Pith. sign in

REVIEW 5 major objections 5 minor 34 references

A Holographic, Hydrodynamic Model of a Schwarzschild Black Hole

T0 review · 5 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A revised metric turns a Schwarzschild black hole's interior into a degenerate fermionic fluid with equation of state $P=\rho/9$.

desk verdict Novel signum-metric and fermionic-fluid construction, but the headline EOS P=ρ/9 is calibrated to Hawking radiation pressure rather than derived; a transparent, speculative paper that deserves a serious referee even though the quantitative claim is not supported as a prediction. read the letter →

arxiv 2505.00732 v1 pith:UK4563KG submitted 2025-04-29 physics.gen-ph

Noah M. MacKay This is my paper · ORCID
classification physics.gen-ph
keywords blackholeinteriorholographicprincipleSchwarzschilddegenerateFermigasequationofstateHawkingradiationPaulipressure2-brane
open problems Quantum Gravity
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 the interior of a Schwarzschild black hole is not an empty void but a holographic, degenerate fermionic medium that is pressure-supported from the horizon. The author draws on the 81% neutrino content of Hawking radiation to motivate a fermionic interior, and revises the Schwarzschild metric with a signum exponent so the region inside the horizon is a linear potential well while the exterior remains Schwarzschild. Running that geometry through a total Einstein-Hilbert plus boundary action and a Hamiltonian analysis yields a Schrödinger-like equation whose hydrodynamic reading gives the equation of state $P=\rho/9$ between internal pressure and black-hole mass density. Ideal-gas counting then predicts roughly 2.8 particles per horizon quantum area, with Fermi energy far above the Hawking temperature, so the medium is genuinely degenerate; the author acknowledges that two fine-tuning coefficients in the chain are fixed by hand rather than derived. If the model is right, a black hole has a structured interior, the horizon rather than the center is the attracting singularity, and Hawking radiation can be viewed as the release of information piled up on the horizon.

What carries the argument

The load-bearing mechanism is the signum-modified metric (3), $\Theta(r)=\mathrm{sgn}(1-r_S/r)$, which sews the Schwarzschild exterior to a linear-well interior so that the horizon, not the center, is the attractive singularity. Around this metric the paper builds a Hamiltonian system: the total action (Einstein-Hilbert plus Gibbons-Hawking-York boundary term) is rewritten with $\nabla_\alpha\nabla^\alpha$ as the kinetic operator and the extrinsic curvature trace as a potential, producing the Schrödinger-like equation (16). The hydrodynamic reading then runs on two balancing identities: $K^2\propto\rho$ ties geometry to mass density, and equating the fluid pressure with Hawking radiation pressure fixes $a=1/12$, which converts $P=\frac43 a\rho$ into the central $P=\rho/9$.

What would settle it

Compute the Einstein tensor of the metric (3) at an interior point $r<r_S$ and check whether it equals a perfect-fluid stress-energy tensor with trace $(-\rho+3P)=0$. If the geometry cannot be sourced by such a fluid, or if the wave equation (30) does not follow from the Hamiltonian, the equation of state and the particle count are unsupported.

Watch

Extended reading notes

Core claim

The central claim is that a non-rotating, chargeless black hole is better described by the signum-modified metric (3), $\Theta(r)=\mathrm{sgn}(1-r_S/r)$, which makes the horizon an attractive $n=4$ black 2-brane and gives the interior a linear-well geometry. From the Einstein-Hilbert action with a Gibbons-Hawking-York boundary term, the author treats the Ricci scalar as a kinetic generator and $K^2\propto\rho$ as a potential, obtaining the Hamiltonian $H_{\rm BH}\simeq (8/\kappa^2)\phi^*(\tfrac12\nabla_\alpha\nabla^\alpha+\tfrac12\kappa^2 a\rho)\phi$ and the Schrödinger-like equation $\nabla_\alpha\nabla^\alpha\phi=8\pi G(\varepsilon-4a\rho)\phi$. Assuming a traceless energy-momentum tensor and a massless, radiation-like medium, and setting the coefficient $a$ by equating internal pressure with Hawking radiation pressure in equilibrium, the paper obtains $P=\rho/9$, a total particle count $N_{\rm tot}=8\pi M^2/(9m_P^2)$, and a Fermi energy $E_F=\hbar/(2GM)(9\pi M/(20m_P))^{2/3}$ that exceeds $k_B T_H$ for massive black holes. The author concludes that the interior is a degenerate fermionic fluid whose Pauli pressure offsets the horizon's inward pull, with Hawking evaporation described as horizon contraction beginning at the Schwarzschild radius.

Load-bearing premise

The load-bearing premise is that a Schwarzschild black hole's interior really is the signum-modified linear-well metric (3), a geometry adopted by construction rather than derived from general relativity or an independent quantum-gravity model.

Editorial extensions

If this is right

  • A Schwarzschild black hole would contain a real, positive-pressure medium rather than a vacuum, with the horizon acting as a confining boundary and the geometric center playing no special role.
  • The pressure $P=\rho/9$ would offset the horizon's inward pull, making the black hole a Pauli-supported, degenerate object whose collapse is resisted internally.
  • Hawking radiation would be naturally read as the escape of matter piled up on the horizon, giving a concrete mechanism by which information could leave the hole.
  • Because the Fermi energy exceeds the Hawking temperature for massive black holes, the medium is degenerate, suggesting Pauli blocking could regulate evaporation rates instead of a simple thermal law.
  • The mass-shell free-fall calculation recovers the Schwarzschild radius as the onset of horizon contraction, so the model is consistent with the standard start of Hawking evaporation.

Reading between the lines

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

  • The equation of state $P=\rho/9$ implies the medium's adiabatic sound speed is $c_s^2=dP/d\rho=1/9$, or one-third the speed of light; computing the full dispersion relation of the interior waves (42) would test that prediction directly.
  • Treating the black hole as a degenerate-fluid object invites a Tolman-Oppenheimer-Volkoff-style structural calculation inside the horizon, which would turn the particle count and pressure into a mass-radius relation with observable evaporation scaling.
  • The same signum-metric construction could be attempted for a rotating black hole; the neutrino-motivated $P=\rho/9$ law would then predict spin-dependent pressure gradients near the horizon that this paper does not explore.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper proposes a holographic hydrodynamic model of the Schwarzschild black hole interior, modeling it as a degenerate, neutrino-dominated fermionic fluid mirrored by the horizon. A signum-modified metric (Eq. 3) is introduced to combine a linear-well interior with the standard Schwarzschild exterior. From the Einstein-Hilbert action with a Gibbons-Hawking-York boundary term, the author constructs a Hamiltonian and a Schrödinger-like equation, and then derives an equation of state P=ρ/9 by matching the resulting pressure to Hawking radiation pressure. The paper also estimates the particle number and Fermi energy of the interior medium, discusses spherical wave solutions of the field equation, and proposes a mass-shell free-fall model of horizon evaporation. The text is clearly organized and the author explicitly acknowledges the preliminary nature of the model and the ad hoc character of its parameters.

Significance. If the central claims were sound, the model would provide a concrete, internally consistent picture of a black hole interior as a holographic degenerate fluid, with potentially testable consequences such as a particle count of roughly 2.8 per horizon quantum area and a Fermi energy far exceeding the Hawking temperature. The paper is honest about its limitations and connects to several established ideas (AdS/CFT, Gibbons-Hawking-York boundary terms, Hawking radiation composition). However, as written the derivation chain contains a circular step and several unsupported premises, so the headline equation of state P=ρ/9 is not established as a prediction. The significance of the contribution is therefore currently limited to a suggestive framework rather than a validated quantitative model.

major comments (5)
  1. [§1.1, Eq. (3)] The modified metric with Θ(r)=sgn(1−r_S/r) is introduced by construction, not derived. The paper does not show that this metric satisfies the Einstein field equations (with or without matter), nor does it follow from the Tangherlini brane metric of Eq. (1) in any explicit way: for n=4, p=2 the brane is the full 3+1 spacetime and the reduction to a spherical 2-brane is not a standard construction. The metric is non-analytic across the horizon, and junction conditions are not analyzed. Since every subsequent result (Hamiltonian, wave equation, equation of state) presupposes this geometry, the central quantitative claims inherit an unvalidated premise.
  2. [§2.2, Eq. (10)] The inference K ∝ sqrt(r_S) from ∂_t K=0 is not valid. The text writes sqrt(1−r/r_S) ∂_t K = 0, which for r<r_S implies ∂_t K = 0, i.e., only that K is time-independent. No radial dependence or proportionality to r_S follows from this condition. The subsequent substitution K^2 = κ^2 a ρ is a new ansatz, and the identification of ρ with M/V_BH is made later. Thus the potential-like term in the Hamiltonian is not derived from the geometry as claimed.
  3. [§3.1, Eqs. (18)-(25)] The derivation of the equation of state P=ρ/9 is circular. The coefficient a is introduced as a free parameter in Eq. (18), and b is fixed by equating the reversible entropy to the Bekenstein-Hawking entropy (Eqs. 21-22). The value a=1/12 is then obtained by equating the holographic pressure (4/3)aρ to the Hawking radiation pressure (Eq. 23). Consequently, Eq. (25) restates the input assumption that the interior pressure equals the radiation pressure at the Hawking temperature; the numerical coefficient 1/9 is not an output of the holographic-hydrodynamic model. In addition, the steps ε=4aρ (Eq. 17) and ε=3P are assumed rather than derived, so the chain from the Schrödinger-like equation to the equation of state is not predictive.
  4. [§2.3, Eqs. (15)-(16)] The passage from the Hamiltonian (15) to the Schrödinger-like equation (16) is not a derivation. It relies on imposing the normalization ⟨φ|φ⟩=1 and on introducing a second copy of φ so that φ*φ acts as a density, but the physical status of φ is never specified. The identification of ε as the energy density eigenvalue is then effectively a definition. The resemblance of Eq. (16) to the contracted Einstein equations is formal, and the subsequent hydrodynamic relations are built on this unestablished equation.
  5. [§4.2, Eqs. (45)-(48)] The parameter j=5/3 is a fitted correction, as the text explicitly states: 'j is a necessary ad hoc correction' and 'its ad hoc nature suggests future work to derive it from first principles.' The mass-shell equation (45) is constructed with j precisely so that the quadratic (47) yields r0=2GM0, so the recovery of the Schwarzschild radius as the onset of evaporation is not an independent check of the model. This, together with the circular derivation of a=1/12, means the model has no numerically predictive content beyond its input assumptions.
minor comments (5)
  1. [§4.1.1, after Eq. (37)] The acronym 'WBK' should be WKB (Wentzel-Kramers-Brillouin), and the same misspelling appears in the paragraph preceding Eq. (37).
  2. [§3.2] The word 'consitituents' is a typo for 'constituents'; similar typographical errors include 'spactime' in §2.1, 'temporial' in §2.2, and 'retieves' in §4.2.
  3. [Eq. (10)] The notation K[m−1]∝sqrt(r_S) is dimensionally awkward: K has dimensions of inverse length while sqrt(r_S) has dimensions of square-root of length, so the proportionality constant must carry nontrivial units; the text does not comment on this.
  4. [References [26], [31]] The paper relies on the author's own arXiv preprints (Refs. [26] and [31]) for the Langevin framework and the Hawking-particle force; these are not peer-reviewed and should be cited in published form if they exist.
  5. [§5, Conclusion] The conclusion states that the model offers 'a potential resolution to the information paradox,' but no information-theoretic mechanism is developed in the paper; this claim goes beyond what the analysis supports.

Circularity Check

3 steps flagged · score 7.0 of 10

The headline EOS P=ρ/9 is calibrated, not predicted: a and j are free parameters fixed by matching known Hawking quantities, and the recovered Schwarzschild radius restates the fitting condition.

  1. fitted input called prediction [Section 3.1, Eqs. (18), (23)-(25)]
    "The goal is to define the fine-tuning parameter a from Eq. (18), by equating the holographic equation of state with the radiation pressure. ... Defining ρ = M/V_BH and explicitly writing V_BH = 4πr_S^3/3, the fine-tuning parameter is calculated to be a simple number: a = 1/12; this quantifies the holographic equation of state (Eq. [18]) as follows, recovering mass density ρ: P = 1/9 ρ."

    The coefficient a is introduced in Eq. (11) as a free fine-tuning parameter with no model-determined value, and Eq. (18) gives P=(4/3)aρ. Section 3.1 then declares that the goal is to fix a by equating this pressure to the Hawking-radiation pressure, with b itself fixed by forcing Eq. (19) to equal the Bekenstein-Hawking entropy. Solving for a therefore imposes P_holographic = P_Hawking as an input; the headline P=ρ/9 is just that matching condition rewritten via ρ=M/V_BH. It is a calibration, not a derivation.

  2. fitted input called prediction [Section 4.2, Eqs. (45)-(48)]
    "Without j in Eq. (45), the model fails to recover the Schwarzschild radius as the onset for BH evaporation, while it is otherwise retrievable with a zero pressure gradient and omitting j. Thus, j is a necessary ad hoc correction, acting as a holographic enhancer to calibrate the force and account for the medium’s pressure, with its value determined algebraically. ... This confirms that BH horizon contraction via Hawking radiation begins at the initial Schwarzschild radius."

    The free parameter j is introduced solely to calibrate the Hawking force against the model's pressure gradient. Eq. (46) determines j=5/3 from the radius-dependent equation, and Eqs. (47)-(48) then solve for r0 and recover the Schwarzschild radius 2GM0. Because j was chosen precisely so that this zero sits at 2GM0, the subsequent confirmation is a restatement of the fitting condition, not an independent check. The author's own text labels j ad hoc.

1 more flagged steps
  1. self citation load bearing [Section 4.2, Eqs. (45)-(46), Refs. [26] and [31]]
    "Timelike Hawking particles, with effective mass m_H = ℏ/(4πGM) [31], excert a force equal but opposite to free-fall: F = ℏ/(16πG^2M^2) [26]."

    Both cited sources are prior arXiv papers by the same author, and they provide the only quantitative inputs (m_H and F) used in the mass-shell model. The consistency check then calibrates the additional parameter j to recover the Schwarzschild radius, so the section's validation is anchored in the author's own prior work plus a fitted constant rather than in an external or independently checkable result. This is secondary to the EOS fit but non-independent.

full rationale

The central quantitative claim P=ρ/9 is not predicted from the holographic Hamiltonian; the parameter a is explicitly introduced as a 'fine-tuning parameter' and then solved for by equating Eq. (18) with the Hawking-radiation pressure. The same pattern repeats in the mass-shell section, where j=5/3 is an 'ad hoc correction' chosen so the model reproduces r0=2GM0, after which reproduction is offered as confirmation. The conclusion itself concedes this: 'the reliance on fine-tuning parameters (e.g. a=1/12 and j=5/3) highlights the model’s preliminary nature, necessitating future work to derive these from first principles rather than from ad hoc assertions.' Additional inputs such as the traceless energy-momentum tensor, ε=3P, and ρ=M/V_BH are assumed rather than derived, and the signum metric of Eq. (3) is introduced 'by design.' These are model assumptions rather than circular steps by themselves, but they reinforce that the headline relation is an input-matching result. Because the main EOS and the mass-shell consistency check both reduce to calibration, the circularity score is 7; the remaining physical framework (metric ansatz, Hamiltonian reduction, approximate wave solutions) provides some independent speculative content, so the score is not 10.

Assumptions & free parameters 2 free parameters · 6 assumptions · 2 invented entities

The central model rests on a handful of assumptions: an ad hoc signum metric, a stationary-bulk simplification of the boundary term, a traceless massless fluid, thermal equilibrium with Hawking radiation, and an ideal-gas application to the interior. The free parameters a and j are fitted to reproduce known results, so the ledger shows that the derivations consume roughly as much input as they output.

free parameters (2)
  • a = 1/12
    Couples extrinsic curvature to mass density via K^2 = κ^2 a ρ (Section 2.2); the value is set in Eqs. (23)-(24) by equating the holographic pressure to the Hawking radiation pressure, so it is fitted to reproduce the expected radiation formula.
  • j = 5/3
    An ad hoc holographic enhancer in the mass-shell free-fall equation (Eq. 45); it is solved algebraically in Eq. (46) so that the onset of horizon contraction occurs at r = 2GM (Eq. 48), which is imposed rather than predicted.
assumptions (6)
  • ad hoc to paper The modified metric (Eq. 3) with Θ(r)=sgn(1-r_S/r) is the physical spacetime for the Schwarzschild black hole exterior and interior.
    Introduced by construction in Section 1.1; it is not derived from field equations or known brane solutions, but all later derivations use it.
  • domain assumption The boundary term simplification assumes a stationary bulk with n^μ ∇_μ K = 0, i.e., K independent of the foliation.
    Used in Eqs. (6)-(10) to turn the Gibbons-Hawking-York term into a volume integral and set K ∝ sqrt(r_S); the step from ∂_t K=0 to K ∝ sqrt(r_S) is not shown.
  • domain assumption The energy-momentum tensor of the holographic medium is traceless, appropriate to a perfect fluid of massless constituents.
    Used in Section 3 (Eq. 17 and following) to set ε=4aρ and P=ε/3; the fluid is treated as massless despite neutrinos being called 'very light'.
  • domain assumption The interior holographic medium is in thermal equilibrium with the Hawking radiation, so T_R = T_BH and the holographic pressure equals the radiation pressure.
    Used in Section 3.1, Eqs. (19)-(23), to solve for a; this equilibrium is assumed, and the equality of interior and exterior pressure is not derived from a microscopic model.
  • domain assumption The interior can be modeled as a degenerate ideal gas with 81% neutrinos and 19% massless bosons.
    Motivated by Page's Hawking radiation composition [25], but treating the Hawking emission composition as direct knowledge of the interior fluid composition is an assumption.
  • standard math Lemma 3.32 of Chow and Knopf gives R_μν = -1/2 ∇^α∇_α g_μν + lower-order terms in harmonic coordinates.
    Cited in Section 2.2 and used to promote R to the Laplace-Beltrami kinetic operator in the Lagrangian.
invented entities (2)
  • black 2-brane horizon in n=4 dimensions
    purpose: The event horizon is treated as a p=2 brane whose interior is a linear-well holographic fluid, making the horizon the attractor rather than the center.
    Introduced in Section 1.1 via the Tangherlini-like metric (Eq. 1) restricted to p=2, n=4; no independent evidence or standard four-dimensional realization is provided.
  • holographic fermionic medium (degenerate neutrino fluid)
    purpose: A dense fluid filling the interior that provides Pauli degeneracy pressure, balances the horizon grip, and yields the equation of state P=ρ/9.
    Posited as the holographic bulk; its existence and composition are inferred from Hawking radiation species, with no direct observational handle.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Holographic, Hydrodynamic Model of a Schwarzschild Black Hole." pith.science (2026). https://pith.science/paper/UK4563KG

@misc{pith2026250500732,
  author       = {Pith},
  title        = {Pith review of: A Holographic, Hydrodynamic Model of a Schwarzschild Black Hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UK4563KG}},
  note         = {Machine review of arXiv:2505.00732}
}
abstract

Schwarzschild (non-rotating and chargeless) black holes are classically understood to be voids of extreme gravitation. In this study, we propose a holographic model for their interiors, envisioning them instead as a hydrodynamic medium. Motivated by the neutrino composition in Hawking radiation (81%), we model the interior as a degenerate fluid, mirrored by the horizon via AdS/CFT duality. A Schwarzschild metric revised with a signum function as the power of the ratio $r_S/r$ distinguishes interior linear-well dynamics from exterior Schwarzschild geometry, rimming the horizon with singularity-like gravitational attraction. A Hamiltonian analysis of the total action leads to formulating a Schr\"odinger-like equation, which offers an alternative representation as the contracted Einstein field equations under a holographic-hydrodynamic framework. This eventually yields an equation of state between holographic pressure and black hole mass density: $P=\rho/9$. Ideal gas analysis reveals a total particle count of $\sim2.8$ times the number of horizon quantum areas, with the Fermi energy far exceeding the Hawking thermal energy, ensuring degeneracy. As our discussion, we explore the mass shell free-fall model of a BH with holographic pressure, and dissect the spherical wave solutions to the Schr\"odinger-like equation describing confined interior fields and freely propagating exterior quanta (i.e., Hawking radiation).

Figures

Figures reproduced from arXiv: 2505.00732 by the authors.

Figure 1
Figure 1. The spacetime manifold for a 2-brane Schwarzschild black hole as depicted by [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. A density plot of the spacetime manifold as depicted by [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Radial log-linear plot of the scalar field solutions (blue solid) with the manifold divergence (purple [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 10 canonical work pages

  1. [1]

    Schwarzschild, Sitzungsber

    K. Schwarzschild, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1916, 189-196 (1916) [arXiv:physics/9905030 [physics]]

    arXiv 1916

  2. [2]

    Schwarzschild, Sitzungsber

    K. Schwarzschild, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1916, 424-434 (1916) [arXiv:physics/9912033 [physics.hist-ph]]

    arXiv 1916

  3. [3]

    Penrose, Phys

    R. Penrose, Phys. Rev. Lett. 14, 57-59 (1965) doi:10.1103/PhysRevLett.14.57

    doi:10.1103/physrevlett.14.57 1965

  4. [4]

    S. W. Hawking, Nature 248, 30-31 (1974) doi:10.1038/248030a0

    doi:10.1038/248030a0 1974

  5. [5]

    S. W. Hawking, Commun. Math. Phys. 43, 199-220 (1975) [erratum: Commun. Math. Phys. 46, 206 (1976)] doi:10.1007/BF02345020

    doi:10.1007/bf02345020 1975

  6. [6]

    J. W. York, Jr., Phys. Rev. Lett. 28, 1082-1085 (1972) doi:10.1103/PhysRevLett.28.1082

    doi:10.1103/physrevlett.28.1082 1972

  7. [7]

    G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752-2756 (1977) doi:10.1103/PhysRevD.15.2752

    doi:10.1103/physrevd.15.2752 1977

  8. [8]

    Kovtun, D

    P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005) doi:10.1103/PhysRevLett.94.111601 [arXiv:hep-th/0405231 [hep-th]]

    arXiv 2005

Show all 34 references
  1. [9]

    Lunin and S

    O. Lunin and S. D. Mathur, Nucl. Phys. B 623, 342-394 (2002) doi:10.1016/S0550-3213(01)00620-4 [arXiv:hep-th/0109154 [hep-th]]

    2002 arXiv

  2. [10]

    Lunin and S

    O. Lunin and S. D. Mathur, Phys. Rev. Lett. 88, 211303 (2002) doi:10.1103/PhysRevLett.88.211303 [arXiv:hep-th/0202072 [hep-th]]

    2002 arXiv

  3. [11]

    S. D. Mathur, Int. J. Mod. Phys. D 11, 1537-1540 (2002) doi:10.1142/S0218271802002852 [arXiv:hep- th/0205192 [hep-th]]

    2002

  4. [12]

    S. D. Mathur, Fortsch. Phys. 53, 793-827 (2005) doi:10.1002/prop.200410203 [arXiv:hep-th/0502050 [hep-th]]

    2005 arXiv

  5. [13]

    S. D. Mathur, Lect. Notes Phys. 769, 3-48 (2009) doi:10.1007/978-3-540-88460-6_1 [arXiv:0803.2030 [hep-th]]

    2009 arXiv

  6. [14]

    S. D. Mathur, [arXiv:0810.4525 [hep-th]]

    arXiv

  7. [15]

    S. D. Mathur, Class. Quant. Grav. 26, 224001 (2009) doi:10.1088/0264-9381/26/22/224001 [arXiv:0909.1038 [hep-th]]

    2009 arXiv

  8. [16]

    S. D. Mathur, Annals Phys. 327, 2760-2793 (2012) doi:10.1016/j.aop.2012.05.001 [arXiv:1205.0776 [hep- th]]. 14

    2012 arXiv

  9. [17]

    F. R. Tangherlini, Nuovo Cim. 27, 636-651 (1963) doi:10.1007/BF02784569

    1963 doi

  10. [18]

    N. A. Obers, Lect. Notes Phys. 769, 211-258 (2009) doi:10.1007/978-3-540-88460-6_6 [arXiv:0802.0519 [hep-th]]

    2009 arXiv

  11. [19]

    Dvali and C

    G. Dvali and C. Gomez, Fortsch. Phys. 61, 742-767 (2013) doi:10.1002/prop.201300001 [arXiv:1112.3359 [hep-th]]

    2013 arXiv

  12. [20]

    Dvali and C

    G. Dvali and C. Gomez, Phys. Lett. B 716, 240-242 (2012) doi:10.1016/j.physletb.2012.08.019 [arXiv:1203.3372 [hep-th]]

    2012 arXiv

  13. [21]

    Dvali and C

    G. Dvali and C. Gomez, Phys. Lett. B 719, 419-423 (2013) doi:10.1016/j.physletb.2013.01.020 [arXiv:1203.6575 [hep-th]]

    2013 arXiv

  14. [22]

    Dvali and C

    G. Dvali and C. Gomez, Eur. Phys. J. C 74, 2752 (2014) doi:10.1140/epjc/s10052-014-2752-3

    2014 doi

  15. [23]

    S. K. Manikandan and A. N. Jordan, Phys. Rev. D 98, no.12, 124043 (2018) doi:10.1103/PhysRevD.98.124043 [arXiv:1811.03209 [gr-qc]]

    2018 arXiv

  16. [24]

    Hayden and J

    P. Hayden and J. Preskill, JHEP 09, 120 (2007) doi:10.1088/1126-6708/2007/09/120 [arXiv:0708.4025 [hep-th]]

    2007 arXiv

  17. [25]

    D. N. Page, Phys. Rev. D 13, 198-206 (1976) doi:10.1103/PhysRevD.13.198

    1976 doi

  18. [26]

    N. M. MacKay, [arXiv:2503.18086 [gr-qc]]

    arXiv

  19. [27]

    R. L. Arnowitt, S. Deser and C. W. Misner, Phys. Rev. 116, 1322-1330 (1959) doi:10.1103/PhysRev.116.1322

    1959 doi

  20. [28]

    B. Chow, D. Knopf. The Ricci Flow: An Introduction (Providence, R.I.: American Mathematical Society,

  21. [29]

    Li, [arXiv:2411.15164 [physics.gen-ph]]

    B. Li, [arXiv:2411.15164 [physics.gen-ph]]

    arXiv

  22. [30]

    Wang and J

    H. Wang and J. Wang, Adv. High Energy Phys. 2020, 9102461 (2020) doi:10.1155/2020/9102461 [arXiv:2006.00433 [hep-th]]

    2020 arXiv

  23. [31]

    N. M. MacKay, [arXiv:2502.08816 [gr-qc]]

  24. [32]

    N. D. Birrell and P. C. W. Davies, Cambridge University Press, 1982, ISBN 978-0-511-62263-2, 978-0- 521-27858-4 doi:10.1017/CBO9780511622632

    1982 doi

  25. [33]

    Earthquakes, Starquakes, and Solar Flares

    P. Bak. “Earthquakes, Starquakes, and Solar Flares. ” In:How Nature Works(New York, N.Y.: Copernicus,

  26. [1996]

    doi:10.1007/978-1-4757-5426-1_5 15

    doi

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.