arxiv: 2605.03428 · v2 · submitted 2026-05-05 · 🌀 gr-qc

Recognition: 3 theorem links

· Lean Theorem

Families of regular spacetimes and energy conditions

Zi-Liang Wang , Emmanuele Battista

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:23 UTC · model grok-4.3

classification 🌀 gr-qc

keywords regular black holesweak energy conditionKretschmann scalarspherically symmetric metricsenergy density profilesBardeen solutionHayward solutionDymnikova solution

0 comments

The pith

A systematic method constructs static spherically symmetric regular spacetimes in general relativity that satisfy the weak energy condition from assumptions on energy density and bounded Kretschmann scalar.

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

The paper develops a method to build regular black hole solutions that avoid singularities while obeying the weak energy condition. It starts from assumptions about how the energy density behaves and requires that the Kretschmann scalar remains finite everywhere. This framework unifies known models like Bardeen and Hayward black holes and generates new families expressed with special functions. The approach also examines when horizons and photon spheres form and how to match to an exterior Schwarzschild region. If successful, it provides a broader catalog of physically plausible regular geometries.

Core claim

By classifying admissible density profiles according to their complexity, the authors recover the Bardeen, Hayward, and Dymnikova models as special cases within a unified framework and derive new closed-form regular geometries involving hypergeometric or incomplete Gamma functions, many of which simplify to algebraic, logarithmic, arctangent, or exponential expressions, all while satisfying the weak energy condition.

What carries the argument

Classification of admissible matter energy density profiles together with the requirement of a bounded Kretschmann scalar, which ensures finiteness of all curvature invariants and completeness of causal geodesics.

Load-bearing premise

Physically reasonable assumptions on the matter energy density together with boundedness of the Kretschmann scalar are sufficient to guarantee regularity and geodesic completeness for the spacetimes considered.

What would settle it

A static spherically symmetric spacetime with bounded Kretschmann scalar that contains a curvature singularity, has incomplete causal geodesics, or violates the weak energy condition despite satisfying the density assumptions.

read the original abstract

We present a systematic method for constructing static, spherically symmetric regular spacetimes in general relativity satisfying the weak energy condition. Our approach relies on physically reasonable assumptions on the matter energy density, together with the boundedness of the Kretschmann scalar. The latter property ensures the finiteness of all curvature invariants and, for the configurations considered, is equivalent to the completeness of causal geodesics. By classifying admissible density profiles according to their complexity, we recover well-known regular black hole solutions such as the Bardeen, Hayward, and Dymnikova models, which are thus naturally embedded in a unified and broader framework. Within this setting, we also derive closed-form expressions for several new families of regular geometries involving hypergeometric or incomplete Gamma functions, which in many cases reduce to elementary functions including algebraic, logarithmic, arctangent, and exponential forms. The emergence of horizons and photon spheres, as well as matching conditions to a Schwarzschild exterior, are also investigated.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A clean classification of regular static spherical spacetimes from density profiles that recovers the standard examples and adds several new closed-form families.

read the letter

The paper's core contribution is a method that starts from assumptions on the matter density profile plus bounded Kretschmann scalar, then derives the metric functions for static spherical symmetry. This recovers the Bardeen, Hayward, and Dymnikova solutions as special cases and produces new families whose curvature functions involve hypergeometric or incomplete gamma expressions, many of which simplify to algebraic, log, arctan, or exponential forms. The authors also track horizon and photon-sphere locations plus the conditions for smooth matching to an exterior Schwarzschild region. That is useful bookkeeping within the regular-black-hole literature and gives explicit, checkable expressions rather than purely numerical or ad-hoc constructions. The weak-energy-condition compliance follows directly once the density is chosen to be positive and decreasing in the right way, and the bounded-curvature assumption is used to guarantee regularity at the center. The limitation is the narrow scope: everything stays inside classical GR and static spherical symmetry, so no rotation, no dynamics, and no quantum corrections. The claim that bounded Kretschmann is equivalent to geodesic completeness for these metrics is stated but would benefit from a short explicit verification for the new families rather than a general appeal. The new solutions are mathematically well-defined but still rest on the same kind of density ansatz that earlier papers used; they enlarge the catalog without changing the underlying physical picture. This work is aimed at researchers who build or classify exact regular solutions in GR. Anyone already working on energy-condition-compliant metrics or looking for closed-form examples will find the unified treatment and the explicit new cases worth reading. The derivations are concrete enough that a referee can check them directly, so the paper deserves peer review.

Referee Report

0 major / 3 minor

Summary. The paper presents a systematic method for constructing static, spherically symmetric regular spacetimes in general relativity that satisfy the weak energy condition. It classifies admissible matter energy density profiles under assumptions of physical reasonableness together with boundedness of the Kretschmann scalar (asserted to guarantee finiteness of all curvature invariants and, for these configurations, completeness of causal geodesics). The classification recovers the Bardeen, Hayward, and Dymnikova solutions as special cases and generates new closed-form families expressed via hypergeometric or incomplete Gamma functions (many reducing to elementary algebraic, logarithmic, arctangent, or exponential forms). The work also examines the emergence of horizons and photon spheres as well as matching conditions to an exterior Schwarzschild geometry.

Significance. If the density-profile classification indeed yields metrics satisfying the weak energy condition everywhere while maintaining bounded curvature, the manuscript supplies a unified, physically motivated framework that embeds known regular black-hole models and produces explicit new examples. This could facilitate systematic exploration of non-singular geometries and their observable features such as photon spheres.

minor comments (3)

[Abstract and §2] Abstract and §2: the asserted equivalence between bounded Kretschmann scalar and geodesic completeness is stated without a self-contained sketch or reference; a brief derivation or citation in the main text would clarify the scope of the claim for the static spherical case.
[§4] §4 (new families): while reductions to elementary functions are noted, explicit mapping of which density-profile parameters produce algebraic versus hypergeometric forms would improve readability and allow readers to reproduce the simpler cases without re-deriving the integrals.
[Figures and Table 1] Figure captions and Table 1: several plots of curvature invariants and energy-condition functions lack axis labels or parameter values; adding these would make the numerical checks of the weak energy condition easier to verify.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation for minor revision. No specific major comments were listed in the report, so there are no individual points requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs families of regular static spherically symmetric spacetimes by starting from external assumptions on admissible matter energy-density profiles (physically reasonable, satisfying WEC) together with the boundedness of the Kretschmann scalar. It then classifies these profiles by complexity, recovers the Bardeen/Hayward/Dymnikova solutions as special cases, and obtains new closed-form families involving hypergeometric or incomplete Gamma functions. The bounded-Kretschmann condition is asserted to guarantee finiteness of all curvature invariants and (for the metrics considered) geodesic completeness; this is presented as a property of the chosen class rather than a derived prediction. No load-bearing self-citations, fitted parameters renamed as predictions, self-definitional steps, or ansatzes smuggled via prior work appear in the derivation chain. The output geometries are generated from the input assumptions without reducing back to them by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on two domain assumptions extracted from the abstract: reasonable energy density profiles that satisfy the weak energy condition and bounded Kretschmann scalar implying geodesic completeness. No free parameters or invented entities are identifiable from the abstract alone.

axioms (2)

domain assumption Physically reasonable assumptions on the matter energy density ensure the weak energy condition holds
Invoked to guarantee positive energy density and satisfaction of WEC for the constructed spacetimes
domain assumption Boundedness of the Kretschmann scalar ensures finiteness of all curvature invariants and completeness of causal geodesics
Stated as equivalent for the static spherically symmetric configurations considered

pith-pipeline@v0.9.0 · 5461 in / 1481 out tokens · 45317 ms · 2026-05-12T04:23:08.601680+00:00 · methodology

Review history (3 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our approach relies on physically reasonable assumptions on the matter energy density, together with the boundedness of the Kretschmann scalar... By classifying admissible density profiles according to their complexity, we recover well-known regular black hole solutions such as the Bardeen, Hayward, and Dymnikova models
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

K = 4(X² + 4Y² + Z²)... the Kretschmann scalar alone is sufficient to fully determine the curvature regularity
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery / embed_strictMono unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

ρ(r) = ρ₀ [1 + (r/rd)^n]^(−ℓ) ... C = 5

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

137 extracted references · 137 canonical work pages · 11 internal anchors

[1]

Lifshitz and I

E. Lifshitz and I. Khalatnikov,Investigations in relativistic cosmology,Advances in Physics 12(1963) 185

work page 1963
[2]

Hawking and G.F.R

S.W. Hawking and G.F.R. Ellis,The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2, 2023), 10.1017/9781009253161

work page doi:10.1017/9781009253161 2023
[3]

Hawking,THE PATH INTEGRAL APPROACH TO QUANTUM GRAVITY, in General Relativity: An Einstein Centenary Survey, pp

S.W. Hawking,THE PATH INTEGRAL APPROACH TO QUANTUM GRAVITY, in General Relativity: An Einstein Centenary Survey, pp. 746–789 (1980)

work page 1980
[4]

Wald,General Relativity, Chicago Univ

R.M. Wald,General Relativity, Chicago Univ. Pr., Chicago, USA (1984), 10.7208/chicago/9780226870373.001.0001

work page doi:10.7208/chicago/9780226870373.001.0001 1984
[5]

Penrose,Gravitational collapse and space-time singularities,Phys

R. Penrose,Gravitational collapse and space-time singularities,Phys. Rev. Lett.14(1965) 57

work page 1965
[6]

Hawking and R

S.W. Hawking and R. Penrose,The Singularities of gravitational collapse and cosmology, Proc. Roy. Soc. Lond. A314(1970) 529

work page 1970
[7]

Senovilla,Singularity Theorems and Their Consequences,Gen

J.M.M. Senovilla,Singularity Theorems and Their Consequences,Gen. Rel. Grav.30(1998) 701 [1801.04912]

work page arXiv 1998
[8]

Mohajan,Singularities in global hyperbolic space-time manifold,Apex Journal of Advanced Sciences & Engineering5(2016) 41

H.K. Mohajan,Singularities in global hyperbolic space-time manifold,Apex Journal of Advanced Sciences & Engineering5(2016) 41

work page 2016
[9]

Curiel, Einstein Stud

E. Curiel,A Primer on Energy Conditions,Einstein Stud.13(2017) 43 [1405.0403]

work page arXiv 2017
[10]

Martin-Moruno and M

P. Martin-Moruno and M. Visser,Classical and semi-classical energy conditions,Fundam. Theor. Phys.189(2017) 193 [1702.05915]

work page arXiv 2017
[11]

Morris and K.S

M.S. Morris and K.S. Thorne,Wormholes in space-time and their use for interstellar travel: A tool for teaching general relativity,Am. J. Phys.56(1988) 395

work page 1988
[12]

Visser,Lorentzian Wormholes

M. Visser,Lorentzian Wormholes. From Einstein to Hawking, Springer, New York, USA (1995)

work page 1995
[13]

Bambi, ed.,Regular Black Holes

C. Bambi, ed.,Regular Black Holes. Towards a New Paradigm of Gravitational Collapse, Springer Series in Astrophysics and Cosmology, Springer (2023), 10.1007/978-981-99-1596-5, [2307.13249]

work page doi:10.1007/978-981-99-1596-5 2023
[14]

C. Lan, H. Yang, Y. Guo and Y.-G. Miao,Regular Black Holes: A Short Topic Review,Int. J. Theor. Phys.62(2023) 202 [2303.11696]

work page arXiv 2023
[15]

Myszkowski, M

M. Myszkowski, M. Damia Paciarini, F. Sannino and V. Vellucci,Regular spacetimes in the effective metric description,Eur. Phys. J. C86(2026) 346 [2506.12620]

work page arXiv 2026
[16]

Carballo-Rubio et al.,Towards a non-singular paradigm of black hole physics,JCAP05(2025) 003, [arXiv:2501.05505]

R. Carballo-Rubio et al.,Towards a non-singular paradigm of black hole physics,JCAP05 (2025) 003 [2501.05505]

work page arXiv 2025
[17]

Bardeen,Non-singular general-relativistic gravitational collapse, inProc

J.M. Bardeen,Non-singular general-relativistic gravitational collapse, inProc. Int. Conf. GR5, Tbilisi, vol. 174, p. 174, sn, 1968

work page 1968
[18]

Dymnikova,Vacuum nonsingular black hole,Gen

I. Dymnikova,Vacuum nonsingular black hole,Gen. Rel. Grav.24(1992) 235

work page 1992
[19]

Formation and evaporation of non-singular black holes

S.A. Hayward,Formation and evaporation of regular black holes,Phys. Rev. Lett.96(2006) 031103 [gr-qc/0506126]

work page Pith review arXiv 2006
[20]

Frolov and A

V.P. Frolov and A. Zelnikov,Quantum radiation from an evaporating nonsingular black hole, Phys. Rev. D95(2017) 124028 [1704.03043]

work page arXiv 2017
[21]

H. Liu, X. Liao and Y. Zhang,Breaking the degeneracy among regular black holes with gravitational lensing,2603.20596

work page arXiv
[22]

Singh, B.K

B. Singh, B.K. Singh and D.V. Singh,Thermodynamics, phase structure of Bardeen massive black hole in Gauss-Bonnet gravity,Int. J. Geom. Meth. Mod. Phys.20(2023) 2350125. – 33 –

work page 2023
[23]

Zeng and Y

H. Zeng and Y. Meng,Images from disk and spherical accretions of Bardeen black hole surrounded by perfect fluid dark matter,Phys. Lett. B876(2026) 140434 [2512.05147]

work page arXiv 2026
[24]

Vertogradov and A

V. Vertogradov and A. Rincon,Energy extraction and evolution of regular black holes: The case of Bardeen spacetime,Phys. Dark Univ.50(2025) 102066 [2508.14489]

work page arXiv 2025
[25]

Konoplya and A

R.A. Konoplya and A. Zhidenko,Dymnikova black hole from an infinite tower of higher-curvature corrections,Phys. Lett. B856(2024) 138945 [2404.09063]

work page arXiv 2024
[26]

Alshammari, S

M. Alshammari, S. Alshammari, S. Khan and M.M. Al-sawalha,Einasto-core generalization of the Dymnikova regular black hole metric,Eur. Phys. J. C85(2025) 1402

work page 2025
[27]

Vertogradov,Gravitational collapse and formation of regular black holes: Dymnikova, Hayward, and beyond,Eur

V. Vertogradov,Gravitational collapse and formation of regular black holes: Dymnikova, Hayward, and beyond,Eur. Phys. J. C85(2025) 839 [2504.19292]

work page arXiv 2025
[28]

Fathi, M

M. Fathi, M. Molina and J.R. Villanueva,Adiabatic evolution of Hayward black hole,Phys. Lett. B820(2021) 136548 [2101.12253]

work page arXiv 2021
[29]

Gohain, K

M.M. Gohain, K. Bhuyan, R. Borgohain, T. Gogoi, K. Bhuyan and P. Phukon,Frolov black hole surrounded by quintessence - I: Thermodynamics, geodesics and shadows,Nucl. Phys. B 1018(2025) 117073 [2412.06252]

work page arXiv 2025
[30]

Bora, D.J

S. Bora, D.J. Gogoi and P.K. Karmakar,Impact of Thermodynamic Corrections on the Stability of Hayward-Anti de Sitter Black Hole Surrounded by a Fluid of Strings,2510.04208

work page arXiv
[31]

Waseem, F

A. Waseem, F. Javed, G. Mustafa, S.K. Maurya, F. Atamurotov and M. Shrahili, Joule–Thomson expansion of Hayward-AdS black hole surrounded by fluid of strings,Annals Phys.480(2025) 170087

work page 2025
[32]

Liang, Z

Q.-Q. Liang, Z. Cai, D. Liu and Z.-W. Long,Observational properties and quasinormal Modes of the Hayward black Hole surrounded by a cloud of strings,2511.02396

work page arXiv
[33]

Naseer, J

T. Naseer, J. Levi Said, R. Altuijri, M.R. Eid and A.-H. Abdel-Aty,Spherically symmetric regular Hayward black hole and its thermodynamic properties: Insights via gravitational decoupling,Int. J. Geom. Meth. Mod. Phys.22(2025) 2540055

work page 2025
[34]

Interior Dynamics of Regular Schwarzschild Black Holes

J. Ovalle,Interior Dynamics of Regular Schwarzschild Black Holes,2509.00816

work page internal anchor Pith review Pith/arXiv arXiv
[35]

OnSchwarzschildblackholesingularityformation,

J. Ovalle, R. Casadio and A. Kamenshchik,Schwarzschild black hole singularity formation, Phys. Rev. D113(2026) 064042 [2603.06451]

work page arXiv 2026
[36]

New Regular Black Hole Solution from Nonlinear Electrodynamics

E. Ayon-Beato and A. Garcia,New regular black hole solution from nonlinear electrodynamics, Phys. Lett. B464(1999) 25 [hep-th/9911174]

work page Pith review arXiv 1999
[37]

The Bardeen Model as a Nonlinear Magnetic Monopole

E. Ayon-Beato and A. Garcia,The Bardeen model as a nonlinear magnetic monopole,Phys. Lett. B493(2000) 149 [gr-qc/0009077]

work page Pith review arXiv 2000
[38]

Regular electrically charged structures in Nonlinear Electrodynamics coupled to General Relativity

I. Dymnikova,Regular electrically charged structures in nonlinear electrodynamics coupled to general relativity,Class. Quant. Grav.21(2004) 4417 [gr-qc/0407072]

work page Pith review arXiv 2004
[39]

Regular black holes with a nonlinear electrodynamics source

L. Balart and E.C. Vagenas,Regular black holes with a nonlinear electrodynamics source, Phys. Rev. D90(2014) 124045 [1408.0306]

work page Pith review arXiv 2014
[40]

Culetu,On a regular charged black hole with a nonlinear electric source,Int

H. Culetu,On a regular charged black hole with a nonlinear electric source,Int. J. Theor. Phys.54(2015) 2855 [1408.3334]

work page arXiv 2015
[41]

Construction of Regular Black Holes in General Relativity

Z.-Y. Fan and X. Wang,Construction of Regular Black Holes in General Relativity,Phys. Rev. D94(2016) 124027 [1610.02636]

work page Pith review arXiv 2016
[42]

Singh, S.G

D.V. Singh, S.G. Ghosh and S.D. Maharaj,Exact nonsingular black holes and thermodynamics,Nucl. Phys. B981(2022) 115854

work page 2022
[43]

Bronnikov,Regular black holes sourced by nonlinear electrodynamics,2211.00743

K.A. Bronnikov,Regular black holes sourced by nonlinear electrodynamics,2211.00743. – 34 –

work page arXiv
[44]

Regular Black Holes in General Relativity from Nonlinear Electrodynamics with de Sitter Cores

A.A. Araújo Filho, E.L.B. Junior, J.T.S.S. Junior, F.S.N. Lobo, J.A.A. Ramos, M.E. Rodrigues et al.,Regular Black Holes in General Relativity from Nonlinear Electrodynamics with de Sitter Cores,2604.20066

work page internal anchor Pith review Pith/arXiv arXiv
[45]

C. Liu, T. Zhu, Q. Wu, K. Jusufi, M. Jamil, M. Azreg-Aïnou et al.,Shadow and quasinormal modes of a rotating loop quantum black hole,Phys. Rev. D101(2020) 084001 [2003.00477]

work page arXiv 2020
[46]

Cadoni, M

M. Cadoni, M. Oi and A.P. Sanna,Effective models of nonsingular quantum black holes,Phys. Rev. D106(2022) 024030 [2204.09444]

work page arXiv 2022
[47]

G.S. M and S. Das,EHT-Constrained Analysis of Shadow Deformation in Quantum-Improved Rotating Non-Singular Magnetic Monopole,2604.04695

work page internal anchor Pith review Pith/arXiv arXiv
[48]

Effective geometrodynamics for renormalization-group improved black-hole spacetimes in spherical symmetry

J. Borissova and R. Carballo-Rubio,Effective geometrodynamics for renormalization-group improved black-hole spacetimes in spherical symmetry,2601.17115

work page internal anchor Pith review Pith/arXiv arXiv
[49]

Vertogradov and A

V. Vertogradov and A. Övgün,Exact regular black hole solutions with de Sitter cores and Hagedorn fluid,Class. Quant. Grav.42(2025) 025024 [2408.02699]

work page arXiv 2025
[50]

Signatures of a de Sitter-core black hole in ringing, transmission and optical appearance

N. Heidari, A.A. Araújo Filho, V. Vertogradov and A. Övgün,Black hole with a de Sitter core: classical and quantum features,2412.05072

work page internal anchor Pith review Pith/arXiv arXiv
[51]

Simpson and M

A. Simpson and M. Visser,Regular black holes with asymptotically Minkowski cores,Universe 6(2019) 8 [1911.01020]

work page arXiv 2019
[52]

Simpson and M

A. Simpson and M. Visser,The eye of the storm: a regular Kerr black hole,JCAP03(2022) 011 [2111.12329]

work page arXiv 2022
[53]

Simpson and M

A. Simpson and M. Visser,Black-bounce to traversable wormhole,JCAP02(2019) 042 [1812.07114]

work page arXiv 2019
[55]

Ditta, T

A. Ditta, T. Xia, R. Ali, G. Mustafa, G. Mustafa and A. Mahmood,Thermal properties of simpson-visser minkowski core regular black holes solution in verlinde’s emergent gravity, Physics of the Dark Universe43(2024) 101418

work page 2024
[56]

H.-W. Hu, C. Lan and Y.-G. Miao,A regular black hole as the final state of evolution of a singular black hole,Eur. Phys. J. C83(2023) 1047 [2303.03931]

work page arXiv 2023
[57]

Estrada and R

M. Estrada and R. Aros,Pure Lovelock gravity regular black holes,JCAP01(2025) 032 [2409.09559]

work page arXiv 2025
[58]

Bueno, P

P. Bueno, P.A. Cano and R.A. Hennigar,Regular black holes from pure gravity,Phys. Lett. B 861(2025) 139260 [2403.04827]

work page arXiv 2025
[59]

Capozziello, S

S. Capozziello, S. De Bianchi and E. Battista,Avoiding singularities in Lorentzian-Euclidean black holes: The role of atemporality,Phys. Rev. D109(2024) 104060 [2404.17267]

work page arXiv 2024
[60]

Zhang,Black holes as portals to an Euclidean realm,2603.25313

F. Zhang,Black holes as portals to an Euclidean realm,2603.25313

work page arXiv
[61]

Borissova, S

J. Borissova, S. Liberati and M. Visser,Timelike convergence condition in regular black-hole spacetimes with (anti–)de Sitter core,Phys. Rev. D112(2025) 104072 [2509.08590]

work page arXiv 2025
[62]

Jacobson,When is g(tt) g(rr) = -1?,Class

T. Jacobson,When is g(tt) g(rr) = -1?,Class. Quant. Grav.24(2007) 5717 [0707.3222]

work page arXiv 2007
[63]

Ovalle,Black holes without Cauchy horizons and integrable singularities,Phys

J. Ovalle,Black holes without Cauchy horizons and integrable singularities,Phys. Rev. D107 (2023) 104005 [2305.00030]

work page arXiv 2023
[64]

Casadio, A

R. Casadio, A. Kamenshchik and J. Ovalle,From black hole mimickers to black holes,Phys. Rev. D109(2024) 024042 [2401.03980]

work page arXiv 2024
[65]

Geroch,What is a singularity in general relativity?,Annals Phys.48(1968) 526

R.P. Geroch,What is a singularity in general relativity?,Annals Phys.48(1968) 526. – 35 –

work page 1968
[66]

G.J. Olmo, D. Rubiera-Garcia and A. Sanchez-Puente,Geodesic completeness in a wormhole spacetime with horizons,Phys. Rev. D92(2015) 044047 [1508.03272]

work page arXiv 2015
[67]

C.B. Owen, N. Yunes and H. Witek,Petrov type, principal null directions, and Killing tensors of slowly rotating black holes in quadratic gravity,Phys. Rev. D103(2021) 124057 [2103.15891]

work page arXiv 2021
[68]

Cherubini, D

C. Cherubini, D. Bini, S. Capozziello and R. Ruffini,Second order scalar invariants of the Riemann tensor: Applications to black hole space-times,Int. J. Mod. Phys. D11(2002) 827 [gr-qc/0302095]

work page arXiv 2002
[69]

Obukhov and F.W

Y.N. Obukhov and F.W. Hehl,On the relation between quadratic and linear curvature Lagrangians in Poincare gauge gravity,Acta Phys. Polon. B27(1996) 2685 [gr-qc/9602014]

work page arXiv 1996
[70]

Steane,Relativity Made Relatively Easy Volume 2: General Relativity and Cosmology, Oxford University Press, Oxford (2021)

A. Steane,Relativity Made Relatively Easy Volume 2: General Relativity and Cosmology, Oxford University Press, Oxford (2021)

work page 2021
[71]

Jackiw and S.Y

R. Jackiw and S.Y. Pi,Chern-Simons modification of general relativity,Phys. Rev. D68 (2003) 104012 [gr-qc/0308071]

work page arXiv 2003
[72]

Carminati and R.G

J. Carminati and R.G. McLenaghan,Algebraic invariants of the Riemann tensor in a four-dimensional Lorentzian space,J. Math. Phys.32(1991) 3135

work page 1991
[73]

Zakhary and C.B.G

E. Zakhary and C.B.G. Mcintosh,A Complete Set of Riemann Invariants,Gen. Rel. Grav.29 (1997) 539

work page 1997
[74]

Santosuosso, D

K. Santosuosso, D. Pollney, N. Pelavas, P. Musgrave and K. Lake,Invariants of the Riemann tensor for class B warped product space-times,Comput. Phys. Commun.115(1998) 381 [gr-qc/9809012]

work page arXiv 1998
[75]

Borissova and R

J. Borissova and R. Carballo-Rubio,Regular black holes from pure gravity in four dimensions, 2602.16773

work page arXiv
[76]

Misner, K.S

C.W. Misner, K.S. Thorne and J.A. Wheeler,Gravitation, W. H. Freeman, San Francisco (1973)

work page 1973
[77]

Nakahara,Geometry, topology and physics(2003)

M. Nakahara,Geometry, topology and physics(2003)

work page 2003
[78]

Poisson,A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics, Cambridge University Press (12, 2009), 10.1017/CBO9780511606601

E. Poisson,A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics, Cambridge University Press (12, 2009), 10.1017/CBO9780511606601

work page doi:10.1017/cbo9780511606601 2009
[79]

Monk, Finite element methods for Maxwell’s equations, Oxford uni- versity press, 2003.doi:10.1093/acprof:oso/9780198508885.001

L. Rezzolla and O. Zanotti,Relativistic Hydrodynamics, Oxford University Press (9, 2013), 10.1093/acprof:oso/9780198528906.001.0001

work page doi:10.1093/acprof:oso/9780198528906.001.0001 2013
[80]

Maeda,Hawking-Ellis type of matter on Killing horizons in symmetric spacetimes,Phys

H. Maeda,Hawking-Ellis type of matter on Killing horizons in symmetric spacetimes,Phys. Rev. D104(2021) 084088 [2107.01455]

work page arXiv 2021
[81]

Klinkhamer, A new type of nonsingular black-hole solution in general relativity , Mod

F.R. Klinkhamer,A new type of nonsingular black-hole solution in general relativity,Mod. Phys. Lett. A29(2014) 1430018 [1309.7011]

work page arXiv 2014

Showing first 80 references.