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arxiv: 2606.31299 · v1 · pith:NMEE3BW7new · submitted 2026-06-30 · 🌀 gr-qc · astro-ph.CO· hep-th

Distance duality relation in symmetric teleparallel gravity

Rocco D'Agostino This is my paper

Pith reviewed 2026-07-01 04:37 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COhep-th
keywords distance duality relationsymmetric teleparallel gravitynonmetricityEtherington reciprocityf(Q) gravityphoton number conservationfine-structure constant
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The pith

The distance duality relation holds under minimal electromagnetic coupling in symmetric teleparallel gravity but is violated dynamically by nonminimal coupling to the nonmetricity scalar.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the Etherington reciprocity relation between luminosity and angular diameter distances when gravity is described by nonmetricity rather than curvature. It establishes that the standard relation survives unchanged provided electromagnetism couples minimally and photon number remains conserved. Introducing a nonminimal interaction between the electromagnetic field and the nonmetricity scalar Q alters the photon current conservation law, producing a redshift-dependent violation whose form depends on the Hubble expansion. The resulting generalized duality formula connects observable distances directly to the expansion rate and to possible variations in the effective fine-structure constant.

Core claim

In symmetric teleparallel gravity the standard Etherington reciprocity relation remains valid when electromagnetism is minimally coupled and the photon number is conserved. In a class of f(Q) theories a nonminimal coupling between the electromagnetic field and the nonmetricity scalar modifies the conservation of the photon number current and thereby induces a dynamical violation of the distance duality relation. On a homogeneous isotropic background in the coincident gauge this yields an explicit generalized formula relating the two distance measures to the Hubble rate, with deviations also tied to changes in the effective fine-structure constant.

What carries the argument

The nonminimal coupling function between the electromagnetic field and the nonmetricity scalar Q, which alters the photon number current conservation.

If this is right

  • Deviations from the standard distance duality can be attributed to dynamical effects of the nonminimal coupling rather than to geometry alone.
  • The generalized formula allows direct use of distance data to constrain the Hubble expansion in these theories.
  • Phenomenologically viable coupling functions produce only small deviations from the standard relation.
  • Observed variations in the effective fine-structure constant can be cross-checked against distance-duality violations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • High-redshift supernova and baryon acoustic oscillation data could be reanalyzed to bound the strength of any nonminimal coupling.
  • Independent laboratory or astrophysical bounds on fine-structure constant drift would provide a separate test of the same coupling function.
  • The same nonminimal-coupling mechanism might be examined in other metric-affine theories to predict analogous violations.

Load-bearing premise

The geometrical optics approximation together with the assumption of a homogeneous and isotropic spacetime background in the coincident gauge.

What would settle it

Precise simultaneous measurements of luminosity distance and angular diameter distance at the same redshift that either match or deviate from the generalized formula in a manner inconsistent with the predicted effect of a specific nonminimal coupling function.

Figures

Figures reproduced from arXiv: 2606.31299 by Rocco D'Agostino.

Figure 1
Figure 1. Figure 1: FIG. 1. Cosmic evolution of the DDR violation parameter [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
read the original abstract

In this work, we investigate the distance duality relation (DDR) in symmetric teleparallel theories, where gravity is mediated by nonmetricity. Starting from the general metric-affine formulation and adopting the geometrical optics approximation, we show that the standard Etherington reciprocity relation remains valid in the presence of nonmetricity when electromagnetism is minimally coupled and the photon number is conserved. We then extend the analysis to a class of $f(Q)$ theories with a nonminimal coupling between the electromagnetic field and the nonmetricity scalar. We demonstrate that such an interaction modifies the conservation of the photon number current, leading to a dynamical violation of the DDR. Focusing on a homogeneous and isotropic spacetime background in the coincident gauge, we derive a generalized DDR formula that directly relates observational distance measures to the Hubble expansion rate. Furthermore, we discuss the link between the deviations from Etherington's relation and variations of the effective fine-structure constant. Specific illustrative examples of the coupling function are also analyzed, showing that phenomenologically viable models predict only small deviations from the standard DDR. Our results provide a unified framework to distinguish between the geometric and dynamical origins of DDR violations, opening new avenues for testing non-Riemannian gravity with future high-precision astrophysical and cosmological observations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims that the standard Etherington distance duality relation (DDR) holds in symmetric teleparallel gravity for minimal electromagnetic coupling (due to photon number conservation), but is dynamically violated in f(Q) theories with nonminimal EM-nonmetricity coupling because the latter modifies the divergence of the photon number current. Starting from the metric-affine formulation and geometrical optics, the authors derive a generalized DDR formula in homogeneous isotropic backgrounds (coincident gauge) that relates distance measures to the Hubble rate, link deviations to variations in the effective fine-structure constant, and analyze specific coupling functions that yield only small deviations.

Significance. If the central derivations hold, the work supplies a concrete distinction between geometric (nonmetricity alone) and dynamical (modified current conservation) origins of DDR violations, which could be tested with future precision cosmology and astrophysics data. The approach of beginning from the general metric-affine action and applying the geometrical optics limit is a strength, as is the explicit reduction to an FLRW-like coincident-gauge background that produces an observationally usable formula.

major comments (2)
  1. [analysis of nonminimal EM-nonmetricity coupling and photon number current] The load-bearing step is the explicit demonstration that the nonminimal coupling term produces a non-vanishing divergence of the photon number current (proportional to the derivative of the coupling function) while the minimal-coupling case yields exact conservation. The manuscript must show all intermediate steps in the metric-affine Maxwell equations, including any contributions from the affine connection to the Hodge star and covariant derivatives, to rule out hidden connection-dependent terms that could affect the continuity equation even when the coupling function is constant.
  2. [derivation of generalized DDR formula] The generalized DDR formula is obtained after adopting the geometrical optics approximation in the general metric-affine formulation together with a homogeneous isotropic background in the coincident gauge. The manuscript should state the precise conditions under which this approximation remains valid when nonmetricity is present and verify that no additional terms arise from the nonmetricity tensor in the ray-tracing or eikonal equation.
minor comments (1)
  1. Notation for the nonminimal coupling function should be introduced once and used consistently; its explicit functional form in the illustrative examples should be stated in the main text rather than only in figures or appendices.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, agreeing that additional explicit steps will strengthen the presentation while maintaining that the central derivations are correct as stated.

read point-by-point responses

  1. Referee: The load-bearing step is the explicit demonstration that the nonminimal coupling term produces a non-vanishing divergence of the photon number current (proportional to the derivative of the coupling function) while the minimal-coupling case yields exact conservation. The manuscript must show all intermediate steps in the metric-affine Maxwell equations, including any contributions from the affine connection to the Hodge star and covariant derivatives, to rule out hidden connection-dependent terms that could affect the continuity equation even when the coupling function is constant.

    Authors: We agree that a fully expanded derivation of the metric-affine Maxwell equations will improve transparency. In the revised manuscript we will insert all intermediate steps, explicitly displaying the action of the affine connection on the Hodge star and on the covariant derivatives. This will confirm that the photon-number current is conserved when the coupling function is constant and acquires a divergence proportional to its derivative in the nonminimal case, with no residual connection-dependent contributions. revision: yes

  2. Referee: The generalized DDR formula is obtained after adopting the geometrical optics approximation in the general metric-affine formulation together with a homogeneous isotropic background in the coincident gauge. The manuscript should state the precise conditions under which this approximation remains valid when nonmetricity is present and verify that no additional terms arise from the nonmetricity tensor in the ray-tracing or eikonal equation.

    Authors: We will add a dedicated paragraph stating the precise validity conditions of the geometrical-optics limit in the presence of nonmetricity. Starting from the metric-affine Maxwell equations we will derive the eikonal equation and demonstrate that, under the coincident-gauge and homogeneous-isotropic assumptions already employed, the nonmetricity tensor does not generate extra terms in the ray-tracing or eikonal equations beyond those already included in the generalized DDR formula. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper starts from the general metric-affine formulation, adopts the geometrical optics approximation, and derives the standard Etherington relation under minimal EM coupling (via photon number conservation) versus dynamical violation under nonminimal f(Q)-EM coupling (via modified current divergence). The generalized DDR formula is then obtained explicitly for homogeneous isotropic spacetime in the coincident gauge, relating distances to the Hubble rate without any reduction to fitted inputs, self-definitions, or load-bearing self-citations. All steps are presented as explicit calculations from the action and conservation laws.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the geometrical optics approximation and the choice of homogeneous isotropic background in coincident gauge; the nonminimal coupling function is introduced as a model choice without independent evidence.

free parameters (1)
  • nonminimal coupling function
    The interaction between the electromagnetic field and the nonmetricity scalar is a free function whose form is chosen for illustrative examples.
axioms (2)
  • domain assumption geometrical optics approximation
    Invoked to derive light propagation and photon number current from the metric-affine formulation.
  • domain assumption homogeneous and isotropic spacetime in coincident gauge
    Used to obtain the explicit generalized DDR formula relating distances to the Hubble rate.

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discussion (0)

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Reference graph

Works this paper leans on

100 extracted references · 1 canonical work pages

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    Under the coincident gauge condition, this parametrization reads I(H 2) = 1 +ϵ e −(H/H0)2 ,(53) which impliesI(H 2

    Exponential coupling Alternatively, we can consider an exponential coupling function of the form I(Q) = 1 +ϵ e −Q/Q0 ,(52) withϵ≪1. Under the coincident gauge condition, this parametrization reads I(H 2) = 1 +ϵ e −(H/H0)2 ,(53) which impliesI(H 2

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    In the coincident gauge, this becomes I(H 2) = 1 +ϵln 1 + H 2 0 H 2 ,(56) which impliesI(H 2

    Logarithmic coupling Finally, we can explore a logarithmic coupling function parameterized as I(Q) = 1 +ϵln 1 + Q0 Q ,(55) whereϵ≪1. In the coincident gauge, this becomes I(H 2) = 1 +ϵln 1 + H 2 0 H 2 ,(56) which impliesI(H 2

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