spoonless | Entries tagged with general relativity

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.

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!

I started my review of wormholes by reading Kip Thorne's famous paper on them from 1989. Thorne is the T in "MTW" a book by Misner, Thorne, and Wheeler called Gravitation, written in 1973 and still one of the most widely used textbooks on general relativity.

I'm not actually sure whether Kip Thorne believes that wormholes are possible--I assume he would at least lean towards "no" but I have no idea. You might think that because he has written important papers on them and because he consulted on a movie that depicts one, he believes they are. But that doesn't follow, because theoretical physicists often explore ideas that they don't think will work out, to see where they will lead and find the limits of existing theories and uncover new questions or problems with them. I didn't search for comments from him so I don't know what his present take on them is or if it has changed any, but in his 1989 paper he doesn't say they are possible, he just outlines what the conditions would have to be in order for them to be possible.

In his paper, he does three main things. The first is to construct a simple example of a stable traversable wormhole geometrically. In other words, he describes what the shape of space and time would have to be, and what distribution of energy would be needed in order to create this shape. (Remember, the basic idea of general relativity is that matter and energy warp space and time; given any distribution of energy you get a well defined shape of space and time.) Unfortunately he finds that the distribution would have to be quite "exotic", meaning it would require a lot of negative energy, a substance which is very different from ordinary matter and energy. The main question is: could such a substance exist or be created somehow, and if so could it exist in large enough quantities to make a wormhole? At the time, little was known about the answer to this question but a lot more work has been done since which is the topic I focused most of the rest of my reading on.

The second important thing he does in his 1989 paper is to show that if it is possible to create even the simplest kind of wormhole that just connects point A to point B in space, then it is also possible to build a time machine out of the wormhole, that could be used for traveling backwards in time.

So while the fact that a prominent very respectable physicist was even discussing the possibility of wormholes must have been very exciting to the sci fi community, what they may not have realized is that both of these results make wormholes less likely, not more likely. The first because he demonstrated that they depend on a substance not known to exist. And the second because time travel has a whole set of causality and consistency problems that come with it. If it were possible to build a wormhole that couldn't be made into a time machine, that would be much more believable than a wormhole that could be made into a time machine. But sadly, it doesn't seem like the first scenario is possible, at least according to Kip Thorne's 1989 results. However, there is some encouraging news here: in 1992 Stephen Hawking conjectured that there may be weird as of yet unknown effects in physics which act to protect against time travel. (He called this the "Chronology Protection Conjecture".) It seems like pure speculation to me, but if Hawking's suggestion is right then there might plausibly be some mechanism that prevents someone traveling through a wormhole if they plan to travel backwards in time. Like, maybe the wormhole suddenly closes up or becomes unstable. However, I don't think he has much reason to believe this is true other than wishful thinking--it would be nice if some kind of wormhole were possible, without having to face all of the obviously troublesome inconsistencies that time travel brings (grandfather paradox, etc.) So he tried to think of any way in which it could be. This is one way of avoiding that problem, but seems unlikely and doesn't do anything to solve the main problem which is a lack of negative energy.