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Relativistic transformation of temperature revisited

T0 review · 1 major / 1 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read Effective temperature of photon, ideal, and electron gases rises with velocity, showing temperature is observer-dependent.

desk verdict The paper calculates Teff from boosted energy density for three gases and finds an increase with velocity, but this follows from defining temperature that way rather than from entropy or the four-vector. read the letter →

arxiv 2606.00521 v1 pith:NO2ZX6I5 submitted 2026-05-30 gr-qc cond-mat.stat-mechphysics.class-ph

Soroor Pouryazdan , Babak Vakili This is my paper · ORCID
classification gr-qccond-mat.stat-mechphysics.class-ph
keywords relativistictemperaturetransformationOtt-Eddingtoninterpretationenergy-momentumtensorphotongasidealelectronobserver-dependentinverse-temperaturefour-vector
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

The paper reexamines conflicting classical proposals for how temperature changes under special relativity. It defines effective temperature Teff for any isotropic system directly from the Lorentz-transformed energy density obtained via the energy-momentum tensor. Explicit calculations for a photon gas, a relativistic ideal gas, and an electron gas all yield the same qualitative result: Teff increases as the relative velocity increases. This pattern favors the Ott-Eddington transformation law and shows that the functional form depends on the equation of state. The work concludes that temperature is not a Lorentz scalar but an observer-dependent quantity best handled by linking it to an inverse-temperature four-vector.

What carries the argument

Effective temperature Teff defined as the temperature a moving observer infers from the Lorentz-transformed energy density of an isotropic system.

What would settle it

A laboratory measurement of the temperature of a gas (photon, ideal, or electron) in a frame moving at relativistic speed, compared directly against the Teff predicted from its boosted energy density.

Watch

Extended reading notes

Core claim

Starting from the energy-momentum tensor of an isotropic system and defining Teff as the temperature inferred by a moving observer from the transformed energy density, analyses of a photon gas, a relativistic ideal gas and an electron gas show that Teff consistently increases with velocity, supporting the Ott-Eddington interpretation while depending on the system's equation of state. These results indicate that temperature is not a Lorentz-invariant scalar but an observer-dependent quantity. A consistent relativistic description emerges when temperature is related to the inverse-temperature four-vector beta, linking operational and invariant viewpoints within a unified thermodynamic framewor

Load-bearing premise

The operational definition of Teff from transformed energy density matches the thermodynamic temperature that would be read by standard thermometers in the moving frame.

Editorial extensions

If this is right

  • Temperature transformation laws differ according to the equation of state of the system under study.
  • The Ott-Eddington result is recovered for the three gases examined rather than the Planck-Einstein result.
  • Temperature cannot be regarded as a Lorentz-invariant scalar quantity.
  • A unified framework is obtained by expressing temperature through the inverse-temperature four-vector beta.

Reading between the lines

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

  • The same energy-momentum approach could be applied to other thermodynamic variables such as pressure or chemical potential in boosted frames.
  • High-energy collider data on boosted particle distributions might provide indirect checks on the predicted rise in Teff.
  • The dependence on equation of state suggests that different relativistic fluids will exhibit quantitatively different temperature transformations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 1 minor

Summary. The manuscript reexamines the relativistic transformation of temperature by starting from the energy-momentum tensor of an isotropic system and defining an effective temperature Teff as that inferred by a moving observer from the Lorentz-transformed energy density. Explicit calculations are performed for a photon gas (yielding Teff ~ u'^{1/4}), a relativistic ideal gas, and an electron gas; in each case Teff increases with velocity, supporting the Ott-Eddington interpretation while depending on the equation of state. The paper concludes that temperature is not a Lorentz scalar but observer-dependent, and proposes a unified description via the inverse-temperature four-vector β^μ that reconciles operational and covariant viewpoints.

Significance. If the operational definition of Teff is shown to coincide with standard thermodynamic temperature, the explicit multi-system calculations would constitute a concrete contribution to resolving the long-standing controversy, by demonstrating both velocity dependence and equation-of-state sensitivity while linking to the covariant β^μ formalism. The provision of results for three distinct systems (photon, ideal, and electron gases) is a positive feature that allows direct comparison across different equations of state.

major comments (1)
  1. [Abstract] Abstract (paragraph on energy-momentum tensor approach) and the subsequent definition of Teff: the central claim that the calculations support the Ott-Eddington interpretation rests on identifying the temperature inferred from the boosted energy density u' with the thermodynamic temperature. The manuscript does not demonstrate that this Teff reproduces the temperature obtained from the maximum-entropy condition or from a comoving thermometer in the boosted frame; without this equivalence the observer-dependence conclusion does not follow from the energy-density transformation alone.
minor comments (1)
  1. The abstract states that temperature 'depends on the system's equation of state' but does not indicate whether this dependence is derived from the functional form T(u) or from an additional assumption about how the EOS transforms under Lorentz boosts.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on energy-momentum tensor approach) and the subsequent definition of Teff: the central claim that the calculations support the Ott-Eddington interpretation rests on identifying the temperature inferred from the boosted energy density u' with the thermodynamic temperature. The manuscript does not demonstrate that this Teff reproduces the temperature obtained from the maximum-entropy condition or from a comoving thermometer in the boosted frame; without this equivalence the observer-dependence conclusion does not follow from the energy-density transformation alone.

    Authors: We appreciate the referee highlighting the need to clarify the status of Teff. Our definition is explicitly operational: Teff is obtained by applying the standard rest-frame relation between energy density and temperature (via the equation of state) to the Lorentz-transformed energy density u' measured by the moving observer. This is the temperature an observer would infer from an energy-density measurement in their own frame. Because the energy-momentum tensor is covariant and the three chosen systems span different equations of state, the resulting velocity dependence directly supports the Ott-Eddington picture within this operational framework. We agree, however, that an explicit demonstration that the same Teff also satisfies the maximum-entropy condition in the boosted frame would strengthen the link to conventional thermodynamics. We will add a short clarifying subsection and a brief discussion of this point in the revised manuscript. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation uses external Lorentz transformations and explicit operational definition

full rationale

The paper begins from the standard energy-momentum tensor and Lorentz transformations (external inputs) and explicitly defines Teff as the temperature inferred by a moving observer from the boosted energy density. It then applies the respective equations of state for photon, ideal, and electron gases to compute the velocity dependence. This produces the claimed observer dependence by direct application of the chosen definition rather than by any reduction of a derived result back to a fitted input or self-citation. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work are present in the abstract or described chain. The central result follows from the operational starting point without circular self-reference.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on the standard Lorentz transformation of the energy-momentum tensor (external) and on the new operational definition of Teff from energy density; no free parameters or invented entities are mentioned in the abstract. The isotropy assumption and the identification of the inferred quantity with thermodynamic temperature are the main unstated premises.

assumptions (2)
  • standard math The energy-momentum tensor of an isotropic system transforms under Lorentz boosts in the standard way.
    Invoked at the start of the analysis to obtain the transformed energy density.
  • domain assumption Effective temperature is defined as the temperature inferred by a moving observer from the transformed energy density.
    This is the load-bearing definition introduced in the abstract; if false, the support for Ott-Eddington does not follow.

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Cite this review

Pith. "Pith review of Relativistic transformation of temperature revisited." pith.science (2026). https://pith.science/paper/NO2ZX6I5

@misc{pith2026260600521,
  author       = {Pith},
  title        = {Pith review of: Relativistic transformation of temperature revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NO2ZX6I5}},
  note         = {Machine review of arXiv:2606.00521}
}
read the original abstract

The relativistic transformation of temperature has long remained controversial, with the classical laws of Planck-Einstein, Ott-Eddington-Moller and Landsberg yielding conflicting results. We reexamine this issue from a relativistic thermodynamic and statistical perspective, starting from the energy-momentum tensor of an isotropic system and defining the effective temperature Teff as that inferred by a moving observer from the transformed energy density. Analyses of a photon gas, a relativistic ideal gas and an electron gas show that Teff consistently increases with velocity, supporting the Ott-Eddington interpretation while depending on the system's equation of state. These results indicate that temperature is not a Lorentz-invariant scalar but an observer-dependent quantity. A consistent relativistic description emerges when temperature is related to the inverse-temperature four-vector beta, linking operational and invariant viewpoints within a unified thermodynamic framework.

Figures

Figures reproduced from arXiv: 2606.00521 by the authors.

Figure 1
Figure 1. Dependence of the effective temperature ratio [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Effective temperature ratio Teff/T versus observer velocity v/c for representa￾tive values of the dimensionless parameter AF (approximating m/T): AF = 0 (m/T ≈ 0), AF = 1 (m/T ∼ 1) and AF = 5 (m/T ≫ 1). Curves are obtained from the relation Teff/T = [γ 2 (1 + v 2/(AF + 3))]1/4 and show the smooth interpolation between the ultra￾relativistic (Teff ≃ T(1 + 1 3 v 2 )) and nonrelativistic (Teff ≃ T(1 + 1 4 v 2 )) limits… view at source ↗

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