Wormholes from beyond

Juliano C. S. Neves

REVIEW 2 major objections 5 minor 1 cited by

Wormholes from beyond

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Two wormhole solutions can arise purely from a higher-dimensional bulk.

desk verdict Useful Nakas-Kanti embeddings of Morris-Thorne and Molina-Neves wormholes, but the claimed global bulk regularity rests on an unflagged modulus drop in the conformal factor. read the letter →

arxiv 2412.11947 v2 pith:MJEPVC3K submitted 2024-12-16 gr-qc hep-th

Juliano C. S. Neves This is my paper · ORCID

classification gr-qchep-th MSC 83E15

keywords braneworldwormholeMorris-ThorneMolina-NevesRandall-SundrumIIinducedgeometryexoticfluidextradimension

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 two known traversable wormhole geometries—the Morris-Thorne wormhole and the Molina-Neves wormhole—can be induced on a four-dimensional brane embedded in a regular five-dimensional anti-de Sitter bulk. On the brane, no fields or particles are needed to support them; the entire support comes from the bulk's exotic fluid and curvature. The paper constructs explicit five-dimensional metrics whose induced metric on the brane solves the effective four-dimensional field equations with the wormhole's Einstein tensor. If correct, this shows that wormhole-like shortcuts could exist in our universe even if the brane itself is empty.

What carries the argument

The carrying device is the Nakas-Kanti uplift, which builds a regular five-dimensional bulk from a desired brane metric. Start with anti-de Sitter space in coordinates where the bulk is conformally flat, then replace the flat factor $-dt^2+d\rho^2$ with a spherically symmetric four-dimensional metric $-A(\rho)dt^2+d\rho^2/B(\rho)+\rho^2 d\Omega_3^2$, warped by the AdS conformal factor. This produces a five-dimensional wormhole whose induced metric on the brane is read off at $y=0$. The link between bulk and brane is the effective four-dimensional field equations, where the projected Weyl tensor and an effective stress tensor built from the bulk fluid combine to give the wormhole's Einstein tensor without any brane sources.

What would settle it

Compute the conformal factor $1+k\rho\cos\chi$ over the full allowed range $\chi\in(0,\pi]$ and $\rho\ge b_0$; if it ever reaches zero in the coordinate chart used for the two wormhole solutions, the metric signature changes there and the bulk is not geodesically complete. A concrete check would be to integrate null geodesics that cross the brane into the $\chi>\pi/2$ side and see if they hit a signature boundary.

Watch

Extended reading notes

Core claim

The central claim is that the Morris-Thorne and Molina-Neves wormholes are not merely four-dimensional solutions but can be obtained as the induced metric on a brane in the Randall-Sundrum II brane-world scenario. Starting from a regular, asymptotically anti-de Sitter five-dimensional spacetime with a conformal factor that warps the extra dimension, the author substitutes the wormhole metric functions into a spherically symmetric bulk ansatz. The resulting five-dimensional metrics are regular everywhere, with the wormhole throat extending into the bulk, and their energy-momentum tensor describes an anisotropic exotic fluid. On the brane at $y=0$, the effective field equations, including the projected Weyl term, yield precisely the Einstein tensor of the corresponding four-dimensional wormhole. Thus the wormhole is supported by bulk influence alone: the brane has no energy-momentum sources ($\tau_{\mu\nu}=0$), and the brane tension is fixed by the bulk curvature parameter.

Load-bearing premise

The argument needs the Z2-symmetric shortcut of replacing $|\cos\chi|$ by $\cos\chi$ in the conformal factor, so that Eq. (6) describes the whole bulk; if this drops a region where the factor changes sign, the bulk may not be regular everywhere.

Editorial extensions

If this is right

If the central claim is right, wormholes in our four-dimensional universe would not require exotic matter living on the brane: the bulk alone can act as the support.
The brane can be asymptotically flat (Morris-Thorne) or asymptotically de Sitter (Molina-Neves), so the same mechanism can supply an effective cosmological constant on the brane.
The five-dimensional bulk remains regular everywhere, avoiding the black-string singularity, so a healthy embedding exists for at least these two wormhole solutions.
Because the wormhole throat extends into the extra dimension, the four-dimensional throat is not the whole story; bulk probes could in principle feel the throat's extra-dimensional extent.

Reading between the lines

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

The same construction could probably induce other wormhole metrics, including charged or rotating ones, by choosing different $A$ and $B$, though the paper does not work those out.
The apparent removal of the absolute value in the conformal factor may hide a coordinate-patch subtlety: for $\chi$ in $\left(\pi/2,\pi\right]$, $1+k\rho\cos\chi$ can vanish, so verifying that the metric really covers both sides of the brane without a signature change is a natural next check.
If these induced wormholes exist, observations of compact objects through gravitational lensing or shadows would need to distinguish a bulk-supported wormhole from a brane-matter wormhole or a black hole.
The exotic fluid is still there—merely displaced into the bulk—so energy-condition violation is shifted, not eliminated; whether this is an improvement depends on one's stance on higher-dimensional physics.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The paper does something concrete and useful: it takes two known 4D wormhole metrics and shows, by explicit construction, that they can be embedded as induced metrics on a brane in a regular asymptotically AdS5 bulk, with no brane-localized matter. The bulk energy-momentum tensor is written out, the Israel junction condition is checked, and the effective brane equations reduce to the Morris-Thorne and Molina-Neves Einstein tensors. That is real work, and the checks are careful.

The main soft spot is the treatment of the conformal factor. Eq. (4) is the RSII metric in spherical coordinates with |cos chi|; Eq. (6) drops the modulus and uses cos chi. For chi in (pi/2, pi], the factor 1 + k rho cos chi vanishes at rho = 1/(k|cos chi|). That point lies inside the parameter range used in the paper (e.g., k=1, b0=0.5, chi=3pi/4 gives rho ~ 1.41), so the five-dimensional metric is singular there, not regular. The original RSII coordinate z = sgn(y)(e^{k|y|}-1)/k is bounded below by -1/k, so the full sphere chi in [0, pi] with rho in [b0, infinity) is not the right coordinate domain. This does not touch the brane physics: at z=0, chi=pi/2, cos chi=0, so the induced metric, the junction condition, and the effective field equations are unaffected. But the claim of a globally regular bulk is not supported as written. The fix is simple: restrict the bulk to the Z2 fundamental domain z>=0 (chi in [0, pi/2]) or keep |cos chi| and the z>-1/k bound throughout.

A second, minor issue: the relation to [45] and [22] is acknowledged but could be spelled out more explicitly. That is a revision-level point, not a flaw.

Bottom line: the paper is a solid consistency construction in the Nakas-Kanti program. The brane-side results are sound; the bulk-side global statement needs a coordinate-domain correction. I would send it to a serious referee, with the request to fix the conformal-factor issue and re-check the regularity claims.

Referee Report

2 major / 5 minor

Summary. This paper extends the Nakas-Kanti brane-world construction to traversable wormholes. Starting from the Randall-Sundrum II metric in spherical coordinates, the author replaces the flat temporal-radial part by a general spherically symmetric ansatz and chooses the metric functions A(ρ) and B(ρ) to match the Morris-Thorne and Molina-Neves wormholes. The resulting five-dimensional geometries are claimed to be regular at the throat and asymptotically anti-de Sitter, and the corresponding bulk energy-momentum tensors are computed. Using the Shiromizu-Maeda-Sasaki effective brane equations and the Israel junction conditions, the paper verifies that the induced brane metric satisfies the wormhole field equations with τμν=0, i.e., without brane-confined matter, and concludes that the wormholes are induced purely by the bulk influence.

Significance. If the construction is properly regularized, the paper provides a useful extension of the Nakas-Kanti program from black holes to wormholes. Its strengths are its explicitness: the bulk metrics, curvature invariants, energy-momentum tensors, and the cancellation of singular terms in the effective brane equations are written out in detail, and the brane junction conditions are checked. The paper is also clear that this is an inverse consistency construction rather than an independent derivation, which is the intended scope of the Nakas-Kanti approach. However, the mathematical gap concerning the modulus in the conformal factor affects the global bulk-regularity claim, and the asymptotic-AdS statement is direction-dependent in the chosen coordinates; these issues need to be resolved before the central claim is fully supported.

major comments (2)

[Sec. II, Eq. (6)] The transition from Eq. (4) to Eq. (6) drops the modulus of cosχ in the conformal factor. This is not a neutral simplification over the stated coordinate range χ∈[0,π] in Eq. (3). On the left side, χ∈(π/2,π], cosχ<0, and the factor 1+kρ cosχ vanishes at ρ=1/(k|cosχ|), which lies inside the domain ρ∈[b0,∞) for representative parameters: with k=1, b0=0.5 and χ=π, the zero occurs at ρ=1. The original metric (4) is regular there, so Eq. (6) is a different spacetime on the negative-z side, not an equivalent rewriting. Consequently, the global regularity statements after Eq. (35) and in Fig. 2, as well as the asymptotic limits (30)-(32) taken for the full bulk, are not established as written. The author should either retain |cosχ| in the bulk metric or explicitly restrict the construction to the Z2 fundamental domain z≥0 (χ∈[0,π/2]) and then extend by reflection, stating the domain whenever 'regular everywhere' is claimed.
[Sec. IVA, Eqs. (30)-(32) and Sec. IVB, Eqs. (59)-(61)] The limits as ρ→∞ are direction-dependent in the (ρ,χ) chart. For χ=π/2 (the brane), the conformal factor in Eq. (6) is unity, so the metric approaches five-dimensional Minkowski space rather than AdS5; the Ricci scalar and Kretschmann limits (30)-(32) therefore do not hold uniformly over χ∈[0,π]. The asymptotically-AdS5 property is properly defined by y→∞ at fixed r in the original coordinates, as in Eqs. (7), (42), and (43). Please clarify that the ρ→∞ limits are taken at fixed χ≠π/2, or restate the asymptotic-AdS claim using the y coordinate or an explicit directional limit.

minor comments (5)

[Sec. II, after Eq. (6)] The text says 'the modulus of y in Eq. (4) could be dropped,' but Eq. (4) contains |cosχ|, not |y|; this should read 'modulus of cosχ' or be rephrased.
[Fig. 2 caption and Sec. IVA] The caption for Fig. 2 should specify the coordinate ranges used in the plot and the parameter values, and should state whether the plotted domain is the Z2 fundamental domain or the full χ∈[0,π] range; otherwise the figure cannot support the 'regular everywhere' statement.
[Secs. IVA and IVB] The off-diagonal bulk energy-momentum components T^r_y and T^y_r are mentioned but never written explicitly; for completeness, they should either be given or the reader should be told that they are omitted for brevity.
[Sec. IVB, Eq. (58)] The relation k² = Λ4d/3 − Λ5d/6 is imposed to make the bulk asymptotically anti-de Sitter; it should be explicitly labeled as a tuning condition or assumption, and its physical or mathematical justification (if any) should be discussed.
[Author affiliation] There is a typographical error in the affiliation: 'Poços de C aldas' should be 'Poços de Caldas'.

Circularity Check

2 steps flagged · score 6.0 of 10

The claimed wormhole-on-brane results are inserted by hand through the A(ρ), B(ρ) ansatz; the bulk regularity and no-brane-matter calculations are independent checks but the wormhole geometry itself is an input, not a prediction.

self definitional [Sec. II (Eq. 6) and Sec. IV.A (Eqs. 26-29, 49-51)]
"Adopting the two metric functions A(ρ) and B(ρ) of known four-dimensional spacetime metrics yields five-dimensional black holes [19–21] or, as we will see, wormholes. In this study, A(ρ) and B(ρ) will be the same of the Morris-Thorne [23] and Molina-Neves [9] wormholes."

Equation (6) is the bulk ansatz with arbitrary A(ρ), B(ρ). The brane is at z=0, i.e. χ=π/2, where the conformal factor (1+kρ cosχ)^{-2} equals 1 and ρ=r. Therefore the induced metric on the brane is exactly the four-dimensional wormhole metric whose functions A and B were inserted into the bulk ansatz. The later computation of Eµν and T(eff)µν and the recovery of the Einstein tensor (Eqs. 49-51) verify that this chosen metric satisfies the effective brane equations, but it does not derive the wormhole from an independent bulk input. The 'Morris-Thorne wormhole obtained on the brane' is equivalent, by construction, to the A, B chosen in Eq. (6).
self definitional [Sec. IV.B (Eqs. 52-56, 87-90)]
"Like the Morris-Thorne case, in order to get a five-dimensional wormhole one should insert (52) into Eq. (6). In this case A(ρ)=e^{2Φ(ρ)} and B(ρ)=e^{2Φ(ρ)}(1−b(ρ)/ρ)."

Equation (52) is the Molina-Neves wormhole, which is the target metric. Inserting its functions into Eq. (6) makes the brane (z=0) metric equal to that same wormhole by construction. The subsequent statement that the Einstein tensor on the brane is the same as obtained by Molina and Neves [9] is therefore a consistency check of the chosen ansatz, not an independent derivation of the wormhole geometry from the bulk fluid. The bulk energy-momentum tensor is then solved from the Einstein equations for that chosen geometry, so the 'bulk influence' is the source required by the ansatz rather than an independently predicted cause.

full rationale

The circularity is confined to the step that presents the wormhole metrics as 'obtained on the brane': because the brane lies at z=0 where the conformal factor in Eq. (6) is unity, the induced metric is literally the A(ρ), B(ρ) functions chosen for the bulk. The paper is transparent about this, saying A and B will be the same as the Morris-Thorne and Molina-Neves wormholes, and it explicitly calls the brane-field-equation check a verification of a necessary condition. That is a constructive consistency argument rather than a first-principles derivation, and the wormhole result is equivalent to its input by construction. However, several nontrivial claims are independent of this circular step: the bulk regularity scalars, the explicit bulk energy-momentum tensor, the Israel-junction demonstration that τµν=0, and the weak-energy violation in the bulk are all computed, not assumed. There is no load-bearing self-citation chain: the Nakas-Kanti approach is cited from independent authors, and the author-overlap citations ([8], [9], [21]) supply target metrics or earlier checks rather than the argument's justification. The Z2/modulus concern raised about Eq. (6) is a mathematical-coordinate-domain issue, not a circularity, and is therefore not scored here. Overall, the central 'wormhole-on-brane' prediction reduces to the ansatz, warranting a partial-circularity score of 6.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The central claim rests on standard brane-world effective equations plus an imposed bulk metric ansatz. The genuinely ad hoc ingredients are the consistency relation (58) and the coordinate-domain assumption needed to keep the conformal factor positive.

free parameters (5)

b0 (Morris-Thorne) = arbitrary (e.g., 0.5 in figures)
Throat radius of the brane wormhole; chosen by hand as the size parameter of the solution.
b0 (Molina-Neves) = arbitrary (e.g., 1.65 in figures), constrained by C = b0(b0^2-r0^2)^(3/2)
Throat radius on the de Sitter brane; sets the integration constant C.
k = related to AdS5 radius; in MN case k^2 = Λ4d/3 - Λ5d/6
Bulk curvature parameter; in Randall-Sundrum II it sets brane tension λ=6k/κ5^2. Chosen by hand, not fitted.
Λ4d = positive (e.g., 1 in figures)
Effective cosmological constant on the brane for the Molina-Neves case; sets rc = sqrt(3/Λ4d) and r0 = sqrt(2/Λ4d).
Λ5d = negative (e.g., -1 or -4 in figures)
Effective five-dimensional cosmological constant simulated by the bulk exotic fluid; must be negative for AdS5 asymptotics.

assumptions (5)

domain assumption The five-dimensional bulk field equations are G_MN = κ5^2 T^(B)_MN without a bare cosmological constant; the bulk fluid simulates the AdS5 asymptotics.
Section II, Eq. (8). The exotic fluid is introduced as the source; there is no independent physical motivation for its form.
standard math The Shiromizu-Maeda-Sasaki effective brane equations (24) are the correct effective field equations, with the brane carrying only tension (τ_μν=0) after applying the Israel junction conditions.
Section III, Eqs. (20)-(24). Standard brane-world result accepted as background.
domain assumption The Z2 symmetry of Randall-Sundrum II allows dropping the modulus in the conformal factor, so Eq. (6) represents the full bulk.
Section II, text after Eq. (5). This is the weakest assumption; requires restricting to z≥0 (or equivalently χ in [0,π/2]) to avoid a conformal factor sign change.
domain assumption The radial coordinate range is ρ in [b0,∞) and r in [0,∞), which ensures B(ρ)≥0 and a Lorentzian signature.
Section IV.A, text after Eq. (38). Needed for the coordinate chart to cover the wormhole exterior.
ad hoc to paper For the Molina-Neves bulk, the relation k^2 = Λ4d/3 - Λ5d/6 (Eq. 58) is imposed so the bulk is asymptotically AdS5.
Section IV.B, Eq. (58). This is a consistency condition that couples brane and bulk parameters; it is not derived from a deeper principle and is specific to this construction.

invented entities (1)

Anisotropic bulk exotic fluid (T^(B)_MN)
purpose: Supports the five-dimensional wormhole and simulates the AdS5 cosmological constant; its presence induces the 4D wormhole on the brane while the brane carries no matter.
The fluid components in Eqs. (39)-(41) and (71)-(73) are computed from the Einstein tensor of the chosen metric, not from an independent matter model. It violates the weak energy condition near the brane and has no external falsifiable handle.

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Cite this review

Pith. "Pith review of Wormholes from beyond." pith.science (2026). https://pith.science/paper/MJEPVC3K

@misc{pith2026241211947,
  author       = {Pith},
  title        = {Pith review of: Wormholes from beyond},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MJEPVC3K}},
  note         = {Machine review of arXiv:2412.11947}
}

read the original abstract

In a brane-world context in which our universe would be a four-dimensional brane embedded into a five-dimensional spacetime or bulk, wormhole geometries are induced on branes. In this article, the Morris-Thorne wormhole and the Molina-Neves wormhole are obtained on the brane using the Nakas-Kanti approach, which starts from a regular five-dimensional spacetime to obtain known black hole and wormhole solutions on the four-dimensional brane. From the bulk perspective, these wormholes are five-dimensional solutions supported by an exotic fluid, but from the brane perspective, such objects are wormholes not supported by any fields or particles that live on the four-dimensional spacetime. Thus, the cause of these wormholes is the bulk influence on the brane.

Figures

Figures reproduced from arXiv: 2412.11947 by the authors.

**Figure 2.** The Kretschmann scalar RMNLK R MNLK for the five-dimensional Morris-Thorne wormhole indicates that spacetime is regular in the extra dimension y. In this graphic, one uses k = 1 and b0 = 0.5. As we can see in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Components of the bulk energy-momentum tensor [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The Kretschmann scalar RMNLK R MNLK for the five-dimensional Molina-Neves wormhole indicates that spacetime is also regular in the extra dimension y. In this graphic, one uses C = Λ4d = 1, Λ5d = −1, k = 1/ √ 2, and b0 = 1.65 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: B(r, y) for the Molina-Neves metric. The black line over the surface indicates that the cosmological horizon r∗ = (rc, yc) extends to the extra dimension y. In this graphic, one adopts Λ4d = 1 and Λ5d = −4, consequently k = C = 1. like the event horizon of black holes …

**Figure 6.** Figure 6: Components of the bulk energy-momentum tensor [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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Embedding of Ellis-Bronnikov wormhole and NED-sourced magnetic/dyonic black strings into RS braneworlds using Local Sum Rules; the dyonic q→0 limit is inconsistent as written.

Reference graph

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Together with Eµν , the tensor T (eﬀ) µν measures the inﬂuence of the bulk ﬁelds on the brane

The new energy-momentum tensor (diagonal tensor) T (eﬀ) µν is some sort of an eﬀective energy-momentum ten- sor on the brane. Together with Eµν , the tensor T (eﬀ) µν measures the inﬂuence of the bulk ﬁelds on the brane. It is deﬁned as T (eﬀ) µν = 2 3k [ T (B) µν + ( T (B) yy − T (B) 4 ) hµν ] ⏐ ⏐ ⏐ ⏐ ⏐ y→ 0 , (25) calculated at y = 0. Therefore, any spa...

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(26) Here I will focus on the simplest case, a case in which spacetime is horizonless and without tidal forces. Thus, the metric functions are Φ( r) = 0 , (27) b(r) = √ b0r, (28) where rthr = b0 is the wormhole throat coordinate in four dimensions. Following the Nakas-Kanti approach, then one inserts the metric (26) with above functions into (6), where in...

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electric

(70) The matter content for the ﬁve-dimensional spacetime is studied from the ﬁeld equations (8) like the previous case. The energy-momentum tensor components in the (t, ρ, θ, φ, χ) coordinates are T (B)t t = 1 κ2 5 { C (ρ2 − r2 0) 5 2 [ (4ρ2 r2 0 − 10 + 9r2 0 2ρ2 ) k cos χ − 3r2 0 ρ3 ] − Λ 5d } , (71) T (B)ρ ρ = 1 κ2 5 { C (ρ2 − r2 0) 3 2 [ 4 r2 0ρ + 3k ...

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(r2 − r2 0) + C2( 2r2 − r2 0 ) 4r3 (r2 − r2 0) 5 2 , (81) Eθ = Eφ = r ( r2 − r2 0 )3 2 6r2 − 9r2 0 + C ( 8r4 − 17r2 0r2 + 6r4 0 ) 6r2 (2r2 − 3r2

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+ C [( 16 − 36k2r2 0 ) r6 − ( 32 − 90k2r2 0 ) r2 0r4 + ( 13 − 54k2r2 0 ) r4 0r2 − 3r6 0 ] 6r2 0r2 (2r2 − 3r2

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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work page Pith review arXiv 2000

[3] [1]

Together with Eµν , the tensor T (eﬀ) µν measures the inﬂuence of the bulk ﬁelds on the brane

The new energy-momentum tensor (diagonal tensor) T (eﬀ) µν is some sort of an eﬀective energy-momentum ten- sor on the brane. Together with Eµν , the tensor T (eﬀ) µν measures the inﬂuence of the bulk ﬁelds on the brane. It is deﬁned as T (eﬀ) µν = 2 3k [ T (B) µν + ( T (B) yy − T (B) 4 ) hµν ] ⏐ ⏐ ⏐ ⏐ ⏐ y→ 0 , (25) calculated at y = 0. Therefore, any spa...

work page

[4] [2]

eﬀective

(26) Here I will focus on the simplest case, a case in which spacetime is horizonless and without tidal forces. Thus, the metric functions are Φ( r) = 0 , (27) b(r) = √ b0r, (28) where rthr = b0 is the wormhole throat coordinate in four dimensions. Following the Nakas-Kanti approach, then one inserts the metric (26) with above functions into (6), where in...

work page

[5] [3]

electric

(70) The matter content for the ﬁve-dimensional spacetime is studied from the ﬁeld equations (8) like the previous case. The energy-momentum tensor components in the (t, ρ, θ, φ, χ) coordinates are T (B)t t = 1 κ2 5 { C (ρ2 − r2 0) 5 2 [ (4ρ2 r2 0 − 10 + 9r2 0 2ρ2 ) k cos χ − 3r2 0 ρ3 ] − Λ 5d } , (71) T (B)ρ ρ = 1 κ2 5 { C (ρ2 − r2 0) 3 2 [ 4 r2 0ρ + 3k ...

work page

[6] [4]

(r2 − r2 0) + C2( 2r2 − r2 0 ) 4r3 (r2 − r2 0) 5 2 , (81) Eθ = Eφ = r ( r2 − r2 0 )3 2 6r2 − 9r2 0 + C ( 8r4 − 17r2 0r2 + 6r4 0 ) 6r2 (2r2 − 3r2

work page

[7] [5]

(82) It is worth emphasizing two points: ﬁrstly, the condition Eµ µ = 0 is satisﬁed

(r2 − r2 0) − C2( 2r2 − r2 0 ) 4r3(r2 − r2 0) 5 2 . (82) It is worth emphasizing two points: ﬁrstly, the condition Eµ µ = 0 is satisﬁed. Secondly, the tensor Eµν is singular for r = rc = √ 3/2 r0 and at the throat, r = b0, where Eq. (69) is valid. This problem will be circumvented when the eﬀective ﬁeld equations were written. Before writing the eﬀective ...

work page

[8] [6]

+ C [( 16 − 36k2r2 0 ) r6 − ( 32 − 90k2r2 0 ) r2 0r4 + ( 13 − 54k2r2 0 ) r4 0r2 − 3r6 0 ] 6r2 0r2 (2r2 − 3r2

work page