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Towards traversable wormholes from force-free plasmas

by Nabil Iqbal, Simon F. Ross

Submission summary

As Contributors: Nabil Iqbal · Simon Ross
Arxiv Link: (pdf)
Date submitted: 2021-06-04 12:39
Submitted by: Iqbal, Nabil
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


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.

Current status:
Editor-in-charge assigned

Author comments upon resubmission

We thank the referees for their careful reading and useful feedback. We are happy that they overall find the paper informative and suitable for publication. We have addressed their concerns, as detailed below.

To Referee 1:

1) FFE seems to be an emergent phenomena when standard electrodynamics interacts with matter. The authors considered the correction to the (total) stress-energy tensor from Alfven waves. Does this take into account the stress-energy tensor of underlying matter?

So, FFE as conventionally formulated assumes that the matter stress energy tensor can be neglected, as that of the electromagnetic field is considered to be higher. The correction we compute in a specific model is a slight deformation away from that limit, so indeed takes into account some of the stress-energy of that matter. We thank the referee for the question, and have clarified this with two comments:

— on p3, we have added the line "In its conventional formulation, it is usually understood that the stress-energy of the electromagnetic field is much higher than that of the charged matter screening the electric field, which can thus be neglected.”

— on p7, we added "Similarly, the stress energy of the Alfven wave can be understood as that of an approximately massless collective scalar field moving in the AdS2 directions; microscopically however this stress energy comes both from the electromagnetic degrees of freedom and from the fermion degrees of freedom bosonized into the field Φ.”

2) It seems that eventually authors impose simple periodic boundary conditions on matter. Is Section 4 necessary then?

We feel that Section 4 is helpful and would like to keep it, as otherwise it may appear that a slight modification of the periodic boundary conditions would be sufficient to evade the later conclusions; in fact it seems to us it is a generic feature of the construction that this is difficult to realize.

3) The conclusion of the paper seems that it is hard to sustain a wormhole with FFE in 4d. In higher dimensions the black hole story is well studied(AdS2 near horizon, 2d Casimir energy, etc), except the dispersion relation for Alfven waves. It would be nice if the author could comment on this.

Actually, in d> 4 FFE is likely to have a very different structure indeed. The lack of phenomenological considerations mean that the case of d > 4 has not been studied by the plasma physics community and remains to be developed. In a more general context, a key role is played by the 1-form symmetry associated with magnetic flux conservation (as emphasized in Ref 21); but in general d this is a (d-3)-form symmetry, and the structure of the theory will be quite different.

Said in a slightly different manner, in d = 5 the magnetically charged black hole must be a black string and not a black hole, and thus things seem different. We feel that this is outside the scope of this paper (which was attempting to address the possibility of creating these wormholes in our own 4d universe).

4) The lengthy calculation in Section 5.2 results in the condition m^2 < 1/(RL). The authors explain this result as: "...AdS kinematics imply that the relevant mass scale is the geometric mean...". I think it would be very helpful for readers to see this kinematic estimate explicitly. Naively, in eq. (41) mass is responsible for .... potential, which is cutoff at ... Since ...the effect of the mass is negligible for ....

[some equations deleted above for formatting reasons; the original comment can be seen in the first submissions page]

We actually do not think there is a direct kinematic argument that shoes that the geometric mean is the relevant scale; rather it appears to follow from a somewhat detailed computation. We agree the original wording was somewhat misleading, and have rephrased the line to read on p16:

"This slightly lengthy computation simply shows that for sufficiently long length scales l a very small mass makes a large difference, and the calculation shows that the relevant mass scale is the geometric mean of l and R.”

To Referee 2:

1) Is it be possible to clarify a little bit the physical setup, to give some intuition to a reader who doesn't know much about FFE? If I understand correctly, the Alfven modes are supposed to describe coherent excitations of the plasma and EM field, so does this mean that the hypothetical wormhole must be filled with plasma? Does this has any physical significance?

As FFE is a rather large subject, we feel that we can’t really do justice to it here and would like to refer readers to the excellent reviews on the subject, e.g. Ref [17]. Regarding the interaction with conventional plasma physics, we have already addressed this on p3 with the line:

"The theory is often considered in situations with a plasma density, but the theory is still useful for describing fluctuations around vacuum electromagnetic backgrounds satisfying the degeneracy condition F ∧ F = 0, if the background magnetic field is strong enough; in response to fluctuations, charges can be easily pair- produced, screening the electric field to zero over long distance scales. We will consider the theory in this setting.”

and further on p8 as

"The black hole is a solution of Einstein gravity coupled to a Maxwell field, but the magnetically charged black holes also satisfy the FFE equations of motion, and can be thought of as solutions of FFE coupled to gravity. As mentioned in the previous section, FFE is usually thought of as a theory of plasmas, but it includes as solutions any degenerate vacuum Maxwell field, and an FFE description is useful if the field is strong enough that fluctuations about this background that would produce electric fields violating the FFE equation are efficiently screened by charges produced by vacuum fluctuations; in this case we expect the low-energy fluctuations to be collective plasma modes (such as Alfven waves) rather than free photon excitations.”

and thus would prefer to not add any further discussion. (We note that FFE is likely valid in other physical situations as well, and in fact its precise domain of validity does not seem to be something that is completely well understood in the current plasma physics literature).

2) The Alfven wave modes are supposed to follow the magnetic field lines in closed loops threading the wormhole. A question that I haven't seen addressed is the following: is it clear that the FFE approximation is also valid in the outside region, far from the black holes? Shouldn't this put a constraint on the outside distance d?

We thank the referee for this good point, which we have now addressed in footnote 8 on p18. The referee is correct that there is a bound on d as well; it is much larger than R (the radius of the throat), but still rather small in everyday terms, which we have now explicitly computed.

3) In [1], a large magnetic field was crucial to have a large number of Casimir modes. This doesn't seem to be the case here where it's only needed for the FFE to be valid. So could we repeat the analysis in a more realistic astrophysical setup? For example by replacing the RN black hole with near-extreme Kerr in a strong magnetic field?

Though the meaning of “large” may be somewhat subjective, we actually still think that the field required is still rather large. i.e. in the approximation we work in, FFE is valid only for B greater than the critical field defined in (15), which is roughly speaking related to having a field large enough that it is reasonable to pair-produce charges from the vacuum. We note that there may have been some confusion here from a typo above Eq (15) where an inequality was accidentally reversed (which we have now corrected to read B > B_{\star}, in agreement with footnote 3). Thus we think it remains a bit unlikely astrophysically.

Finally, Referee 3 did not ask for any further corrections.

Again, we thank all the referees for their careful reading and insightful questions, which we feel have improved the paper.

List of changes

The point-by-point list is available above in our responses to the referee comments.

Reports on this Submission

Anonymous Report 2 on 2021-6-30 (Invited Report)


The authors have satisfactorily answered my comments.

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Anonymous Report 1 on 2021-6-27 (Invited Report)


Dear Authors,
Thank you to your clarifications. The only remaining question I have is about Casimir energy in AdS2. More specifically, eq. (66).
Parameter $\epsilon$ was first introduced right below eq. (41). So I believe you have a typo in the definition of $\epsilon$ above eq. (44).
Unless I am very much mistaken, in the limit $\Delta \rightarrow 1, m R \rightarrow 0$ and with $\epsilon=R/l$, the quantity $\alpha_- \epsilon^{1-2\Delta}/\alpha_+$ goes to $m l$. This implies that in eq. (66) it should be $1/l^2$ instead of $1/(l R)$. The same conclusion can be reached in a very simple way, by looking at eq. (41) and requiring that the $m/\cos^2$ potential is negligible.

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Author:  Nabil Iqbal  on 2021-07-09

(in reply to Report 1 on 2021-06-27)
reply to objection

Dear referee,

In response to your question below:

Thank you to your clarifications. The only remaining question I have is about Casimir energy in AdS2. More specifically, eq. (66). Parameter ϵ was first introduced right below eq. (41). So I believe you have a typo in the definition of ϵ above eq. (44).

We don’t quite understand this comment; in particular, there is no definition of \epsilon above eq. (44).

Unless I am very much mistaken, in the limit Δ →1, mR → 0 and with ϵ = R/l, the quantity alpha_- epsilon^(1- 2 Delta)/alpha_+ goes to ml

We actually disagree with this statement; in particular, from Eq. (50), in the limit of small m, alpha_- goes like m^2 R^2, so alpha_- epsilon^(1- 2 Delta)/alpha_+ = alpha_-/epsilon = m^2 R l, and not ml. (We find it possible the referee made an incorrect Taylor expansion of Eq. (50), thinking erroneously that \alpha_- \sim (m R).)

This implies that in eq. (66) it should be 1/l^2 instead of 1/(lR). The same conclusion can be reached in a very simple way, by looking at eq. (41) and requiring that the m/cos^2 potential is negligible.

Based on the above argument, eq (66) is correct as written. As far as we can see there is no extremely simple argument to establish this scaling; obtaining it requires our explicit calculation. The referee's proposed bound would be obtained by evaluating the potential at the boundary — i.e. looking at the potential at \sigma = \frac{\pi}{2} - \epsilon — however this gives a result which is too strong, just as evaluating the potential at a generic point in the interior results in a bound m^2 < 1/R^2, which is too weak.

Anonymous on 2021-07-16

(in reply to Nabil Iqbal on 2021-07-09)
reply to objection

Dear Authors, thank you for the clarifications.

there is no definition of $\epsilon$ above eq. (44).

I am sorry, I meant $\epsilon$ above eq. (66).

I agree with your reasoning regarding $\alpha_\pm$. However, I still have some doubts regarding the Casimir energy computation. My intuition is that for finding when the mass is irrelevant it is not necessary to find the exact spectrum, simply identifying what is the relevant expansion parameter is enough. If $m^2 r l$ is the right expansion parameter, contrary to naive $m l$ then this is a separate interesting result. Especially since there have been lots of discussions regarding wormholes recently.

To wit, you have found an explicit correction to energies, eq. (65)(also I believe it should be $\omega=\alpha_- + 2r+ \delta \omega$ above it). For all non-zero $r$ this correction is indeed small if $m^2 r l$ is small. However, for $r=0$, there is dangerous $\Gamma(\alpha_-) \approx 1/\alpha_- = 1/mr$. So the correction to $\omega=0$ is not necessarily small. I presume this correction should be found from the exact quantization condition (58). If this correction turns out to be irrelevant for Casimir energy then it will clearly demonstrate that $m^2 r l$ is the right parameter.

Anonymous on 2021-09-13

(in reply to Nabil Iqbal on 2021-07-09)

Dear Authors,
My concern is that in eq. (65) for $r=0$ mode the Gamma function $\Gamma(\alpha_- + r)$ actually behaves as
$1/\alpha_-$ which cancels overall $\alpha_-$ so the correction is not small anymore. It becomes of order $\epsilon^{-1}$.

Anonymous on 2021-08-09

(in reply to Nabil Iqbal on 2021-07-09)

The referee is correct that there is a typo above eq (66), where epsilon should be R/l, not l/R.

However, in their other comment, they appear to have confused the mode number r with R, the curvature radius of the AdS2. (At small m, \alpha_{-} scales like mR, not mr). Our claim is that the relevant expansion parameter is m^2 R l. As far as we can see this is equally valid for the case with r=0.

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