# Towards traversable wormholes from force-free plasmas

### Submission summary

 As Contributors: Nabil Iqbal · Simon Ross Arxiv Link: https://arxiv.org/abs/2103.01920v3 (pdf) Date accepted: 2022-02-04 Date submitted: 2021-10-07 15:49 Submitted by: Iqbal, Nabil Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### Abstract

The near-horizon region of magnetically charged black holes can have very strong magnetic fields. A useful low-energy effective theory for fluctuations of the fields, coupled to electrically charged particles, is force-free electrodynamics. The low energy collective excitations include a large number of Alfven wave modes, which have a massless dispersion relation along the field worldlines. We attempt to construct traversable wormhole solutions using the negative Casimir energy of the Alfven wave modes, analogously to the recent construction using charged massless fermions. The behaviour of massless scalars in the near-horizon region implies that the size of the wormholes is strongly restricted and cannot be made large, even though the force free description is valid in a larger regime.

Published as SciPost Phys. 12, 086 (2022)

In this resubmission we correct some small typos and make small clarifications requested by the referee.

### List of changes

Before equations (63) and (66) we correct the mode-numbering of the frequencies in the inline formula for \omega.

We add footnote 7 on page 6 to explain a potential subtlety in the lowest mode.

### Submission & Refereeing History

#### Published as SciPost Phys. 12, 086 (2022)

Resubmission 2103.01920v3 on 7 October 2021
Submission 2103.01920v1 on 25 March 2021

## Reports on this Submission

### Anonymous Report 1 on 2021-10-22 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2103.01920v3, delivered 2021-10-22, doi: 10.21468/SciPost.Report.3725

### Report

Dear Authors,

Thank you for answering my questions. The result about the Casimir energy in AdS is very important and can have a big significance for the field of wormhole study in general. This is why I want to make sure it is indeed determined by $m^2 R l$ as you claim.
Now I completely agree with your results for the even spectrum of periodic BC.
However, I noticed that that the Dirichlet spectrum, eqns. (63), (64) have a problem: for large $r$, the corrections $\delta \omega$ behave as $r^{1-2 \alpha_-}$, so it grows faster than the original, linear in $r$, answer. So the approximation breaks down. How does it affect the final answer for the Casimir energy?
Also it would be useful to check if the same happens with the odd spectrum for periodic BC.

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### Author:  Nabil Iqbal  on 2021-12-05  [id 2010]

(in reply to Report 1 on 2021-10-22)

Dear referee,

We thank you for the detailed reading.

Actually, the large-r growth of $\delta\omega$ seen in Eq (63) is not physical, and is an artifact of assuming that the frequency shift $\delta\omega$ is *small*; for a fixed value of $\epsilon$, this ceases to be a good assumption for sufficiently large $r$, and thus one should not linearize the Gamma functions in Eq (62) when finding the frequency shift.

There is of course no issue with numerically solving the original defining equation Eq. (62) numerically. We have done this below for the first 100 zeros with ($\epsilon^{2\Delta-1} = 0.01$,$m^2R^2 = 0.05$). These parameters have been picked to show both that the two expressions agree for small $r$, (where the domain of validity increases as $\epsilon$ is made smaller), but also that for large $r$ the growth of the *exact* answer saturates to be much slower than linear and there is no breakdown of the Casimir energy calculation. (See Figure in file "pic.pdf", which should be attached to this comment).

We stress that this has no impact on our answers, which do not require the precise $r$-dependence in (63) and really use only the overall scaling with $\epsilon$ in Eq (63). As $\epsilon$ goes to zero, the approximation used to obtain (63) remains valid for larger and larger $r$, and the leading non-analytic dependence on \epsilon indeed arises from Eq (63).

As this is all rather technical (and the considerations are only important if one takes the particular order of limits where $\epsilon$ is held fixed, UV cutoff on r is taken to infinity), we do not think that this requires a revision in the paper.

### Anonymous on 2022-02-06  [id 2162]

(in reply to Nabil Iqbal on 2021-12-05 [id 2010])

The fact that the correction (63) grows with $r$ stems from a simple fact that the authors expanded the hypergeometric function in eq. (61). Taking $m=0$ in eq. (62) produces $\sin(\pi \omega/2)- \epsilon \omega \cos(\pi \omega/2)=0$ instead of $\sin((\pi/2-\epsilon) \omega)=0$. For large $\omega$ the difference becomes important. Using the full hypergeometric function, one can easily check numerically that the proper set of eigenenergies is $\omega = (2r + 1 + \alpha_+)/(1-2 \epsilon/\pi)$ instead of $(2r + 1 + \alpha_+)$.
So I agree that it is enough to require $m^2 R l$ to be small, as the Authors have claimed.
One simple way to derive this requirement is to apply standard quantum-mechanical first-order perturbation theory to the eigenproblem (41), treating $m^2 R^2/\cos(\sigma)^2$ as a perturbation(previously I suggested comparing this term to $1$, but this is obviously too crude).

I do not have any further questions and I recommend the paper for publication.

### Anonymous on 2022-01-23  [id 2118]

(in reply to Nabil Iqbal on 2021-12-05 [id 2010])

Dear Authors,
Thank you for a detailed reply.
Your graph indicates that for large $r$ the correction becomes finite even for small $\epsilon$. I did the same calculation for smaller $\epsilon$ and got a similar result(please see the attached plot). It means that for large $r$ the approximation does break down: the shift $\delta \omega$ becomes a constant(my numerical experiments suggest this constant is 1)
This shifts Casimir energy by an order 1 amount, as the sum $\sum_{r=0}^\infty r$ and $\sum_{r=0}^{\infty}(r+c)$ are different. So I disagree with your claim that it does not affect your result: Casimir energy is sensitive to precise $r$ dependence.