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REVIEW 4 major objections 5 minor 78 references

Doppler, gravitational and cosmological redshifts

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper argues that the Hubble tension disappears when a variable speed of light is used inside an expanding black-hole universe, correcting the remote Hubble constant from 67.4 to 74.2 km/s/Mpc.

desk verdict A clean Doppler derivation and a defensible point about momentum conservation, wrapped around a speculative Hubble-tension fix that fails on a sign error in the interior potential. read the letter →

arxiv 2502.00087 v2 pith:CIJAS6O7 submitted 2025-01-31 physics.gen-ph

Klaus Wilhelm , Bhola N. Dwivedi This is my paper · ORCID
classification physics.gen-ph PACS 98.80.-k04.20.-q
keywords redshiftDopplereffectgravitationalcosmologicalvariablespeedoflightHubbletensionexpandingblackholeenergy-momentumconservation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that all three redshift mechanisms—Doppler, gravitational, and cosmological—must be derived from energy and momentum conservation together, not energy alone. The Doppler and gravitational sections are worked examples of that program; the novel result sits in the cosmological section. There, the authors propose that the observable universe lies near the center of an expanding black hole, with the cosmic microwave background emitted near the Schwarzschild radius and traveling inward. Because light moves more slowly inside the gravitational well, distances computed with a constant speed of light are wrong at large scales. Correcting the distance scale raises the remote Hubble constant from 67.4 to 74.2 km/s/Mpc, matching local measurements and leaving a single H0 near 71 km/s/Mpc.

What carries the argument

The load-bearing object is the variable-speed-of-light law $c(r)=c_0\left(1+\frac{2U}{c_0^2-U}\right)$ of Eq. (30), combined with the assumed interior gravitational potential $U(r)=U_S(r/r_S)^2=-c_0^2(r/r_S)^2$ inside an expanding black hole. Photons emitted at the Schwarzschild radius travel toward the center through this potential, and integrating $C\,dt$ over the path fixes the CMB travel time at 0.432 of the universe's age. The travel-time calculation is what converts an apparent 1600 Mpc into a true 1450 Mpc, producing the $1600/1450$ correction to the remote Hubble constant. For the gravitational redshift, the key mechanism is a momentum adjustment $x$ at emission, determined by balancing the photon momentum before and after it interacts with an atom in the gravitational field; for Doppler shifts, it is the momentum conservation equation that yields $\nu'=\gamma\nu(1-\beta\cos\vartheta)$ and the associated aberration formula.

What would settle it

Measure the Hubble constant from independent distance indicators across a wide redshift range without assuming a constant light speed. The scenario predicts the inferred $H_0$ should drift lower with distance because the travel-time correction grows; a flat distance-independent $H_0$ would refute it. More sharply, one could use the paper's $c(r)$ and $U(r)$ to predict the angular position of the first CMB acoustic peak and check it against the measured value.

Watch

Extended reading notes

Core claim

The central claim is that the Hubble tension is an artifact of assuming a constant speed of light inside a cosmos that behaves like an expanding black hole. The paper's Eq. (30) gives $c(r)=c_0\left(1+\frac{2U}{c_0^2-U}\right)$, and its Eq. (36) gives the interior potential $U(r)=U_S(r/r_S)^2$; combining them and integrating the photon path fixes the CMB travel time from the Schwarzschild radius to the center at 0.432 of the universe's age. At 1600 Mpc the mean photon speed works out to $2.72\times 10^8$ m/s, so the same travel time at $c_0$ corresponds to only 1450 Mpc. The paper multiplies the remote $H_0$ of 67.4 km/s/Mpc by $1600/1450$ to obtain 74.2 km/s/Mpc, in line with local estimates near 73, and concludes that the expansion rate is constant at about 70.92 km/s/Mpc. For the gravitational redshift, energy and momentum conservation require a small momentum adjustment at emission, encoded in a photon speed that depends on the local potential; for the Doppler shift, momentum conservation yields $\nu'=\gamma\nu(1-\beta\cos\vartheta)$ and the aberration formula.

Load-bearing premise

The cosmological conclusion collapses if we are not near the center of an expanding black hole with the CMB emitted at its Schwarzschild radius, or if the variable-speed-of-light law is wrong; the paper offers no independent evidence for either.

Editorial extensions

If this is right

  • If the correction is right, the observed split between $H_0\approx 67.4$ and $H_0\approx 73$ disappears; one expansion rate near 70.92 km/s/Mpc fits both distant and nearby measurements.
  • The age and radius of the universe become internally consistent: expanding at $c_0$ for $4.3508\times 10^{17}$ s gives a radius of 4227 Mpc, with $H_0=70.92$ km/s/Mpc.
  • No new dark-energy-like ingredient is needed to explain the Hubble tension; the apparent discrepancy is a distance calibration error.
  • Gravitational-redshift and Doppler observations already cited in the paper (Pound–Rebka, solar lines, Ives–Stilwell) remain consistent because the speed correction is negligible in weak fields.
  • The CMB is explained as radiation emitted near the Schwarzschild surface that has been traveling inward ever since, with its redshift arising from the Doppler recession of the emitting plasma.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the scenario is right, the apparent Hubble constant should be a function of distance before correction: measurements at larger distances should read systematically lower, so reanalyzing distance-ladder data for such a trend is a direct test.
  • The same travel-time correction would change angular-diameter distances and the apparent brightness of standard candles, so the scenario predicts that Type Ia supernova distance moduli deviate from the standard cosmic expansion in a specific, distance-dependent way.
  • Because the paper's photon keeps constant energy while its speed changes, the photon momentum changes along the path; this should leave a trace in the redshift–temperature relation of the CMB or in arrival-time differences of multi-messenger events, giving an observable signature beyond the $H_0$ correction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper responds to Ortiz and Ibarra-Castor (2024) by arguing that any unified treatment of Doppler, gravitational, and cosmological redshifts must enforce both energy and momentum conservation. Section 2 derives the relativistic Doppler formula and aberration law from momentum/energy balance for a source at rest and moving receivers, and this derivation is algebraically consistent and standard in content. Section 3 builds a variable-speed-of-light model of the gravitational redshift, in which the emitting atom is assumed unable to sense the local gravitational potential or light speed, and derives an exact speed law c(r) from a momentum-adjustment hypothesis. Section 4 then proposes that the observable universe lies inside an expanding black hole, that the CMB is emitted near the Schwarzschild radius and travels inward, and that the resulting distance-dependent light speed converts a remote H0 measurement of 67.4 km/s/Mpc into a corrected value near 74.2 km/s/Mpc, thereby 'resolving' the Hubble tension. The paper is candid that this cosmological scenario is hypothetical, but the quantitative claim rests on the specific interior potential and on an unexplained tuning of the emission time.

Significance. If the Section 4 calculation were correct, the paper would offer a striking, testable resolution of the Hubble tension and a concrete prediction for a distance-dependent effective speed of light. The Doppler-section derivation is a useful, self-contained reminder that momentum conservation fixes the relativistic Doppler formula and aberration angle. The paper is also transparent about the speculative nature of the expanding-black-hole scenario and about its reliance on the authors' earlier variable-speed-of-light work. However, the numerical Hubble-tension claim is the main new quantitative result, and it is not robust: the interior potential in Eq. (36) is inconsistent with Newton's shell theorem as used in the text, the key parameter X=0.432 is not derived, and the application of the correction to the Planck/CMB determination of H0 conflates a sound-horizon measurement with a 1600 Mpc distance-ladder measurement. These problems make the central new claim unsupported, even though the pedagogical Doppler part is sound.

major comments (4)
  1. [§4, paragraph beginning 'We now have to understand...'] The interior gravitational potential is inconsistent with the shell-theorem reasoning in footnote 12. For a constant-density sphere with U(0)=0, the potential is U(r)=+(GN M/(2 rS^3)) r^2 = +(c0^2/2)(r/rS)^2, not U(r)=US(r/rS)^2 = -c0^2(r/rS)^2. The sign is reversed and the coefficient is off by a factor of two. As a result, the gradient of Eq. (36) corresponds to a repulsive acceleration, which contradicts the positive-mass sphere assumed. Since Eq. (30) feeds U into c(r), and the travel-time integral over 1600 Mpc yields the mean speed 2.72e8 m/s used in Eq. (37), the claimed correction factor 1600/1450 is an artifact of an incorrect potential and would change, and likely reverse, if the potential were corrected.
  2. [§4] The value X=0.432 is stated without showing the integration or the assumed expansion law rS(T). The text says only that the path C dt is integrated in steps of 1/10^4 of the age of the Universe and that X=0.432 is 'found'. The result depends on the chosen rS(T) and on the interior mass-density evolution [rS/rS(T)]^3, neither of which is specified quantitatively. Without these inputs, the travel time and the resulting 1600 Mpc/1450 Mpc ratio cannot be reproduced or independently checked.
  3. [§4] The correction factor is applied as if H0=67.4 km/s/Mpc were a distance measurement at 1600 Mpc. In the standard cosmology, the Planck value H0≈67.4 is derived from the CMB power spectrum through the sound horizon and early-universe parameters, not from a local distance-redshift relation at 1600 Mpc. The simple rescaling 67.4×1600/1450 is therefore not a valid way to transform the Planck H0 into a local-ladder value. This conceptual mismatch invalidates the claimed 'resolution' of the Hubble tension even if the travel-time arithmetic were correct.
  4. [Eq. (30)] The variable-speed-of-light law used throughout the cosmological section is derived from the nonstandard postulate that the emitting atom cannot sense the local gravitational potential or light speed, and that the photon's energy is constant while its momentum and wavelength adjust. This assumption is not independently tested or derived from a known theory; it is a modeling choice. In addition, the printed Eq. (30) has a dimensional inconsistency unless the 'U' in the denominator is understood as U/c0^2, which is not stated in the text. Since the numerical result in Section 4 depends directly on this law, the central claim inherits the unverified status of the assumption.
minor comments (5)
  1. [Abstract] There is a typographical error: 'physical e ects' should read 'physical effects'.
  2. [Eq. (37)] The arithmetic is slightly off: 67.4 × 1600 / 1450 = 74.37, not 74.2. The ratio corresponding to 74.2 would be about 1.1009, not 1.1034.
  3. [§4] The statement that an increase of rS(T) 'has no direct effect on the gravitational potential, because of the shell theorem' is confusing, since the following sentence introduces a density decrease [rS/rS(T)]^3 that does change the potential. Please clarify the intended time dependence and its effect on U(r) at fixed r.
  4. [Fig. 2] The caption refers to multiple line styles ('solid', 'short dashed', 'long dashed', 'dash-dot', 'dash-dot-dot-dot') but does not tie them to the axes and curves in a way that is easy to parse. Labeling each curve with its equation number would improve readability.
  5. [References] The reference 'St. John (1917)' appears in the reference list but is not cited in the text; conversely, 'St. John (1928)' is cited but the reference entry gives the pages 195-239 for the 1928 paper. Please check consistency of citation and reference-list entries.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the H0 correction is model-derived rather than fitted, though Section 3 leans on the authors' prior self-citations.

full rationale

The H0 correction in Eq. (37) is not equivalent to any fitted input. X=0.432 is solved in Section 4 as a boundary condition requiring CMB photons emitted at the Schwarzschild radius to arrive at the present age, not chosen to reproduce H0; the mean speed 2.72e8 m/s and the ratio 1600/1450 follow from the assumed interior potential Eq. (36) and the variable-speed law Eq. (30). The input 67.4 km/s/Mpc is an external measured value, so the output 74.2 is a model-derived prediction rather than a restatement of an input. The variable-speed law is re-derived in the text from energy and momentum consistency (Eqs. 26 and 29) and is externally supported by the Shapiro-delay and light-deflection references, so the self-citations to Wilhelm and Dwivedi (2014, 2019, 2020) are present but not load-bearing. The main vulnerabilities are physical rather than circular: the interior potential Eq. (36) does not follow correctly from Newton's shell theorem (the coefficient and sign are inconsistent with a uniform-density sphere), and the black-hole geometry is speculative. Under the required standard of exhibiting an equation that reduces to its own input by construction, no circular step can be identified.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The central cosmological claim rests on the authors' nonstandard variable-speed-of-light law from their earlier work, an ad hoc emission equation, and a postulated black-hole geometry. The only tuned number is X=0.432, although the unspecified growth of rS(T) carries additional freedom. No independent measurement anchors these choices.

free parameters (2)
  • X, fraction of the present age at which CMB photons leave the Schwarzschild radius = 0.432
    Chosen in Section 4 so the integrated photon travel time ends at the present age; not constrained by independent data.
  • Expansion law rS(T) and interior mass-density evolution = not specified
    Section 4 assumes rS grows with time and density scales as [rS/rS(T)]^3, but no explicit law is given; the travel time depends on it.
assumptions (6)
  • domain assumption A photon in a static gravitational field does not change its energy, while its speed and momentum change (Okun et al. 2000).
    Invoked in Section 3 to build the momentum-adjustment derivation and the variable speed of light.
  • ad hoc to paper The momentum-matching equation (p0 - x)c0 = (p0 + x)c(r) governs the emission adjustment.
    This is Eq. (26), the core of the gravitational-redshift derivation, and is not derived from a more basic principle.
  • ad hoc to paper The emitting atom cannot sense the gravitational potential U or the local speed of light c(r).
    Stated in Section 3 before Eq. (25); it is necessary for the clock argument and is not independently justified.
  • ad hoc to paper The observable universe sits inside an expanding black hole, with Earth near the center.
    Postulated in Section 4 to explain the CMB arrival and the Hubble tension; no independent evidence is given.
  • domain assumption Newton's shell theorem inside the black hole gives U(r) = US (r/rS)^2.
    Used in Eq. (36) for the interior potential; it is a standard classical result whose extension to a GR black hole is assumed.
  • domain assumption The expansion front moves at c0 and the present age is T0 = 4.3508e17 s from Planck.
    Used in Section 4 to set the radius r0 = c0 T0 and to derive H0 = 70.92 km/s/Mpc.
invented entities (1)
  • Expanding black hole containing the Universe
    purpose: Provides the geometry in which CMB photons travel inward through a variable-speed-of-light medium and yields the distance correction for H0.
    The scenario is fitted to the CMB arrival time and the universe's age; no falsifiable prediction outside the model is given.

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Cite this review

Pith. "Pith review of Doppler, gravitational and cosmological redshifts." pith.science (2026). https://pith.science/paper/CIJAS6O7

@misc{pith2026250200087,
  author       = {Pith},
  title        = {Pith review of: Doppler, gravitational and cosmological redshifts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CIJAS6O7}},
  note         = {Machine review of arXiv:2502.00087}
}
read the original abstract

Ortiz and Ibarra-Castor (2024) have presented a "Generalized redshift formula" taking account of only energy conservation considerations. Contrary to their claim, we emphasize to invoke both energy and momentum considerations in order to deduce all three types of redshift (Doppler, gravitational and cosmological). We formulate our views on the three physical e ects in a consistent manner in addition to addressing the lack of relevant references in Ref. (Ortiz and Ibarra-Castor, 2024)

Discussion (0). Continue with ORCID to comment.

Reference graph

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