REVIEW 4 major objections 4 minor 23 references
Black hole entropy from trace dynamics and non-commutative geometry
T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The paper argues that black hole entropy is thermodynamic entropy of entangled 'atoms of space-time-matter' and equals the standard area law.
desk verdict The paper offers a genuinely new route from trace-dynamics microstates to Bekenstein-Hawking entropy, but the derivation's key step—identifying the post-localisation Hamiltonian with the Euclidean action—is asserted rather than shown, so the result inherits its content from a known identity. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the 'atom of space-time-matter' (STM atom), a Grassmann operator $q=q_B+q_F$ whose bosonic part $q_B$ carries the non-commutative geometry through the Dirac operator $D_B$ and whose fermionic part $q_F$ localises into classical spacetime markers. Its dynamics is set by a trace Lagrangian whose Hamiltonian appears, after localisation, as the Einstein-Hilbert plus point-source action. The calculation is carried by the canonical-ensemble machinery of trace dynamics, namely the Lagrange multiplier $\tilde{\tau}$, the conserved charge associated with global unitary invariance, and the partition function over eigenvalues, together with the spectral-action identity that turns a sum of squared Dirac eigenvalues into $\int\sqrt{g}R$.
What would settle it
If experiments on spontaneous localisation push the collapse time for a nucleon beyond $10^{17}$ seconds, macroscopic superpositions would persist and the mechanism that produces classical black holes from entangled atoms would be ruled out, as the paper itself notes. A direct calculation of the partition function with $N_0\neq N$ or with $\tilde{\tau}$ different from Planck time would also show whether deviations from $\mathrm{Area}/4L_P^2$ appear; without such a calculation the equality is not yet a prediction.
Extended reading notes
Core claim
On the paper's own terms, the discovery is that after spontaneous localisation the coarse-grained trace Hamiltonian of the entangled STM atoms is the classical gravitational action, and the canonical partition function of the same Hamiltonian yields $S_E = \int d^4x\sqrt{g}\,[c^4/(16\pi G)R + c^2\sum_i m_i\delta^3(x-x_0)] = \mathrm{Area}/(4L_P^2)$. The steps are: the inverse-temperature Lagrange multiplier $\tilde{\tau}$ is set to Planck time, the exponent in the partition function becomes unity because the $H_{ni}$ are far below Planck energy, and the number of microstates $N_0$ is taken to equal the number of atoms $N$. The Euclidean action equality for Schwarzschild is then imported from the literature, giving the Bekenstein-Hawking value.
Load-bearing premise
The whole calculation rides on treating the 'It is clear that' step, that the coarse-grained trace Hamiltonian after localisation is the classical Euclidean gravitational action, as exact, and on choosing the inverse-temperature Lagrange multiplier to be the Planck time and the microstate count $N_0$ to equal the atom count $N$; these are imported assumptions, not derived results.
Editorial extensions
If this is right
- If the central claim holds, the Bekenstein-Hawking area law is a counting statement: $A/4L_P^2$ counts entangled STM-atom microstates, giving an explicit statistical origin for black hole entropy.
- Spontaneous localisation and black hole evaporation become two sides of one fluctuation-dissipation process: collapse drags entangled atoms out of equilibrium and Hawking radiation returns them toward equilibrium.
- Classical general relativity emerges as a thermodynamic condensate of the underlying non-commutative dynamics, so the gravitational field itself is not an object to be quantised.
- A sufficiently massive entangled system necessarily forms a black hole in this framework, with a sharp transition at Planck mass between a classical black-hole phase and a quantum phase.
Reading between the lines
- If the imported equality (the Euclidean action equals area/4) is replaced by a derivation within the same framework, including the Euclidean time-temperature compactification, the argument would close a loop the paper leaves open; a reader should look for that derivation as the natural next step.
- The derivation's freedom in choosing $N_0=N$ suggests a testable consistency condition: computing $S_E$ for $N_0\neq N$ should produce corrections that remain compatible with observed black hole thermodynamics; otherwise the equality is fine-tuned.
- Because the framework currently omits non-gravitational forces, its prediction that every massive collapse produces a black hole is likely an artifact of that truncation; adding gauge fields may produce ordinary macroscopic objects whose entropy is not area-limited.
- One could test the broader idea on other horizons: applying the same trace-dynamics ensemble to de Sitter or Rindler horizons would predict whether their entropies also equal the corresponding Euclidean actions within the same assumptions.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a theory of non-commutative matter-gravity, combining trace dynamics with Connes' non-commutative geometry, and uses it to argue that a Schwarzschild black hole is formed by spontaneous localisation of many entangled STM (space-time-matter) atoms. The central new claim is that the statistical entropy of the microstates of these atoms, computed via a canonical ensemble in trace dynamics, equals the classical Euclidean gravitational action, which in turn equals the Bekenstein-Hawking entropy Area/4L_P^2 (Eqs. 59-61). The paper also sketches a fluctuation-dissipation argument relating spontaneous localisation to Hawking radiation.
Significance. If the derivation were sound, this would be a significant contribution: it would provide a concrete microstate origin for black hole entropy in a candidate quantum gravity framework and would connect spontaneous collapse models to gravitational thermodynamics. The paper is creative and builds on established formalisms (Adler's trace dynamics, Connes' spectral action), and it explicitly makes falsifiable predictions. However, the central entropy calculation in Section V rests on several unproved identifications and normalisation choices; as it stands, Eq. (60) is essentially a restatement of the assumed Hamiltonian-to-action correspondence rather than a derived result. The paper itself concedes that the assumptions require future rigorous justification.
major comments (4)
- [Section V, Eq. (50)] The load-bearing identification H → (1/τ_Pl) ∫ d⁴x√g [c⁴/(16πG)R + c²Σ m_i δ³(x−x₀)] is asserted without derivation. The preceding section derives the classical limit of the trace Lagrangian/action (Eqs. 40-47), but the trace Hamiltonian in Eq. (49) is obtained from that Lagrangian by a Legendre transform; its classical limit is not shown to coincide with the action. For a relativistic particle the Hamiltonian is energy, not action, and no operator-level argument is supplied to bridge this gap. The phrase "It is clear that..." is the only support for Eq. (50), and this equation is the foundation of the entropy result: Eq. (60) is exactly Eq. (50) after the normalisation choices in Eqs. (58)-(59).
- [Section V, Eqs. (58)-(59)] The statistical derivation is not a genuine microstate count. The Lagrange multiplier τtilde is set to the Planck time by hand, the Boltzmann exponent exp(−τtilde H) is approximated to unity, and the number of microstates N₀ is assumed to equal the number of atoms N. None of these steps is derived; they are choices that force the final coefficient. Moreover, N₀ cancels between the partition function and the entropy expression, so the result is independent of the actual number of microstates. The entropy is therefore not computed from a counting of states but is injected through the normalisation and the identification in Eq. (50).
- [Section V, Eqs. (59)-(61)] The argument is circular in the sense that the Euclidean action is both the input and the output. The statistical mechanics is bypassed: after setting τtilde = τ_Pl and approximating the exponential to unity, the entropy reduces to the classical action times constants. The independent input is the cited Euclidean-action result in Eq. (61), which is then identified with Bekenstein-Hawking entropy. The paper does not show that the trace-dynamics partition function, evaluated from a well-defined set of microstates, produces the black-hole action; it assumes the correspondence in Eq. (50) and then reads off the action.
- [Section V and Section VI] The concluding remarks state that the calculation involves "certain assumptions, which we hope to address rigorously in future work." This admission is accurate, but the assumptions are not local technicalities: they include the key step Eq. (50), the arbitrary choice of τtilde, and the normalisation N = N₀. Because these are the core of the derivation and are not justified in the manuscript, the claimed microstate derivation of Bekenstein-Hawking entropy is not established. The paper is a proposal or an estimate, not a derivation, despite the abstract's claim to "show" the result.
minor comments (4)
- [General] There are numerous typographical errors and awkward phrasings (e.g., "ant-self-adjoint" in Section II, "Spont aneous" in the Section II heading, "finit" in an earlier version). The paper would benefit from careful proofreading.
- [Section IV, Eqs. (40)-(47)] The transition from the trace action to the Einstein-Hilbert plus point-particle action involves several plausible but not fully specified assumptions, such as replacing 1/L³ by a delta function and identifying the trace over eigenvalues with an integral over √g R. A more precise derivation of the coefficients would strengthen the paper.
- [Section V, Eq. (53)] The introduction of the fugacity-like term −ηN in the partition function is not explained or used subsequently; it appears only in the definition of Z and then drops out. This should be clarified or removed.
- [Section V, Eq. (61)] The paper cites a specific reference for the Euclidean action of a Schwarzschild black hole, but the result is standard and more commonly attributed to Gibbons and Hawking. The citation should be checked for accuracy and completeness.
Circularity Check
The black-hole entropy result reduces to the asserted H→action equality (Eq. 50); normalisation choices and the cited action-area identity supply the Area/4 coefficient.
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self definitional
[Section V, Eqs. (50) and (59)–(60)]
"It is clear that upon spontaneous localisation of a large number of STM atoms, their net trace Hamiltonian will reach the same classical limit as the trace Lagrangian; the latter limit having been shown above. Thus we conclude the important result that, upon spontaneous localisation, H−→ 1/τP l ∫ d4x√g [ c4/16πGR + c2 ∑ i miδ3(x− x0) ] (50)"
The entropy is then computed from this Hamiltonian: Eq. (58) gives SEn ≈ τPl N0^{-1} Σ_i Hni, and Eq. (59) replaces Σ_i Hni by the Euclidean action. After assuming N=N0, Eq. (60) makes the total entropy exactly the integral in Eq. (50). Thus the statistical output is the action inserted at Eq. (50) by assertion; no microstate count is performed. The preceding section derived the classical limit of the trace Lagrangian, not of the Legendre-transformed canonical Hamiltonian, and no operator-level proof of Eq. (50) is supplied.
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other
[Section V, Eqs. (53), (58)–(60)]
"Since the Hni are much smaller than 1/τ˜_Pl the exponent above can be set to unity to a very good approximation, and we get the contribution to the entropy from the n-th atom to be, with N0 being the number of states, SEn =τP lN−1 0 ∑ i Hni ... if we assume that N equals N0 to a very high accuracy, the black hole entropy is then given by SE = ∫ d4x√g [ c4/16πGR + c2 ∑ i miδ3(x− x0) ] (60)"
The partition-function evaluation is forced by normalisation choices rather than by a computed density of states. The paper sets τ˜=τPl, approximates exp(−τ˜H)≈1, and then postulates N=N0 to cancel the 1/N0 prefactor in Eq. (59). Each of these choices is needed to make the final coefficient exactly one; if τ˜ were not τPl or N/N0 were not unity, S_E would not equal the action. The Planck-temperature identification and the equality N=N0 are imported inputs, not consequences of the STM action principle.
1 more flagged steps
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renaming known result
[Section V, Eqs. (60)–(61)]
"Next, we recall that we are working in a Euclidean 4-d space-time. Now, in this case, it is known from the literature that for a Schwarzschild black hole sourced by a point mass source at the centre, the Euclidean gravitational action is precisely equal to one-fourth the black-hole area: see Eqn. (2.7) in [19] ∫ d4x√g [ c4/16πGR ] = Area/4L2 P (61)"
Once Eq. (60) has made S_E equal to the Euclidean action by construction, Eq. (61) is imported from the literature to identify that action with Area/4L_P^2. The final Bekenstein-Hawking value is therefore the known action-area identity restated after renaming the quantity as microstate entropy. The claimed microstate origin is not an independent derivation of Area/4; it reduces to the external identity once Eq. (50) is granted.
full rationale
The paper's derivation chain is: write the STM partition function Z = ∫dµ exp(−Tr λ˜C − τ˜H); approximate −τ˜ ∂ log Z/∂τ˜ ≈ τ˜⟨H⟩; set τ˜=τPl and exp(−τ˜H)≈1; replace H by (1/τPl) times the classical Euclidean action via Eq. (50); assume N=N0; then S_E equals the Euclidean action; finally cite [19] to set that action to Area/4L_P^2. The only step that could supply a microstate origin is Eq. (50), but it is asserted with the phrase 'It is clear that' and is not derived from the canonical Hamiltonian. The Legendre transform from the trace Lagrangian to H is not carried out, and in ordinary mechanics H is energy, not action. Hence Eq. (60) is the assumed input re-labelled as entropy, not a computed microstate count. The paper itself concedes this: 'This has involved certain assumptions, which we hope to address rigorously in future work.' No load-bearing self-citation chain is present; the circularity is internal to the derivation. The result is not completely circular, because the trace-dynamics statistical framework and the spectral-action identity (43) could in principle support a genuine derivation if Eq. (50) were proven. But as written, the central claim reduces by construction, so the score is 8.
Assumptions & free parameters
free parameters (3)
- Inverse-temperature Lagrange multiplier τtilde =
τtilde = τ_Pl (Planck time)
- Microstate count N0 =
N0 = N (number of STM atoms)
- Action constant C0 =
C0 = ħ
assumptions (6)
- domain assumption The statistical thermodynamics of trace dynamics remains valid after coupling to gravity via Connes time.
- domain assumption The spectral action with χ(u)=u and the heat-kernel expansion reproduces the Einstein-Hilbert action after spontaneous localisation.
- ad hoc to paper Spontaneous localisation localises the fermionic part of each STM atom to size L_I=L³/L_P² and creates a Schwarzschild black hole.
- ad hoc to paper The trace Hamiltonian's post-localisation classical limit is the Euclidean gravitational action with point sources (Eq. 50).
- domain assumption Euclidean time is compactified with period 8πGM/c³ so the Euclidean action equals Area/4L_P².
- standard math Wodzicki residue and heat-kernel coefficient relate the slash integral of D^{-2} to the scalar curvature.
invented entities (1)
-
Atoms of space-time-matter (STM atoms)
Cite this review
Pith. "Pith review of Black hole entropy from trace dynamics and non-commutative geometry." pith.science (2026). https://pith.science/paper/K27BWJW3
@misc{pith2026190902434,
author = {Pith},
title = {Pith review of: Black hole entropy from trace dynamics and non-commutative geometry},
year = {2026},
howpublished = {\url{https://pith.science/paper/K27BWJW3}},
note = {Machine review of arXiv:1909.02434}
}
abstract
Spontaneous localisation is a falsifiable, phenomenological, mechanism for explaining the absence of macroscopic position superpositions, currently being tested for in the laboratory. The theory of trace dynamics provides a possible theoretical origin for spontaneous localisation. We have recently proposed how to employ non-commutative geometry to include gravity in trace dynamics, and suggested the emergence of classical space-time geometry via spontaneous localisation. In our theory, which we call non-commutative matter gravity, a black hole arises from the spontaneous localisation of an entangled state of a large number of `atoms of space-time-matter [STM]'. Prior to localisation, the non-commutative curvature of an STM atom is described by the spectral action of non-commutative geometry. By using the techniques of statistical thermodynamics from trace dynamics, we show that the gravitational entropy of a Schwarzschild black hole results from the microstates of the entangled STM atoms and is given (subject to certain assumptions) by the classical Euclidean gravitational action. This action, in turn, equals the Bekenstein-Hawking entropy (Area/$4{L_P}^2$) of the black hole. We argue that spontaneous localisation is related to black-hole evaporation through the fluctuation-dissipation theorem.
Figures
Reference graph
Works this paper leans on
-
[19]
The euclidean gravitational action as black hole entropy, singularities, and space-time voids,
Carlos Castro, “The euclidean gravitational action as black hole entropy, singularities, and space-time voids,” J. Math. Phys. 49, 042501 DOI: 10.1142/S0217751X08039669 (2008)
-
[1]
Unified dynamics for microscopic and macroscopic systems,
Gian Carlo. Ghirardi, Alberto Rimini, and Tullio Weber, “Unified dynamics for microscopic and macroscopic systems,” Phys. Rev. D 34, 470–491 (1986)
work page 1986
-
[2]
Gian Carlo Ghirardi, Philip Pearle, and Alberto Rimini, “Markov processes in hilbert space and continuous spontaneous localization of systems of identical particles,” Phys. Rev. A 42, 78–89 (1990)
work page 1990
-
[3]
Combining stochastic dynamical state-vector reduction with spontaneous lo- calization,
Philip Pearle, “Combining stochastic dynamical state-vector reduction with spontaneous lo- calization,” Phys. Rev. A 39, 2277–2289 (1989)
work page 1989
-
[4]
Angelo Bassi and Gian Carlo Ghirardi, “Dynamical reduction models,” Phys. Rep. 379, 257– 426 (2003)
work page 2003
-
[5]
Models of wave function collapse, underlying theories, and experimental tests,
A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, “Models of wave function collapse, underlying theories, and experimental tests,” Rev. Mod. Phys. 85, 471 (2013)
work page 2013
-
[6]
Opto-mechanical tests of collapse models
Matteo Carlesso and Mauro Paternostro, “Opto-mechanical tests of collapse models,” , arXiv:1906.11041 (2019)
work page Pith review arXiv 2019
-
[7]
The problem of time and the problem of quantum measurement,
T. P. Singh, “The problem of time and the problem of quantum measurement,” in Re-thinking time at the interface of physics and philosophy , (arXiv:1210.81110), edited by T. Filk and A. von Muller (Berlin-Heidelberg:Springer, 2015)
Show all 23 references
-
[8]
[arxiv:gr-qc/0510042],
T. P. Singh, “[arxiv:gr-qc/0510042],” Bulg. J. Phys. 33, 217 (2006 [arXiv:gr-qc/0510042])
2006 arXiv
-
[9]
Proposal for a new qantum theory of gravity iii: Equations for quantum gravity, and the origin of spontaneous localisation,
Palemkota Maithresh and Tejinder P. Singh, “Proposal for a new qantum theory of gravity iii: Equations for quantum gravity, and the origin of spontaneous localisation,” arXiv:1908.04309 Zeitschrift fur Naturforschung A 10.1515/zna-2019-0267 (2019)
2019 arXiv
-
[10]
Adler, Quantum theory as an emergent phenomenon (Cambridge University Press, 2004)
Stephen L. Adler, Quantum theory as an emergent phenomenon (Cambridge University Press, 2004)
2004
-
[11]
Generalized quantum dynamics,
S. L. Adler, “Generalized quantum dynamics,” Nucl. Phys. B 415, 195 (1994)
1994
-
[12]
Pearle, Stud
P. Pearle, Stud. Hist. Philos. Mod. Phys. 36, 716 (2005 [arXiv:quant-ph/0602078])
2005 arXiv
-
[13]
Visions in mathematics - gafa 2000 special volume, part ii,
A. Connes, “Visions in mathematics - gafa 2000 special volume, part ii,” (Springer, 2000) Chap. Non-commutative geometry 2000, pp. 481 Eds. N. Alon and J. Bourgain and A. Connes and M. Gromov and V. Milman, arXiv:math/0011193
2000 arXiv
-
[14]
Eigenvalues as dynamical variables,
Giovanni Landi, “Eigenvalues as dynamical variables,” Lect. Notes Phys. 596, 299 (2002 gr- 32 qc/9906044)
2002
-
[15]
Space-time from collapse of the wave-function,
Tejinder P. Singh, “Space-time from collapse of the wave-function,” Zeitschrift f¨ ur Natur- forschung A 74, 147 arXiv.org:1809.03441 (2019)
2019 arXiv
-
[16]
Gravitation and the noise needed in objective reduction models,
S. L. Adler, “Gravitation and the noise needed in objective reduction models,” (arXiv:1401.0353 [gr-qc] 2014)
2014 arXiv
-
[17]
Connecting the dots: Mott for emulsions, collapse models, colored noise, frame dependence of measurements, evasion of the “free will theorem
Stephen L. Adler, “Connecting the dots: Mott for emulsions, collapse models, colored noise, frame dependence of measurements, evasion of the “free will theorem”,” Foundations of Physics 48, 1557 arXiv:1807.11450v3 (2018)
2018 arXiv
-
[18]
General relativity in terms of dirac eigenvalues,
Giovanni Landi and Carlo Rovelli, “General relativity in terms of dirac eigenvalues,” Phys. Rev. Lett. 78, 3051 arXiv:gr–qc/9612034 (1997)
1997
-
[20]
Proposal for a new qantum theory of gravity: Karolyhazy uncertainty relation, planck scale foam, and holography,
Tejinder P. Singh, “Proposal for a new qantum theory of gravity: Karolyhazy uncertainty relation, planck scale foam, and holography,” arXiv:1910.06350 [gr-qc] (2019)
2019 arXiv
-
[21]
Dark energy as a large scale quantum gravitational phenomenon,
Tejinder P. Singh, “Dark energy as a large scale quantum gravitational phenomenon,” arXiv:1911.02955 [gr-qc] (2019)
2019 arXiv
-
[22]
Spontaneous quantum gravity,
Tejinder P. Singh, “Spontaneous quantum gravity,” arXiv:1912.03266 (2019)
2019 arXiv
-
[23]
From quantum foundations to quantum gravity,
Tejinder P. Singh, “From quantum foundations to quantum gravity,” arXiv:1909.06340 [quant-ph] (2019). 33
2019 arXiv
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