2–3, pp. 5–15, PDF integrate out torsion and nonmetricity under explicit derivative and operator-dimension rules, obtaining metric contact interactions.	Classical, local, analytic action with at most two derivatives and stated matter gauge symmetries.	The selection rules are assumptions, not a theorem covering propagating affine sectors or arbitrary higher-dimensional operators.
Generic Ricci-based and higher-curvature extensions	Beltrán Jiménez and Delhom 2020, §§III–VII, pp. 5–21, PDF identify ghosts tied to broken projective symmetry, a projective mode, and a two-form sector; they also exhibit constrained exceptions.	Nonlinear Ricci-based models, decoupling checks, matter couplings, and several restricted geometries.	Strong negative evidence against generic safety is not a no-go theorem for every constrained or symmetry-protected model.
Extended projective symmetry	Barker and Zell 2024, pp. 1–4, Eqs. (8)–(15), PDF construct the most general extended-projective-invariant class at most quadratic in field strengths and reduce it to a graviton plus one pseudoscalar.	Particular double-vector symmetry, specified fermion couplings, and terms no higher than quadratic in curvature, torsion, and nonmetricity.	A promising constructive subclass does not establish universal stability, ultraviolet completion, or uniqueness beyond its operator class.

Competing positions and discriminating checks

The conservative position restricts model building to projectively invariant matter, algebraic connection sectors, or subclasses whose complete constraints are demonstrably healthy. It treats generic curvature-squared propagation as unsafe until the nonlinear modes, residues, and characteristic surfaces are controlled. The permissive position looks for additional symmetries or constrained geometries that retain selected affine particles; extended projective symmetry is a concrete example rather than a blanket license to add connection operators.

The apparent contrast is resolved only after fixing the theory class. A ghost in a broken-projective Ricci model does not refute an algebraic Einstein–Cartan contact interaction, while a healthy scalar in one extended-symmetry class does not rescue a generic connection kinetic term. A discriminating calculation must specify the action and matter metric, count constraints on a generic background, inspect pole residues or the nonlinear principal symbol, and verify that the same result survives the proposed solution branch below its cutoff. A complete classification remains an open question for broad regions of metric–affine theory space.

Empirical likelihoods and numerical bounds belong to Tests of Gravity rather than this construction page. A stable low-energy quantum-gravity EFT also requires separate power counting and ultraviolet evidence; classical elimination of an auxiliary connection does not supply either.

What could change these conclusions

The conclusions could change after an independent nonlinear analysis changes the generic constraint count, a symmetry theorem enlarges or excludes a healthy operator class, a matter-coupling calculation alters the projective trace condition, or a cutoff analysis invalidates a claimed propagating branch.

Common pitfalls

Identifying every affine component as a particle. Independence in the variational principle does not imply propagation. Algebraic equations, gauge freedom, and constraints can remove most or all connection components.

Breaking projective symmetry silently. A non-projective matter current or quadratic invariant changes the trace equation. It must be treated as a physical model choice and checked for new modes.

Exercises

In four dimensions suppose a matter model has pure dilation hypermomentum

$$\Delta_\lambda{}^{\mu\nu} =\frac14\delta^\nu_\lambda J^\mu,$$

with $J^\mu\ne0$. Can it consistently couple to a purely Einstein–Hilbert Palatini gravitational sector? Compare it with a pure spin current satisfying $\Delta^\mu{}_{\lambda\nu}=\Delta^\mu{}_{[\lambda\nu]}$.

Solution

Tracing the dilation source gives $\Delta_\nu{}^{\mu\nu}=(1/4)\delta^\nu_\nu J^\mu=J^\mu$. The Palatini tensor has the identity $P_\nu{}^{\mu\nu}=0$, so its traced field equation would demand $J^\mu=0$, contrary to the premise. The coupling is inconsistent unless the gravitational action or a justified constraint breaks the projective degeneracy. A pure spin current is antisymmetric in its internal index pair, so its trace vanishes; it passes this projective test, although its remaining connection equation and stability still have to be solved.

References

• Barker, W., and S. Zell, “Consistent Particle Physics in Metric-Affine Gravity from Extended Projective Symmetry,” arXiv:2402.14917 [hep-th] (2024). Stable record. Open PDF
• Beltrán Jiménez, J., and A. Delhom, “Instabilities in Metric-Affine Theories of Gravity with Higher-Order Curvature Terms,” European Physical Journal C 80 (2020) 585. DOI. Open PDF
• Iosifidis, D., Metric-Affine Gravity and Cosmology: Aspects of Torsion and Non-Metricity in Gravity Theories, PhD thesis, Aristotle University of Thessaloniki (2019). Stable record. Open PDF
• Rigouzzo, C., and S. Zell, “Coupling Metric-Affine Gravity to the Standard Model and Dark Matter Fermions,” Physical Review D 108 (2023) 124067. DOI. Open PDF