wormholes, part IV: the Casimir Effect
In part 3, I mentioned there was a difference between the standard local energy conditions which were originally proposed in classical General Relativity and the "averaged" conditions. But I went off on a tangent about quantum inequalities and quantum interest, and never got around to connecting this back with the averaged conditions or defining what they are.
The local energy conditions original proposed in GR apply to every point in spacetime. Since general relativity is a theory about the large-scale structure of the universe, the definition of a "point" in spacetime can be rather loose. For the purposes of cosmology, thinking of a point as being a ball of 1km radius is plenty accurate enough. You won't find any significant curvature of spacetime that's smaller than that, so whether it's exactly 0 in size or 1km in size doesn't matter. But for quantum mechanics, it matters a lot because it's a theory of the very small scale structure of the universe. There, the difference between 0 and 1km is huge, in fact so huge that even anything the size of a millimeter is already considered macroscopic.
So if you're going to ask whether quantum field theory respects the energy conditions proposed in general relativity, you have to get more precise with your definitions of these energy conditions. The question isn't "can energy be negative at a single point in spacetime?" but "can the average energy be negative in some macroscopic region of space over some period of time long enough for anyone to notice?" The actual definition of the AWEC (averaged weak energy condition) is: energy averaged along any timelike trajectory through spacetime is always zero or positive. A timelike trajectory basically means the path that a real actual observer in space who is traveling at less than the speed of light could follow. From the reference frame of this observer, this just means the energy averaged at a single point over all time. The ANEC (averaged null energy condition) is similar but for "null" trajectories through spacetime. Null trajectories are the paths that photons and other massless particles follow--all particles that move at the speed of light. A real observer could not follow this trajectory, but you can still ask what the energy density averaged over this path would be.
From what I understand, the quantum energy inequalities are actually a bit stronger than these averaged energy conditions. The AWEC basically says that if there is a negative energy spike somewhere, then eventually there has to be a positive energy spike that cancels it out. The QEI's say that not only does this have to be true, but the positive spike has to come very soon after the negative spike--the larger the spikes are, the sooner.
However, you may notice that the QEI's (and the averaged energy conditions) just refer to averaging over time. What about space? Personally, I don't fully understand why Kip Thorne and others focused on whether the average over time is violated but didn't seem to care about the average over space. Because the average over space seems important for constructing wormholes too--if you can't generate negative energy more than a few Planck lengths in width, then how would you ever expect to get enough macroscopic negative energy to support and stabilize a wormhole that someone could actually travel through?
I haven't mentioned the Casimir Effect yet, which is a big omission as it's one of the first things people will cite as soon as you ask them how they think someone could possibly build a traversable wormhole. Do the quantum inequalities apply to the Casimir Effect? Yes and no.
As I understand them, the quantum inequalities don't actually limit the actual absolute energy density, they limit the difference between the energy density and the vacuum energy density. Ordinarily, vacuum energy density is zero or very close to it. (It's actually very slightly positive because of dark energy, also known as the cosmological constant, but this is so small it doesn't really matter for our purposes.) The vacuum energy is pretty much the same everywhere in the universe on macroscopic scales. So ordinarily, if a quantum energy inequality tells you that you can't have an energy density less than minus (some extremely small number) then this also places a limit on the absolute energy density. But this is not true in the case of the Casimir Effect. Because the Casimir Effect lowers the vacuum energy in a very thin region of space below what it normally is. This lowered value of the energy (which is slightly negative) can persist for as long as you want in time. But energy fluctuations below that slightly lowered value are still limited by the QEI's.
This seems like really good news for anyone hoping to build a traversable wormhole--it's a way of getting around the quantum energy inequalities, as they are usually formulated. However, if you look at how the Casimir Effect actually works you see a very similar limitation on the negative energy density--it's just that it is limited in space instead of limited in time.
The Casimir Effect is something that happens when you place 2 parallel plates extremely close to each other. It produces a very thin negative vacuum energy density in the region of space between these plates. To get any decent amount of negative energy, the plates have to be enormous but extremely close together. It's worth mentioning that this effect has been explained without any reference to quantum field theory (just as the relativistic version of the van der Waals force). As far as I understand, both explanations are valid they are just two different ways of looking at the same effect. The fact that there is a valid description that doesn't make any reference to quantum field theory lends weight to the conclusion that despite it being a little weird there is no way to use it to do very weird things that you couldn't do classically like build wormholes. However, I admit that I'm not sure what happens to the energy density in the relativistic van der Waals description--I'm not sure there is even a notion of vacuum energy in that way of looking at it, as vacuum energy itself is a concept that exists only in quantum field theory (it's the energy of the ground state of the quantum fields).
Most of what I've read on quantum inequalities has come from Ford and Roman. They seem very opposed to the idea that traversable wormholes would be possible. I've also read a bit by Matt Visser, who seems more open to the possibility. The three of them, as well as Thorne, Morris, and Hawking seem to be the most important people who have written papers on this subject. Most other people writing on it write just a few papers here or there, citing one of them. Visser, Ford, and Roman seem to have all dedicated most of their careers to understanding what the limits on negative energy densities are and what their implications are for potentially building wormholes, time machines, or other strange things (like naked singularities--"black holes" that don't have an event horizon).
There are a few more things I'd like to wrap up in the next (and I think--final) part. One is to give some examples of the known limitations on how small and how short lived these negative energy densities can be, and what size of wormhole that would allow you to build. Another is to mention Alcubierre drives (a concept very similar to a wormhole that has very similar limitations). Another is to try to enumerate which averaged energy conditions are known for sure to hold in quantum field theory and in which situations, comparing this with which conditions would need to be violated to make various kinds of wormholes. And finally, to try to come up with any remotely realistic scenario for how this might be possible and give a sense for the extremely ridiculous nature of things that an infinitely advanced civilization would need to be able to do in order for that to happen practically, from a technological perspective.
The local energy conditions original proposed in GR apply to every point in spacetime. Since general relativity is a theory about the large-scale structure of the universe, the definition of a "point" in spacetime can be rather loose. For the purposes of cosmology, thinking of a point as being a ball of 1km radius is plenty accurate enough. You won't find any significant curvature of spacetime that's smaller than that, so whether it's exactly 0 in size or 1km in size doesn't matter. But for quantum mechanics, it matters a lot because it's a theory of the very small scale structure of the universe. There, the difference between 0 and 1km is huge, in fact so huge that even anything the size of a millimeter is already considered macroscopic.
So if you're going to ask whether quantum field theory respects the energy conditions proposed in general relativity, you have to get more precise with your definitions of these energy conditions. The question isn't "can energy be negative at a single point in spacetime?" but "can the average energy be negative in some macroscopic region of space over some period of time long enough for anyone to notice?" The actual definition of the AWEC (averaged weak energy condition) is: energy averaged along any timelike trajectory through spacetime is always zero or positive. A timelike trajectory basically means the path that a real actual observer in space who is traveling at less than the speed of light could follow. From the reference frame of this observer, this just means the energy averaged at a single point over all time. The ANEC (averaged null energy condition) is similar but for "null" trajectories through spacetime. Null trajectories are the paths that photons and other massless particles follow--all particles that move at the speed of light. A real observer could not follow this trajectory, but you can still ask what the energy density averaged over this path would be.
From what I understand, the quantum energy inequalities are actually a bit stronger than these averaged energy conditions. The AWEC basically says that if there is a negative energy spike somewhere, then eventually there has to be a positive energy spike that cancels it out. The QEI's say that not only does this have to be true, but the positive spike has to come very soon after the negative spike--the larger the spikes are, the sooner.
However, you may notice that the QEI's (and the averaged energy conditions) just refer to averaging over time. What about space? Personally, I don't fully understand why Kip Thorne and others focused on whether the average over time is violated but didn't seem to care about the average over space. Because the average over space seems important for constructing wormholes too--if you can't generate negative energy more than a few Planck lengths in width, then how would you ever expect to get enough macroscopic negative energy to support and stabilize a wormhole that someone could actually travel through?
I haven't mentioned the Casimir Effect yet, which is a big omission as it's one of the first things people will cite as soon as you ask them how they think someone could possibly build a traversable wormhole. Do the quantum inequalities apply to the Casimir Effect? Yes and no.
As I understand them, the quantum inequalities don't actually limit the actual absolute energy density, they limit the difference between the energy density and the vacuum energy density. Ordinarily, vacuum energy density is zero or very close to it. (It's actually very slightly positive because of dark energy, also known as the cosmological constant, but this is so small it doesn't really matter for our purposes.) The vacuum energy is pretty much the same everywhere in the universe on macroscopic scales. So ordinarily, if a quantum energy inequality tells you that you can't have an energy density less than minus (some extremely small number) then this also places a limit on the absolute energy density. But this is not true in the case of the Casimir Effect. Because the Casimir Effect lowers the vacuum energy in a very thin region of space below what it normally is. This lowered value of the energy (which is slightly negative) can persist for as long as you want in time. But energy fluctuations below that slightly lowered value are still limited by the QEI's.
This seems like really good news for anyone hoping to build a traversable wormhole--it's a way of getting around the quantum energy inequalities, as they are usually formulated. However, if you look at how the Casimir Effect actually works you see a very similar limitation on the negative energy density--it's just that it is limited in space instead of limited in time.
The Casimir Effect is something that happens when you place 2 parallel plates extremely close to each other. It produces a very thin negative vacuum energy density in the region of space between these plates. To get any decent amount of negative energy, the plates have to be enormous but extremely close together. It's worth mentioning that this effect has been explained without any reference to quantum field theory (just as the relativistic version of the van der Waals force). As far as I understand, both explanations are valid they are just two different ways of looking at the same effect. The fact that there is a valid description that doesn't make any reference to quantum field theory lends weight to the conclusion that despite it being a little weird there is no way to use it to do very weird things that you couldn't do classically like build wormholes. However, I admit that I'm not sure what happens to the energy density in the relativistic van der Waals description--I'm not sure there is even a notion of vacuum energy in that way of looking at it, as vacuum energy itself is a concept that exists only in quantum field theory (it's the energy of the ground state of the quantum fields).
Most of what I've read on quantum inequalities has come from Ford and Roman. They seem very opposed to the idea that traversable wormholes would be possible. I've also read a bit by Matt Visser, who seems more open to the possibility. The three of them, as well as Thorne, Morris, and Hawking seem to be the most important people who have written papers on this subject. Most other people writing on it write just a few papers here or there, citing one of them. Visser, Ford, and Roman seem to have all dedicated most of their careers to understanding what the limits on negative energy densities are and what their implications are for potentially building wormholes, time machines, or other strange things (like naked singularities--"black holes" that don't have an event horizon).
There are a few more things I'd like to wrap up in the next (and I think--final) part. One is to give some examples of the known limitations on how small and how short lived these negative energy densities can be, and what size of wormhole that would allow you to build. Another is to mention Alcubierre drives (a concept very similar to a wormhole that has very similar limitations). Another is to try to enumerate which averaged energy conditions are known for sure to hold in quantum field theory and in which situations, comparing this with which conditions would need to be violated to make various kinds of wormholes. And finally, to try to come up with any remotely realistic scenario for how this might be possible and give a sense for the extremely ridiculous nature of things that an infinitely advanced civilization would need to be able to do in order for that to happen practically, from a technological perspective.