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REVIEW 3 major objections 5 minor 2 references

The Toll of the Tolman Effect: On the Status of Classical Temperature in General Relativity

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The Tolman effect splits temperature into three rival concepts, none of which recovers all of classical thermodynamics.

desk verdict A well-sourced, clearly argued philosophy paper in which the real contributions—Einstein's 1912 priority and the TL/TG tradeoff—hold up, while the third temperature TWL is an honestly flagged sketch rather than a fatal flaw. read the letter →

arxiv 2507.10529 v1 pith:ES3TDNDN submitted 2025-07-14 physics.hist-ph gr-qc

Eugene Y. S. Chua , Craig Callender This is my paper · ORCID
classification physics.hist-phgr-qc
keywords Tolmaneffectrelativisticthermodynamicstemperaturegeneralrelativitygravitationalredshiftlocalequilibriumframeequivalenceprinciplefragmentationoftime
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 the Tolman effect, the prediction that an equilibrium system stretched across a varying gravitational potential has a temperature gradient, destroys the idea of a single natural temperature in general relativity. It identifies at least three candidate temperatures: the local temperature $T_L$ measured by a nearby thermometer, the global temperature $T_G=T_L\sqrt{g_{00}}$ built from a static global time, and a new 'wahre-local' temperature $T_{\rm WL}$ defined only inside local equilibrium frames. Each candidate preserves some roles of classical temperature, and each sacrifices others. The paper concludes that equilibrium thermodynamics has no unique successor in the relativistic domain, and that this fragmentation of temperature is the expected companion of the fragmentation of time.

What carries the argument

The load-bearing structure is the Tolman relation $T\sqrt{g_{00}}=\text{const}$, paired with the distinction between 'pocket temperature' (the local, thermometer-readable $T_L$) and 'wahre temperature' (the globally defined $T_G$). The paper then introduces the local equilibrium frame: a local inertial frame at a point of an extended system in which the system is at rest, in thermodynamic equilibrium, and has $g_{00}\approx1$. The wahre-local temperature $T_{\rm WL}$ is defined only with respect to such frames, which is what lets it preserve the laws of thermodynamics within its limited domain.

What would settle it

A concrete check would be to measure local temperatures at two heights in a tall equilibrium column: a violation of $T\sqrt{g_{00}}=\text{const}$ would refute the effect, while a demonstration that local equilibrium frames are unavailable for some realistic extended system would leave $T_{\rm WL}$ undefined.

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Extended reading notes

Core claim

The central claim, stated sympathetically, is that there is no one natural successor to classical thermodynamics in general relativity. The Tolman relation $T\sqrt{g_{00}}=\text{const}$ can be read in three incompatible ways: as a genuine gradient in the local temperature $T_L$, as a constant global temperature $T_G$, or as evidence that temperature should be indexed to local equilibrium frames through $T_{\rm WL}$. The paper adds historical support for the effect, showing that Einstein derived it in 1912 from mass-energy equivalence and the equivalence principle, and argues that the effect is best understood through clocks rather than energy loss. Because equilibrium thermodynamics requires a choice of time, and general relativity offers several natural time concepts, the classical temperature concept fragments rather than extending uniquely.

Load-bearing premise

The account of the wahre-local temperature assumes that every point of an extended system has a local inertial frame in which the system is at rest, in thermodynamic equilibrium, and well described by ordinary thermodynamics, with no precise criterion for when this frame is available.

Editorial extensions

If this is right

  • If the paper is right, the local temperature $T_L$ keeps empirical thermometry meaningful but breaks the zeroth law, the first law as usually stated, and the Clausius form of the second law.
  • The global temperature $T_G=T_L\sqrt{g_{00}}$ restores the laws whenever a static global time exists, but it is not what any local observer measures, so it buys the theory at the cost of empirical accessibility.
  • The wahre-local temperature $T_{\rm WL}$ preserves both local thermodynamics and local measurability, but only inside local equilibrium frames where $g_{00}\approx1$; beyond those frames it is simply undefined.
  • The Tolman effect cannot be turned into a perpetual-motion machine, because gravity acts universally on every connecting rod and wire one would use to exploit the gradient.

Reading between the lines

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

  • One can read the paper as implying that any attempt to build a relativistic statistical mechanics with a single Gibbs temperature implicitly chooses a clock; the choice is not forced by nature.
  • The taxonomy suggests a testable extension: in systems with large gravitational potential differences, reports of whether a distant body is 'hotter' or 'colder' should be meaningful only relative to a stated clock frame.
  • The same three-way split may carry over to uniformly accelerated frames in flat spacetime and to analogue-gravity systems, where proper-time gradients mimic the Tolman effect.
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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

3 major / 5 minor

Summary. The paper is a conceptual and historical study of the Tolman effect, the relativistic result that a system in thermal equilibrium in a static gravitational field has a local temperature satisfying T√g00 = const. It defends five claims: (1) Einstein derived the effect in 1912 before Tolman, and his derivation shows its robustness; (2) reading the effect through the local temperature TL breaks down much of classical thermodynamics, including the Zeroth Law, the First Law, and one version of the Second Law, though without permitting a perpetuum mobile because gravity couples universally to all heat-transport devices; (3) a global temperature TG = T√g00 restores the usual thermodynamic laws at the cost of empirical inaccessibility; (4) the effect is better understood through the behavior of clocks than through energy loss; and (5) one can sketch a third 'wahre-local' temperature TWL defined relative to local equilibrium frames, which preserves thermodynamics locally but only there. The paper concludes that classical temperature fragments into at least three concepts in general relativity and that there is no single natural successor to classical thermodynamics in that domain.

Significance. If the central thesis is accepted, the paper makes a valuable contribution to the philosophy of physics by giving a clear conceptual map of a relativistic effect that is often confined to cosmology. Its historical reconstruction of Einstein's 1912 derivation is careful and well sourced, and the argument that the universality of gravity blocks a Maxwell-demon exploit of the Tolman effect is an original and illuminating point. The paper is also commendably explicit about its own limitations: the authors repeatedly describe TWL as a sketch and do not claim it is the true successor. That honesty is welcome. However, the third temperature concept is not developed to the same standard as the TL/TG analysis, and the central 'fragmentation' claim in Section 7 leans on TWL being a well-defined alternative. As it stands, TWL is stipulated rather than derived, so the paper's strongest conclusion is only conditionally supported.

major comments (3)
  1. [§6]
  2. [§6]
  3. [§7]
minor comments (5)
  1. [§3]
  2. [§3]
  3. [§5]
  4. [§6]
  5. [§5]

Circularity Check

1 steps flagged · score 4.0 of 10

Main Tolman/TG analysis is externally grounded; Section 6's TWL 'preservation of classical thermodynamics' is definitional rather than derived, with modest self-citation of Chua (2023).

  1. self definitional [Section 6, 'Truly Local Temperature?', pp. 18-19 of arXiv:2507.10529v1]
    "However, it is a particular local inertial frame in which the system, at that point, also satisfies certain thermodynamic conditions and is in thermodynamic equilibrium with the environment, if only instantaneously; only then does it make sense to assign a temperature to this point of the system. ... Since one can always choose a local inertial frame ... and any system that can be ascribed a temperature should satisfy appropriate thermodynamic conditions, we can always define TWL relative to such a local equilibrium frame ... Indeed we preserve all of classical thermodynamics relative to TWL."

    TWL is defined only relative to a local equilibrium frame, and that frame was just specified as one in which the system satisfies thermodynamic conditions and is in equilibrium. The conclusion that 'we preserve all of classical thermodynamics relative to TWL' is therefore a restatement of the defining condition, not a derived result. The existence assertion likewise assumes the availability of a frame in which ordinary thermodynamics applies, which is the contested issue for extended systems. The circularity is local and softened by the authors' explicit caveat that TWL is a sketch and not the 'true' or 'best' successor; nevertheless, the advertised benefit of recovering the thermodynamic laws is built into the definition.

full rationale

The paper's main physical and historical content is not circular. The Tolman relation T√g00 = const is imported from Einstein (1912), Tolman (1930), Tolman and Ehrenfest (1930), and more recent work (Santiago and Visser 2019; Kovtun 2023), and the TL versus TG analysis uses that externally established relation to exhibit tradeoffs between empirical accessibility and preservation of thermodynamic laws. No parameter is fitted and no quantity is silently renamed as a prediction. The consilience framework is cited to the first author's prior paper (Chua 2023), but the present paper's arguments about TL, TG, the clock interpretation, and the breakdown of the thermodynamic laws are developed independently of that citation, so this is at most a minor self-citation. The one genuinely definitional step is the TWL proposal: TWL is defined relative to local equilibrium frames, and those frames are specified as frames in which the system already satisfies thermodynamic conditions and is in equilibrium. Consequently, the claim that TWL preserves all of classical thermodynamics is true by construction rather than by derivation. Because this step is a local, explicitly disclosed sketch and not the core of the externally grounded Tolman analysis, the overall circularity is moderate rather than severe, and a score of 4 is appropriate.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central analysis rests on the standard Tolman-Ehrenfest law and the static spacetime assumption for TG. The TWL proposal additionally requires local equilibrium frames, which are assumed without a formal existence criterion. No numerical free parameters are fitted; the main freedom is the choice of temperature convention, which is the paper's subject rather than a hidden parameter.

assumptions (5)
  • domain assumption Tolman-Ehrenfest equilibrium law T√g00 = const for static spacetimes
    Imported from Tolman (1930) and Tolman and Ehrenfest (1930); the paper does not rederive it and uses it as the starting point for all three temperature concepts.
  • domain assumption Existence of a global timelike Killing field for static spacetimes
    Needed to define global energy UG and global temperature TG; the paper notes this context limitation in Section 4.
  • domain assumption The equivalence principle justifies transferring results from uniformly accelerating frames to static gravitational fields
    Used in Section 2 to connect Einstein's scalar-theory derivation to general relativity and to motivate Tolman temperature gradients in accelerated gases.
  • ad hoc to paper Existence of local equilibrium frames at each point of an extended system
    Assumed in Section 6 to define TWL; no precise characterization of when thermodynamic equilibrium conditions hold is given.
  • domain assumption The Minus First Law identifies the temperature that governs relaxation to equilibrium as physically fundamental
    Used in Section 4 to argue that TG is the boss; this is a philosophical preference, not a theorem.
invented entities (1)
  • wahre-local temperature TWL
    purpose: Defines temperature only relative to a local equilibrium frame, so that comparisons across varying gravitational potential are not made.
    Introduced in Section 6 as a third temperature concept. It is a definition or interpretive proposal, not a physical entity; it has no falsifiable consequences beyond the already-known Tolman effect, and its advertised benefits are built into its construction.

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Cite this review

Pith. "Pith review of The Toll of the Tolman Effect: On the Status of Classical Temperature in General Relativity." pith.science (2026). https://pith.science/paper/ES3TDNDN

@misc{pith2026250710529,
  author       = {Pith},
  title        = {Pith review of: The Toll of the Tolman Effect: On the Status of Classical Temperature in General Relativity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ES3TDNDN}},
  note         = {Machine review of arXiv:2507.10529}
}
read the original abstract

The Tolman effect is well-known in relativistic cosmology but rarely discussed outside it. That is surprising because the effect -- that systems extended over a varying gravitational potential exhibit temperature gradients while in thermal equilibrium -- conflicts with ordinary classical thermodynamics. In this paper we try to better understand this effect from a foundational perspective. We make five claims. First, as Tolman knew, it was Einstein who first discovered the effect, and furthermore, Einstein's derivation helps us appreciate how robust it is. Second, the standard interpretation of the effect in terms of 'local temperature' leads to the breakdown of much of classical thermodynamics. Third, one can rescue thermodynamics by using Einstein's preferred interpretation in terms of the 'wahre Temperatur' -- what we'll call global temperature -- but it too has some costs. Fourth, the effect is perhaps best understood in terms of clocks as opposed to energy loss. Fifth, inspired by a proposal of Einstein's elsewhere, we sketch an interpretation of the effect in terms of a third novel temperature, which we call the 'wahre-local temperature'. On this view, temperature -- and thermodynamics -- is defined only in relation to local clocks. In sum, we view the fragmentation of temperature in thermodynamics as a natural and expected result of the fragmentation of time in general relativity.

Figures

Figures reproduced from arXiv: 2507.10529 by the authors.

Figure 1
Figure 1. TL vs TG. TL is constant at each layer (of constant g00), but not between layers of varying g00. TG is constant throughout. Fourth, we try to form a better understanding of the Tolman effect. Motivated by a separate interpretive issue with the gravitational redshift, we identify an "energy interpret￾ation" and a "clock interpretation" of the Tolman effect. Both are given by Einstein. We provide some reasons in favor… view at source ↗
Figure 2
Figure 2. The principle of equivalence: a system in a uniformly accelerating frame is [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Adapted from Santiago and Visser ([2019]). where U is the internal energy, Q the heat and W work. The d are inexact differentials, which indicates that energy may take the form of heat or work, respectively. This equa￾tion must now include sources of energy from mass-energy equivalence and gravitational energy. But the “spirit” of the law still holds, as the law is only "broken" in the sense that more sources of ene… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Ice cube meets the Tolman effect. the simplest and most powerful generalizations of the facts of ordinary thermodynam￾ics. After all, it saves the compact and powerful laws at the price of only a little com￾plexity, and it varies with the core physical tendency underly…

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Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [1]

    Balazs, N. L. [1958]: ‘On relativistic thermodynamics’,Astrophysical Journal, 128, pp. 398–

    work page 1958

  2. [405]

    Thermodynamics without Time

    Balazs, N. L. and J. M. Dawson [Feb. 1965]: ‘On thermodynamic equilibrium in a gravit- ational field’, Physica, 31, pp. 222–32, ISSN : 0031-8914. 17 Rovelli and Smerlak ([2011]) and Haggard and Rovelli ([2013])’s “thermal-time" proposal defines a universal time step at equilibrium, relative to Gibbs states ρ representing systems in thermal equi- librium a...

    work page Pith review arXiv 2024

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Reviewed August 6, 2026 · model on record in the stance chip above.