# Wormhole Metrics and the Density of Negative Energy

I've noticed quite a bit of the literature on wormholes, warp drives, etc. try to reduce the overall quantity of negative mass-energy their spacetimes require, but I have seen little discussion regarding the density of this "exotic matter" in the literature.

Regarding warp drives, say the Alcubierre Metric, it seems a simple parameter (σ) is related to the region of negative energy and how thick or thin it is. It isn't hard for me to imagine how this relates to the density of the exotic matter. Wormholes on the other hand, I have difficulty finding the density of the exotic matter required. The math to General Relativity is beyond me, so I cannot perform the calculations myself.

Question: what is the (typical) density of the negative mass-energy in wormhole metrics, and setting aside the issue of Quantum Inequalities (QI), can the density be altered substantially like with the Alcubierre Metric without the wormhole pinching off? Does altering the density affect the overall quantity of the negative mass-energy required, as it does with the Alcubierre Metric?

My first question to this site poses many sub-questions. However, I am now in a position to answer each of them to my satisfaction, and wish to share my results with the community. The generic answer to my question is no: there is no such simple parameter like σ from the Alcubierre Metric. If one will recall, in Alcubierre's recent synopsis of his metric (2017) he states the simple relation as:

$$E ≈ −v²R²σ$$

Where E required is related to the velocity v squared (in the starship's frame of reference), the radius R of the warp bubble squared, and one over the bubble wall thickness (σ). Looking through the pivotal papers on wormholes physics, Morris-Thorne (1988) and Visser (1989), it's clear no such simple relationship exists. I ask:

What is the (typical) density of the negative mass-energy in wormhole metrics . . .

Seems to vary, but given the issue of Quantum Inequalities (where if one wants to exploit the Casimir Effect to produce the negative mass-energy needed to hold open a wormhole, there are severe constraints), the densities are pushed to the Planck scale. This appears near the end of the paper by Morris-Thorne (1988). Yet I asked:

. . . setting aside the issue of Quantum Inequalities (QI), can the density be altered substantially like with the Alcubierre Metric without the wormhole pinching off?

In short yes, but it won't help much at all with the requirements. In Krasnikov's paper (2003) he thinks he has found a class of situations to avoid QIs for wormholes, but as Krasnikov states:

So, to support a human-sized wormhole of this type it would suffice E−tot ≈ 10^−2 M⊙ of exotic matter. This trifling, in comparison with (9), energy is about the energy of a supernova. QI, if it holds, does not change this estimate in any way.

So, when it comes to the issue of QI, it doesn't affect the total mass-energy needed. As Visser notes in 1989, in order to have any would-be traveler avoid the negative mass-energy region, the densities are pushed to at least that of cosmic strings (albeit cosmic strings with negative mass and negative tension, which no GUT proposes a manner by which those can exist: only positive mass / positive tension ones).

From all of the above, my final sub-question can now be easily answered:

Does altering the density affect the overall quantity of the negative mass-energy required, as it does with the Alcubierre Metric?

Not really. Visser (1989) and Krasnikov (2003) both calculate that negative mass-energies on the scale of Jupiter are required (-1.898 x 10^27 kg) regardless of QI considerations, with densities at least on the scale of cosmic strings (Visser). Perhaps we could lax the density consideration by ignoring Visser, but then we have a traversable wormhole in theory, but not in practice since any would be traveler would then have to pass through the negative mass-energy region in the wormhole's throat, which will have a tension ~10^37 dyn/cm² (Morris-Thorne, 1988). Safe to say, this would be a very bad idea.

Since I have worked with wormholes, modified gravity and all that (unfortunately), let me give you a somewhat mid-sized explanation of what these things are.

The point of energy conditions is to make sure that the stress-tensor does not contribute in a way so that gravitation is repulsive. This has many important reasons, one of them being, like you may have possibly noted, to preserve the causal structure of the spacetime. With wormholes, at least in pure GR it is not possible to do "exotic" physics without violating all the energy conditions. However, in modified gravity, like $$f(R/T)$$ or Lovelock gravity, it is possible to have a wormhole preserving (some) energy conditions. In the pure GR case, you have some weird things with exactly negative contributions from $$T_{\mu \nu }$$, and calculations of this volumetrically can also be done. However, see below for the questions you ask.

what is the (typical) density of the negative mass-energy in wormhole metrics, and setting aside the issue of Quantum Inequalities (QI)?

As far as I am aware of, there is no "typical" density of exotic contributions to wormhole metrics. And I am not aware of quantum inequalities -- perhaps you mean a version of energy conditions? It is possible to calculate how much of such a contribution is present by using something like the averaged null energy condition and turning it into a volume quantifier. See M. Visser, S. Kar, N. Dadhich. Phys. Rev. D 90, 201102 (2003).

. . . setting aside the issue of Quantum Inequalities (QI), can the density be altered substantially like with the Alcubierre Metric without the wormhole pinching off?

Again -- when you modify the amounts of exotic contribution, this affects the energy conditions. This applies in general and is not specific to any solution, and this tells us how the "throat" of the wormhole topology is, defining the characteristic "shape function" in Morris-Thorne wormhole solutions. Now this assumes spherical symmetry, but a shape function works for any wormhole metric -- see recent papers on the arXiv to see many an example of this.

Wormholes on the other hand, I have difficulty finding the density of the exotic matter required. The math to General Relativity is beyond me, so I cannot perform the calculations myself.

Well, this is more or less a simple volumetric calculation and some simple tweaks with the metric. The general idea though, is that the ANEC is violated, and the amounts by which this is so can be calculated, telling us how the overall topology of the wormhole is affected and how such amounts are localised. In modified gravity, you don't have to have violations of (some of) the energy conditions in the first place -- you can simply add geometric corrections to things like $$f(R/T)$$ terms in the modified Einstein-Hilbert action and the modified matter Lagrangian.

• Interesting, wasn’t aware of how modified gravity addressed wormhole physics differently. Thank you for those insights. Regarding Quantum Inequalities, I’d check Alcubierre’s 2017 paper cited in my answer to my own question (he has a whole section on it). While you examine that, let me sleep on what you’ve said here and get back to you tomorrow. Commented Sep 26, 2023 at 5:39
• Thank you for your answer, it seems to confirm the literature review I’ve done so far (some of which is cited in my answer to myself). I’ll broaden my reading further on the shape-function to more recent articles, as you suggest. Unless I’m misunderstanding you, it seems my answer to myself basically has it right, just not as elegant as your response. Thanks again for your time! It is appreciated. Commented Sep 26, 2023 at 18:32
• @Hokon Well, I am not sure about the GUT bit in your answer. And like I cited (and you mentioned as well), while the amounts of "exotic" matter required to violate ANEC vary in general, just an infinitesimal amount suffices to do so. So the "minimum" would be calculated like in the paper by Visser et al. If I may, I would suggest (just a general recommendation) that while these are important things numerically in GR, most of these (like tweaking amounts of ANEC violation, background modified theory, etc.) are no longer real places to work in. I unfortunately know from experience. Commented Sep 27, 2023 at 3:43
• @Hokon There are a number of good papers on these things, you may have come across them. However, most of the works simply change modified gravity or shape functions and declare that as a "new" result. This issue is plagued in the gr-qc arXIv, so if you are actually considering working on these things, maybe this isn't the best place to get started (or in general anytime) unless it is really a good result. Again; just a general opinion you may ignore, but better to keep in mind. Commented Sep 27, 2023 at 3:46
• @Hokon I made a wormholes room here. Commented Sep 27, 2023 at 4:12