Spherically symmetric and static black bounces with multiple horizons, throats, and anti-throats in four dimensions

Manuel E. Rodrigues; Marcos V. de S. Silva

REVIEW 3 major objections 5 minor 50 references

Spherically symmetric and static black bounces with multiple horizons, throats, and anti-throats in four dimensions

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper constructs a new family of regular, asymptotically flat black-bounce spacetimes whose metric functions can oscillate, giving multiple horizons, wormhole throats, and anti-throats, with a partly phantom scalar field and a…

desk verdict Genuinely new family of smooth black-bounce metrics with multiple horizons and throats, but the claimed scalar+NED source has a likely single-valuedness problem in L(F), so the solution-to-action claim is not established. read the letter →

arxiv 2502.00502 v2 pith:AZWCDAWX submitted 2025-02-01 gr-qc

Manuel E. Rodrigues , Marcos V. de S. Silva This is my paper · ORCID

classification gr-qc

keywords blackbounceregularholewormholethroatmultiplehorizonsphantomscalarfieldnonlinearelectrodynamicsmagneticmonopoleenergyconditions

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

Black bounce spacetimes are regular alternatives to black holes in which a wormhole throat sits inside an event horizon. This paper proposes a new family of such spacetimes in four dimensions where the metric function and the area radius oscillate as the radial coordinate changes. The defining choice, $A(x)=1-\frac{2M\cos(a_0/\sqrt{a^2+x^2})}{\sqrt{a^2+x^2}}$ and $r^2(x)=x^2\cos^2(a_0/\sqrt{a^2+x^2})+a^2$, reduces to the standard black bounce when the extra parameter is set to zero and to Schwarzschild when both parameters vanish, while at infinity it always approaches Schwarzschild. Depending on the parameters, the geometry shows several horizons and several local minima and maxima of the area, i.e. throats and anti-throats, and a curvature invariant stays finite, so the spacetime is regular. The paper also identifies a matter source, a scalar field with a kinetic term that changes sign plus a nonlinear electromagnetic field with magnetic charge, and shows that the energy conditions are violated only in parts of the spacetime.

What carries the argument

The load-bearing structure is the pair of metric functions $A(x)=1-\frac{2M\cos(a_0/\sqrt{a^2+x^2})}{\sqrt{a^2+x^2}}$ and $r^2(x)=x^2\cos^2(a_0/\sqrt{a^2+x^2})+a^2$, with $M$, $a$, and $a_0$ as parameters. The cosine factors make both $A(x)$ and the area $4\pi r^2(x)$ oscillate with $x$, so zeros of $A(x)$ give horizons and extrema of the area give throats and anti-throats, while the constant $a^2$ term prevents $r$ from reaching zero. The source construction is carried by the identity $h(\varphi)\,\varphi'^2=-r''/r$, which fixes the scalar kinetic coupling once $\varphi$ is chosen as $\arctan(x/a)$, and then determines the electrodynamics functions $L(x)$ and $L_F(x)$; the consistency of those functions is asserted through the relation $L_F\,dF/dr-dL/dr=0$.

What would settle it

Substitute the metric functions and $\varphi=\arctan(x/a)$ into the four field equations and evaluate the residuals on a dense grid of $x$ across several oscillations for representative parameters; if any residual fails to vanish numerically, or if the consistency relation $L_F\,dF/dr-dL/dr=0$ breaks down away from the plotted points, the proposed source does not generate the spacetime. A second check is to scan allowed parameters for a divergence of the curvature invariant at finite $x$, which would contradict the regularity claim.

Watch

Extended reading notes

Core claim

The central claim is that the two functions $A(x)=1-\frac{2M\cos(a_0/\sqrt{a^2+x^2})}{\sqrt{a^2+x^2}}$ and $r^2(x)=x^2\cos^2(a_0/\sqrt{a^2+x^2})+a^2$ define a new exact family of static, spherically symmetric, asymptotically flat black-bounce solutions. The oscillatory cosines are what produce multiple zeros of $A(x)$, each a horizon, and multiple extrema of the area $4\pi r^2(x)$, each a throat or an anti-throat; the parameter $a$ keeps the area radius from reaching zero, which removes the central singularity. In the limits $a_0=0$ and $a_0=a=0$ the solution returns to the usual black-bounce metric and to Schwarzschild. The paper claims the spacetime is generated by a scalar field with a kinetic coupling that changes sign between canonical and phantom behavior, together with a nonlinear electrodynamics magnetic monopole, and it supports the identification through plots and the consistency condition $L_F\,dF/dr-dL/dr=0$.

Load-bearing premise

The load-bearing premise is that the reverse-engineered matter source is globally consistent: with the scalar field fixed as $\varphi=\arctan(x/a)$, there is a single-valued potential $V(\varphi)$ and a single-valued nonlinear electrodynamics Lagrangian $L(F)$ that satisfy all field equations everywhere, a point supported only by an asserted consistency relation and graphical checks rather than explicit formulas or a verification.

Editorial extensions

If this is right

The two parameters $a$ and $a_0$ tune the number and position of horizons, throats, and anti-throats, so physical predictions can be mapped as the causal structure changes.
At large distances the metric is Schwarzschild to leading order, so weak-field tests are preserved, while strong-field phenomena can differ because of the extra structure.
The source is a partially phantom scalar plus a magnetic monopole, so the spacetime is presented as a genuine solution of an Einstein-scalar-nonlinear-electrodynamics action rather than only a regularized metric.
Energy conditions are violated only in some regions, contrary to typical black bounces, so the model provides an example where wormhole-like features do not require global exotic matter.
The quasi-local mass can become negative while approaching $M$ at infinity, giving the geometry a nonstandard interior energy content.

Reading between the lines

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

The paper does not prove that the scalar potential and the electrodynamics Lagrangian are single-valued; the plotted Lagrangian develops cusps between oscillation branches, so a full action-level description may require branch choices or a modified formulation.
A direct numerical test of the field equations with the proposed source, including the oscillatory region, would either confirm or break the construction; such a check is not included.
The multiple horizons invite horizon-by-horizon thermodynamic quantities such as surface gravity, entropy, and temperature, and a generalized first law, none of which the paper computes.
If the family is extended to rotating or time-dependent metrics, the oscillatory structure should persist and could leave distinctive imprints in shadows and quasinormal-mode spectra.

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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

You should know that this paper's geometry is the strong part and its field-source construction is the weak part. The metric in Eq. (2) is genuinely new, explicit, and reduces cleanly to Simpson-Visser and Schwarzschild. The cosine modulations in A(x) and r^2(x) do produce multiple horizons, throats, and anti-throats, and the asymptotic analysis of the Kretschmann scalar makes the regularity claim plausible for a > 0. The quasi-local mass and energy-condition plots are honest and useful. So the paper is a real addition to the black-bounce program, though incremental over the authors' prior multi-throat work.

The problem is in Sec. III. The authors reverse-engineer a scalar field phi = arctan(x/a) and then derive h(x), L(x), and LF(x) from the field equations. They assert the consistency relation (27) but do not prove it, and more importantly they admit that because F(x) oscillates, L as a function of F splits into two curves joined at cusps. That is a red flag: an action-based L(F) must be single-valued. If L(F) is multi-valued, the action (11) is not well-defined, and the metric cannot be claimed as a solution of that action in any global sense. The consistency relation is local along the radial profile and does not resolve the branch ambiguity. So the central physical source claim is conditional at best, even though the geometry stands on its own as a regular metric.

Minor soft spot: global regularity is argued from asymptotics and a single numerical plot, not a complete proof. Given the explicit functions, that is fixable and not a deep flaw.

Serious refereeing should go ahead: the geometry is explicit and checkable, and the source-sector question is exactly what a good referee should push on. I would recommend sending it to review with a request to clarify whether a single-valued L(F) exists on the relevant range, and to prove or explicitly verify relation (27) rather than assert it. The paper, as written, is a useful metric family with an unfinished source story.

Referee Report

3 major / 5 minor

Summary. The paper introduces a new family of static, spherically symmetric metrics, Eq. (2), with A(x) = 1 - 2M cos(a0/sqrt(a^2+x^2))/sqrt(a^2+x^2) and r^2(x) = x^2 cos^2(a0/sqrt(a^2+x^2)) + a^2. It claims that these are regular black bounce spacetimes with multiple horizons, throats, and anti-throats, that they reduce asymptotically to Schwarzschild, and that they are generated by a partially phantom scalar field together with a nonlinear electrodynamics magnetic monopole. The analysis is largely graphical: horizon/throat structure and the Kretschmann scalar are illustrated, the source functions are displayed as plots, energy conditions are evaluated through an anisotropic fluid, and the quasi-local mass is discussed. The paper does not provide explicit formulas for V(phi) or L(F), and it asserts rather than proves the consistency of the reverse-engineered matter sector.

Significance. If the construction were complete, the family would be a useful addition to the black bounce program: it is an explicit deformation of Simpson-Visser with richer causal structure and potentially new observational signatures. The geometrical part of the claim is credible because for a>0 the metric functions are smooth, r^2(x) >= a^2 > 0, the spacetime is asymptotically flat, and the oscillatory coefficients can plausibly generate multiple horizons and area extrema. However, the central physical claim is that Eq. (2) is a solution of the action (11) with a scalar field and a single-valued NED Lagrangian; that claim is not established and appears to fail on the manuscript's own admission that L(F) splits into two different curves. The paper is explicit about its reverse-engineering method, which is a strength, but the source sector remains incomplete. The significance is therefore prospective rather than established.

major comments (3)

[Sec. III, Eq. (27) and text after Fig. 6] The action (11) requires a single-valued Lagrangian L(F). The paper states that oscillations in F(x) generate two different curves in the Lagrangian, with transitions through a cusp; this is an explicit admission that no global single-valued function L(F) exists. The consistency relation (27) is a local identity along the one-parameter curve x and cannot exclude multi-valuedness on disconnected level sets of F. Therefore the claim that the metric (2) is generated by the scalar-field plus NED action (11) is not supported. The authors need to exhibit L(F) as a single-valued function and verify (27) globally, or explicitly restrict the solution to a domain on which F is monotonic, which would forfeit the multiple-throat structure.
[Sec. III, Eqs. (19)-(26)] The scalar potential V(phi) and the NED Lagrangian L(F) are never given as explicit or parametric functions; V is only displayed graphically in Fig. 5 and L in Fig. 6. A plot is not a definition of a function, and without formulas for V and L, together with their domains, the field equations (19)-(22) cannot be checked. The asserted consistency relation (27) is also presented without derivation or numerical verification. This is a load-bearing gap because the central claim is that Eq. (2) solves the field equations of action (11).
[Sec. II, Eqs. (2)-(7)] The counting of horizons, throats, and anti-throats is only demonstrated for one numerical example in Fig. 1 (and a second parameter set later in the energy-conditions section). The abstract and introduction claim multiple horizons and throats, but no parameter ranges or existence statements are proven. Since the metric functions are explicit, a rigorous count should be possible; the present claims are purely graphical.

minor comments (5)

[Sec. II, Eq. (4)] The symbol A is used both for the metric function A(x) and for the area 4 pi r^2(x), which is confusing; a different symbol such as script-A should be used for the area.
[Sec. II, Fig. 1 caption] The caption refers to the 'area of a spherical surface with radius x', but the horizontal coordinate is x and the areal radius is r(x); the caption should say 'with areal radius r(x)'.
[Sec. II, Eq. (3)] The paper only discusses the asymptotic region x going to +infinity, but since r(x) is approximately |x|, there is also an asymptotically flat region as x goes to -infinity; the two-end structure should be stated explicitly.
[Sec. II, Eq. (2)] At x=0, r(0)=a>0, so x=0 is a wormhole throat rather than a 'center' of the spacetime; the terminology should be adjusted accordingly.
[Sec. II, Eq. (5)] The regularity argument would be stronger and simpler if the authors noted that for a>0, r^2(x) >= a^2 > 0 and A(x), r(x) are smooth on R, so the Kretschmann scalar (5) is smooth on R; the asymptotic expansions then suffice to establish boundedness.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the metric family is an explicit ansatz, and the matter sources are openly reverse-engineered from the field equations; the few self-citations supply method rather than load-bearing support.

full rationale

The derivation is self-contained in the sense relevant to circularity. Section II defines the metric functions A(x) and r^2(x) as an explicit ansatz (Eq. 2); horizons, throats, and anti-throats are read directly from A(x)=0 and extrema of r^2(x), while regularity is analyzed from the standard Kretschmann formula. Section III specifies the action (11) and then solves the field equations backwards: phi=arctan(x/a) is chosen explicitly, h(phi) is obtained from Eq. (26), and V and L are determined from Eqs. (23)-(24). The paper itself acknowledges this is 'a type of reverse engineering' (Sec. V), so no fitted quantity is relabeled as a prediction. The self-citations to [13], [25], and [47] are methodological: [13] supplies the Kretschmann and quasi-local-mass formulas, [25] supplies the monotonic-scalar-field construction, and [47] supplies the throat/anti-throat language. None is used as an external uniqueness theorem that forces the metric. The main unproven step is Eq. (27), the consistency relation L_F dF/dr - dL/dr = 0, which is asserted without derivation, and the paper admits that oscillations in F make L(F) split into two curves joined at a cusp; this is a real correctness risk for the existence of a single-valued NED Lagrangian, but it is not a circular reduction of the result to its inputs. The regularity claim is supported only asymptotically and graphically, which is likewise an evidentiary limitation rather than circularity. Score 2 reflects the presence of same-author citations in the construction chain; the central metric and source-construction claims do not reduce to those citations.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central geometric construction rests on the hand-chosen metric parameters a and a0, with M and q as input source parameters. No new particles, forces, or dimensions are introduced; the scalar+NED action is itself assumed. The main hidden load is the global consistency of the reverse-engineered source functions.

free parameters (4)

a = 0.1, 0.2, 1 in figures
Regularization length chosen by hand; sets the throat radius at x=0 via r^2(0)=a^2 and controls the oscillations of the metric functions.
a0 = 4 in most figures
Parameter controlling the number of horizons, throats, and anti-throats; chosen to make A(x) and r(x) oscillate.
M = 10 in Figs. 1-2, 1 in Figs. 3 and 7
ADM mass, an input from the Schwarzschild limit; values chosen for plots, not fitted to observations.
q = 0.2 in Fig. 6
Magnetic charge in the nonlinear electrodynamics source; chosen for plots.

assumptions (5)

domain assumption Einstein equations with the action (11) containing a scalar field with kinetic coupling h(phi) and NED term L(F)
The paper assumes this action is the correct framework for sourcing the metric.
standard math Metric ansatz (1) with g_xx=-1/A and r^2(x)>=a^2>0
Standard line element for static spherically symmetric spacetimes; smoothness of the coordinate functions is assumed.
domain assumption Bronnikov reverse-engineering method determines sources from a given metric
Used throughout Sec. III, relying on prior work [24,25] without re-derivation.
ad hoc to paper Choice phi=arctan(x/a) is admissible and yields a well-defined h(phi)
Imposed in Sec. III to integrate Eq. (25); no independent justification is given.
domain assumption Kretschmann scalar boundedness at x=0 and infinity plus smoothness implies global regularity
Used in Secs. II and V; geodesic completeness and other curvature invariants are not checked.

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Cite this review

Pith. "Pith review of Spherically symmetric and static black bounces with multiple horizons, throats, and anti-throats in four dimensions." pith.science (2026). https://pith.science/paper/AZWCDAWX

@misc{pith2026250200502,
  author       = {Pith},
  title        = {Pith review of: Spherically symmetric and static black bounces with multiple horizons, throats, and anti-throats in four dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AZWCDAWX}},
  note         = {Machine review of arXiv:2502.00502}
}

read the original abstract

Black bounce spacetimes usually arise from the Simpson-Visser regularization method. This type of metric presents a wormhole throat inside an event horizon. In this paper, we presented new classes of black bounce spacetime solutions, which have multiple horizons, throats, and anti-throats. These solutions are variants of black holes and wormholes, based on modifications of the Schwarzschild and Simpson-Visser metrics. The metric function allows for multiple horizons and throats, and the asymptotic behavior recovers the Schwarzschild solution. The article considers a scalar field coupled to nonlinear electrodynamics, generating solutions with a partially phantom scalar field and a magnetic monopole. The energy conditions can be satisfied or violated depending on the region of spacetime, analyzed through an anisotropic fluid. The regularity of the spacetime is ensured by the analysis of the Kretschmann scalar.

Figures

Figures reproduced from arXiv: 2502.00502 by the authors.

**Figure 2.** Graphical representation of the Kretschmann scal [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Hernandez-Misner-Sharp quasi-local mass for the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Behavior of the functions h(φ(x) and φ(x) in terms of the radial coordinate. We are considering a0 = 4, M = 10, and a = 0.2. In this way, the electromagnetic expressions will only be determined once we have the expressions for the functions related to the scalar field.…

**Figure 5.** Figure 5: Potential associated with the scalar field as a func [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: Behavior of the electromagnetic functions in term [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: Graphical representation of the energy condition [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

Discussion (0). Continue with ORCID to comment.

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