arxiv: 2605.12076 · v1 · submitted 2026-05-12 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Causality Violating Solutions in Curvature-Squared Gravity

J. C. R. de Souza , A. F. Santos , R. Bufalo

Authors on Pith no claims yet

Pith reviewed 2026-05-13 04:52 UTC · model grok-4.3

classification 🌀 gr-qc

keywords curvature-squared gravityclosed timelike curvesGodel metriccausality violationWeyl tensorenergy conditionsmodified gravity

0 comments

The pith

Curvature-squared gravity permits Gödel and Gödel-type metrics as solutions that allow closed timelike curves, with all Weyl tensor contributions removed.

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

The paper tests whether closed timelike curves can exist in curvature-squared gravity by substituting Gödel, Gödel-type, and axially symmetric metrics into the derived field equations. It finds that the first two families satisfy the equations exactly and thereby permit CTCs, while every term involving the Weyl tensor drops out of the solutions. An axially symmetric metric is then used to isolate the Weyl tensor's role, showing that it contributes to the energy density and modifies the weak energy condition. A sympathetic reader would care because this demonstrates that adding quadratic curvature terms can relax the causality restrictions found in Einstein gravity for the same geometries.

Core claim

The authors establish that the Gödel and Gödel-type metrics are exact solutions of the curvature-squared field equations, so closed timelike curves are allowed, and that every contribution involving the Weyl tensor is removed from these solutions. For the axially symmetric cosmological solution the Weyl tensor remains and contributes directly to the energy density, thereby altering the weak energy condition.

What carries the argument

The field equations obtained by varying the curvature-squared action, applied to the Gödel, Gödel-type, and axially symmetric metrics, with the Weyl tensor terms canceling identically in the first two cases.

If this is right

Closed timelike curves are permitted for Gödel-type geometries in curvature-squared gravity.
The Weyl tensor plays no role in determining the energy density or pressure for these Gödel solutions.
Axially symmetric metrics receive additional contributions from the Weyl tensor that affect the weak energy condition.
Causality-violating features can be studied in this model without interference from conformal curvature terms.

Where Pith is reading between the lines

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

Similar metrics might be checked in other quadratic or higher-derivative gravity theories to see whether Weyl cancellation is generic.
The absence of Weyl contributions could simplify the analysis of energy conditions in non-causal spacetimes within modified gravity.
One could look for observational signatures by embedding these solutions in asymptotically flat or expanding backgrounds.

Load-bearing premise

The chosen Gödel, Gödel-type, and axially symmetric metrics are exact solutions to the field equations derived from the curvature-squared action.

What would settle it

Direct substitution of the Gödel metric into the curvature-squared field equations yielding nonzero residual terms or showing that the Weyl tensor contributions do not cancel.

Figures

Figures reproduced from arXiv: 2605.12076 by A. F. Santos, J. C. R. de Souza, R. Bufalo.

read the original abstract

In this paper, we consider some causality violating solutions in the curvature-squared gravity in order to examine whether closed timelike curves (CTCs) are allowed in these models. These aspects are studied in terms of the G\"odel, G\"odel-type and axially symmetric cosmological solutions. We observe that the G\"odel and G\"odel-type metrics are causal solutions of the model so that CTCs are now allowed and, surprisingly, every contribution involving the Weyl tensor is removed from the solutions. Hence, in order to study the effect (if any) of the Weyl tensor (an conformal symmetry) into CTCs a third metric is considered. In this case, we obtain contributions due to the Weyl tensor to the energy density and led to modifications of the weak energy condition.

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

The paper shows Gödel and Gödel-type metrics satisfy the quadratic gravity equations with Weyl terms canceling out, allowing CTCs, while the axial case restores Weyl dependence.

read the letter

The main takeaway is that standard Gödel and Gödel-type metrics work as exact solutions in curvature-squared gravity, with every Weyl-tensor contribution dropping out so that closed timelike curves remain allowed. The authors then introduce an axially symmetric metric precisely to bring the Weyl tensor back in, where it modifies the energy density and the weak energy condition. This is a straightforward check of known metrics inside a known higher-order action. The work does a clean job of setting up the three cases and reporting the outcomes for causality and energy conditions. The observation that Weyl terms vanish in the first two metrics is presented clearly, and the choice of the third metric to isolate conformal effects is sensible. What is actually new is limited to this specific application and the reported cancellation; the metrics and the action are both taken from prior literature. The soft spot is verification. The central claim rests on these metrics satisfying the fourth-order field equations identically, yet the abstract gives no explicit expressions for the substituted curvatures or the resulting equations. Without those steps or a clear statement of the coefficient ranges, it is hard to judge whether the cancellation is generic or tied to particular parameter choices. The stress-test note on unverified substitution is fair on the evidence shown. This paper is for people already working on Gödel-type solutions or causality in modified gravity. A reader looking for concrete checks of quadratic actions against rotating cosmologies will find the cases useful. It deserves peer review so referees can inspect the curvature substitutions and confirm the solutions hold exactly.

Referee Report

2 major / 2 minor

Summary. The paper examines causality-violating solutions in curvature-squared gravity using the Gödel, Gödel-type, and axially symmetric cosmological metrics. It claims that the first two metrics are exact solutions to the fourth-order field equations of the model, thereby permitting closed timelike curves (CTCs), and that all Weyl-tensor contributions cancel identically in these cases. The axially symmetric metric is then introduced to isolate nonzero Weyl effects, which are shown to modify the energy density and the weak energy condition.

Significance. If the central claims are verified, the work would demonstrate that standard causality-violating metrics from general relativity remain solutions in quadratic curvature gravity, with a nontrivial cancellation of Weyl terms that is not generic. This could inform studies of conformal invariance and energy conditions in higher-order gravity. The choice to add the axially symmetric case specifically to restore Weyl dependence is a constructive step toward isolating the effect.

major comments (2)

[Abstract and solution sections] Abstract and the sections presenting the solutions: the manuscript states that the Gödel and Gödel-type metrics satisfy the field equations derived from the curvature-squared action and that Weyl contributions vanish identically, yet no explicit form of the fourth-order field equations is written, no curvature components (Riemann, Ricci, or Weyl) are computed for these metrics, and no substitution verifying that the equations hold identically is provided. This verification is load-bearing for the claim that CTCs are allowed.
[Axially symmetric solutions] Axially symmetric metric analysis: the claim that Weyl-tensor contributions appear in the energy density and modify the weak energy condition is asserted without showing the explicit decomposition of the field equations into Ricci and Weyl parts or the resulting expressions for the energy density. Without these steps it is unclear whether the modifications are generic or depend on specific parameter choices in the action.

minor comments (2)

[Abstract] The abstract and introduction should explicitly state the precise form of the curvature-squared action (including the coefficients of the R² and R_{μν}R^{μν} terms) so that readers can reproduce the field equations.
[Gödel-type solutions] Notation for the Gödel-type metric parameters and the range of the rotation parameter should be introduced once and used consistently when discussing CTC existence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We agree that the explicit derivations and verifications are essential to support the central claims and will revise the manuscript to include them in full.

read point-by-point responses

Referee: [Abstract and solution sections] Abstract and the sections presenting the solutions: the manuscript states that the Gödel and Gödel-type metrics satisfy the field equations derived from the curvature-squared action and that Weyl contributions vanish identically, yet no explicit form of the fourth-order field equations is written, no curvature components (Riemann, Ricci, or Weyl) are computed for these metrics, and no substitution verifying that the equations hold identically is provided. This verification is load-bearing for the claim that CTCs are allowed.

Authors: We agree that the explicit verification is load-bearing and currently insufficient in the manuscript. In the revised version we will first write the complete fourth-order field equations obtained by varying the curvature-squared action. We will then compute the Riemann, Ricci and Weyl tensors for the Gödel and Gödel-type metrics in the chosen coordinates, substitute them directly into the field equations, and show that the equations are satisfied identically with every Weyl contribution canceling. These steps will be presented in a new subsection so that the allowance of closed timelike curves is fully substantiated. revision: yes
Referee: [Axially symmetric solutions] Axially symmetric metric analysis: the claim that Weyl-tensor contributions appear in the energy density and modify the weak energy condition is asserted without showing the explicit decomposition of the field equations into Ricci and Weyl parts or the resulting expressions for the energy density. Without these steps it is unclear whether the modifications are generic or depend on specific parameter choices in the action.

Authors: We accept that the lack of explicit decomposition leaves the role of the Weyl tensor unclear. In the revision we will decompose the field equations into their Ricci and Weyl sectors for the axially symmetric metric. We will derive the explicit expression for the energy density, isolate the Weyl-dependent terms, and show their effect on the weak energy condition. We will also examine the dependence of these modifications on the free parameters of the action and state whether the observed violations are generic or parameter-specific. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results follow from direct substitution of known metrics into derived field equations.

full rationale

The paper derives the fourth-order field equations from the curvature-squared action and substitutes the standard Gödel, Gödel-type, and axially symmetric metrics (taken from prior literature) to check satisfaction and compute curvature contributions. The vanishing of Weyl-tensor terms in the first two cases and their appearance in the third are presented as outcomes of this explicit calculation rather than inputs or definitions. No self-citation is invoked to justify uniqueness or to force the metrics as solutions; the central claims rest on the algebraic reduction of the field equations for these specific spacetimes, which is independently verifiable and not equivalent to the inputs by construction. This constitutes a standard, self-contained analysis in modified gravity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, new axioms, or invented entities are stated. The work relies on standard assumptions of general relativity and the variational principle for the quadratic action.

axioms (1)

domain assumption The spacetime metrics satisfy the vacuum or matter-filled field equations obtained by varying the curvature-squared action.
Invoked when the authors state that the Godel and Godel-type metrics are solutions of the model.

pith-pipeline@v0.9.0 · 5433 in / 1144 out tokens · 24724 ms · 2026-05-13T04:52:10.221565+00:00 · methodology

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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

We observe that the Gödel and Gödel-type metrics are causal solutions of the model so that CTCs are now allowed and, surprisingly, every contribution involving the Weyl tensor is removed from the solutions.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

the field equations are given by … +α(2∇ρ∇σCρμνσ + Cρμνσ Rρσ) + …

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

47 extracted references · 47 canonical work pages · 1 internal anchor

[1]

Perfect fluid The energy-momentum tensor of a perfect fluid is defined in eq. (14), in which the four-velocity is defined as usualuµ =δ µ 0 = (1,0,0,0), while the corresponding covector is uµ = 1,0, 4ωsinh 2 mr 2 m2 ,0 ! .(30) 9 Thus, the non-vanishing components of the energy-momentum tensor are T00 =ρ , T02 = 4ωρ m2 sinh2 mr 2 , T11 =T 33 =p , T22 = 16ω...

work page
[2]

Massless scalar field Another source known to engender interesting features in the Gödel-type universe is a scalar field [9, 10]. The energy-momentum tensor that describes the massless scalar field is given by T µν =∂ µϕ ∂νϕ−g µν 1 2 ∂αϕ ∂αϕ+V(ϕ) .(32) For simplicity, we assume that the scalar field depends only on the variablez, that is,ϕ=ϕ(z). The non-v...

work page 2023
[3]

How isotropic is the Universe?,

D. Saadeh, S. M. Feeney, A. Pontzen, H. V. Peiris and J. D. McEwen, “How isotropic is the Universe?,” Phys. Rev. Lett.117(2016) no.13, 131302 arXiv:1605.07178 [astro-ph.CO]

work page arXiv 2016
[4]

The distribution of galaxy rotation in JWST Advanced Deep Extragalactic Survey,

L. Shamir, “The distribution of galaxy rotation in JWST Advanced Deep Extragalactic Survey,” Mon. Not. Roy. Astron. Soc.538(2025) no.1, 76-91 arXiv:2502.18781 [astro-ph.CO]

work page arXiv 2025
[5]

Non-gaussianity detections in the Bianchi vii(h) corrected wmap 1-year data made with directional spherical wavelets,

J. D. McEwen, M. P. Hobson, A. N. Lasenby and D. J. Mortlock, “Non-gaussianity detections in the Bianchi vii(h) corrected wmap 1-year data made with directional spherical wavelets,” Mon. Not. Roy. Astron. Soc.369(2006), 1858-1868 arXiv:astro-ph/0510349

work page arXiv 2006
[6]

Can rotation solve the Hubble Puzzle?,

B. E. Szigeti, I. Szapudi, I. F. Barna and G. G. Barnaföldi, “Can rotation solve the Hubble Puzzle?,” Mon. Not. Roy. Astron. Soc.538(2025) no.4, 3038-3041 arXiv:2503.13525 [gr-qc]

work page arXiv 2025
[7]

Black holes in Godel universes and pp waves,

E. G. Gimon and A. Hashimoto, “Black holes in Godel universes and pp waves,” Phys. Rev. Lett.91 (2003), 021601 arXiv:hep-th/0304181

work page arXiv 2003
[8]

General Non-extremal Rotating Charged Godel Black Holes in Minimal Five-Dimensional Gauged Supergravity,

S. Q. Wu, “General Non-extremal Rotating Charged Godel Black Holes in Minimal Five-Dimensional Gauged Supergravity,” Phys. Rev. Lett.100(2008), 121301 arXiv:0709.1749 [hep-th]

work page arXiv 2008
[9]

Extended uncertainty principle inspired black hole in a Gödel Universe,

R. C. Pantig, “Extended uncertainty principle inspired black hole in a Gödel Universe,” Phys. Lett. B 868(2025), 139722 arXiv:2506.00295 [gr-qc]

work page arXiv 2025
[10]

An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation,

K. Gödel, “An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation,” Rev. Mod. Phys.21(1949), 447-450

work page 1949
[11]

On the Homogeneity of Riemannian Space-Times of Godel Type,

M. J. Reboucas and J. Tiomno, “On the Homogeneity of Riemannian Space-Times of Godel Type,” Phys. Rev. D28(1983), 1251-1264

work page 1983
[12]

Isometries of Homogeneous Godel-type space- times,

A. F. F. Teixeira, M. J. Reboucas and J. E. Aman, “Isometries of Homogeneous Godel-type space- times,” Phys. Rev. D32(1985), 3309-3311

work page 1985
[13]

A NOTE ON GODEL TYPE SPACE-TIMES,

M. J. Reboucas, J. E. Aman and A. F. F. Teixeira, “A NOTE ON GODEL TYPE SPACE-TIMES,” J. Math. Phys.27(1986), 1370-1372 19

work page 1986
[14]

On physical foundations and observational effects of cosmic rotation,

Y. N. Obukhov, “On physical foundations and observational effects of cosmic rotation,” Published in ’Colloquium on Cosmic Rotation’, Eds. M. Scherfner, T. Chrobok and M. Shefaat (Wissenschaft und Technik Verlag: Berlin, 2000) pp. 23-96 arXiv:astro-ph/0008106

work page internal anchor Pith review arXiv 2000
[15]

f(R) Theories Of Gravity

T. P. Sotiriou and V. Faraoni, “f(R) Theories Of Gravity,” Rev. Mod. Phys.82(2010), 451-497 arXiv:0805.1726 [gr-qc]

work page Pith review arXiv 2010
[16]

Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models

S. Nojiri and S. D. Odintsov, “Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models,” Phys. Rept.505(2011), 59-144 arXiv:1011.0544 [gr-qc]

work page Pith review arXiv 2011
[17]

Nojiri, S

S. Nojiri, S. D. Odintsov and V. K. Oikonomou, “Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution,” Phys. Rept.692(2017), 1-104 arXiv:1705.11098 [gr-qc]

work page arXiv 2017
[18]

Introduction to Modified Gravity,

A. Petrov, J. R. Nascimento and P. Porfirio, “Introduction to Modified Gravity,” 2nd ed. Springer, 2023, arXiv:2004.12758 [gr-qc]

work page arXiv 2023
[19]

Gödel-type solutions within the f(R,Q) gravity,

F. S. Gama, J. R. Nascimento, A. Y. Petrov, P. J. Porfirio and A. F. Santos, “Gödel-type solutions within the f(R,Q) gravity,” Phys. Rev. D96(2017) no.6, 064020 arXiv:1707.03440 [hep-th]

work page arXiv 2017
[20]

Gödel-type solutions within the f(R, Q, P) gravity,

J. R. Nascimento, A. Y. Petrov, P. J. Porfirio and R. N. da Silva, “Gödel-type solutions within the f(R, Q, P) gravity,” Eur. Phys. J. C83(2023) no.4, 331 arXiv:2302.09935 [gr-qc]

work page arXiv 2023
[21]

Gödel-type spacetimes in f(Q) gravity,

A. M. Silva, M. J. Rebouças and N. A. Lemos, “Gödel-type spacetimes in f(Q) gravity,” Int. J. Mod. Phys. D33(2024) no.16, 2450060 arXiv:2407.04750 [gr-qc]

work page arXiv 2024
[22]

Reine Infinitesimalgeometrie,

H. Weyl, “Reine Infinitesimalgeometrie,” Math. Z.2(1918) no.3-4, 384-411

work page 1918
[23]

Making the Case for Conformal Gravity,

P. D. Mannheim, “Making the Case for Conformal Gravity,” Found. Phys.42(2012), 388-420 arXiv:1101.2186 [hep-th]

work page arXiv 2012
[24]

Strong Coupling Approach to the Quantization of Conformal Gravity,

M. Kaku, “Strong Coupling Approach to the Quantization of Conformal Gravity,” Phys. Rev. D27 (1983), 2819

work page 1983
[25]

On the Cosmological Constant Problem,

I. Antoniadis and N. C. Tsamis, “On the Cosmological Constant Problem,” Phys. Lett. B144(1984), 55-60

work page 1984
[26]

Quantization of higher derivative theories of gravity,

D. G. Boulware, “Quantization of higher derivative theories of gravity,” inQuantum Theory of Gravity, ed. S. M. Christensen (Adam Hilger, Bristol, 1984), p. 267

work page 1984
[27]

Hamiltonian analysis of curvature-squared gravity with or without conformal invariance,

J. Klusoň, M. Oksanen and A. Tureanu, “Hamiltonian analysis of curvature-squared gravity with or without conformal invariance,” Phys. Rev. D89(2014) no.6, 064043 arXiv:arXiv:1311.4141 [hep-th]

work page arXiv 2014
[28]

Probing the small distance structure of canonical quantum gravity using the conformal group,

G. ’t Hooft, “Probing the small distance structure of canonical quantum gravity using the conformal group,” arXiv:1009.0669 [gr-qc]

work page arXiv
[29]

Einstein Gravity from Conformal Gravity,

J. Maldacena, “Einstein Gravity from Conformal Gravity,” arXiv:1105.5632 [hep-th]

work page arXiv
[30]

Massive Gravity from Higher Derivative Gravity with Boundary Conditions,

M. Park and L. Sorbo, “Massive Gravity from Higher Derivative Gravity with Boundary Conditions,” JHEP01(2013), 043 arXiv:1210.7733 [hep-th]

work page arXiv 2013
[31]

Renormalization of Higher Derivative Quantum Gravity,

K. S. Stelle, “Renormalization of Higher Derivative Quantum Gravity,” Phys. Rev. D16(1977), 953-969

work page 1977
[32]

Renormalizable asymptotically free quantum theory of gravity,

E. S. Fradkin and A. A. Tseytlin, “Renormalizable asymptotically free quantum theory of gravity,” Nucl. Phys. B201(1982), 469-491 20

work page 1982
[33]

Structure of Kerr black hole spacetimes in Weyl conformal gravity,

M. Y. Asuncion, K. Horne, R. Kusano and M. Dominik, “Structure of Kerr black hole spacetimes in Weyl conformal gravity,” Class. Quant. Grav.42(2025) no.21, 215005 arXiv:2507.01379 [gr-qc]

work page arXiv 2025
[34]

Weyl geometric gravity black holes in light of the Solar System tests,

M. Khodadi and T. Harko, “Weyl geometric gravity black holes in light of the Solar System tests,” Eur. Phys. J. C85(2025) no.11, 1325 arXiv:2509.15838 [gr-qc]

work page arXiv 2025
[35]

Black holes in higher-derivative Weyl conformal gravity,

L. A. Lessa, C. F. B. Macedo and M. M. Ferreira, “Black holes in higher-derivative Weyl conformal gravity,” arXiv:2509.21495 [gr-qc]

work page arXiv
[36]

Broken Weyl Gravity: Dynamical Torsion and Generalized Proca,

L. F. Sarmiento and I. M. Rodríguez, “Broken Weyl Gravity: Dynamical Torsion and Generalized Proca,” arXiv:2510.08778 [hep-th]

work page arXiv
[37]

Singularities and Causality Violation,

F. J. Tipler, “Singularities and Causality Violation,” Annals Phys.108(1977), 1-36

work page 1977
[38]

On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes,

M. Sáncez, “On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes,” Diff. Geom. Applications24(2006) 21-32 arXiv:math/0502322

work page arXiv 2006
[39]

Is a Topology Change After a Big Rip Possible?,

V. K. Oikonomou, “Is a Topology Change After a Big Rip Possible?,” Int. J. Geom. Meth. Mod. Phys. 16(2019) no.03, 1950048 arXiv:1805.11945 [gr-qc]

work page arXiv 2019
[40]

Finite-time cosmological sin- gularities and the possible fate of the Universe,

J. de Haro, S. Nojiri, S. D. Odintsov, V. K. Oikonomou and S. Pan, “Finite-time cosmological sin- gularities and the possible fate of the Universe,” Phys. Rept.1034(2023), 1-114 arXiv:2309.07465 [gr-qc]

work page arXiv 2023
[41]

Dynamical Systems Perspective of Cosmological Finite-time Singularities inf(R)Gravity and Interacting Multifluid Cosmology,

S. D. Odintsov and V. K. Oikonomou, “Dynamical Systems Perspective of Cosmological Finite-time Singularities inf(R)Gravity and Interacting Multifluid Cosmology,” Phys. Rev. D98(2018) no.2, 024013 arXiv:1806.07295 [gr-qc]

work page arXiv 2018
[42]

Rescaled Einstein-Hilbert Gravity from f(R) Gravity: Inﬂation, Dark Energy and the Swampland Criteria,

V. K. Oikonomou, “Rescaled Einstein-Hilbert Gravity fromf(R)Gravity: Inflation, Dark Energy and the Swampland Criteria,” Phys. Rev. D103(2021) no.12, 124028 arXiv:2012.01312 [gr-qc]

work page arXiv 2021
[43]

An axially symmetric spacetime with causality violation

B. B. Hazarika, “An axially symmetric spacetime with causality violation”, Phys. Scr.96, 075208 (2021)

work page 2021
[44]

Type III Spacetime with Closed Timelike Curves

F. Ahmed, “Type III Spacetime with Closed Timelike Curves”, PROGRESS IN PHYSICS12, 329 (2016),http://www.ptep-online.com

work page 2016
[45]

A type N radiation field solution withΛ<0in a curved spacetime and closed time-like curves

F. Ahmed, “A type N radiation field solution withΛ<0in a curved spacetime and closed time-like curves”, Eur. Phys. J. C78, 385 (2018)

work page 2018
[46]

TAUB-NUT SPACE AS A COUNTEREXAMPLE TO ALMOST ANYTHING,

C. W. Misner, “TAUB-NUT SPACE AS A COUNTEREXAMPLE TO ALMOST ANYTHING,” In J. Ehlers (ed.). Relativity Theory and Astrophysics I: Relativity and Cosmology. Lectures in Applied Mathematics. Vol. 8. American Mathematical Society. pp. 160–169

work page
[47]

The Large Scale Structure of Space-Time,

S. W. Hawking and G. F. R. Ellis, “The Large Scale Structure of Space-Time,” Cambridge University Press, 2023,

work page 2023