---
id: "20261026-0230-outside-arxiv261032569-cartographing-gravity"
title: "Cartographing Gravity: Operational Maps, the Conatus of Proper Time, and the Potentia of Matter (de Haro & Elizalde, 2026)"
type: "capture"
status: "promoted"
origin: "batch"
writer_model: "claude-sonnet-5"
promoted_to: ["30-notes/claim-de-haro-elizalde-2026-reconstruct-linearized-gr-from-conatus-and-potentia.md","30-notes/claim-de-haro-elizalde-2026-invoke-lovelock-theorem-to-force-einstein-equations.md","30-notes/claim-de-haro-elizalde-2026-borrow-conatus-potentia-from-spinoza.md","30-notes/claim-de-haro-elizalde-2026-attribute-einsteins-two-step-route-to-geometric-reading-of-ds.md","40-entities/entity-lovelock-theorem.md (concept hub, created)","40-entities/entity-jaume-de-haro.md (person hub, created)","40-entities/entity-sebastian-de-haro.md (namesake-disambiguation line appended)","50-questions/question-corroborate-de-haro-proper-time-reconstruction-of-gr-independent-source.md (corroboration question routed)"]
not_promoted: ["Soldner/Einstein 1911 half-deflection coincidence (Further lead) — a well-known physics-history point the capture explicitly says needs a dedicated history-of-science source, not this paper; not this paper's claim to own. Left as a lead.","Synge's 'Equivalence Principle should be buried with honours' (Further lead) — quotable, but needs Synge's own 1960 text read directly (no network in promotion organ); nice-to-have, not load-bearing for any kept claim. Left as a lead.","Wheeler aphorism contrast ('space tells matter how to move…') (Further lead) — a substantivism-vs-relationalism thread worth connecting someday, but no atomic claim here worth its own note; left as a lead.","Tartaglia 1999 rotating-disk disagreement (Further lead) — a live technical disagreement the paper cites but does not resolve; folded as a watch_flag into the Einstein-two-step claim-note rather than given its own note.","Hilbert's Sixth Problem / 1915–1917 axiomatic program (Further lead) — contextual framing for the paper's target, not a standalone claim; left as a lead.","The de Haro self-citation dependency chain (Further lead) — not skipped but ROUTED: became the corroboration question above, since it is the load-bearing structural doubt under the two physics claim-notes.","Entity candidate Baruch Spinoza — NOT promoted to a hub this session; recurs across two unrelated clusters (lens-grinding dish + conatus/potentia) but both appearances are passive, so flagged for the Warden in seek-flags rather than built. Bias-against-flood.","Entity candidates Emilio Elizalde, Max Abraham, Marcel Grossmann, J.L. Synge, Thomas Hobbes, David Lovelock — left as mentions (named in claim-notes / on the Lovelock's-theorem hub); single-mention co-authors/cited figures, no separate standing. Lovelock's load-bearing contribution is captured as the concept hub entity-lovelock-theorem instead of a person page.","Entity candidate 'Conatus' (concept) — the Spinozist/Hobbesian disambiguation is captured inside claim-de-haro-elizalde-2026-borrow-conatus-potentia-from-spinoza; no separate concept hub, as it does not yet recur beyond this paper."]
date_created: "2026-10-26T00:00:00.000Z"
provenance: "batch run 2026-10-26"
derived_from: []
tags: ["physics","general-relativity","philosophy-of-physics","operationalism","spinoza","lovelock-theorem","equivalence-principle","outside"]
source_url: "https://arxiv.org/abs/2610.32569"
source_sha: "83d26470fa57f8fe6ec2d4df44588769eb05457759064e4b35a8968b0d993fe8"
source_author: "Jaume de Haro; Emilio Elizalde"
source_date: "2026-10-23T00:00:00.000Z"
source_title: "Cartographing Gravity: Operational Maps, the Conatus of Proper Time, and the Potentia of Matter"
source_venue: "arXiv (physics.hist-ph / gr-qc); also published as Symmetry 2026, 18(10), 1692 (MDPI), DOI 10.3390/sym18101692"
source_tier: 1
seek_code_commit: "d40ee39"
---


A 30-page physics.hist-ph / gr-qc preprint (submitted 23 Oct 2026, authors Jaume de Haro, Universitat Politècnica de Catalunya, and Emilio Elizalde, ICE/CSIC-IEEC Barcelona) that tries to rebuild the weak-field structure of General Relativity — and, by appeal to a 1971 uniqueness theorem, Einstein's field equations specifically — from operational/dynamical starting points rather than from spacetime geometry. The arXiv record cross-links a peer-reviewed version in *Symmetry* (MDPI), 18(10), 1692. The abstract page and the PDF were both fetched and archived cleanly (verified TLS); no safety signals fired on either.

> [!note] Seek's commentary
> The paper's own framing is unusually candid about its limits: it explicitly disclaims deriving GR "without prior assumptions" or establishing "an unrestricted uniqueness theorem," calling its contribution "primarily conceptual and foundational" rather than a new physical result. The physics (linearized GR in harmonic gauge, Lovelock's theorem) is standard and not new — it's openly building on the authors' own earlier paper [[claim below, ref 5]]. What's novel is the vocabulary: borrowing "conatus" and "potentia" from Spinoza to redescribe inertia and gravity as, respectively, a body's striving and matter's capacity, rather than geometric facts about a manifold.

## Claim: The paper reconstructs linearized General Relativity from "conatus" (proper-time extremization) and "potentia" (gravitational modulation of clock rate), treating geometry as a derived synthesis rather than a starting postulate

Claim type: technical-mechanism (how the authors say their derivation proceeds). Source: the paper itself, Tier 1 (primary author's own venue, also peer-reviewed in *Symmetry*) — clears the Tier 1–2 floor required for mechanism claims.

The paper identifies the "conatus" of a free massive particle with its tendency to follow a worldline of stationary (locally maximal) proper time, and identifies gravitation with the "potentia" of matter to modify the local rate of proper time. Starting from the universality of Galilean free fall and the Lorentz-invariant retarded potential (attributed to Max Abraham, 1911), the authors derive a static weak-field invariant, Lorentz-boost it to cover moving sources, and state that "the resulting linearized field equations coincide with those of General Relativity in the harmonic gauge."

source_quote: "The conatus of a massive free particle is identified with its natural tendency to follow a worldline of stationary proper time, which, under the appropriate local conditions, corresponds to a maximum of its proper time, whereas gravitation is interpreted as the potentia of matter to modify the local rate of proper time."
source_url: https://arxiv.org/pdf/2610.32569
source_sha: 66d699cca1cf27e09985542598e79d2c709c5a79e28e98338cbde4f86410f96b
source_tier: 1

The paper explicitly says this part is not new: "Part of the technical construction presented in this essay builds directly on our previous work [5]... These results are therefore not claimed as new here." Ref [5] is J. de Haro, "A Dynamical Approach to General Relativity Based on Proper Time," *Universe* 12(3), 79 (2026) — a single-author peer-reviewed paper by the same Jaume de Haro (see Further leads on the single-source question).

## Claim: The paper argues the nonlinear completion is forced to be Einstein's field equations specifically — not merely "something GR-like" — by Lovelock's 1971 theorem, applied under covariance, second-order, and energy-momentum-conservation assumptions

Claim type: specific technical-mechanism claim (what forces the specific form of the field equations). Source: the paper itself, Tier 1.

The move from linearized to full nonlinear GR is not argued from a new derivation in this paper; it is argued by invoking an existing uniqueness result, D. Lovelock, "The Einstein Tensor and Its Generalizations," *Journal of Mathematical Physics* 12, 498–501 (1971) [ref 6 in the paper]. The authors state that once general covariance, locality, and second-order field equations are imposed together with general energy-momentum conservation, Lovelock's theorem singles out the Einstein tensor (up to a cosmological-constant term, which the authors' Minkowski-reference-structure choice excludes) as the unique divergence-free covariant structure — i.e., Einstein's field equations are not postulated but are the only option left standing within that class of theories.

source_quote: "It is at this point that the Einstein tensor emerges as the unique divergence-free covariant structure within the class of theories satisfying the assumptions of Lovelock's theorem, up to a cosmological term. With the Minkowski reference structure adopted here, the cosmological term is excluded, leaving Einstein's field equations as the resulting non-linear completion."
source_url: https://arxiv.org/pdf/2610.32569
source_sha: 66d699cca1cf27e09985542598e79d2c709c5a79e28e98338cbde4f86410f96b
source_tier: 1

Note this is explicitly framed by the authors as a restricted, assumption-dependent result, not an unconditional proof: "The essay does not claim to derive General Relativity without prior assumptions, nor to establish an unrestricted uniqueness theorem."

## Claim: "Conatus" and "potentia" are imported by the authors directly from Spinoza's *Ethics*, naming inertia (conatus) and gravitation (potentia) respectively, with the authors noting conatus also has a rival, non-Spinozist lineage in Hobbes

Claim type: definitional/historical (what a term means in this text's settled usage, and whose usage it traces to). Tier 3–4 would be acceptable for this kind of claim, but the source clears Tier 1 directly — this is the primary document's own self-reported terminological choice, including its own citations to the philosophical sources.

The paper calls the proper-time extremization "the physical manifestation of the Spinozian classical conatus — the innate endeavor of matter to persevere in its own state — translated here into the relativistic domain," and later restates: gravitation "is interpreted as a physical manifestation of Spinoza's potentia — the capacity of matter to modulate the local rate of proper time. By contrast, the motion of freely falling particles is governed by their conatus — the intrinsic endeavor to optimize experienced proper time along a worldline." The paper's reference list cites B. Spinoza, *The Collected Works of Spinoza, Vol. I* (Princeton University Press, 1985), including the *Ethics* (orig. 1677), as the source for both terms, alongside H.A. Wolfson's *The Philosophy of Spinoza* (Harvard, 1934) as secondary gloss. The paper separately notes a different philosophical lineage for "conatus" in Thomas Hobbes, *De Corpore* (1655), which the authors describe as defining conatus physically as "infinitesimal motion" — a distinct, non-Spinozist sense of the same word — and in Descartes' *Principia Philosophiae* (1644).

source_quote: "This extremization represents the physical manifestation of the Spinozian classical conatus—the innate endeavor of matter to persevere in its own state—translated here into the relativistic domain."
source_quote_2: "It is interpreted as a physical manifestation of Spinoza's potentia—the capacity of matter to modulate the local rate of proper time. By contrast, the motion of freely falling particles is governed by their conatus—the intrinsic endeavor to optimize experienced proper time along a worldline."
source_url: https://arxiv.org/pdf/2610.32569
source_sha: 66d699cca1cf27e09985542598e79d2c709c5a79e28e98338cbde4f86410f96b
source_tier: 1

## Claim: The paper argues Einstein's 1907–1916 route to GR's geometric formulation proceeded through two separate heuristic devices (an elevator thought experiment, then the rotating-disk/Ehrenfest paradox) specifically because he treated the invariant ds as a Gaussian geometric "line element" rather than operationally as proper time, and that starting from the operational view collapses the two steps into one

Claim type: historical/historiographical claim about Einstein's conceptual development. This is interpretive and load-bearing for the paper's narrative (not a settled, uncontested textbook fact), so it is treated at the Tier 1-2 floor rather than resting on Tier 3-4: the source here is the primary document itself making its own historiographical argument, which clears Tier 1. This claim describes what de Haro & Elizalde argue, not an independently confirmed fact about Einstein's actual psychology.

The paper's account: in his 1912 paper ("Lichtgeschwindigkeit und Statik des Gravitationsfeldes," *Annalen der Physik* 38:355–369), Einstein used the elevator form of the Equivalence Principle to conclude gravity modifies only the temporal component g00 of ds², which the authors say "predicted only half the observed deflection of light." To fix the spatial sector, Einstein then separately invoked the rotating-disk/Ehrenfest-paradox argument. De Haro & Elizalde attribute this two-step, patched-together route to Einstein's having inherited a geometric (Gaussian surface-theory) reading of ds — influenced by Marcel Grossmann — rather than Minkowski's 1908 operational reading of ds as literally the proper time elapsed on a clock. They argue that starting from the operational reading dissolves the need for two separate heuristic arguments.

source_quote: "Why did Einstein feel compelled to proceed in this convoluted manner? The root of the problem lies in how he conceptualized the invariant ds between 1907 and 1916. Throughout this crucial period, unlike Minkowski, Einstein did not make the operational interpretation of ds as the differential of proper time the central organizing principle of his construction. Instead, heavily influenced by Marcel Grossmann and the legacy of 19th-century mathematics, he viewed ds through the lens of Gaussian surface geometry."
source_quote_2: "This two-step reasoning—using the elevator to motivate the temporal sector and the rotating disc to motivate the spatial sector—eventually led Einstein toward a fully geometric formulation of gravitation."
source_url: https://arxiv.org/pdf/2610.32569
source_sha: 66d699cca1cf27e09985542598e79d2c709c5a79e28e98338cbde4f86410f96b
source_tier: 1

## Further leads

- Soldner/Einstein 1911 half-deflection coincidence: the paper cites J.G. von Soldner's 1801 corpuscular light-deflection calculation (*Astronomisches Jahrbuch für das Jahr 1804*, 161–172) as numerically matching Einstein's 1911 pre-GR deflection prediction (both give half the correct GR value) — a well-known physics-history point, worth independently checking against a dedicated history-of-science source rather than this paper alone. — source: arXiv:2610.32569, refs [18]–[19].
- The paper's dependency chain: this work states it builds directly on J. de Haro, "A Dynamical Approach to General Relativity Based on Proper Time," *Universe* 12(3), 79 (2026), and points to a forthcoming book, J. de Haro, *Gravitation and Cosmology: Dynamics, Geometry, and the Ontology of Spacetime*, Fundamental Theories of Physics 226, Springer (in press, 2026) — worth checking whether/how many vault notes come to rest on this single author's chain of self-citations. — source: arXiv:2610.32569, refs [5], [35].
- J.L. Synge's remark that the Equivalence Principle should be "buried with honours" (Synge, *Relativity: The General Theory*, North-Holland, 1960) — vivid, quotable critique of the EP as a historical relic, worth checking Synge's actual text directly. — source: arXiv:2610.32569, ref [26].
- The paper contrasts its field-theoretic reading of vacuum GR solutions with Wheeler's aphorism "Space tells matter how to move; matter tells space how to curve" (Misner, Thorne & Wheeler, *Gravitation*, 1973), arguing the aphorism "appears as conceptually incomplete" once the gravitational pseudo-tensor's self-sourcing role is made explicit — a substantivism-vs-relationalism debate that could connect to the vault's existing hole-argument/background-independence material. — source: arXiv:2610.32569, ref [38].
- A. Tartaglia, "Lengths on rotating platforms," *Foundations of Physics Letters* 12, 17–28 (1999), is cited as giving a competing operational analysis of the Ehrenfest rotating-disk paradox in which the circumference-to-diameter ratio stays at 2π for a co-rotating observer, against this paper's own claim of a non-Euclidean ratio in the external observer's frame — a live technical disagreement worth its own look. — source: arXiv:2610.32569, ref [28].
- Hilbert's 1900 Sixth Problem and his 1915–1917 axiomatic program for gravitation (cited refs [43]-[44], Hilbert 1916; Brading & Ryckman 2008) are invoked by the paper as the other half of the "axiomatize physics like mathematics" tendency it is reacting against — a thread connecting to Einstein–Hilbert priority historiography. — source: arXiv:2610.32569, refs [43]-[44].

## Entity candidates

- David Lovelock — person — the foundational figure the paper's whole uniqueness claim rests on: "Lovelock's theorem" (Lovelock, *J. Math. Phys.* 12, 498–501, 1971) is cited as what forces the nonlinear completion to be specifically Einstein's field equations rather than some other covariant theory; currently unknown to the vault.
- Baruch Spinoza — person — already known in the vault (1 mention, in an unrelated claim about a grinding dish) but newly load-bearing here as the explicit philosophical source of both organizing concepts of this paper, "conatus" and "potentia" (cited directly to Spinoza's *Ethics*).
- Jaume de Haro — person — co-author (Universitat Politècnica de Catalunya); explicitly flagged as NOT the same person as "Sebastian de Haro" already in the vault (entity-sebastian-de-haro, 10 mentions, background-independence/black-hole-information cluster) — a namesake collision worth keeping distinct in retrieval.
- Emilio Elizalde — person — co-author (ICE/CSIC and IEEC, Barcelona); also author of a cited 2021 Springer history-of-cosmology book (ref [1]) relevant to this paper's historical framing.
- Max Abraham — person — physicist whose 1911 retarded-potential paper ("Sulla teoria della gravitazione") the authors use as the dynamical content for gravitational interaction, in place of Einstein's accelerated-frame Equivalence Principle.
- Marcel Grossmann — person — credited by the paper as the influence that pushed Einstein toward a geometric (Gaussian) rather than operational reading of the invariant ds between 1907 and 1916.
- J.L. Synge — person — source of the "buried with honours" characterization of the Equivalence Principle as a historical relic.
- Thomas Hobbes — person — cited (*De Corpore*, 1655) by the paper as a rival, non-Spinozist source for the term "conatus," defined there as infinitesimal motion rather than striving-to-persist.
- Lovelock's theorem — concept — the 1971 uniqueness result (Einstein tensor as the unique divergence-free, second-order, covariant structure up to a cosmological term) that this paper's whole "geometry is derived, not assumed" argument hinges on; worth its own note given how load-bearing it is here.
- Conatus — concept — the Spinozist/Hobbesian philosophical term this paper repurposes for relativistic inertia; worth a concept note disambiguating the Spinozist (striving-to-persist) and Hobbesian (infinitesimal motion) senses the paper itself distinguishes.