On the Principle of Relativity of Inertia in both General and Entangled Relativities

Olivier Minazzoli

REVIEW 3 major objections 5 minor 26 references

On the Principle of Relativity of Inertia in both General and Entangled Relativities

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Entangled Relativity claims that matter-free spacetime is impossible and therefore satisfies Einstein's principle of relativity of inertia.

desk verdict Minazzoli's Machian pitch for Entangled Relativity is philosophically engaging, but the key claim that L_m=0 makes the path integral undefined is asserted, not shown, and the paper's own caveats leave the door open. read the letter →

arxiv 2505.04667 v1 pith:RHVEIRMK submitted 2025-05-07 gr-qc

Olivier Minazzoli This is my paper · ORCID

classification gr-qc PACS 01.65.+g04.50.Kd04.60.Gw

keywords EntangledRelativityMach'sprincipleofinertiapathintegralgravityvacuumspacetimesmatter-curvaturecouplingemergentPlanckconstantgeneralrecovery

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 Entangled Relativity, a theory of gravity defined by a path integral whose phase is $-\frac{i}{2\epsilon^2}\int d^4_g x\, L_m^2/R$, makes matter-free spacetime structurally impossible. Because the action is built from the matter Lagrangian $L_m$ in the numerator and the Ricci scalar $R$ in the denominator, the author claims that $Z_{\mathrm{ER}}$ cannot be defined when $L_m=\varnothing$; in that sense, an empty universe is not even an option. If this is right, Entangled Relativity satisfies Einstein's principle of the relativity of inertia by construction, whereas General Relativity does not, since it admits vacuum solutions such as Minkowski, Schwarzschild, and Kerr. The paper further claims that Entangled Relativity needs only two universal dimensionful constants, reduces to General Relativity and to standard quantum field theory in appropriate limits, and yields $G\propto\hbar$.

What carries the argument

The load-bearing object is the path integral $Z_{\mathrm{ER}}$ with action $S=-\frac{1}{2\epsilon^2}\int d^4_g x\, L_m^2/R$. The matter Lagrangian $L_m$ appears quadratically in the numerator and the Ricci scalar $R$ in the denominator, so the theory has no purely gravitational sector: the same function that defines the dynamics is singular when matter is absent. In the classical limit, the ratio $\kappa=-R/L_m$ becomes an extra scalar gravitational field, governed by $3\kappa^2\Box\kappa^{-2}=\kappa(T-L_m)$, and an equivalent Einstein-dilaton phase $\Theta=\frac{1}{\epsilon^2}\int d^4_g x\, \frac{1}{\kappa}\left(\frac{R}{2\kappa}+L_m\right)$ makes this extra degree of freedom manifest. This machinery is what converts Einstein's demand that no $g_{\mu\nu}$-field exist without matter from a selected principle into a built-in feature.

What would settle it

Compute or regulate $Z_{\mathrm{ER}}$ for $L_m=0$: the integrand becomes $\exp(0)=1$, so a cutoff or measure may give a constant rather than a singularity. If one can construct a consistent $L_m=0$ sector, for example via a limiting procedure from $L_m\to 0$ that keeps the integral finite, then Entangled Relativity does not preclude vacua. Alternatively, find any classical solution of Eqs. (3)--(7) with $L_m=0$ and $R=0$ everywhere; that would be a vacuum solution within the theory.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that the non-linear coupling between matter and curvature in Entangled Relativity precludes the existence of spacetime devoid of matter: the path integral $Z_{\mathrm{ER}}=\int [Dg]\prod_i [Df_i]\exp\!\left(-\frac{i}{2\epsilon^2}\int d^4_g x\, \frac{L_m^2}{R}\right)$ is asserted to be undefined in the case $L_m=\varnothing$, so vacuum spacetimes are not merely absent from the solution set but are not even configurations of the theory. Classically, the theory's field equations involve an extra scalar degree of freedom $\kappa=-R/L_m$; whenever $L_m=T$ on shell, this degree of freedom is not sourced and the theory reproduces General Relativity minimally coupled to matter. When gravity is neglected, the path integral reduces to the standard quantum field theory path integral, with $\kappa\epsilon^2=c\hbar$, so the Planck quantum of action and Newton's constant are not independent inputs.

Load-bearing premise

The load-bearing premise is that the path integral defining Entangled Relativity has no meaning when the matter Lagrangian vanishes everywhere; if a global $L_m=0$ configuration can be assigned a finite or even well-defined value, the theory would permit vacuum spacetimes and the central claim collapses.

Editorial extensions

If this is right

Minkowski, Schwarzschild, Kerr, and other vacuum solutions of General Relativity would have no counterparts in Entangled Relativity, removing the clearest anti-Machian models from the theory.
The theory has only two independent dimensionful constants, the causal constant $c$ and the squared quantum of energy $\epsilon^2$; Newton's constant and Planck's constant emerge through $\kappa\epsilon^2=c\hbar$, so $G\propto\hbar$.
In a universe well approximated by dust plus radiation, $L_m=T$ on shell, the extra degree of freedom $\kappa$ is not sourced, and Entangled Relativity reproduces General Relativity to very good accuracy.
In the weak-gravity limit, the path integral reduces to the standard quantum field theory path integral, so collider physics is recovered.
Existing phenomenological bounds constrain the variation of the emerging $\hbar$: a few percent between the center of dense neutron stars and distant observers, and $\delta\hbar/\hbar\approx 2.5\times 10^{-12}$ between the Sun's surface and a remote observer.

Reading between the lines

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

The no-vacuum conclusion is only as strong as the claim that $Z_{\mathrm{ER}}$ has no $L_m=0$ sector; a regulated evaluation of the path integral with $L_m=0$ could assign it a constant value, which would open a quantum vacuum sector even if classical vacuum solutions remain excluded.
The same matter-curvature ratio structure suggests a family of theories, parametrized by powers of $L_m/R$, that share the no-vacuum feature; testing whether the property survives quantization and is stable under radiative corrections would clarify whether it is a feature of the action or an artifact of the classical limit.
The paper's derivation of $G\propto\hbar$ implies that gravitational and quantum couplings are not separately tunable; a sufficiently precise measurement of a time- or position-dependent fine-structure constant or gravitational constant would be a direct test, though no current experiment has the claimed sensitivity.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

You should know two things about this paper before reading it. First, it is a conference-style proceedings piece, not a full research article. Second, the central claim—that Entangled Relativity precludes vacuum spacetimes because the path integral cannot be defined for L_m=0—is presented as fact but is actually an unproven assertion. The reader's report and the stress test are right to press on this.

What the paper does well: the historical review of Einstein's relativity of inertia, especially the role of the cosmological constant, is clear and useful. The 'dual ontology' critique of GR—that the theory allows inertia to be both relational and absolute depending on the solution—is a legitimate philosophical observation. The paper also gives a compact, honest summary of ER's phenomenology: recovery of GR and QFT in appropriate limits, no free parameters, and the prediction that ℏ varies with the dilaton field κ. Those results come from the author's earlier published work, and citing them is fine.

The soft spot is the foundation. Eq. (2) with L_m=0 yields integrand exp(0)=1 over most metrics; the action is zero, not singular. The paper asserts that Z_ER 'cannot even be defined' in that case, but gives no argument that all possible definitions or regularizations fail. For a formal path integral, undefinedness is not an obvious physical consequence. The author's own Section 3.3 concedes that whether local L_m=0 can occur 'needs further investigation'; a global vacuum is the same question in the limit. The charged black hole example only shows that L_m and R can vanish together at the same rate, not that an exact vacuum is excluded. So the Machian conclusion rests on a premise that is both unproven and internally softened.

The philosophical framing does not rescue the technical gap. If the undefinedness claim fails, the preclusion of vacuum spacetimes fails, and the comparison to Einstein's principle loses its uniqueness.

Who is this for? Readers interested in Mach's principle and alternative gravity will find the historical and conceptual parts worth a look. But do not cite it for the no-vacuum claim. I would send it to a referee, because the claim is consequential and can be tested by a precise argument; a qualified referee could demand a proof or force the author to retract the strong wording. But the paper as it stands does not support its headline conclusion.

Referee Report

3 major / 5 minor

Summary. The paper argues that General Relativity fails Einstein's principle of the relativity of inertia because it admits vacuum spacetimes and thus permits a dual ontology of inertia, as illustrated by Kerr black holes. It then presents Entangled Relativity (ER), defined by the path integral in Eq. (2) with action proportional to ∫ L_m²/R, and claims that because Z_ER cannot be defined when L_m vanishes globally, ER precludes the existence of spacetime devoid of matter, thereby satisfying Mach's principle. The paper further claims that ER recovers General Relativity and standard quantum field theory in appropriate limits, uses only two universal dimensionful constants, has no free parameters, and implies a relation G ∝ ℏ. Section 3.3 discusses a local vacuum limit and concedes that whether L_m=0 can occur locally needs further investigation.

Significance. If the central claim were established, ER would be a conceptually interesting candidate for a Machian theory of gravity, and the paper's historical synthesis of Einstein's writings on Mach's principle is a useful contribution. The paper also makes an explicit, falsifiable prediction—no exact vacuum spacetimes—and offers a concrete alternative action with nontrivial phenomenology. However, the paper's headline result rests entirely on an unproven assertion about the path integral in Eq. (2), and the theory is constructed in such a way that the 'no vacuum' property is essentially built into the chosen Lagrangian. The significance is therefore highly conditional on a rigorous demonstration that the formal integral is genuinely undefined for L_m=0 and that this undefinedness translates into a physical exclusion.

major comments (3)

[Section 3, Eq. (2)] The load-bearing claim that 'Z_ER cannot even be defined if L_m=∅' is asserted without proof and appears questionable. For any metric with R ≠ 0, the exponent in Eq. (2) is 0/R = 0, so the integrand is exp(0) = 1; only at R=0 does one encounter an indeterminate 0/0. A path integral can often be defined on the R ≠ 0 sector or by a limiting/regularization procedure, and the paper gives no argument that all such definitions fail. Since the entire Machian conclusion depends on this assertion, the central claim is unsupported.
[Section 3.3] The paper concedes that whether L_m=0 can occur locally 'needs further investigation,' yet a global vacuum is the all-space version of the same question. The supplied example of a charged black hole demonstrates only that L_m ∝ E² and R can approach zero at the same rate, not that an exact L_m=0 configuration is excluded. The trace equation (8) is expressed in terms of κ = -R/L_m, which is singular when L_m=0, so the claim that the equation 'remains well-behaved' in the (L_m, T) → 0 limit does not establish that exact vacuums are impossible. Thus, the conclusion that ER 'precludes the existence of spacetime devoid of matter' is not established by the arguments presented.
[Section 3.2, Eq. (11)] The claimed recovery of standard quantum field theory rests on setting κ ε² = cℏ, where κ is a dynamical scalar field introduced in Eq. (9). This is a matching condition rather than a derivation, and it implies that ℏ varies with κ; the sense in which 'standard' QFT is recovered therefore requires substantial qualification. Moreover, the derivation is cited to the author's own prior work [17,18] rather than shown, and no independent check is given. This weakens the secondary claim that ER is more economical or parameter-free than GR.

minor comments (5)

[Throughout] There are numerous rendering artifacts such as '/emdash.cyr' and missing spaces (e.g., 'satisify,' 'Interrestingly,' 'stationnary') that should be corrected in a revised manuscript.
[Section 3] The notation 'L_m = ∅' is nonstandard and unclear; since L_m is a function (or functional), the intended statement is 'L_m ≡ 0' or 'L_m = 0 identically.'
[Section 2.4, Eq. (1)] The statement that 'solutions devoid of matter fields cannot exist as long as L_m ≠ ∅' is confusing, because the standard path integral in Eq. (1) with L_m=0 still contains the Einstein-Hilbert term and would still admit classical vacuum solutions in the pure-gravity sector.
[Eq. (2)] The sign of the exponent in Eq. (2) differs from that in Eq. (1); the author should clarify whether this is intentional and whether the imaginary-time convention is consistent.
[References] Reference [18] is listed twice with the same arXiv identifier, and several references lack complete bibliographic details; these should be cleaned up.

Circularity Check

2 steps flagged · score 6.0 of 10

Central 'no vacuum' claim reduces to a domain condition of the defining path integral and is then assumed as a premise in Section 3.3.

self definitional [Section 3, Eq. (2) and following paragraph]
"ZER = ∫ [Dg]∏i [Dfi] exp(−i/2ǫ² ∫ d⁴_g x L_m²(f,g)/R(g)) ... Indeed, Z_ER cannot even be defined if L_m = ∅."

The paper's central claim that Entangled Relativity precludes matter-free spacetime rests directly on the assertion that the defining path integral is undefined when L_m=0. Because Z_ER is the definition of the theory, this assertion is a domain restriction: the theory is only defined for configurations with L_m≠0. The 'prediction' that no vacuum exists is therefore the theory's domain condition restated as a physical result, with no independent derivation. Section 3.3 later concedes the local L_m→0 limit 'needs further investigation,' so the asserted undefinedness is not established as a theorem. The Machian success claim reduces by construction to the chosen form of the Lagrangian ansatz.
other [Section 3.3, first sentence]
"Since Lm ⁄= ∅ in Eq. (2), Entangled Relativity inherently requires matter fields to permeate the entirety of spacetime."

This sentence uses 'Lm ≠ ∅ in Eq. (2)' as an established premise, but Eq. (2) merely writes the action with L_m in the numerator and does not by itself forbid L_m=0. The non-vanishing of L_m is exactly the load-bearing conclusion that was to be demonstrated, so the argument assumes the answer to the question it purports to settle. The immediate follow-up admits that whether L_m can locally vanish 'needs further investigation,' which further undermines the global 'Since' and exposes the question-begging structure.

full rationale

The paper's main original argument is that Entangled Relativity satisfies Einstein's relativity of inertia because it precludes vacuum spacetimes. That preclusion is not independently derived; it is tied entirely to the assertion that the path integral defining the theory, Eq. (2), is undefined when L_m=∅. If accepted, this makes the no-vacuum property a direct consequence of the definition's domain rather than an empirical prediction. Section 3.3 then explicitly assumes 'Lm ≠ ∅ in Eq. (2)' as a premise, which is question-begging. The paper also contains non-circular technical content: the classical field equations, the recovery of General Relativity when L_m=T, and the recovery of quantum field theory in the weak-gravity limit are presented with derivational steps and are supported by prior work, including self-citations. Those self-citations are not themselves circular chains because they refer to calculational results that can be checked independently. However, the central Machian claim, which is the paper's headline result, reduces by construction to a property of the chosen Lagrangian, and the paper's own Section 3.3 caveat leaves the key mathematical assertion unresolved. This warrants a partial circularity score of 6 rather than a higher score, because the GR and QFT recovery sections retain independent content and the circularity is concentrated in the no-vacuum argument.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central claim of the paper rests on an ad hoc choice of the action (L_m²/R) and on cited prior work by the same author. There are no new free parameters fitted to data, but the theory itself is built on a non-standard coupling with no independent justification in this paper.

assumptions (4)

ad hoc to paper The Entangled Relativity action is S = -i/(2ε²) ∫ d⁴_g x L_m²/R (Eq. 2), chosen without derivation from an underlying principle.
The entire no-vacuum claim follows from this specific choice; the paper offers no independent motivation for this form.
domain assumption L_m is the standard model matter Lagrangian with the standard coupling to the metric.
The arguments assume the usual form of L_m and the comma-to-semicolon rule (cited in Section 2.4).
domain assumption The Einstein-dilaton action (Eq. 9) is equivalent to the ER action when L_m ≠ 0.
Equivalence is asserted, not derived in this paper; it is needed for the QFT recovery step in Section 3.2.
ad hoc to paper Recovering standard QFT requires setting κ ε² = c ℏ (Eq. 11).
This matching condition converts ε into the reduced Planck energy and introduces ℏ as a derived quantity; it is a normalization choice, not a first-principles result.

invented entities (1)

κ (dilaton scalar field)
purpose: Represents the extra gravitational degree of freedom in ER; its variation gives κ = -R/L_m as in Eq. (5).
The paper gives no new falsifiable prediction for κ; prior compact-object bounds are cited but not reproduced here, so there is no independent handle within this paper.

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Cite this review

Pith. "Pith review of On the Principle of Relativity of Inertia in both General and Entangled Relativities." pith.science (2026). https://pith.science/paper/RHVEIRMK

@misc{pith2026250504667,
  author       = {Pith},
  title        = {Pith review of: On the Principle of Relativity of Inertia in both General and Entangled Relativities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RHVEIRMK}},
  note         = {Machine review of arXiv:2505.04667}
}

read the original abstract

Entangled Relativity is a novel theory of relativity that offers a more economical approach than General Relativity. It successfully recovers both General Relativity and standard quantum field theory within a specific (yet generic) limit. Furthermore, Entangled Relativity precludes the existence of spacetime devoid of the matter that permeates it. Consequently, I argue that Entangled Relativity is not only preferable from the standpoint of Occam's razor, due to its economical nature, but it also aligns more closely with Einstein's original vision for a satisfactory theory of relativity.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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Einstein A. Prinzipielles zur allgemeinen Rela- tivit¨ atstheorie // Annalen der Physik. /emdash.cyr 1918. /emdash.cyr V. 360, no. 4. /emdash.cyr P. 241–244. /emdash.cyr Translation available at https://einsteinpapers.press.princeton.edu/vol7-trans/49

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Mach’s criticism of Newton and Einstein’s reading of Mach: The stimulating role of two misunderstandi ngs // Mach’s Principle: From Newton’s Bucket to Quantum Gravity / Ed

Borzeszkowski H., Wahsner R. Mach’s criticism of Newton and Einstein’s reading of Mach: The stimulating role of two misunderstandi ngs // Mach’s Principle: From Newton’s Bucket to Quantum Gravity / Ed. 10 by Julian B. Barbour, Herbert Pﬁster. /emdash.cyr Boston University : Birkh¨ aser,

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