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REVIEW 3 major objections 6 minor 78 references

A nonlocal quantum field theory's ultraviolet regulator, applied to the energy-momentum tensor of a point mass, replaces the Schwarzschild curvature singularity with a regular de Sitter core.

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2026-08-02 08:14 UTC pith:YSY6ZMG3

load-bearing objection A clean, honest re-derivation of the Gaussian-sourced regular black hole in a nonlocal-gravity costume; the central claim outruns the equations actually solved. the 3 major comments →

arxiv 2607.13061 v1 pith:YSY6ZMG3 submitted 2026-07-07 physics.gen-ph

De Sitter Cores from Nonlocal Quantum Field Theories

E. J. Thompson This is my paper
classification physics.gen-ph
keywords nonlocal quantum field theoryentire-function regulatoreffective sourceGaussian smearingde Sitter coreregular black holesingularity resolutionstrong energy condition
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper argues that the same entire-function regulator that makes nonlocal quantum field theories finite at loop level also, through the effective-source representation of the Einstein equations, maps a point mass into a smooth Gaussian energy density. Solving the static, spherically symmetric equations then gives a metric that is exactly Schwarzschild far away and becomes the static patch of de Sitter space near the center, with finite curvature at r = 0. If correct, the ultraviolet completion does double duty: it tames quantum loops and removes the classical singularity of the point-mass solution, with the nonlocality scale setting the core's size and curvature. The paper also derives the effective pressures, the violation of the strong energy condition inside the core, horizon structure including an extremal zero-temperature endpoint, and the leading corrections to black-hole entropy.

Core claim

The central claim is that in the static Einstein-form representation of an entire-function regulated nonlocal gravity, the bare point-mass density ρ = M δ^(3)(x) is replaced by an effective Gaussian density ρ_eff(r) = M Λ_G^3/(4π)^{3/2} exp(−Λ_G^2 r^2/4). Integrating this into the Misner–Sharp mass function gives m(r) = M[erf(Λ_G r/2) − (Λ_G r/√π) exp(−Λ_G² r²/4)], which behaves as m(r) ≈ M Λ_G³ r³/(6√π) for small r. Consequently the metric function f(r) = 1 − 2G_N m(r)/r approaches 1 − Λ_eff r²/3 with Λ_eff = G_N M Λ_G³/√π, which is the static patch of de Sitter space; the Ricci scalar at the center is finite, R(0) = 4Λ_eff, and all curvature invariants are regular. The paper emphasizes tha

What carries the argument

The key mechanism is the entire-function regulator F(□/Λ_G²) appearing in the nonlocal kinetic terms, specialized to the exponential form. In the effective-source representation G_μν = κ S_μν, with S_μν = F²(□/Λ_G²) T_μν, the regulator acts as a nonlocal smearing operator. For static configurations it reduces to the heat-kernel exponential exp(∇²/Λ_G²); acting on a delta-function density it produces the Gaussian profile via its Fourier transform. This same operator is what makes loop amplitudes UV-finite, so the paper uses the very same function to dress the source rather than introducing a new smoothing prescription.

Load-bearing premise

The result hinges on the step where the covariant nonlocal regulator is replaced by a flat-space Laplacian exp(∇²/Λ_G²) and the linearized derivation of the effective source is then used to solve the full nonlinear Einstein equations in the strong-field core; if the true covariant form or the neglected higher-curvature terms (R², R_μνR^{μν}, R_μνρσR^{μνρσ}) shift the metric significantly near r ~ 1/Λ_G, the de Sitter core may be an artifact of this truncation.

What would settle it

Solve the full nonlocal gravitational field equations, not the Einstein-form truncation, for a static point source while keeping the covariant d'Alembertian and the higher-curvature terms from the derivative expansion (Eq. 46), then compute the metric at r = 0. If R(0) diverges or the metric does not approach f(r) ≈ 1 − (G_N M Λ_G³/(3√π)) r², the claimed de Sitter core is not a property of the full nonlocal theory. A less expensive check is to estimate the contributions of the αR² + βR_μνR^{μν} + γR_μνρσR^{μνρσ} terms at the extremal coupling G_N M Λ_G ≈ 1.9; if they are order-one, the truncat

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The nonlocality scale Λ_G sets both the cutoff for loop integrals and the size and curvature scale of the black-hole core: the central density is ρ_eff(0) ~ M Λ_G³ and the effective cosmological constant is Λ_eff = G_N M Λ_G³/√π.
  • The strong energy condition is violated only inside the radius r < 2/Λ_G, so the resolution of the singularity does not require violating energy conditions at macroscopic distances.
  • For a given mass M, the horizon structure has three regimes: no horizon for small M, a degenerate extremal horizon at M_0 ≈ 1.904/(G_N Λ_G), and two horizons for larger M; the extremal configuration has zero Hawking temperature.
  • For large black holes the surface-gravity temperature reduces to the Schwarzschild value up to exponentially small corrections, while corrections to the Wald entropy from higher-curvature terms are power-suppressed as O(1/(Λ_G r_h)^2).
  • The Gaussian profile is a concrete realization of a general result: any smooth effective density with finite central value ρ_0 yields a de Sitter core with f(r) ≈ 1 − (8πG_N ρ_0/3) r²; the regulator fixes this central value in terms of M and Λ_G.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the same entire-function regulator is applied to point charges in nonlocal QED, a similar smearing would produce a finite electric field at the origin, potentially making self-energy calculations finite in a way parallel to the gravitational core; the paper does not discuss this.
  • The existence of the de Sitter core suggests that sufficiently light compact objects would have no horizon at all, since the extremal mass is ~1.9/(G_N Λ_G), which could imply a mass gap for black hole formation from nonlocal effects; the paper does not develop this dynamical collapse argument.
  • The zero-temperature extremal endpoint is a natural remnant candidate, but the paper explicitly does not establish its dynamical stability; a testable extension would be to study perturbation spectra of the core to see whether the inner horizon is stable or subject to mass inflation.
  • The effective-source representation is derived from a linearization about Minkowski; a nontrivial extension would be to solve the full covariant nonlocal field equations without replacing □ by ∇² to test whether the de Sitter core survives beyond the weak-field short-distance approximation.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript considers nonlocal quantum field theories with an entire-function regulator F(□/Λ_G²), normalized by F(0)=1, in which the regulated gravitational field equations can be algebraically rewritten in Einstein form G_μν = κ S_μν with S_μν = F²(□/Λ_G²) T_μν. For a static point mass, the paper takes the regulator in its effective-source form as exp(∇²/Λ_G²) with the flat spatial Laplacian, obtaining a Gaussian effective energy density ρ_eff(r) = MΛ_G³/(4π)^{3/2} exp(−Λ_G²r²/4). Solving the Einstein equations with this source yields the exact mass function (17), an asymptotically Schwarzschild exterior, and a core of the form f(r)≈1−Λ_eff r²/3, i.e. a static patch of de Sitter space with Λ_eff = G_N M Λ_G³/√π. The paper derives the effective pressures, shows the strong energy condition is violated for r≲2/Λ_G, analyzes horizons including an extremal configuration (r₀≈3.02/Λ_G, M₀≈1.90/(G_NΛ_G)), and computes the semiclassical surface-gravity temperature. It explicitly disclaims dynamical collapse, stability, and full Wald entropy derivations from the complete nonlocal action.

Significance. If established, the result would provide a concrete UV-motivated regular black-hole model in which the same exponential form factor that renders loop amplitudes finite also smears the classical point-mass source into a regular de Sitter core. The algebraic derivations are clean and checkable: Eq. (11) is a standard Fourier transform, the mass function (17) integrates correctly, the extremal root x₀≈1.5112 is correctly found, and the temperature formula (36) reduces to the Schwarzschild value for large horizons. The paper is also commendably candid about its limitations, stating explicitly that it does not solve the full nonlocal initial-value problem, does not prove stability, and does not derive a first-principles Wald entropy. However, the significance is tempered by two issues: the paper itself concedes that any smooth density with finite positive central value yields a de Sitter core, so the existence of the core is not specific to nonlocality; and, more importantly, the flat-space replacement used to define the smeared source is applied in a strong-curvature regime where the full nonlocal field equations include curvature-squared terms that are never estimated. The central phy

major comments (3)
  1. [§2 (Eq. 6) and §3 (Eqs. 20–23)] The load-bearing step is Eq. (6), where F²(□/Λ_G²)|_static is replaced by exp(∇²/Λ_G²) with the flat spatial Laplacian, described as a 'weak-field short-distance regime.' This replacement is then used to obtain the Gaussian density (11) and mass function (17), which are fed into the fully nonlinear Einstein equations. At the extremal solution quoted in Eq. (44), G_N M Λ_G ≈ 1.904, so the core curvature is R(0) = 4G_N M Λ_G³/√π ≈ 4.3 Λ_G². Thus R/Λ_G² is of order unity, not small, and the flat Laplacian cannot be assumed to approximate the covariant d'Alembertian without a quantitative error estimate. Because the Gaussian profile is the entire basis for the mass function and hence for the de Sitter core, this is not a minor inconsistency: without a bound on the error in replacing □ by the flat Laplacian, the result is a property of the truncated Einstein-form system with a flat-space Gaus
  2. [§5 (Eq. 46)] The derivative expansion of the nonlocal action written in Eq. (46) contains αR² + βR_μνR^{μν} + γR_μνρσR^{μνρσ} terms at order Λ_G^{-2}, with coefficients determined by the regulator. These terms are absent from the solved equations, and the paper uses Eq. (46) only to estimate the Wald entropy correction at the horizon, Eqs. (49)–(50). At the core, where R/Λ_G² is O(1), these curvature-squared terms are a priori as large as the Einstein-Hilbert term, so they can shift the core geometry by order one. The central claim that the regulator 'develops a regular de Sitter core' requires either solving the field equations including these corrections or at least estimating their back-reaction on R(0). As written, the de Sitter core is a solution of the Einstein-form truncation, not a demonstrated solution of the full nonlocal theory whose regulator is invoked.
  3. [§3, paragraph after Eq. (22)] The paper explicitly states that any smooth effective density with finite positive central value gives a de Sitter core at small r, with the role of the regulator being to fix the profile and the relation of Λ_eff to Λ_G. This concession undermines the abstract's phrasing that the solution 'develops a regular de Sitter core' as a consequence of the nonlocal regulator. The existence of the core is generic for regular spherically symmetric sources; the regulator-specific content is the Gaussian shape and Eq. (21). The conclusion should be reworded to make clear that the paper establishes a UV-motivated example of a known mechanism, not that nonlocality itself is the cause of the core. This distinction is important for assessing the novelty and should be reflected in the abstract.
minor comments (6)
  1. [§3, first sentence] Typo: 'I we consider a static point mass' should read 'We consider a static point mass'.
  2. [§4, Eqs. (37)–(38)] Eq. (38) repeats Eq. (37) almost verbatim; the duplication should be removed or one of the equations should be referenced rather than restated.
  3. [§5, Eq. (40)] The notation F(x) for the horizon mass function collides with the regulator F(□/Λ_G²) used throughout. Use a different symbol, e.g. M(x) or Φ(x), to avoid confusion.
  4. [§5, Eq. (50) and surrounding text] Minor grammar: 'So correction is controlled by...' should be 'So the correction is controlled by...'.
  5. [Acknowledgements] Typo: 'refree' should be 'referee'.
  6. [References] Reference [4] and [79] are both Wald's General Relativity textbook; reference [45] and [80] are both Wald's Black Hole Entropy Is the Noether Charge. These should be consolidated to avoid duplication.

Circularity Check

0 steps flagged

No significant circularity: the de Sitter core is computed from the smeared point source, not assumed; self-citations are contextual and not load-bearing.

full rationale

The paper's central derivation is self-contained and not circular. It starts from a specified entire-function regulator with F(0)=1, applies it to a delta-function point mass, obtains the Gaussian effective density (Eq. 11), solves the Einstein equations (Eqs. 14-22), and finds a de Sitter core by a small-r Taylor expansion. The output (ρ_eff, m(r), f(r), Λ_eff) is derived, not fitted or assumed. Λ_G and M are inputs; Λ_eff = G_N M Λ_G^3/√π is a computed combination, not a parameter fitted to the result. The paper explicitly acknowledges that any smooth density with finite central value would give a de Sitter core, but it does not hide this: it states that the regulator's role is to fix the profile and the curvature scale, not the generic existence of the core. That admission is a limitation on the specificity of the claim, not a circularity. The main weaknesses are correctness/validity concerns, not circularity: the flat-space replacement (Eq. 6) is used in a regime where the core curvature is not weak, and the derivative-expansion terms in Eq. (46) are not included in the solved equations. These are outside the circularity rubric. The heavy self-citation in the reference list is contextual; the load-bearing equations in this paper do not reduce to any self-cited uniqueness theorem or prior result.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The paper introduces no new particles, forces, or dimensions. The only new inputs are the free scale ΛG, the ad hoc choice of exponential regulator, and the assertion that the Einstein-form effective-source model is equivalent to the nonlocal dynamics in the regime of interest — an assertion justified only at linear order and contradicted in spirit by the curvature-squared terms of the paper's own action (46). The 'de Sitter core' is a geometric consequence, not an invented entity.

free parameters (2)
  • ΛG (nonlocality scale)
    The UV scale entering every physical output: core width ~2/ΛG, Λ_eff = GN M ΛG³/√π, extremal mass M0 ≈ 1.9/(GN ΛG). No experiment or independent constraint fixes it; all 'predictions' scale with it.
  • α, β, γ (curvature-squared coefficients, Eq. 46)
    Determined by the small-□ expansion of the form factor but not specified numerically; used only for the Wald-correction estimate, not the core.
axioms (5)
  • ad hoc to paper Regulated field equations take the form F^{-2}(□/ΛG²) G_μν = κ T_μν (Eq. 2)
    Proposed dynamics, not derived from an action in the paper; the paper's own action expansion (46) contains curvature-squared terms incompatible with this simple form.
  • ad hoc to paper S_μν = F²(□)T_μν is an equivalent representation of the dynamics beyond linear order, and the flat Laplacian can replace □ for static sources (Eqs. 3–6)
    Justified only via linearization about Minkowski (Eqs. 4–5); applied to full nonlinear solutions. This is the load-bearing regime assumption.
  • domain assumption Exponential regulator F² = exp(∇²/ΛG²) is the relevant choice (Eq. 6)
    Any entire function with F(0)=1 gives UV finiteness; the Gaussian/de Sitter outcome specifically follows from the exponential choice, so the qualitative result selects the regulator rather than following from the whole class.
  • standard math Standard static Schwarzschild-gauge metric and tt-component relation m'(r) = 4πr²ρ_eff (Eqs. 14–16)
    Standard GR toolkit, unproblematic.
  • standard math Surface-gravity Hawking temperature T = f'(r_h)/(4π) (Eq. 34)
    Standard semiclassical result; the paper correctly flags it as an effective diagnostic rather than the full nonlocal temperature.

pith-pipeline@v1.3.0-alltime-deepseek · 10197 in / 27864 out tokens · 252124 ms · 2026-08-02T08:14:45.374951+00:00 · methodology

0 comments
read the original abstract

Nonlocal quantum field theories achieve perturbative ultraviolet finiteness by inserting gauge- and diffeomorphism-covariant entire-function regulators into all kinetic terms. These operators can be viewed as nonlocal smearing maps acting on sources in the Einstein equations. In this paper I show that, when the same entire-function regulator responsible for UV-finite loops is applied to the energy-momentum tensor of a point mass, the resulting static, spherically symmetric solution develops a regular de~Sitter core in place of a curvature singularity.

discussion (0)

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Reference graph

Works this paper leans on

78 extracted references · 4 canonical work pages

  1. [1]

    Relativistic Cosmology. I,

    A. K. Raychaudhuri, “Relativistic Cosmology. I,” Phys. Rev. 98, 1123–1126 (1955)

    1955

  2. [2]

    Gravitational Collapse and Space-Time Singu- larities,

    R. Penrose, “Gravitational Collapse and Space-Time Singu- larities,” Phys. Rev. Lett.14, 57–59 (1965)

    1965

  3. [3]

    S. W. Hawking and G. F. R. Ellis,The Large Scale Struc- ture of Space-Time(Cambridge University Press, Cambridge, 1973)

    1973

  4. [4]

    R. M. Wald,General Relativity(University of Chicago Press, Chicago, 1984)

    1984

  5. [5]

    J. W. Moffat, Finite nonlocal gauge field theory, Phys. Rev. D41, 1177–1184 (1990), doi:10.1103/PhysRevD.41.1177

    doi:10.1103/physrevd.41.1177 1990

  6. [6]

    J. W. Moffat, Ultraviolet complete quantum gravity, Eur. Phys. J. Plus126, 53 (2011), doi:10.1140/epjp/i2011-11053-5, arXiv:1008.2482

    Pith/arXiv arXiv 2011

  7. [7]

    Black holes in an ultraviolet complete quantum gravity,

    L. Modesto, J. W. Moffat, and P. Nicolini, “Black holes in an ultraviolet complete quantum gravity,” Phys. Lett. B695, 397–400 (2011), arXiv:1010.0680 [gr-qc], doi:10.1016/j.physletb.2010.11.046

    Pith/arXiv arXiv 2011

  8. [8]

    E. T. Tomboulis, Superrenormalizable gauge and gravitational theories, arXiv:hep-th/9702146

    Pith/arXiv arXiv

  9. [10]

    Holomorphic Uni- fied Field Theory of Gravity and the Standard Model,

    J. W. Moffat and E. J. Thompson, “Holomorphic Uni- fied Field Theory of Gravity and the Standard Model,” arXiv:2506.19161 [hep-th]. Eur. Phys. J. C85(2025) no.10, 1157 doi:10.1140/epjc/s10052-025-14907-2

    arXiv 2025

  10. [11]

    On the in- variant and geometric structure of the holomor- phic unified field theory,

    J. W. Moffat and E. J. Thompson, “On the in- variant and geometric structure of the holomor- phic unified field theory,” Axioms 2026, 15(1), 43; https://doi.org/10.3390/axioms15010043 arXiv:2510.06282 (2025)

    arXiv 2026

  11. [12]

    Finite nonlocal holomor- phic unified quantum field theory,

    J. W. Moffat and E. J. Thompson, “Finite nonlocal holomor- phic unified quantum field theory,” arXiv:2507.14203 (2025)

    Pith/arXiv arXiv 2025

  12. [13]

    On the Standard Model mass spectrum and interactions in the holomorphic unified field theory,

    J. W. Moffat and E. J. Thompson, “On the Standard Model mass spectrum and interactions in the holomorphic unified field theory,” Eur. Phys. J. Plus 141, 343, arXiv:2508.02747 (2026)

    arXiv 2026

  13. [14]

    Embedding SL(2,C)/Z 2 in complex Riemannian geometry,

    J. W. Moffat and E. J. Thompson, “Embedding SL(2,C)/Z 2 in complex Riemannian geometry,” arXiv:2506.19158 (2025)

    Pith/arXiv arXiv 2025

  14. [15]

    Comment on a “Comment on ‘Standard Model Mass Spectrum and Interactions In The Holomorphic Unified Field Theory

    J. W. Moffat and E. J. Thompson, “Comment on a “Comment on ‘Standard Model Mass Spectrum and Interactions In The Holomorphic Unified Field Theory” arXiv:2508.08510 (2025)

    Pith/arXiv arXiv 2025

  15. [16]

    On Gauge-Invariant Entire-Function Regulators and UV Finiteness in Non- Local Quantum Field Theory,

    J. W. Moffat and E. J. Thompson, “On Gauge-Invariant Entire-Function Regulators and UV Finiteness in Non- Local Quantum Field Theory,” arXiv:2511.11756v7 [hep-th], Annalen der Physik, 538, 4, e70207, https://doi.org/10.1002/andp.70207, (2026). arXiv:2511.11756 [hep-th] (2025)

    Pith/arXiv arXiv 2026

  16. [17]

    On the Complexified Spacetime Manifold Mapping of AdS to dS,

    J. W. Moffat and E. J. Thompson, “On the Complexified Spacetime Manifold Mapping of AdS to dS,” arXiv:2511.11658 [gr-qc] (2025)

    arXiv 2025

  17. [18]

    Nonlocal Gauge Theories,

    N. V. Krasnikov, “Nonlocal Gauge Theories,” Theor. Math. Phys.73, 1184–1190 (1987)

    1987

  18. [19]

    To- wards Singularity- and Ghost-Free Theories of Gravity,

    T. Biswas, E. Gerwick, T. Koivisto, and A. Mazumdar, “To- wards Singularity- and Ghost-Free Theories of Gravity,” Phys. Rev. Lett.108, 031101 (2012), arXiv:1110.5249

    Pith/arXiv arXiv 2012

  19. [20]

    Super-renormalizable Quantum Gravity,

    L. Modesto, “Super-renormalizable Quantum Gravity,” Phys. Rev. D86, 044005 (2012), arXiv:1107.2403

    Pith/arXiv arXiv 2012

  20. [21]

    Nonlocal Quantum Gravity: A Review,

    L. Modesto and L. Rachwa l, “Nonlocal Quantum Gravity: A Review,” Int. J. Mod. Phys. D26, 1730020 (2017)

    2017

  21. [22]

    Renormalization of Gauge Theories,

    C. Becchi, A. Rouet, and R. Stora, “Renormalization of Gauge Theories,” Ann. Phys. (N.Y.)98, 287–321 (1976)

    1976

  22. [23]

    Gauge Invariance in Field Theory and Sta- tistical Physics in Operator Formalism,

    I. V. Tyutin, “Gauge Invariance in Field Theory and Sta- tistical Physics in Operator Formalism,” Lebedev Institute preprint 39 (1975), arXiv:0812.0580

    Pith/arXiv arXiv 1975

  23. [24]

    Top Quark Bound States in Finite and Holomorphic Quantum Field Theories,

    E. J. Thompson, “Top Quark Bound States in Finite and Holomorphic Quantum Field Theories,” arXiv:2507.16831 (2025)

    Pith/arXiv arXiv 2025

  24. [25]

    Z 2-Stable Dark Matter via Broken SU(5) Gauge Bosons,

    E. J. Thompson, “Z 2-Stable Dark Matter via Broken SU(5) Gauge Bosons,” arXiv:2511.19462v1 [hep-ph] (2025)

    arXiv 2025

  25. [26]

    Nonlocal Quantum Field Theory and Quantum Entanglement,

    R. Landry, J. W. Moffat “Nonlocal Quantum Field Theory and Quantum Entanglement,” Eur. Phys. J. Plus 139, 71, arXiv:2309.06576v3 [hep-th] (2024)

    Pith/arXiv arXiv 2024

  26. [27]

    Nonlo- cal regularizations of gauge theories

    D. Evens, J. W. Moffat, G. Kleppe, and R. Woodard “Nonlo- cal regularizations of gauge theories” Phys. Rev. D, 43, 499, (1991)

    1991

  27. [28]

    G. V. Efimov, ”Non-local Quantum Theory of the Scalar Field”, Commun. Math. Phys. 5, 42 (1967)

    1967

  28. [29]

    Nonlocal Yang-Mills

    G. Kleppe and R. P. Woodard, “Nonlocal Yang-Mills”, Nucl. Phys. B388 (1992) 81-112, arXiv:9203016v1 [hep-th]

    1992

  29. [30]

    Ultraviolet Complete Quantum Field Theory and Gauge Invariance

    J. W. Moffat, “Ultraviolet Complete Quantum Field Theory and Gauge Invariance”, arXiv:1104.5706v1 [hep-th]

    Pith/arXiv arXiv

  30. [31]

    Ultraviolet Complete Quantum Field The- ory and Particle Model

    J. W. Moffat, “Ultraviolet Complete Quantum Field The- ory and Particle Model”, Eur. Phys. J. Plus, 134:443 (2019), arXiv:1812.01986v4 [physics.gen-ph]

    Pith/arXiv arXiv 2019

  31. [32]

    Model of Boson and Fermion Particle Masses

    J. W. Moffat, “Model of Boson and Fermion Particle Masses”, Eur. Phys. J. Plus, 136:601 (2021), arXiv:2009.10145v2 [hep- ph]

    Pith/arXiv arXiv 2021

  32. [33]

    Finite Quantum Field Theory and the Renormalization Group

    M. A. Green and J. W. Moffat, “Finite Quantum Field Theory and the Renormalization Group”, Eur. Phys. J. Plus, 136, 919 (2021), arXiv:2012.04487 [hep-th]

    Pith/arXiv arXiv 2021

  33. [34]

    J. W. Moffat, Proc. of the 1st Karl Schwarzschild Meeting on Gravitational Physics, Springer, 2016, p.229, arXiv:1407.2086

    Pith/arXiv arXiv 2016

  34. [35]

    Conformally-flat, nonsingular static metric in infinite deriva- tive gravity,

    G. Buoninfante, A. S. Koshelev, and S. Mazumdar, “Conformally-flat, nonsingular static metric in infinite deriva- tive gravity,” JCAP06, 014 (2018), arXiv:1802.00399

    Pith/arXiv arXiv 2018

  35. [36]

    Mass-gap for Black Hole Formation in Ghost-free Gravity,

    V. P. Frolov, “Mass-gap for Black Hole Formation in Ghost-free Gravity,” Phys. Rev. Lett.115, 051102 (2015), arXiv:1505.00492

    Pith/arXiv arXiv 2015

  36. [37]

    Vacuum Nonsingular Black Hole,

    I. Dymnikova, “Vacuum Nonsingular Black Hole,” Gen. Rel- ativ. Gravit.24, 235–242 (1992)

    1992

  37. [38]

    Non-singular General-Relativistic Gravita- tional Collapse,

    J. M. Bardeen, “Non-singular General-Relativistic Gravita- tional Collapse,” inProceedings of the International Confer- ence GR5, Tbilisi, USSR (1968)

    1968

  38. [39]

    Formation and Evaporation of Nonsingular Black Holes,

    S. A. Hayward, “Formation and Evaporation of Nonsingular Black Holes,” Phys. Rev. Lett.96, 031103 (2006), arXiv:gr- qc/0506126

    arXiv 2006

  39. [40]

    Heat Kernel Expansion: User’s Manual,

    D. V. Vassilevich, “Heat Kernel Expansion: User’s Manual,” Phys. Rept.388, 279–360 (2003), arXiv:hep-th/0306138

    Pith/arXiv arXiv 2003

  40. [41]

    Noncommutative Geometry Inspired Schwarzschild Black Hole,

    P. Nicolini, A. Smailagic, and E. Spallucci, “Noncommutative Geometry Inspired Schwarzschild Black Hole,” Phys. Lett. B 632, 547–551 (2006), arXiv:gr-qc/0510112

    Pith/arXiv arXiv 2006

  41. [42]

    Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse,

    C. W. Misner and D. H. Sharp, “Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse,” Phys. Rev.136, B571–B576 (1964)

    1964

  42. [43]

    On the Gravitational Field of a Mass Point According to Einstein’s Theory,

    K. Schwarzschild, “On the Gravitational Field of a Mass Point According to Einstein’s Theory,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 189–196 (1916). 7

    1916

  43. [44]

    The Thermodynamics of Black Holes,

    R. M. Wald, “The Thermodynamics of Black Holes,”Living Rev. Relativ.4, 6 (2001), doi:10.12942/lrr-2001-6, arXiv:gr- qc/9912119

    arXiv 2001

  44. [46]

    Some Properties of the Noether Charge and a Proposal for Dynamical Black Hole Entropy,

    V. Iyer and R. M. Wald, “Some Properties of the Noether Charge and a Proposal for Dynamical Black Hole Entropy,”Phys. Rev. D50, 846–864 (1994), doi:10.1103/PhysRevD.50.846

    doi:10.1103/physrevd.50.846 1994

  45. [47]

    Spherical black holes with regular center: a re- view of existing models including a recent realization with Gaussian sources,

    S. Ansoldi, “Spherical black holes with regular center: a re- view of existing models including a recent realization with Gaussian sources,” arXiv:0802.0330 [gr-qc]

    Pith/arXiv arXiv

  46. [48]

    Thermodynam- ics of regular black hole,

    Y. S. Myung, Y.-W. Kim, and Y.-J. Park, “Thermodynam- ics of regular black hole,”Gen. Relativ. Gravit.41, 1051– 1067 (2009), doi:10.1007/s10714-008-0690-9, arXiv:0708.3145 [gr-qc]

    Pith/arXiv arXiv 2009

  47. [49]

    Thermodynamics of a class of regular black holes with a generalized un- certainty principle,

    R. V. Maluf and J. C. S. Neves, “Thermodynamics of a class of regular black holes with a generalized un- certainty principle,”Phys. Rev. D97, 104015 (2018), doi:10.1103/PhysRevD.97.104015, arXiv:1801.02661 [gr-qc]

    Pith/arXiv arXiv 2018

  48. [50]

    Regular Rotating Black Holes: A Review,

    R. Torres, “Regular Rotating Black Holes: A Review,” arXiv:2208.12713 [gr-qc]

    arXiv

  49. [51]

    Black Holes and Entropy,

    J. D. Bekenstein, “Black Holes and Entropy,” Phys. Rev. D 7, 2333–2346 (1973)

    1973

  50. [52]

    The Four Laws of Black Hole Mechanics,

    J. M. Bardeen, B. Carter, and S. W. Hawking, “The Four Laws of Black Hole Mechanics,” Commun. Math. Phys.31, 161–170 (1973)

    1973

  51. [53]

    Particle Creation by Black Holes,

    S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys.43, 199–220 (1975)

    1975

  52. [54]

    Past-Future Asymmetry of the Gravitational Field of a Point Particle,

    D. Finkelstein, “Past-Future Asymmetry of the Gravitational Field of a Point Particle,” Phys. Rev.110, 965–967 (1958)

    1958

  53. [55]

    Maximal Extension of Schwarzschild Metric,

    M. D. Kruskal, “Maximal Extension of Schwarzschild Metric,” Phys. Rev.119, 1743–1745 (1960)

    1960

  54. [56]

    Internal Structure of Black Holes,

    E. Poisson and W. Israel, “Internal Structure of Black Holes,” Phys. Rev. D41, 1796–1809 (1990)

    1990

  55. [57]

    Inner Structure of a Charged Black Hole: An Exact Mass-Inflation Solution,

    A. Ori, “Inner Structure of a Charged Black Hole: An Exact Mass-Inflation Solution,” Phys. Rev. Lett.67, 789–792 (1991)

    1991

  56. [58]

    H. S. Reall,Part III: Black Holes(lecture notes, 2014)

    2014

  57. [59]

    Black holes, white holes, and near- horizon physics,

    R. Gaur and M. Visser, “Black holes, white holes, and near- horizon physics,” JHEP05(2024) 172, arXiv:2304.10692 [gr- qc]

    Pith/arXiv arXiv 2024

  58. [60]

    Death of White Holes,

    D. M. Eardley, “Death of White Holes,” Phys. Rev. Lett.33 (1974) 442. doi:10.1103/PhysRevLett.33.442

    work page doi:10.1103/physrevlett.33.442 1974

  59. [61]

    The death of white holes,

    C. Barrab` es, W. Israel, and E. Poisson, “The death of white holes,” Phys. Rev. D47(1993) 2383. doi:10.1103/PhysRevD.47.2383

    work page doi:10.1103/physrevd.47.2383 1993

  60. [62]

    Stability of white holes revisited,

    I. Nikitin, “Stability of white holes revisited,” arXiv:1811.03368 [gr-qc] (2018)

    Pith/arXiv arXiv 2018

  61. [63]

    Do transient white holes have a place in Nature?,

    C. Barcel´ o, R. Carballo-Rubio, L. J. Garay, and G. Jannes, “Do transient white holes have a place in Nature?,” J. Phys. Conf. Ser.600(2015) 012033. doi:10.1088/1742- 6596/600/1/012033

    doi:10.1088/1742- 2015

  62. [64]

    Internal structure of black holes,

    E. Poisson and W. Israel, “Internal structure of black holes,” Phys. Rev. D41(1990) 1796. doi:10.1103/PhysRevD.41.1796

    doi:10.1103/physrevd.41.1796 1990

  63. [65]

    Inner structure of a charged black hole: An ex- act mass-inflation solution,

    A. Ori, “Inner structure of a charged black hole: An ex- act mass-inflation solution,” Phys. Rev. Lett.67(1991) 789. doi:10.1103/PhysRevLett.67.789

    work page doi:10.1103/physrevlett.67.789 1991

  64. [66]

    Planck stars, White Holes, Rem- nants and Planck-mass quasi-particles. The quantum grav- ity phase in black holes’ evolution and its manifestations,

    C. Rovelli and F. Vidotto, “Planck stars, White Holes, Rem- nants and Planck-mass quasi-particles. The quantum grav- ity phase in black holes’ evolution and its manifestations,” arXiv:2407.09584 [gr-qc] (2024)

    Pith/arXiv arXiv 2024

  65. [67]

    White holes as remnants: a surpris- ing scenario for the end of a black hole,

    E. Bianchi, M. Christodoulou, F. D’Ambrosio, H. M. Hag- gard, and C. Rovelli, “White holes as remnants: a surpris- ing scenario for the end of a black hole,” Class. Quant. Grav. 35(2018) 225003, arXiv:1802.04264 [gr-qc]. doi:10.1088/1361- 6382/aae550

    Pith/arXiv arXiv 2018

  66. [68]

    Closer look at white hole remnants,

    A. Barrau, D. B´ enisty, C. G´ erard, M. Lemos, and F. Tupin, “Closer look at white hole remnants,” Phys. Rev. D103 (2021) 043532. doi:10.1103/PhysRevD.103.043532

    work page doi:10.1103/physrevd.103.043532 2021

  67. [69]

    Geometry of the black- to-white hole transition within a single asymptotic region,

    M. Han, C. Rovelli, and F. Soltani, “Geometry of the black- to-white hole transition within a single asymptotic region,” Phys. Rev. D107(2023) 064011, arXiv:2302.03872 [gr-qc]. doi:10.1103/PhysRevD.107.064011

    Pith/arXiv arXiv 2023

  68. [70]

    The Case For Black Hole Remnants: A Review,

    Y. C. Ong, “The Case For Black Hole Remnants: A Review,” arXiv:2412.00322 [gr-qc] (2024)

    Pith/arXiv arXiv 2024

  69. [71]

    Action integrals and partition functions in quantum gravity,

    G. W. Gibbons and S. W. Hawking, “Action integrals and partition functions in quantum gravity,” Phys. Rev. D15, 2752–2756 (1977). doi:10.1103/PhysRevD.15.2752

    doi:10.1103/physrevd.15.2752 1977

  70. [72]

    On black hole entropy,

    T. Jacobson, G. Kang, and R. C. Myers, “On black hole entropy,” Phys. Rev. D49, 6587–6598 (1994). doi:10.1103/PhysRevD.49.6587 arXiv:gr-qc/9312023

    Pith/arXiv arXiv 1994

  71. [73]

    Dirty black holes: Entropy as a sur- face term,

    M. Visser, “Dirty black holes: Entropy as a sur- face term,” Phys. Rev. D48, 5697–5705 (1993). doi:10.1103/PhysRevD.48.5697 arXiv:hep-th/9307194

    Pith/arXiv arXiv 1993

  72. [74]

    Wald entropy for ghost-free, infinite derivative theories of gravity,

    A. Conroy, A. Mazumdar, and A. Teimouri, “Wald entropy for ghost-free, infinite derivative theories of gravity,” Phys. Rev. Lett.114, 201101 (2015). doi:10.1103/PhysRevLett.114.201101 arXiv:1503.05568 [hep-th]

    Pith/arXiv arXiv 2015

  73. [75]

    Defo- cusing of null rays in infinite derivative gravity,

    A. Conroy, A. S. Koshelev, and A. Mazumdar, “Defo- cusing of null rays in infinite derivative gravity,” Phys. Rev. D90, 123525 (2014). doi:10.1103/PhysRevD.90.123525 arXiv:1408.6205 [gr-qc]

    Pith/arXiv arXiv 2014

  74. [76]

    Entropy of black hole in the infinite- derivative gravity,

    Y. S. Myung, “Entropy of black hole in the infinite- derivative gravity,” Phys. Rev. D95, 106003 (2017). doi:10.1103/PhysRevD.95.106003 arXiv:1702.00915 [gr-qc]

    Pith/arXiv arXiv 2017

  75. [77]

    Black holes in non-local gravity,

    L. Buoninfante, B. L. Giacchini, and T. de Paula Netto, “Black holes in non-local gravity,” inHandbook of Quantum Gravity, edited by C. Bambi, L. Modesto, and I. L. Shapiro, Springer, Singapore (2024). doi:10.1007/978-981-99-7681-2 36 arXiv:2211.03497 [gr-qc]

    Pith/arXiv arXiv 2024

  76. [78]

    Nonlocal gravity inDdimensions: Propagators, entropy, and a bouncing cosmology,

    A. Conroy, A. Mazumdar, S. Talaganis, and A. Teimouri, “Nonlocal gravity inDdimensions: Propagators, entropy, and a bouncing cosmology,” Phys. Rev. D92, 124051 (2015). doi:10.1103/PhysRevD.92.124051 arXiv:1509.01247 [hep-th]

    Pith/arXiv arXiv 2015

  77. [79]

    R. M. Wald,General Relativity, University of Chicago Press, Chicago (1984)

    1984

  78. [80]

    Black hole entropy is the Noether charge,

    R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D48, R3427–R3431 (1993). doi:10.1103/PhysRevD.48.R3427 arXiv:gr-qc/9307038

    Pith/arXiv arXiv 1993