Light propagation and intensity transport in metric-affine geometry

Technical Summary: Light Propagation and Intensity Transport in Metric–Affine Geometry

Problem Statement
In General Relativity (GR), spacetime geometry is fully determined by the metric, and the affine connection is uniquely fixed as the Levi–Civita connection. This ensures a unique notion of parallel transport and inertial motion, where free-falling particles follow metric geodesics. However, in broader geometric frameworks such as metric–affine gravity, teleparallel gravity, and symmetric teleparallel gravity, the affine connection is an independent object characterized by curvature, torsion, and non-metricity. In these non-Riemannian geometries, the equivalence between geodesics (extremizing the spacetime interval) and autoparallels (parallel transport of the tangent vector) breaks down.

A critical ambiguity arises when coupling electromagnetism to such geometries. Unlike in GR, where the connection is metric-compatible and torsion-free, the presence of independent torsion and non-metricity allows for multiple, inequivalent prescriptions for defining the electromagnetic field strength and covariant derivatives. Standard approaches that arbitrarily choose a coupling prescription (e.g., replacing partial derivatives with full covariant derivatives in the field strength definition) often break fundamental symmetries, such as $U(1)$ gauge invariance or projective invariance of the connection. This work addresses the lack of a systematic, symmetry-guided framework for constructing electromagnetic sectors in metric–affine geometries and investigates the resulting propagation of electromagnetic waves in the geometric-optics (GO) limit.

Methodology
The authors adopt a symmetry-based approach to constrain the class of admissible electromagnetic extensions, rather than imposing a specific coupling a priori. The construction of viable models is guided by two fundamental principles:

1. Preservation of Standard $U(1)$ Gauge Invariance: This requires the electromagnetic field strength tensor to remain the standard Levi–Civita Faraday tensor, $\mathring{F}_{\mu\nu} = \mathring{\nabla}_\mu A_\nu - \mathring{\nabla}_\nu A_\mu$, ensuring the theory remains well-defined independently of the affine structure's torsion.
2. Projective Invariance: The affine connection possesses a gauge redundancy under projective transformations ($\Gamma^\alpha_{\mu\nu} \to \Gamma^\alpha_{\mu\nu} + \delta^\alpha_\nu \xi_\mu$). Physical couplings must be invariant under this transformation, meaning the electromagnetic sector can only depend on the trace-free parts of torsion ($T^f$) and non-metricity ($Q^f$), or scalar invariants constructed from them.

Based on these principles, the authors introduce two representative models:

• Scalar-Coupled Model: The Maxwell Lagrangian is modified by a scalar prefactor $\Xi$ constructed from projectively invariant invariants of torsion and non-metricity ($L_{\Xi} = -\frac{1}{4} \Xi \mathring{F}_{\mu\nu}\mathring{F}^{\mu\nu}$).
• Constitutive-Tensor Model: The kinetic term is modified by a rank-four constitutive tensor $\chi_{\mu\nu\alpha\beta}$, acting as an anisotropic effective medium ($L_{\chi} = -\frac{1}{4} \chi_{\mu\nu\alpha\beta} \mathring{F}^{\mu\nu}\mathring{F}^{\alpha\beta}$). This tensor is built from projectively invariant combinations of torsion and non-metricity.

The authors then perform a geometric-optics analysis using the ansatz $A_\mu = a_\mu e^{i\omega\Psi}$, expanding the equations of motion in powers of the large frequency parameter $\omega$. They derive the leading-order dispersion relations (governing wavefronts) and the next-to-leading order transport equations (governing amplitude and intensity evolution).

Key Contributions and Results

• Scalar-Coupled Model:

  • Light Cone: At leading order ($O(\omega^2)$), the dispersion relation remains $g_{\mu\nu}k^\mu k^\nu = 0$. The non-Riemannian geometry induces only a conformal rescaling of the effective inverse metric ($g^{\text{eff}}_{\mu\nu} = \Xi g_{\mu\nu}$). Consequently, photon trajectories coincide with the standard null geodesics of the background metric; the light cone is not deformed, and no birefringence occurs at this order.
  • Intensity Transport: At next-to-leading order ($O(\omega)$), the scalar prefactor modifies the transport equation. The conserved quantity along a ray bundle is not the squared amplitude $A^2$, but a "dressed" intensity $I = \Xi A^2$. This implies that while the canonical flux is conserved, an observer interpreting the signal via standard Maxwell normalization would infer an apparent flux renormalization.

• Constitutive-Tensor Model:

  • Light Cone and Dispersion: The leading-order propagation is governed by a polarization-dependent eigenvalue problem. The scalar sector of the constitutive tensor generates an effective inverse metric that is not simply conformal to the background metric; it includes anisotropic corrections from torsion and non-metricity. This allows for the deformation of the light cone itself.
  • Birefringence and Mode Mixing: The tensorial structure of $\chi_{\mu\nu\alpha\beta}$ introduces genuine polarization-dependent propagation. The two transverse polarization modes generally propagate with different speeds and directions, leading to vacuum birefringence.
  • Amplitude Transport: The transport equation involves a mixing matrix $M^{(i)}_{(j)}$ acting on the polarization vector. In a generic basis, the polarization modes mix as they propagate. However, in the local eigenbasis of the constitutive map, the transport reduces to a scalar law similar to the scalar model, with a mode-dependent eigenvalue $\Xi_\chi$.
  • Effective Currents: The derivative of the constitutive tensor ($\mathring{\nabla}\chi$) acts as an effective current, describing an exchange of energy and momentum between the electromagnetic field and the non-Riemannian geometry. This can lead to amplification or damping of wave amplitudes and, potentially, the transmutation of photons into torsion/non-metricity modes.

Significance and Claims
The paper establishes the formal basis for connecting metric–affine geometry with electromagnetic observables. It demonstrates that torsion and non-metricity can affect light propagation in distinct ways absent in standard GR:

1. Modifying the effective light cone (anisotropic deformation).
2. Altering the relationship between field amplitude and intensity (flux renormalization).
3. Inducing polarization-dependent propagation and birefringence.
4. Generating mode mixing and potential photon-number non-conservation.

The authors position this work as the first in a series intended to bridge non-Riemannian geometry with phenomenology. They explicitly state that the quantitative impact of these effects is model-dependent and that the current work focuses on identifying consistent couplings and deriving propagation equations. Future papers in this series will apply this framework to concrete backgrounds (e.g., black holes) to investigate specific signatures such as black-hole imaging, birefringent lensing, and polarization observables. The paper does not claim to have solved these phenomenological problems but rather provides the necessary theoretical machinery to do so.