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wormholes, part III: negative energy

So what is this thing called negative energy (also called "exotic matter", and could it exist somewhere, or if it doesn't exist naturally, is there a way we could somehow generate it?

The two main theories of fundamental physics today are General Relativity and Quantum Field Theory. General Relativity was developed as a way to understand the large scale structure of the universe (cosmology, astrophysics, etc), while quantum field theory was developed as a way to understand the small scale structure (quantum mechanics, subatomic particles, etc.) Putting the two together is still a work in progress and string theory so far seems to be the only promising candidate, but it is far from complete.

General Relativity by itself is usually referred to as a "classical" theory of physics, since it doesn't involve any quantum mechanics. But there has been a lot of work using a "semi-classical" theory called Quantum Field Theory in Curved Spacetime. This is basically quantum field theory but where the space the quantum fields live in is allowed to be slightly curved as opposed to perfectly flat. Because this doesn't work once the curvature becomes too strong, it's not a full theory of quantum gravity, and is only regarded as an approximation. But it has been good enough to get various interesting results (for example, the discovery of Hawking radiation).

In General Relativity by itself (usually referred to by string theorists as "classical GR"), there are a number of "energy conditions" which were conjectured early on, specifying what kinds of energy are allowed to exist. The main ones are the strong energy condition, the dominant energy condition, the weak energy condition, and the null energy condition. As I understand it, all of these are satisfied by classical physics. If there were no quantum mechanics or quantum field theory, then it would be easy to say that wormholes are impossible, since negative energy is not even a thing. But in quantum field theory, the situation is much more subtle. In Kip Thorne's 1989 paper he finds that a variant of the weak energy condition (AWEC = averaged weak energy condition) is the one which would need to be violated in order to construct his wormhole. I've seen more recent papers which focus more on ANEC (averaged null energy condition) though, so perhaps there have been wormhole geometries since discovered which only require violation of the null energy condition.

I'm not going to explain what the difference is between all of these different energy conditions. But I should explain the difference between the "averaged" conditions and the regular ("local") conditions. The weak energy condition says that the energy density measured by every ordinary observer at a particular location in space must be zero or positive. The surprising thing about quantum field theory is that this, as well as all of the other local conditions (local means at a particular point) are violated. In other words, in quantum field theory, negative energy is very much "a thing".

But hold your horses for a second there! Because the thing about quantum field theory is that, there are loads of different examples of weird things that can happen on short time scales and at short distances that cannot happen macroscopically. For example, virtual particles exist that travel faster than the speed of light, masses can be imaginary, and energy is not even strictly conserved (there are random fluxuations due to quantum uncertainty). There are particles and antiparticles being created out of the vacuum and then annihilated all the time (quantum foam). There are bizarre things called "ghosts" that can have negative probability (which I won't go into). But when you look at the macroscopic scale, none of these weird effects show up--through very delicate mathematics, they all cancel out and you end up having very normal looking physics on the large scale. It's like if you look at the individual behavior at the microscopic level, everything is doing something completely weird and bizarre. But if you take an average of what's happening, it all gets smoothed out and you have very solid reliable macroscopic properties: energy is conserved, probabilities are positive, everything moves at less than the speed of light, etc. These things have been proven and are well understood. So given everything I know about how quantum field theory works, my intuition would be that something similar happens for negative energy: it's the kind of thing that could happen momentarily on the microscopic scale, but would never be the kind of thing one would expect to see on the macroscopic scale. And that's the main reason I've always told people I don't think wormholes are possible, despite not having reviewed most of the relevant literature related to it until this month.

After reviewing the literature, I have seen that over the past 20 years, the case that negative energy cannot exist macroscopically in our universe has grown stronger. Since the mid 90's the focus has shifted from energy conditions to what are known as "quantum energy inequalities" or QEI's. I read a couple review papers on QEI's, and will try to summarize in my next part. The gist of it is that while negative energy can happen locally, there are limits which can be placed on how negative that energy can be. And the limits depend on what timescale you're looking at. If you want a very negative energy, you will only find that on a very short timescale. If you want only a little bit of negative energy, you might find it on a longer time scale. But once you get to timescales like a second or more, the amount of negative energy you can have at a point is indistinguishably different from zero. There is a related idea called "quantum interest". Quantum interest refers to the fact that: given any negative energy spike there will be some compensating positive energy spike in the near future to compensate it (and make it average out to zero). And the time you have to wait to have this "payback" in the energy balance is shorter the larger the initial spike.

Gotta run for now, but I still have more to summarize on this. To be continued in part IV!

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