pith. sign in

arxiv: 2412.18961 · v3 · submitted 2024-12-25 · 🌀 gr-qc

Rotating Traversable Wormholes with a Throat-Localized Conical Dressing and Two Conical Cosmic-String Cores

Vedant Subhash This is my paper

Pith reviewed 2026-05-23 07:20 UTC · model grok-4.3

classification 🌀 gr-qc
keywords traversable wormholecosmic stringnull energy conditionconical singularityrotating wormholescalar perturbationsaxisymmetric metricthroat dressing
0
0 comments X

The pith

A rotating traversable wormhole is built with throat-localized conical cosmic-string cores that saturate the radial null energy condition.

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

The paper develops a single stationary axisymmetric metric for a traversable wormhole that includes a conical factor localized precisely at the throat. This factor generates two genuine conical tips at the poles of every angular cross-section, which are interpreted as cosmic-string cores along the rotation axis. Evaluation of the exact radial null energy condition at the throat shows that these ideal string cores saturate the condition rather than violate it. The necessary exoticity for traversability is therefore supplied by the smooth throat sector together with the localized dressing. Analysis of scalar perturbations on the same background reveals that NEC violation remains throat-centered while the conical dressing induces nonaxisymmetric angular-channel mixing.

Core claim

In the metric written in proper radial distance, the radial null-energy-condition evaluated at the throat is saturated by the conical string cores, so that the required exotic matter is furnished by the smooth throat sector and the localized conical dressing; the same background yields an exact axisymmetric Schrödinger equation for scalar perturbations and a coupled nonaxisymmetric system generated by the dressing.

What carries the argument

The throat-localized conical factor inserted into the stationary axisymmetric metric, which produces genuine conical tips at the poles while preserving a single consistent background geometry throughout.

If this is right

  • The ideal string cores saturate the radial NEC at the throat.
  • Exoticity for traversability is supplied entirely by the smooth throat sector and the localized dressing.
  • Scalar perturbations exhibit NEC violation that remains centered at the throat.
  • The localized conical dressing produces coupled nonaxisymmetric angular-channel mixing in the perturbation equations.
  • Numerical integration of the perturbation system confirms the dynamical signature of the dressed throat.

Where Pith is reading between the lines

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

  • The saturation property suggests that known cosmic-string geometries can be grafted onto wormhole throats without increasing the total exotic-matter budget.
  • Nonaxisymmetric channel mixing may produce distinctive signatures in null geodesics or in the spectrum of waves traversing the wormhole.
  • The construction supplies a concrete background on which to test whether the conical cores remain stable once back-reaction from the exotic throat matter is included.
  • Extension to charged or spinning matter fields could reveal whether the saturation of the radial NEC persists beyond the vacuum string case.

Load-bearing premise

A single consistent background geometry can be used throughout with the conical factor localized exactly at the throat while still producing genuine conical tips at the poles of each angular cross-section.

What would settle it

A direct evaluation of the radial null energy condition at the throat in which the conical cores are found to violate rather than saturate the condition, or a metric calculation showing that the poles fail to exhibit genuine conical deficits.

Figures

Figures reproduced from arXiv: 2412.18961 by Vedant Subhash.

Figure 1
Figure 1. Figure 1: illustrates the perturbation h[r, θ] due to the two cosmic strings. The plot is presented in polar coordinates, highlighting the localization of the perturbations around θ = π/4 and θ = 3π/4. FIG. 1. Localized perturbations h[r, θ] due to two cosmic strings. The color scale represents the magnitude of the perturbation ϵh[r, θ], with brighter regions indicating stronger effects. The perturbations are concen… view at source ↗
read the original abstract

A stationary axisymmetric traversable wormhole with a throat-localized conical factor is developed. The conical factor produces two genuine conical tips at the poles of each angular cross-section, interpreted as cosmic-string cores along the rotation axis. A single consistent background geometry is used throughout. The metric is written in proper radial distance \(l\), and the exact radial null-energy-condition (NEC) is derived and evaluated at the throat. It is shown that the ideal string cores saturate, rather than violate, the radial NEC, so the required exoticity is supplied by the smooth throat sector together with the localized dressing. Scalar perturbations are then studied on the same background. The exact axisymmetric sector, its Schr\"odinger form, and the coupled nonaxisymmetric system generated by the localized conical dressing are obtained. Numerical results show that NEC violation is throat-centered, while the clearest dynamical signature of the dressed throat is nonaxisymmetric angular-channel mixing.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs a stationary axisymmetric traversable wormhole using a single background metric in proper radial coordinate l, with a conical factor localized at the throat that generates two genuine conical cosmic-string cores along the axis. It derives the exact radial null-energy-condition (NEC) and evaluates it at the throat, claiming that the ideal string cores saturate (rather than violate) the radial NEC, so that the required exoticity is supplied exclusively by the smooth throat sector together with the localized dressing. Scalar perturbations are then analyzed on the same background, yielding an exact axisymmetric sector in Schrödinger form, a coupled nonaxisymmetric system induced by the dressing, and numerical results showing throat-centered NEC violation together with nonaxisymmetric angular-channel mixing as the principal dynamical signature.

Significance. If the metric construction and NEC attribution are internally consistent, the work supplies a concrete example in which the string cores contribute neutrally to the radial NEC while the throat and dressing furnish the necessary violation. This separation, together with the perturbation analysis that isolates the dynamical effect of the localized dressing, would be a useful addition to the literature on traversable wormholes and energy-condition requirements. The use of a single consistent background rather than piecewise matching is a methodological strength.

major comments (2)
  1. [Metric ansatz and construction (described in abstract and §II)] The central claim that the string cores saturate the radial NEC while the throat supplies the violation rests on the conical factor being localized exactly at l=0 yet still producing genuine conical deficits at the poles of every l=const angular cross-section throughout the spacetime. The manuscript must demonstrate explicitly (via the curvature scalars or the deficit angle computation) that the localization does not confine the conical contribution to the throat alone; otherwise the cores cease to be ideal strings along the full axis and the NEC attribution at the throat no longer isolates the string contribution cleanly.
  2. [NEC derivation] §III (NEC derivation): the exact radial NEC expression is stated to have been derived and evaluated at the throat, but the intermediate steps from the metric components to the final NEC formula are not reproduced. Without these steps it is impossible to verify that the string-core contribution indeed saturates rather than violates the condition.
minor comments (2)
  1. [Perturbation analysis] The numerical method, grid resolution, and boundary conditions used for the perturbation results should be stated explicitly so that the reported angular-channel mixing can be reproduced.
  2. [Abstract and §II] Notation for the conical factor strength and the rotation parameter should be introduced once and used consistently; the abstract refers to both without defining symbols.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments identify points where additional explicit derivations would strengthen the presentation. We address each below and will incorporate the requested material in a revised version.

read point-by-point responses

  1. Referee: [Metric ansatz and construction (described in abstract and §II)] The central claim that the string cores saturate the radial NEC while the throat supplies the violation rests on the conical factor being localized exactly at l=0 yet still producing genuine conical deficits at the poles of every l=const angular cross-section throughout the spacetime. The manuscript must demonstrate explicitly (via the curvature scalars or the deficit angle computation) that the localization does not confine the conical contribution to the throat alone; otherwise the cores cease to be ideal strings along the full axis and the NEC attribution at the throat no longer isolates the string contribution cleanly.

    Authors: We agree that an explicit verification is required. The conical factor is introduced as a multiplicative dressing localized at l=0, but the resulting geometry produces a constant angular deficit at the poles for every fixed-l slice. In the revision we will add (i) the deficit-angle calculation obtained from the proper circumference-to-radius ratio on l=const surfaces for l≠0, confirming that the deficit remains uniform along the axis, and (ii) the relevant curvature scalars (in particular the components that encode the conical singularity) evaluated away from the throat, demonstrating that the string cores extend along the full axis. These additions will make the separation between the neutral string contribution and the throat-supplied exoticity fully transparent. revision: yes

  2. Referee: [NEC derivation] §III (NEC derivation): the exact radial NEC expression is stated to have been derived and evaluated at the throat, but the intermediate steps from the metric components to the final NEC formula are not reproduced. Without these steps it is impossible to verify that the string-core contribution indeed saturates rather than violates the condition.

    Authors: We accept that the intermediate algebra was omitted. In the revised §III we will insert the complete derivation: starting from the metric components in proper radial coordinate l, we compute the Einstein tensor, contract with the radial null vector to obtain the radial NEC, and isolate the separate contributions arising from the conical dressing and from the smooth throat sector. The resulting expression will show explicitly that the conical cores contribute exactly zero to the NEC violation (i.e., they saturate the bound), while the throat and localized dressing supply the required negative term. This expanded derivation will allow direct verification of the claimed attribution. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper starts from an explicit metric ansatz incorporating a throat-localized conical factor, writes the line element in proper radial coordinate l, and computes the radial NEC directly from the Einstein tensor components of that metric. The statement that ideal string cores saturate the radial NEC while exoticity is supplied by the smooth throat plus dressing is an evaluation of those derived expressions, not a redefinition or a fit renamed as a prediction. No self-citation chain, uniqueness theorem, or ansatz smuggling is invoked to justify the central separation; the construction is self-contained against the chosen geometry.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 2 invented entities

The model rests on the Einstein equations applied to a stationary axisymmetric metric ansatz that includes an ad-hoc conical factor localized at the throat; the conical factor and the interpretation of its tips as cosmic-string cores are introduced without independent evidence outside the construction itself.

free parameters (2)
  • conical factor strength
    Parameter introduced to produce genuine conical tips at the poles; its specific value is chosen so that the tips behave as cosmic-string cores.
  • rotation parameter
    Controls the stationary axisymmetric character of the wormhole; value selected to maintain consistency with the conical dressing.
axioms (2)
  • standard math Einstein field equations hold with the given metric ansatz
    Used to obtain the exact radial NEC from the metric components.
  • domain assumption The conical factor can be localized exactly at the throat while preserving a single consistent background geometry
    Invoked when the metric is written in proper radial distance l and the conical tips are required to be genuine.
invented entities (2)
  • throat-localized conical dressing no independent evidence
    purpose: Produces two conical tips interpreted as cosmic-string cores along the rotation axis
    Postulated directly in the metric construction; no independent evidence supplied.
  • cosmic-string cores no independent evidence
    purpose: Account for the conical tips at the poles of angular cross-sections
    Interpretation assigned to the geometry; no separate observational or theoretical justification given.

pith-pipeline@v0.9.0 · 5693 in / 1786 out tokens · 26738 ms · 2026-05-23T07:20:24.388166+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

40 extracted references · 40 canonical work pages

  1. [1]

    The studies link cylindrical geometries with energy condition violations, suggesting cosmic strings can act as natural candidates for the worm- hole throat

    explore the dynamics of rotating wormholes, emphasizing their stability and traversabil- ity under exotic matter conditions. The studies link cylindrical geometries with energy condition violations, suggesting cosmic strings can act as natural candidates for the worm- hole throat. Their application in observational cosmology is also discussed. These studi...

    work page

  2. [2]

    This requires careful adjustments to the angular metric components to accommodate the presence of the strings

    Metric Modification: Building upon Teo’s metric, I introduced cosmic strings into the geometry. This requires careful adjustments to the angular metric components to accommodate the presence of the strings

    work page

  3. [3]

    NEC Violation Analysis : The methodology for evaluating NEC violations is sim- ilar to Teo’s analysis of null vectors and the stress-energy tensor components at the wormhole throat

    work page

  4. [4]

    This process involved the use of a toy model to decouple and solve the equation

    Scalar Perturbation Analysis: Following Sung-Won Kim’s approach, I derived and simplified the scalar wave equation within the context of the modified metric. This process involved the use of a toy model to decouple and solve the equation. THE MODIFIED METRIC The spacetime metric, incorporating the effects of cosmic strings, is expressed as: ds2 = −N(r, θ)...

    work page

  5. [5]

    The plot is presented in polar coordinates, highlighting the localization of the perturbations around θ = π/4 and θ = 3π/4

    Visualization of Perturbations Figure 1 illustrates the perturbation h[r, θ] due to the two cosmic strings. The plot is presented in polar coordinates, highlighting the localization of the perturbations around θ = π/4 and θ = 3π/4. FIG. 1. Localized perturbations h[r, θ] due to two cosmic strings. The color scale represents the magnitude of the perturbati...

    work page

  6. [6]

    This ensures that the cosmic string effects are localized near the wormhole throat

    Interpretation of the Plot The plot demonstrates several key features of the perturbation function: 8 • Radial Localization : The perturbation magnitude peaks around r0, with a rapid falloff determined by σr. This ensures that the cosmic string effects are localized near the wormhole throat. • Angular Localization : The perturbation is symmetrically distr...

    work page

  7. [7]

    A regularized form is: b(r) = r0 1 − ϵ(r − r0)2 r2 0 , where ϵ > 0

    Regularized Parameter Forms To avoid divergence at the throat, b(r) should approach r0 smoothly while ensuring r − b(r) ̸= 0 for small deviations from r0. A regularized form is: b(r) = r0 1 − ϵ(r − r0)2 r2 0 , where ϵ > 0. This ensures: b(r0) = r0, ∂b(r) ∂r = −2ϵ(r − r0) r2 0 , ∂2b(r) ∂r2 = −2ϵ r2 0 . Cosmic String Factor f(r, θ): To avoid f(r, θ) → 0, we...

    work page

  8. [8]

    For example, N(r)2 ∼ ϵ(r − r0)2 at the throat, eliminating divergences

    Regularized Contributions: Terms involving N(r), such as Term 3, are regularized by the choice b(r) = r0 − ϵ(r − r0)2, ensuring all terms are finite. For example, N(r)2 ∼ ϵ(r − r0)2 at the throat, eliminating divergences

    work page

  9. [9]

    Notably: Term 17: −8f(r0, θ) (1 − ϵh(r0, θ))2 ω(r0, θ)2 sin(2θ) ∂f (r0,θ) ∂θ K(r0, θ)2N(r0)2 , Term 20: 2(r0 − b(r0))ω(r0, θ) sin3(θ) (−2λα cos(θ)) N(r0)2

    Negative NEC Contributions: Several terms contribute negatively due to their dependence on derivatives of f(r, θ), h(r, θ), and ω(r, θ). Notably: Term 17: −8f(r0, θ) (1 − ϵh(r0, θ))2 ω(r0, θ)2 sin(2θ) ∂f (r0,θ) ∂θ K(r0, θ)2N(r0)2 , Term 20: 2(r0 − b(r0))ω(r0, θ) sin3(θ) (−2λα cos(θ)) N(r0)2 . At r = r0, the regularized N(r0) amplifies these negative contributions

    work page

  10. [10]

    This is evident in terms like Term 11 and Term 17, where derivatives of h(r, θ) combine with sinusoidal terms, producing localized negative energy density contributions

    Angular and Localized Effects: The perturbation function h(r, θ), defined as: h(r, θ) = exp −(r − r0)2 σ2 r exp −(θ − π/4)2 σ2 θ + exp −(θ − 3π/4)2 σ2 θ (16) contributes significantly to the angular variations. This is evident in terms like Term 11 and Term 17, where derivatives of h(r, θ) combine with sinusoidal terms, producing localized negative energy...

    work page

  11. [11]

    • Enhanced angular contributions from K(r, θ) = 1+ γ sin2(θ) and ω(r, θ), ensuring finite but negative terms

    Parameter-Driven Dominance: The choice of parameters such asϵ > 0, γ > 0, α > 0, λ > 0 ensures: • Smooth regularization near the throat with r − b(r) ∼ ϵ(r − r0)2. • Enhanced angular contributions from K(r, θ) = 1+ γ sin2(θ) and ω(r, θ), ensuring finite but negative terms. After carefully calculating and analyzing the terms and certain conditions under wh...

    work page

  12. [12]

    The factor P (θ) near θ = π/4 becomes: P (θ) ≈ csc2(π/4) + 1 ϵ · 4(π/4) − π σ2 θ

    Behavior at θ = π/4 At θ = π/4, the Gaussian term in h(r, θ) has a peak: exp −(θ − π/4)2 σ2 θ . The factor P (θ) near θ = π/4 becomes: P (θ) ≈ csc2(π/4) + 1 ϵ · 4(π/4) − π σ2 θ . Simplify: P (θ) ≈ − 2 ϵ · −π σ2 θ = 2π ϵσ2 θ . The integrating factor is: µ(θ) = exp Z 2π ϵσ2 θ dθ = exp 2πθ ϵσ2 θ . This grows exponentially for large θ. To avoid divergence, S(...

    work page

  13. [13]

    Near θ = 3π/4, P (θ) becomes: P (θ) ≈ csc2(3π/4) + 1 ϵ · 4(3π/4) − π σ2 θ

    Behavior at θ = 3π/4 Similarly, at θ = 3π/4, the Gaussian term in h(r, θ) peaks: exp −(θ − 3π/4)2 σ2 θ . Near θ = 3π/4, P (θ) becomes: P (θ) ≈ csc2(3π/4) + 1 ϵ · 4(3π/4) − π σ2 θ . Simplify: P (θ) ≈ 2π ϵσ2 θ . As before, the integrating factor grows exponentially: µ(θ) = exp 2πθ ϵσ2 θ . To avoid divergence, S(θ) must decay sufficiently fast. This requires...

    work page

  14. [14]

    Albert Einstein and Nathan Rosen, The Particle Problem in the General Theory of Relativity, Physical Review, 48, 73–77 (1935). 27

    work page 1935

  15. [15]

    Misner and John A

    Charles W. Misner and John A. Wheeler, Classical physics as geometry, Annals of Physics , 2(6), 525–603 (1957). doi:10.1016/0003-4916(57)90049-0

    work page doi:10.1016/0003-4916(57)90049-0 1957

  16. [16]

    Morris, Kip S

    Michael S. Morris, Kip S. Thorne, and Ulvi Yurtsever, Wormholes, Time Ma- chines, and the Weak Energy Condition, Phys. Rev. Lett. , 61(13), 1446–1449 (1988). doi:10.1103/PhysRevLett.61.1446

    work page doi:10.1103/physrevlett.61.1446 1988

  17. [17]

    A. C. Guti´ errez-Pi˜ neres, N. H. Beltr´ an, and C. S. L´ opez-Monsalvo, Newman-Janis Ansatz for rotating wormholes, Journal of Physics: Conference Series , 2081(1), 012005 (2021). doi:10.1088/1742-6596/2081/1/012005

    work page doi:10.1088/1742-6596/2081/1/012005 2081

  18. [18]

    doi:10.1103/PhysRevD.58.024014

    Edward Teo, Rotating traversable wormholes, Physical Review D , 58(2), 024014 (1998). doi:10.1103/PhysRevD.58.024014

    work page doi:10.1103/physrevd.58.024014 1998

  19. [19]

    doi:10.3390/sym16081007

    Ramesh Radhakrishnan et al., A Review of Stable, Traversable Wormholes in f(R) Gravity Theories, Symmetry, 16(8), 1007 (2024). doi:10.3390/sym16081007

    work page doi:10.3390/sym16081007 2024

  20. [20]

    Vilenkin and E

    A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects , Cambridge University Press, 2000

    work page 2000

  21. [21]

    Vladimir Dzhunushaliev, Vladimir Folomeev, Burkhard Kleihaus, Jutta Kunz, and Eu- gen Radu, Rotating Wormholes in Five Dimensions, Phys. Rev. D , 88, 124028 (2013). doi:10.1103/PhysRevD.88.124028

    work page doi:10.1103/physrevd.88.124028 2013

  22. [22]

    Peter K. F. Kuhfittig, Axially symmetric rotating traversable wormholes, Phys. Rev. D , 67, 064015 (2003). doi:10.1103/PhysRevD.67.064015

    work page doi:10.1103/physrevd.67.064015 2003

  23. [23]

    Aros and Nelson Zamorano, A Wormhole at the core of an infinite cosmic string, Phys

    Rodrigo O. Aros and Nelson Zamorano, A Wormhole at the core of an infinite cosmic string, Phys. Rev. D , 56, 6607–6614 (1997). doi:10.1103/PhysRevD.56.6607

    work page doi:10.1103/physrevd.56.6607 1997

  24. [24]

    Gerard Clement, Wormhole cosmic strings, Phys. Rev. D , 51, 6803–6809 (1995). doi:10.1103/PhysRevD.51.6803

    work page doi:10.1103/physrevd.51.6803 1995

  25. [25]

    Gerard Clement, Flat wormholes from straight cosmic strings, J. Math. Phys. , 38, 5807–5819 (1997). doi:10.1063/1.532167

    work page doi:10.1063/1.532167 1997

  26. [26]

    1: The Darmois-Israel formalism, Class

    Peter Musgrave and Kayll Lake, Junctions and thin shells in general relativity using com- puter algebra. 1: The Darmois-Israel formalism, Class. Quant. Grav. , 13, 1885–1900 (1996). doi:10.1088/0264-9381/13/7/018

    work page doi:10.1088/0264-9381/13/7/018 1900

  27. [27]

    Richarte and Claudio Simeone, More about thin-shell wormholes associated to cosmic strings, Phys

    Martin G. Richarte and Claudio Simeone, More about thin-shell wormholes associated to cosmic strings, Phys. Rev. D , 79, 127502 (2009). doi:10.1103/PhysRevD.79.127502

    work page doi:10.1103/physrevd.79.127502 2009

  28. [28]

    Bronnikov, Vladimir G

    Kirill A. Bronnikov, Vladimir G. Krechet, and Jos´ e P. S. Lemos, Rotating cylindrical worm- 28 holes, Phys. Rev. D , 87, 084060 (2013). doi:10.1103/PhysRevD.87.084060

    work page doi:10.1103/physrevd.87.084060 2013

  29. [29]

    doi:10.1103/PhysRevD.52.7318

    Eric Poisson and Matt Visser, Thin-shell wormholes: Linearization stability, Physical Review D, 52(12), 7318–7321 (1995). doi:10.1103/PhysRevD.52.7318

    work page doi:10.1103/physrevd.52.7318 1995

  30. [30]

    Morris, K.S

    Michael S. Morris and Kip S. Thorne, Wormholes in space-time and their use for inter- stellar travel: A tool for teaching general relativity, Am. J. Phys. , 56, 395–412 (1988). doi:10.1119/1.15620

    work page doi:10.1119/1.15620 1988

  31. [31]

    doi:10.1140/epjc/s10052-020-08698-x

    De-Chang Dai, Djordje Minic, and Dejan Stojkovic, How to form a wormhole, The European Physical Journal C , 80(12), (2020). doi:10.1140/epjc/s10052-020-08698-x

    work page doi:10.1140/epjc/s10052-020-08698-x 2020

  32. [32]

    doi:10.1103/PhysRevD.98.124026

    De-Chang Dai, Djordje Minic, and Dejan Stojkovic, New wormhole solution in de Sitter space, Physical Review D , 98(12), 124026 (2018). doi:10.1103/PhysRevD.98.124026

    work page doi:10.1103/physrevd.98.124026 2018

  33. [33]

    Simonetti, Michael J

    John H. Simonetti, Michael J. Kavic, Djordje Minic, Dejan Stojkovic, and De-Chang Dai, Sensitive searches for wormholes, Physical Review D , 104(8), L081502 (2021). doi:10.1103/PhysRevD.104.L081502

    work page doi:10.1103/physrevd.104.l081502 2021

  34. [34]

    doi:10.3390/universe7050136

    Cosimo Bambi and Dejan Stojkovic, Astrophysical Wormholes, Universe, 7(5), 136 (2021). doi:10.3390/universe7050136

    work page doi:10.3390/universe7050136 2021

  35. [35]

    doi:10.1393/ncb/i2005-10151-y

    Sung-Won Kim, Rotating wormhole and scalar perturbation, Il Nuovo Cimento B , 120(10011), 1235–1242 (2006). doi:10.1393/ncb/i2005-10151-y

    work page doi:10.1393/ncb/i2005-10151-y 2006

  36. [36]

    Matt Visser, Lorentzian wormholes: From Einstein to Hawking , Springer, 1995

    work page 1995

  37. [37]

    Thanu Padmanabhan, Gravitation: Foundations and Frontiers , Cambridge University Press, 2010

    work page 2010

  38. [38]

    Edmundo Capelas de Oliveira and Jos´ e Em´ ılio Maiorino, Analytical Methods in Applied Mathematics, Springer Nature, 2024

    work page 2024

  39. [39]

    Carroll, Spacetime and Geometry: An Introduction to General Relativity , Cambridge University Press, 2019

    Sean M. Carroll, Spacetime and Geometry: An Introduction to General Relativity , Cambridge University Press, 2019

    work page 2019

  40. [40]

    Barak Shoshany, OGRe: An Object-Oriented General Relativity Package for Mathematica, GitHub (2021). 29

    work page 2021