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REVIEW 3 major objections 3 minor 19 references

Leibniz Equivalence, Newton Equivalence, and Substantivalism

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Active diffeomorphisms change the physical situation, and all changed situations are equally possible.

desk verdict A useful logical point about substantivalism and a clear exposition of active diffeomorphisms, but the §8 escape from Earman-Norton's indeterminism argument does not survive contact with vacuum GR. read the letter →

arxiv 1908.04326 v1 pith:X2CAM6MS submitted 2019-08-12 physics.hist-ph

Oliver Davis Johns This is my paper · ORCID
classification physics.hist-ph
keywords activediffeomorphismspassiveLeibnizEquivalenceNewtonholeargumentsubstantivalismgeneralcovariancesymmetryandidentity
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 introduces Newton Equivalence as an interpretation of active diffeomorphisms: a diffeomorphism that moves manifold points and pushes forward fields changes the physical situation being modeled, and general covariance then makes the new situation equally possible. The paper contrasts this with Leibniz Equivalence, which identifies all actively diffeomorphic models with a single physical situation. It argues that active diffeomorphisms are symmetry operations, not re-descriptions, and that localized ones can be observed from an untransformed region. On this reading, the generalized hole argument's two objections to substantivalism, the verificationist dilemma and the indeterminism dilemma, both fail. If the paper is right, denying Leibniz Equivalence does not commit a researcher to substantivalism, and the substantivalism debate becomes separable from the interpretation of active diffeomorphisms.

What carries the argument

The load-bearing distinction is between passive and active diffeomorphisms. A passive diffeomorphism changes the local coordinates used to represent manifold objects while leaving the objects themselves unchanged; an active diffeomorphism leaves the coordinate system fixed and pushes the manifold objects forward to new objects. The interpretation of that push-forward is what separates the two principles: Leibniz Equivalence treats the pushed-forward objects as a new representation of the same physical situation, while Newton Equivalence treats them as a different, equally possible situation. A second mechanism is the matched-versus-unmatched hole diffeomorphism: when the active map is the identity outside a source-free hole, it changes the metric without changing the source, but when it is not matched to the source region it changes both solution and source, which the paper reads as defining a different experiment.

What would settle it

Take a vacuum solution of the general-relativity field equation with a compactly supported active diffeomorphism that is the identity outside a hole and changes the metric inside; if two distinct metrics related by such a map both satisfy the same field equation with the same zero source and the paper's spurious-solution criterion cannot single out one, the source-matching escape fails and the indeterminism question is reinstated.

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Extended reading notes

Core claim

The central claim is that an active diffeomorphism produces genuinely different manifold objects, functions, vector fields, and metrics, and those different objects model genuinely different physical situations. Because the model is generally covariant, the transformed situation obeys the same laws and is therefore as possible as the original; it is a different experiment that could happen, not the same experiment in different coordinates. The paper argues this reading matches standard physics practice, which distinguishes symmetry from identity: a rotated magnet on a table is a new physical configuration even though rotation is a symmetry. It also argues that a localized active diffeomorphism leaves an untransformed reference region behind, so the change is observable in principle. The generalized hole argument's indeterminism charge is said to fail because its hole diffeomorphisms are not matched to a source-free region and therefore alter the source term, turning the alleged indeterminism into the ordinary prediction of a different experiment.

Load-bearing premise

The paper's escape from the indeterminism dilemma assumes that the hole diffeomorphisms considered by the generalized hole argument are not matched to a source-free region; when the source already vanishes inside the hole, the diffeomorphism leaves the source unchanged and the dilemma remains open.

Editorial extensions

If this is right

  • If Newton Equivalence is correct, a generally covariant model and its active diffeomorphic images describe a family of distinct possible experiments, so counting possible worlds should not identify diffeomorphic models.
  • The substantivalism question decouples from active diffeomorphisms: rejecting Leibniz Equivalence no longer forces one into substantivalism, since Newton Equivalence allows agnosticism.
  • The verificationist dilemma collapses for localized active diffeomorphisms, because an untransformed region can serve as a reference system that registers the change.
  • The indeterminism dilemma fails for unmatched hole diffeomorphisms, since changing the source changes the experiment; only a source-matched hole diffeomorphism reproduces the original hole-argument worry.
  • Standard practice in theoretical and experimental physics, treating symmetries as relating distinct but equally allowed situations, is preserved.

Reading between the lines

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

  • A consequence the paper does not develop: in vacuum general relativity, where the source vanishes everywhere, every hole diffeomorphism leaves the source unchanged; on the paper's own source-matching test, its escape from the indeterminism dilemma does not apply to vacuum spacetimes.
  • By analogy, Newton Equivalence suggests a general stance toward gauge symmetries: symmetry-related configurations could be physically distinct yet equally possible, which would change how gauge redundancy is handled in quantization.
  • A testable extension would compare active diffeomorphic images of a laboratory configuration: if the apparatus and the table are both transformed, no reference remains to register the change, so the observability argument likely applies only to localized, not global, diffeomorphisms.
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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 / 3 minor

Summary. The paper distinguishes passive from active diffeomorphisms and introduces "Newton Equivalence" as an interpretation of active diffeomorphisms: the different manifold objects produced by an active diffeomorphism model different physical situations, but general covariance makes all of these situations equally possible. This is contrasted with "Leibniz Equivalence," the Earman-Norton claim that diffeomorphic models represent the same physical situation. The paper argues that Newton Equivalence is consistent with current physics practice, that Leibniz Equivalence rests on an unjustified extrapolation from Leibnizian relativism, and that accepting Newton Equivalence allows one to remain agnostic about substantivalism while escaping Earman and Norton's verificationist and indeterminism dilemmas.

Significance. If the central argument were sound, the paper would provide a clearly articulated alternative to Leibniz Equivalence and a direct challenge to the standard hole-argument conclusions about substantivalism. The paper is genuinely useful as a conceptual taxonomy: its definitions of active and passive diffeomorphisms are standard, its uncontested-points list in Section 4 is helpful, and the two examples in Section 5 make the abstract distinction concrete. The paper also deserves credit for stating the points of agreement between the two interpretations before arguing for one. However, the significance is substantially limited by the fact that the paper's escape from the indeterminism dilemma rests on a premise that fails in the vacuum case and on an unstated, self-cited criterion for rejecting "spurious" solutions.

major comments (3)
  1. [§8, especially the paragraph containing footnote 34] The paper's escape from the Earman-Norton indeterminism dilemma is load-bearing and is not secure. The paper claims that "unmatched" active hole diffeomorphisms change the source term and therefore describe a different experiment, so that Earman-Norton's dilemma is merely the correct action of a symmetry principle. But the hole argument is standardly posed, and Einstein originally posed it, for a source-free region in which T_μν = 0. In that case every active diffeomorphism supported in the hole leaves the source identically zero, as the paper itself concedes in footnote 34. Therefore the distinction between "matched" and "unmatched" diffeomorphisms does not do the work assigned to it: an "unmatched" diffeomorphism supported in a vacuum hole still leaves the source unchanged. The indeterminism question thus remains live for the case that motivated the hole argument, and the paper's conclusion that the Earman-Norton indeterminism dilemma "does not generalize Einstein's version" is unsupported.
  2. [§7.4 and the conclusion of §8] The fallback escape from indeterminism is delegated to the author's own forthcoming paper, Johns (2019), without stating or deriving the criterion by which a metric solution is rejected as "spurious." The paper says it "may be possible" to reject as spurious a metric solution whose local coordinates violate a desired symmetry, and then uses this possibility to justify uniqueness. But no invariant, non-question-begging criterion is given here. Since φ*g is by construction isometric to g, any criterion based on invariant or symmetry-invariant quantities will see the two metrics as the same; a criterion that uses the local coordinate expression would appear to reintroduce a preferred coordinate system of exactly the kind general covariance denies. Until this criterion is stated and defended, the appeal to spurious solutions cannot carry the weight placed on it in §8.
  3. [§8, Earman-Norton's initial-data formulation] The paper's "different experiment" response misses the structure of the Earman-Norton dilemma as an initial-data argument. The hole diffeomorphism is the identity outside the hole, so all fields on a Cauchy surface outside the hole are identical in the two models, while the future inside the hole differs. The source inside the future hole is part of the future state of that model, not an externally fixed experimental setup. Calling the changed source a "different experiment" therefore does not defuse the indeterminism charge: the two models have the same initial data but different futures, which is precisely the form of indeterminism at issue.
minor comments (3)
  1. [§5.1 and §3.1] Example 1 says the rotation generated by the vector field (0,-x2,x1,0) is "assumed to be the identity everywhere except in the apparatus," but a global rotation by τ = π/4 is not the identity outside a bounded region. A localized diffeomorphism must be constructed with a bump function or similar device; as stated, the example is internally inconsistent.
  2. [§8, final paragraph of the indeterminism subsection] The phrase "Earman-Newton treatment" appears to be a typo for "Earman-Norton treatment." Please correct it.
  3. [§7.5] The sentence beginning "These extrapolations will appeal to a researcher..." appears to mean "will be appealing to" rather than "will appeal to" in the sense of making a request; consider rewording for clarity.

Circularity Check

2 steps flagged · score 4.0 of 10

Section 8's hole-argument escape relies on a source-matching dichotomy that collapses in vacuum GR, with the fallback imported from the author's own forthcoming Johns (2019) spurious-solution result.

  1. self citation load bearing [Section 8, footnotes 34 and 35]
    "Footnote 34: "Outside the hole, the active diffeomorphism is the identity and hence does not change the source there. Inside the hole, the source is identically zero and hence is not transformed, since zero tensors transform to zero tensors regardless of the active diffeomorphism applied." Footnote 35: "Without it, the active diffeomorphism also modifies the source term and Einstein's proof fails. See Section 4 of Johns (2019).""

    The matched/unmatched dichotomy on which the escape from Earman-Norton indeterminism depends collapses in the vacuum case: footnote 34 concedes that a zero source in the hole is unchanged by any diffeomorphism, so in vacuum GR every hole diffeomorphism vacuously matches the source. The paper's claim that Earman-Norton's 'unmatched' diffeomorphisms change the source and therefore describe a different experiment does not apply to that case, which is exactly the case Einstein and Earman-Norton posed. The only remaining exit named in the text is the same-author citation in footnote 35, 'See Section 4 of Johns (2019)', where the spurious-solution rejection is said to live.

  2. uniqueness imported from authors [Section 7.4, "History"]
    "It has been suggested by Johns (2019) that there may be a less drastic escape from Einstein's dilemma. The local coordinates used to write Einstein's field equation do not have any physical meaning until after a metric solution is found that defines their relation to physical quantities. Therefore, it may be possible to reject as spurious a metric solution that gives a physical meaning to its local coordinates that violates a desired symmetry, for example spherical symmetry for the Schwarzschild solution."

    The 'unique result' is imported solely from Johns (2019), a same-author forthcoming paper, and is treated as if it were an established mathematical fact. The 'desired symmetry' criterion is not derived in this manuscript; it is an ansatz used to reject diffeomorphic images of a solution. Because any image metric produced by a hole diffeomorphism is isometric to the original, invariant criteria cannot separate the two, while a coordinate-dependent symmetry criterion reintroduces preferred coordinates of the kind general covariance denies. The paper does not show that the criterion follows from general covariance or from any independent principle; it simply cites the author's own prior work.

full rationale

The paper's central interpretive thesis—that active diffeomorphisms change the physical situation while general covariance makes the new situation equally possible—is not a disguised tautology: it is supported by independent examples in Sections 5 and 6 and by the standard physics distinction between symmetry and identity. There is no data fitting, and no quantity is defined in terms of the conclusion. The circularity burden is concentrated in the hole-argument escape. Section 8's matched/unmatched dichotomy collapses for vacuum general relativity, where the source is identically zero and therefore unchanged by any diffeomorphism, as footnote 34 concedes; the only exit on offer is the same author's forthcoming Johns (2019) spurious-solution result, invoked in footnotes 35 and in Section 7.4 as if established. That self-citation is load-bearing because the conclusion that Newton Equivalence escapes radical local indeterminism depends on rejecting the isometric images generated by hole diffeomorphisms as 'spurious', a rejection not derived in this manuscript. Accordingly, the score is 4: some load-bearing self-citation, but the central interpretive claim retains independent content.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities are present. The central claim rests on six axioms, two of which are close to the paper's own interpretive conclusion (general covariance implies distinct-but-equally-possible situations) or to a contested reading of Earman and Norton (the unmatched hole diffeomorphism). The load-bearing assumptions are philosophical rather than mathematical.

assumptions (6)
  • domain assumption A well-defined distinction between manifold objects (physical) and coordinate objects (representational) is accepted as the correct way to interpret differential geometric models.
    The entire argument in §§2 and 6.2 rests on this separation, which is standard in physics but is not an uncontroversial philosophical position.
  • domain assumption General covariance implies that any two models related by an active diffeomorphism are equally possible physical situations.
    Introduced in the abstract and §6; this is close to being the content of Newton Equivalence itself, so it functions as a postulate rather than a derived consequence.
  • standard math In pre-general-relativistic physics with a fixed metric, only isometric active diffeomorphisms are permissible.
    Used in §3 with citations to Lee (1997); needed for Example 1, where rotations are allowed because they are isometries of the fixed Euclidean metric.
  • domain assumption Localized active diffeomorphisms can be applied to a subregion while leaving the surrounding reference system untouched.
    Developed in §7.3 and used in Example 1; this is required to render active diffeomorphisms observable, and it presumes a physically meaningful boundary between transformed and untransformed regions.
  • standard math A zero energy-momentum tensor transforms to itself under any diffeomorphism, so source-free holes are invariant.
    Noted in footnote 34; used in the discussion of Einstein's original hole argument.
  • domain assumption Earman and Norton's generalized hole diffeomorphism is not matched to a source-free region and therefore changes the source.
    Stated in §8; this is an interpretive claim about the Earman-Norton paper and is the paper's main ground for saying their indeterminism argument fails.

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Cite this review

Pith. "Pith review of Leibniz Equivalence, Newton Equivalence, and Substantivalism." pith.science (2026). https://pith.science/paper/X2CAM6MS

@misc{pith2026190804326,
  author       = {Pith},
  title        = {Pith review of: Leibniz Equivalence, Newton Equivalence, and Substantivalism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X2CAM6MS}},
  note         = {Machine review of arXiv:1908.04326}
}
read the original abstract

Active diffeomorphisms map a differentiable manifold to itself. They transform manifold points and objects without changing the system of local coordinates used to represent those objects. What has been called Leibniz Equivalence is the assertion that, although active diffeomorphisms do change manifold objects, they do not change what is called the "physical situation" being modeled by those objects. This paper introduces the contrasting idea of Newton Equivalence, which asserts that the different values of manifold objects produced by active diffeomorphisms do model different physical situations. But due to the assumption of general covariance, these different physical situations are all equally possible. They represent physically different situations all of which could happen. This paper compares these two interpretations of active diffeomorphisms, and comments on their importance in the substantivalism debate.

Figures

Figures reproduced from arXiv: 1908.04326 by the authors.

Figure 5.1
Figure 5.1. a, b, and c [PITH_FULL_IMAGE:figures/full_fig_p009_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. a, b, and c [PITH_FULL_IMAGE:figures/full_fig_p010_5_2.png] view at source ↗
Figure 5
Figure 5. to be mounted in a closed room sitting on a turntable, then if some [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: that cannot be reduced to Leibnizian arguments about translation [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

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