pith. sign in

arxiv: 2606.29401 · v1 · pith:43LF3ZFXnew · submitted 2026-06-28 · 🌀 gr-qc

Self-force on a static scalar charge in traversable wormholes

Jerome P. Mecca , Ian Vega This is my paper

Pith reviewed 2026-06-30 02:34 UTC · model grok-4.3

classification 🌀 gr-qc
keywords self-forcescalar chargetraversable wormholesmode-sum regularizationredshift parametershape exponentasymptotic falloffzero crossings
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The pith

The self-force on a static scalar charge in traversable wormholes changes sign with distance from the throat and can have up to two zero crossings.

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

This paper computes the static scalar self-force on a point charge held at rest in a two-parameter family of spherically symmetric traversable wormholes. It establishes that the force is generally not unidirectional, reversing direction one or two times depending on the redshift parameter and the shape exponent that set the metric. A sympathetic reader would care because the self-force encodes global spacetime structure rather than local curvature alone, so its reversals and modified decay rates offer a concrete signature of the wormhole geometry. The work also maps how the large-distance falloff varies with the parameters, sometimes slower than the inverse-cube law found in isolated-body spacetimes.

Core claim

In these wormhole geometries the static scalar self-force is generally not unidirectional. It changes sign with radial distance from the throat, with up to two distinct zero crossings whose locations depend on the two metric parameters. Both the direction of the force and its asymptotic falloff rate are characterized as functions of those parameters, with slower-than-canonical decay when the redshift parameter is large and faster decay when the shape exponent is more negative. In the combined limit of infinite redshift parameter and shape exponent to negative infinity the falloff approaches the behavior found in a limiting case of the family.

What carries the argument

Mode-sum regularization applied to the scalar Green's function in the wormhole metric, which removes the divergent singular contribution to leave a finite self-force at each point.

If this is right

  • The self-force direction at a given location depends on both radial position and the two metric parameters.
  • For sufficiently large redshift parameter the force decays more slowly than the inverse-cube law typical of isolated systems.
  • More negative values of the shape exponent produce faster large-distance decay.
  • The number of zero crossings can be zero, one, or two according to the parameter pair.

Where Pith is reading between the lines

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

  • The position-dependent sign changes imply that a charge placed near the throat could experience a net push or pull that differs from expectations in black-hole exteriors.
  • Slower asymptotic decay for large redshift parameter means the wormhole structure continues to influence the force at greater distances than in standard spacetimes.
  • Parameter-tuned reversals suggest that self-force measurements on test charges could in principle constrain the redshift and flaring properties of a wormhole.

Load-bearing premise

The mode-sum regularization procedure remains valid and yields a physically meaningful finite self-force throughout the exterior region of these wormhole spacetimes.

What would settle it

A direct numerical evaluation of the self-force at a radial coordinate where the calculation predicts a zero crossing, to check whether the force actually changes sign there.

Figures

Figures reproduced from arXiv: 2606.29401 by Ian Vega, Jerome P. Mecca.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Embedding diagram showing the characteristic flaring-out of the wormhole geometry for various shape exponents [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Falloff behavior of the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Behavior of scalar self-force on a static particle in the vicinity of a wormhole with shape exponent [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Behavior of scalar self-force on a static particle in the vicinity of a wormhole with shape exponent [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Behavior of the roots of vanishing self-force as a function of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. In (a), the red and blue markers represent the self-force obtained for [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Three-dimensional plot with corresponding density plot of the self-force on a static scalar charge in the Konoplya-Zhidenko two [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Behavior of the falloff rate [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Check of fitting for the power-law distribution against the [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Number of self-force crossings as a function of the wormhole [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
read the original abstract

The self-force acting on a charged particle is sensitive to the global structure of curved spacetime and can serve as a probe of geometry beyond local curvature. We compute the static scalar self-force on a point charge in the two-parameter family of spherically symmetric wormholes introduced by Konoplya and Zhidenko, members of the broader Morris-Thorne class of traversable wormholes. Using mode-sum regularization, we analyze its dependence on the shape exponent $q$, which controls the throat geometry, and the redshift parameter $p$, which determines the redshift function and tidal strength. We find that the self-force is generally not unidirectional: it can change sign with radial distance from the throat, with up to two distinct zero crossings depending on $(p,q)$. We provide a systematic characterization of how both the direction and large-distance falloff depend on the wormhole parameters. For sufficiently large $p$, the force can decay at a slower rate than the canonical $\sim r^{-3}$ behavior typical of isolated-body spacetimes, with stronger flaring (more negative $q$) leading to more rapid decay. In the combined limit $p \to \infty$ and $q \to -\infty$, the asymptotic falloff approaches that of the static scalar self-force in the Ellis wormhole.

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 paper computes the static scalar self-force on a point charge in the two-parameter Konoplya-Zhidenko family of traversable wormholes using mode-sum regularization. It reports that the force is generally not unidirectional, exhibiting up to two zero crossings whose locations depend on the shape exponent q and redshift parameter p, and characterizes the direction and large-r falloff (which can be slower than r^{-3} for large p, or approach the Ellis-wormhole limit for p o∞ and q o- au).

Significance. If the regularization procedure is shown to be complete, the result would demonstrate that global topology can produce qualitatively new self-force phenomenology (sign changes and modified asymptotics) not present in isolated-body spacetimes, providing a concrete diagnostic for wormhole geometries.

major comments (2)
  1. [regularization procedure and numerical implementation] The central numerical claims rest on the mode-sum regularization yielding a finite, physically meaningful self-force throughout the exterior. The manuscript applies the standard procedure without explicit demonstration that the retarded Green's function contributions from paths crossing the throat are either absent or correctly incorporated; standard mode-sum derivations assume a single asymptotically flat exterior, and the two-sided Konoplya-Zhidenko metric requires verification that no additional transmission or image terms are needed (see the method description and the numerical results section).
  2. [numerical results] No error-bar estimates, convergence tests with respect to mode cutoff, or explicit checks that all divergent pieces are subtracted for every (p,q) pair are reported; such diagnostics are required to confirm that the reported sign changes and zero crossings are not artifacts of incomplete regularization.
minor comments (2)
  1. [abstract and §4] The abstract states the falloff can be slower than the canonical r^{-3}; the manuscript should clarify whether this refers to the leading 1/r^3 term or a slower power, and provide the explicit asymptotic expression.
  2. [introduction] Notation for the wormhole parameters p and q should be defined at first use with reference to the metric line element.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting important points regarding the regularization procedure and numerical validation. We address each major comment below and will incorporate the suggested improvements in the revised version.

read point-by-point responses

  1. Referee: [regularization procedure and numerical implementation] The central numerical claims rest on the mode-sum regularization yielding a finite, physically meaningful self-force throughout the exterior. The manuscript applies the standard procedure without explicit demonstration that the retarded Green's function contributions from paths crossing the throat are either absent or correctly incorporated; standard mode-sum derivations assume a single asymptotically flat exterior, and the two-sided Konoplya-Zhidenko metric requires verification that no additional transmission or image terms are needed (see the method description and the numerical results section).

    Authors: We agree that an explicit verification is warranted given the two-sided nature of the geometry. In the revised manuscript we will expand the method section with a dedicated subsection demonstrating that the mode decomposition is performed over the full radial coordinate that spans both asymptotic regions. Because the metric is smooth and regular across the throat, the retarded Green's function is obtained by solving the wave equation globally with outgoing boundary conditions at both spatial infinities; no additional image or transmission terms arise beyond those already captured by the mode sum. We will include a short derivation confirming that the standard Detweiler-Whiting regularization coefficients remain valid in this setting and that throat-crossing contributions are automatically accounted for by the global solution. revision: yes

  2. Referee: [numerical results] No error-bar estimates, convergence tests with respect to mode cutoff, or explicit checks that all divergent pieces are subtracted for every (p,q) pair are reported; such diagnostics are required to confirm that the reported sign changes and zero crossings are not artifacts of incomplete regularization.

    Authors: We acknowledge that the current numerical results section lacks these diagnostics. In the revised manuscript we will add (i) error estimates derived from the tail of the mode sum, (ii) convergence plots versus mode cutoff for representative values of (p,q) including the cases exhibiting sign changes, and (iii) explicit verification that the regularization parameters A, B, and C have been correctly subtracted for every parameter pair shown. These additions will confirm that the reported zero crossings and modified asymptotic decay are robust against truncation and regularization artifacts. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical evaluation on fixed background

full rationale

The paper computes the static scalar self-force via mode-sum regularization applied to the fixed Konoplya-Zhidenko wormhole metric. No parameters are fitted to the target self-force data, no self-referential definitions equate inputs to outputs, and no load-bearing steps reduce to self-citations or ansatze imported from the authors' prior work. The reported sign changes and falloff behaviors are direct outputs of the numerical procedure on the given spacetime, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The computation rests on the standard mode-sum regularization technique for scalar fields in static spherically symmetric spacetimes and on the assumption that the Konoplya-Zhidenko metric is a valid traversable wormhole background; no new free parameters or invented entities are introduced beyond the two metric parameters p and q already present in the cited family.

axioms (1)
  • domain assumption Mode-sum regularization subtracts all divergent contributions leaving a finite, physically meaningful self-force on the static charge.
    Invoked by the choice of computational method in the abstract.

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discussion (0)

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Reference graph

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    × 10-8 ρ/b0 ε 0 1 2 3 4 5 0.00 0.02 0.04 0.06 0.08 0.10 ρ/b0 (b0/Q)2Fself (b) FIG. 2. (a) Falloff behavior of the l-mode components of the self-force on the scalar charge located at ρ/b0 = 1 along with results of the subtraction of A, B and D regularization parameters; (b) comparison of the self-force obtained using Taylor’s exact expression, and by the m...

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    The self-force on a static scalar charge Q has an exact form that was obtained by Taylor [40]

    Ellis wormhole(p= 0, q=−1) The ultrastatic wormhole (Φ = 0) belonging to this class is the well-known Ellis wormhole. The self-force on a static scalar charge Q has an exact form that was obtained by Taylor [40]. For the case of a minimum coupling, the self-force is given by F Ellis ρ = Q2b0 ρ π(ρ 2 +b 2 0)2 = Q2 πb2 0 y (1 +y 2)2 ,(44) hence f(y;p= 0, q=...

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    × 10-4 10-2 ρ/b0 |(b0 /Q)2 Fself | 0 1 2 3 4 5 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 ρ/b0 (b0/Q)2Fself (c)1/2≲p≲1 (q=−1, σ= +1)                                                                                                                  ...

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    Ultrastatic wormholes In the previously considered cases of ultrastatic wormholes with shape exponent q=−1 and q= 1/3 , the self-force computed is found to be purely repulsive and attractive, respec- tively. Given this, we speculate that no ultrastatic wormhole produces a self-force profile with crossings. By examining other throat profiles, we verify tha...

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    Generic(q, p)wormholes As p is increased to unity, self-force crossing starts to appear again in Fig. 6(c) for q= 1/2 . The self-force acting on the scalar charge in the vicinity of the throat is repulsive and be- comes attractive as the charge is placed farther away, while its magnitude increases with a more negative q value. In Fig. 6(d), for the case q...

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