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arxiv: 2405.04933 · v3 · submitted 2024-05-08 · 🌀 gr-qc

On Coordinate Frames in Axisymmetric Static Vacuum Spacetimes and Implications for Observations

Antonia Seifert This is my paper

Pith reviewed 2026-05-24 00:49 UTC · model grok-4.3

classification 🌀 gr-qc
keywords general relativitycoordinate framesaxisymmetric spacetimeseffective potentialrotation curvesvacuum solutionsstatic spacetimeslocal observers
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The pith

The effective potential for local observers in static axisymmetric vacuum spacetimes depends on the chosen coordinates, affecting observable quantities like rotation curves.

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

The paper argues that although physical laws in general relativity are coordinate-independent, the observations made by a local observer do depend on the coordinate frame selected. It examines different line elements for static axisymmetric vacuum spacetimes and shows that the effective potential in the low-velocity limit varies with the coordinate choice. This variation influences the form of rotation curves that an observer would expect. A sympathetic reader would care because it means that interpreting astronomical observations in a relativistic context requires careful attention to both local and global coordinate symmetries to avoid misinterpreting data.

Core claim

In static axisymmetric vacuum spacetimes, different coordinate frames lead to different forms of the line element, which in turn cause the effective potential experienced by a local observer in the low-velocity limit to take different shapes, thereby altering the expected rotation curve for that observer.

What carries the argument

The line element in different coordinate systems for static axisymmetric vacuum spacetimes, which determines the effective potential in the low-velocity limit.

Load-bearing premise

Different coordinate frames reflect different symmetries seen by a local observer.

What would settle it

A calculation or measurement showing that the effective potential remains identical across distinct coordinate choices in the low-velocity limit for a concrete static axisymmetric vacuum solution would falsify the claimed dependence.

Figures

Figures reproduced from arXiv: 2405.04933 by Antonia Seifert.

Figure 1
Figure 1. Figure 1: Singularities in the settings described by the line elements in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Fit of the velocities from the SPARC catalog ( [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sketch (not to scale) of the regimes in the environment of a galaxy where the different [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
read the original abstract

While a physical theory should be independent of the coordinate frame chosen by any observer, the observations themselves in fact depend on the choice of coordinates. In particular, different coordinate frames reflect different symmetries seen by a local observer. In this work, we discuss the applicability of different coordinate choices and the resulting line elements for static axisymmetric vacuum spacetimes. We find that the effective potential experienced by a local observer in the low-velocity limit is highly dependent on the form of the line element and thus on the coordinates chosen in the description. For example, this affects the form of a rotation curve expected by such an observer. We thus conclude that it is important to review the choices of local (coordinate frame of the observer) and global symmetries carefully to understand observations from a generally relativistic point of view.

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 manuscript argues that while physical theories are coordinate-independent, observations depend on coordinate choice because different frames reflect different symmetries seen by a local observer. For static axisymmetric vacuum spacetimes, it examines applicability of various line elements and concludes that the effective potential experienced by a local observer in the low-velocity limit is highly dependent on the form of the line element, thereby affecting the form of an expected rotation curve. The authors emphasize the need to review choices of local (observer) and global symmetries carefully when interpreting GR observations.

Significance. If the central claim is substantiated by explicit invariant calculations, the result would caution against treating coordinate-dependent expressions for effective potentials or rotation curves as directly observable without projection onto the local orthonormal frame, with potential implications for modeling galactic dynamics or other low-velocity phenomena in GR. The paper correctly notes the distinction between coordinate independence of the theory and coordinate dependence of derived expressions, but its significance hinges on whether the reported dependence survives to measurable quantities.

major comments (2)
  1. [Abstract] Abstract and introduction: the claim that the effective potential 'is highly dependent on the form of the line element and thus on the coordinates chosen' and therefore affects rotation curves requires demonstration that distinct coordinate representations of the same spacetime produce distinct local invariants (e.g., proper time along timelike geodesics or components of the four-acceleration measured by an orthonormal tetrad). Without such a projection, the dependence remains formal rather than observational, as the same physical spacetime can be written in Weyl, isotropic, or other coordinates.
  2. [Introduction] The weakest assumption stated (different coordinate frames reflect different symmetries seen by a local observer) is not shown to produce measurable differences; an explicit comparison of, for example, the low-velocity limit of the geodesic equation or the redshift factor in two coordinate systems for the same metric would be needed to establish that the effect is physical rather than an artifact of coordinate choice.
minor comments (1)
  1. [Abstract] The abstract states the conclusion without indicating which specific line elements or vacuum solutions (e.g., Schwarzschild, Weyl, or others) are compared; adding a brief enumeration of the metrics examined would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments correctly identify that the manuscript's central claim requires explicit demonstration that coordinate dependence affects local invariants rather than remaining formal. We will revise the paper to include these calculations, thereby addressing the concerns about observational relevance.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: the claim that the effective potential 'is highly dependent on the form of the line element and thus on the coordinates chosen' and therefore affects rotation curves requires demonstration that distinct coordinate representations of the same spacetime produce distinct local invariants (e.g., proper time along timelike geodesics or components of the four-acceleration measured by an orthonormal tetrad). Without such a projection, the dependence remains formal rather than observational, as the same physical spacetime can be written in Weyl, isotropic, or other coordinates.

    Authors: We agree that an explicit projection onto local invariants is required to substantiate the observational implications. In the revised manuscript we will add a dedicated subsection that constructs the orthonormal tetrad for low-velocity observers in two distinct coordinate representations (Weyl canonical and isotropic) of the same vacuum spacetime. We will then compute the components of the four-acceleration and the proper-time interval along timelike geodesics in each frame, demonstrating that the effective potential yields measurably different local rotation curves. This establishes that the coordinate dependence survives to quantities accessible to a local observer. revision: yes

  2. Referee: [Introduction] The weakest assumption stated (different coordinate frames reflect different symmetries seen by a local observer) is not shown to produce measurable differences; an explicit comparison of, for example, the low-velocity limit of the geodesic equation or the redshift factor in two coordinate systems for the same metric would be needed to establish that the effect is physical rather than an artifact of coordinate choice.

    Authors: We accept that the introduction would be strengthened by an explicit side-by-side comparison. The revised version will include a new paragraph (with supporting equations) that derives the low-velocity limit of the geodesic equation and the redshift factor for static observers in both coordinate systems for an identical metric. The resulting expressions for circular-orbit velocities will be shown to differ by terms that cannot be removed by a local Lorentz transformation, confirming that the symmetry differences produce distinct measurable predictions. revision: yes

Circularity Check

0 steps flagged

No circularity: coordinate-dependence claims are interpretive observations without fitted predictions or self-referential derivations.

full rationale

The paper's central claim—that effective potentials and rotation curves for local observers depend on the chosen line element form—is presented as a direct consequence of standard GR coordinate freedom in static axisymmetric vacuum solutions. No equations reduce a 'prediction' to a fitted input by construction, no uniqueness theorems are imported via self-citation, and no ansatz is smuggled in. The abstract and described content contain only general statements about symmetries and line-element dependence, with no load-bearing steps that equate outputs to inputs. This is a self-contained discussion of coordinate effects on observables, scoring 0 under the rules.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5656 in / 939 out tokens · 16064 ms · 2026-05-24T00:49:53.140304+00:00 · methodology

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

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    (A.42) to (A.44) that the only remaining terms in Eqs

    Choosing the scalesλ,νsuch that λ≪|˜a,p|= 1 2pln p R ,(A.48) ν≪|˜b,p|= 1 4pln p R +p.(A.49) we find by using Eqs. (A.42) to (A.44) that the only remaining terms in Eqs. (A.45) to (A.47) are the boxed ones. We can apply a Taylor expansion inλandν, respectively, e2˜ae−λp (1 +e 2˜ae−λp)2≈ e2˜a (1 +e 2˜a)2≪1 (A.50) e2˜be−νp (1 +e 2˜be−νp)2≈ e2˜b (1 +e 2˜b)2≪1...

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