Relativistic transformation of temperature revisited

Babak Vakili; Soroor Pouryazdan

REVIEW 1 major objections 1 minor 25 references

Effective temperature of photon, ideal, and electron gases rises with velocity, showing temperature is observer-dependent.

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2026-06-28 18:32 UTC pith:NO2ZX6I5

load-bearing objection 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. the 1 major comments →

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

Relativistic transformation of temperature revisited

Soroor Pouryazdan , Babak Vakili This is my paper

classification gr-qc cond-mat.stat-mechphysics.class-ph

keywords relativistic temperature transformationOtt-Eddington interpretationenergy-momentum tensorphoton gasrelativistic ideal gaselectron gasobserver-dependent temperatureinverse-temperature four-vector

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

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.

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

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.

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.

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 this falsifier — get emailed when new claim-graph text bears on it.

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.

Where Pith is reading between the lines

These are 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.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

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

The one thing to take from this is that the authors calculate an effective temperature Teff from the Lorentz-transformed energy density for a photon gas, a relativistic ideal gas, and an electron gas, and in each case Teff goes up as the boost velocity increases. This lines up with the Ott-Eddington view and makes clear that the outcome depends on the equation of state.

They begin with the energy-momentum tensor for isotropic systems, apply the transformation, and pull out Teff using the standard energy-temperature relation for each case. The calculations are explicit enough to see the pattern across systems, and they link the result to the inverse-temperature four-vector to bridge the operational and invariant pictures.

This adds concrete numbers to a long-running debate rather than just restating positions. The dependence on the EOS comes through directly from the different relations used for each gas.

The soft spot is the definition of Teff. It is set up as what a moving observer infers from the new energy density, assuming the same functional form holds after the boost. If thermodynamic temperature is instead defined via entropy or the four-vector in the instantaneous rest frame, this Teff may not be the quantity that thermometers would register or that follows from maximum entropy. The paper treats it as operational, but that choice drives the increase and the observer-dependence claim.

Readers already working on relativistic statistical mechanics or the temperature transformation problem will get the most from the explicit cases. It is worth sending to peer review so the derivations can be checked in detail and the connection to prior work on the same definition can be assessed.

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)

[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)

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

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 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.

axioms (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.

pith-pipeline@v0.9.1-grok · 5666 in / 1621 out tokens · 18840 ms · 2026-06-28T18:32:23.167989+00:00 · methodology

0 comments

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 Babak Vakili, Soroor Pouryazdan.

**Figure 2.** Figure 2: Effective temperature ratio Teff/T versus observer velocity v/c for representative 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 ultrarelativistic (Teff ≃ T(1 + 1 3 v 2 )) and nonrelativistic (Teff ≃ T(1 + 1 4 v 2… view at source ↗

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 7 canonical work pages · 2 internal anchors

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[1] [1]

R. C. Tolman, Relativity, Thermodynamics and Cosmology , Oxford, Clarendon Press (1934)

1934

[2] [2]

¨Uber das Relativit¨ atsprinzip und die aus demselben gezoge nen Fol- gerungen

A. Einstein, “ ¨Uber das Relativit¨ atsprinzip und die aus demselben gezoge nen Fol- gerungen”, Jahrbuch der Radioaktivit¨ at und Elektronik 4 (1907) 411

1907

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Zur Dynamik bewegter Systeme

M. Planck, “Zur Dynamik bewegter Systeme”, Annalen der Phys. 26 (1908) 1

1908

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Ott, ”Lorentz transformation of heat and temperature ”, Zeitschrift Phys

H. Ott, ”Lorentz transformation of heat and temperature ”, Zeitschrift Phys. 175 (1963) 70

1963

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A. S. Eddington, The Mathematical Theory of Relativity , Cambridge University Press (1924)

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2004

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N. G. van Kampen, ”Relativistic thermodynamics of movin g systems”, Phys. Rev. 173 (1968) 295

1968

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Israel, ”Nonstationary irreversible thermodynamic s: A causal relativistic theory”, Ann

W. Israel, ”Nonstationary irreversible thermodynamic s: A causal relativistic theory”, Ann. Phys. 100 (1976) 310

1976

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G. L. Sewell, ”Quantum ﬁelds on manifolds: PCT and gravi tationally induced thermal states”, Ann. Phys. 141 (1982) 201

1982

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T. K. Nakamura, ”Lorentz Transform of Black Body Radiat ion Temperature”, Euro- phys. Lett. 88 (2009) 20004

2009

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Cubero et al

D. Cubero et al. , ”Thermal equilibrium and statistical thermometers in spe cial rela- tivity”, Phys. Rev. Lett. 99 (2007) 170601

2007

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J. Dunkel and P. H¨ anggi, ”Relativistic Brownian motio n”, Phys. Rep. 471 (2009) 1 (arXiv: 0812.1996 [cond-mat.stat-mech])

work page internal anchor Pith review Pith/arXiv arXiv 2009

[14] [14]

Mareˇ s, P

J. Mareˇ s, P. Hub ´ ık and V.ˇSpiˇ cka, ”On relativistic transformation of temperature” , Fortschr. Phys. 65 (2017) 1

2017

[15] [15]

J. A. Heras and M. G. Osorno, ”A note on the relativistic t emperature”, Eur. Phys. J. Plus 137 (2022) 423 (arXiv: 2204.12572 [physics.class-ph])

work page arXiv 2022

[16] [16]

X. Hao, S. Li and L. Zhao, ”Relativistic transformation of thermodynamic parame- ters and reﬁned Saha equation”, Commun. Theor. Phys. 75 (2023) 035601 (arXiv: 2105.07313 [gr-qc])

work page arXiv 2023

[17] [17]

Wallace, ”On relativistic thermodynamics”, Philos

D. Wallace, ”On relativistic thermodynamics”, Philos. Sci. 92 (2025) 688 (arXiv: 2307.00354 [cond-mat.stat-mech])

work page arXiv 2025

[18] [18]

Sato, ”Kinetic theory of relativistic temperature i n self-gravitating systems”, (arXiv: 2411.19405 [gr-qc])

N. Sato, ”Kinetic theory of relativistic temperature i n self-gravitating systems”, (arXiv: 2411.19405 [gr-qc])

work page arXiv

[19] [19]

Far ´ ıas, V

C. Far ´ ıas, V. A. Pinto and P. S. Moya, ”What is the temperature of a moving body?”, Sci. Rep. 7 (2017) 17657 15

2017

[20] [20]

E. Y. S. Chua, ”T Falls Apart: On the Status of Classical T emperature in Relativity”, Philos. Sci. 90 (2023) 5 (arXiv: 2303.14847 [physics.hist-ph])

work page arXiv 2023

[21] [21]

Hakim, ”Relativistic Stochastic Processes”, J

R. Hakim, ”Relativistic Stochastic Processes”, J. Math. Phys. 9 (1968) 1805

1968

[22] [22]

Hakim, ”Remarks on Relativistic Statistical Mechan ics

R. Hakim, ”Remarks on Relativistic Statistical Mechan ics. I”, J. Math. Phys. 8 (1967) 1315

1967

[23] [23]

V. E. Ambrus and I. I. Cotaescu, ”Maxwell-Juttner distr ibution for rigidly-rotating ﬂows in spherically symmetric spacetimes using the tetrad f ormalism”, Phys. Rev. D 94 (2016) 085022 (arXiv: 1605.07043 [hep-th])

work page internal anchor Pith review Pith/arXiv arXiv 2016

[24] [24]

Livadiotis, ”Modeling anisotropic Maxwell–J¨ uttn er distributions: derivation and properties”, Ann

G. Livadiotis, ”Modeling anisotropic Maxwell–J¨ uttn er distributions: derivation and properties”, Ann. Geophys. 34 (2016) 1145

2016

[25] [25]

Das Maxwellsche Gesetz der Geschwindigk eitsverteilung in der Rela- tivtheorie

F. J¨ uttner, “Das Maxwellsche Gesetz der Geschwindigk eitsverteilung in der Rela- tivtheorie”, Annalen der Phys. 339 (1911) 856 16

1911