There has been a lot of discussion around here lately about what space is and what a vacuum is, etc, so I thought you people may be interested in this. Recently I read an article (0909.1408) regarding Einstein's hole argument in a quasi-classical context. Now, I'm not really sure that I believe in their results, but I hadn't heard of this "Einstein's hole argument" before this and I found it quite interesting, particularly considering its implications (at least in Einstein's view) regarding the physical reality of space. It is basically something Einstein ran into while writing down his field equations, mainly he was a little unsettled by the fact that a gravitational field is not uniquely defined by its sources. The wiki example is spacetime empty except for the sun, say. One can make a coordinate transformation such that the sun looks perfectly unchanged in the new coordinates but the metric is very different. At first Einstein was worried this meant that general coordinate invariance was not a good principle to base his theory on, since it didn't seem to make unique predictions about gravity and therefore wasn't deterministic, but of course everything is actually ok because it turns out that the physical predictions of theory, i.e. predictions about how objects move about relative to each other, are the same no matter what tricky transformations you do. This was a big thing at the time since principles like gauge invariance hadn't been discovered. Anyway Einstein interpreted this to mean that it was meaningless to talk about space (gravity) without referring to the things in it, since the coordinate invariance of space made its independent physical reality questionable. Prior to this is seemed perfectly natural to have such a thing as a universe devoid of matter, being only space, but Einstein now denied such a thing could exist, saying that if there was no matter there would be no space. I was wondering if anyone had any interesting insights about this. I keep confusing myself as to whether it is a deep insight or something really obvious (in retrospect at least), since I have become a bit too used to thinking that doing general coordinate transformations on space is a perfectly normal thing to do. Also I feel like I might be missing some subtlety of the hole argument since I'm not sure why Einstein was surprised to find this, since he had set out to find a theory invariant under general coordinate transformations in the first place.
Yes, this is one of two opposing views of the nature of space and time. The view of Einstein is called the relativistic view, which is usually attributed to Anaxagoras and Aristotle, who lived more than 2000 years before Einstein. The other is the one held by Newton, called the substantial view, attributed to Leucippus and Democritus. It is an amazing story that these two views were taking turns to be dominant throughout history. I have not heard of this story about Einstein being worried about nonuniqueness of solutions in his equations; I would think that his original motivation was to have such nonuniqueness to implement Mach's principle.
I think that is the modern usage of the hole argument, but I believe Einstein originally tried to use it to show that all generally covariant theories were nonsense because they violated determinism. He was having trouble trying to make a generally covariant theory so he started to think that it was the wrong way to go.
Ahh, ok, here is perhaps a more simple version of the argument (which I stole from the stanford encyclopedia of philosophy http://plato.stanford.edu/entries/spacetime-holearg) than my convoluted story: (As background, in this case the are two metric and matter fields distributed over a set of spacetime events, such that they are the same except for a "hole" region) 1. If one has two distributions of metric and matter fields related by a hole transformation, manifold substantivalists must maintain that the two systems represent two distinct physical systems. 2. This physical distinctness transcends both observation and the determining power of the theory since: * The two distributions are observationally identical. * The laws of the theory cannot pick between the two developments of the fields into the hole. 3. Therefore the manifold substantivalist advocates an unwarranted bloating of our physical ontology and the doctrine should be discarded. Originally Einstein saw 2 -"The laws of the theory cannot pick between the two developments of the fields into the hole" as the problem (since he implicitly assumed a substantivalist viewpoint) and decided to try and use the argument to toss out all generally covariant theories. Perhaps he didn't realise the observational equivalence of the two fields at the time. Anyway apparently it wasn't until later he realised something along the lines of the modern version above.
I think you confusing two different things here. One is general covariance, and now I am convinced Einstein had no problem with it and there is no problem of determinism stemming from general covariance. The other is the problem of determinism, which can be violated for spacetimes with Cauchy horizon. Today, one of the biggest challenges in mathematical general relativity is to prove that if things like Cauchy horizon exists they must be hidden behind an even horizon, making them not influence the ``normal'' part of the universe.
Not meaning to take away from the rest of the discussion, as I too find it quite interesting, but I want to point out that gauge invariance was already a known topic in the field of electromagnetism, i.e. there's some gauge freedom available in choosing the electric and magnetic potentials. I suppose such a result could have caught Einstein by surprise in the sense that he wouldn't have expected gauge invariance to arise in GR, but the concept itself should have been known to him, being the guru of classical EM that he was.
I don't know if I understand most of the argument, but this is an interesting thought. As the universe expands and matter becomes very spread out, would it mean that space contracts; can space disappear if there is no matter in it? If there is no matter then it seems kind of pointless to even think of space; but I bet there is a more mathematical reason to it as well.
I think this was a deeply philosophical issue for Einstein at the time; it wasn't obvious to him (or anyone else, apparently Einstein managed to convince even Hilbert for a while that general covariance led to distasteful conclusions). The issue was actually to do with active coordinate transformations I believe. Passive coordinate transformations were fine, you're just relabelling space a bit, but if you consider the transformations actively then you are actually transforming one space into another space. Mathematically they are equivalent of course but I think the physical meaning of this in a generally covariant theory was not understood. Due to the implicit acceptance of substantivalism it seemed that two spaces related by an active coordinate transformation were actually different spaces, despite the fact that the spaces look the same to observers in them. And since the spaces are not distinguished observably they are not distinguished by the equations of motion either so one cannot say which space the universe will "actually" evolve into. To the substantivalist this constitutes a loss of determinism, in some abstract sense, since they ascribe a physical meaning to each of these spaces separately, while the relativist is quite happy to say that if everything looks the same in each space, then they ARE the same, so there is no loss of determinism. Yeah that is a seperate issue to the philosophical one we are worrying about here. I don't think it was realised by anybody at the time that the EM gauge invariance was a similar phenomenon to spacetime covariance, especially since the EM field itself remained unchanged under the gauge transformations. In the spacetime version the gravitational field appears to totally change under coordinate transformations, which makes it a lot more conceptually difficult to handle. It is not a trivial thing to identify what aspects of the spacetime manifold remain unchanged under coordinate transformations, to connect these things with properties that are physically observable and to realise that this implies that all those different gravitational fields are actually equivalent, just viewed by different observers. I mean it is quite messed up, in my opinion, that here on the Earth we perceive a gravitational field, and feel a force due to gravity, but if we just go for a skydive we can make the gravitational field vanish, and feel absolutely no force due to gravity.
Ok, I misunderstood you before. My understanding is that in some sense manifold has very much redundancy in describing points in physical spacetime and you need to quotient out by diffeomorphisms to get close to what "really is" out there. Of course this is in line with the gauge idea. Please comment on this. I like details.
Yes this is general idea. Part of the argument is over what it is that one labels as spacetime. For example a manifold consists of a set of points and a metric field, do you call the whole thing spacetime, or just the set of points? We include the whole manifold these days, but this doesn't necessarily appear the natural thing to do at first. The metric field carries energy and momentum, so it doesn't fit the empty "container" picture of spacetime that existed prior to general relativity. However it is also hard to consider just the set of points, or spacetime events, as the "container" since you can't describe the locations of anything in it without the metric. So yes, we can only define a spacetime up to a class of metrics equivalent under diffeomorphisms, which is very much in line with the relativist view (although the older relativists never thought about general coordinate transformations, they only considered ordinary translations and rotations etc, or the symmetries of Euclidean space).
Here is an interesting thought I had- Suppose that you have an empty sphere, or a shell, that emits gravitational waves. This sphere is it's own universe and is not affected by any outside forces. The inside of the shell is a vacuum. What is the curvature of space-time inside of the shell? I don't think a simple euclidean space will suffice in explaining this.. If "Einstein now denied such a thing could exist, saying that if there was no matter there would be no space" is taken literally there should be more space close to the edges of this void than there is at the center..
Well a euclidean space has no curvature. I'm not actually sure what the spacetime you describe would be like. From the newtonian perspective -at least if the mass distribution of the shell is perfectly spherically symmetric- the gravitational field inside the shell is zero, but I don't actually know if there are any corrections to that due to GR. I'd guess no, in which case the space inside the shell would be flat. This isn't really what he was saying. I don't really know what you mean by the concept of "more space close to the edges of the void" either so I can't really comment on it. Einstein wasn't saying that parts of space with nothing in them at the time didn't exist, we was contemplating the state of existence of a universe totally devoid of matter. I haven't heard/read about arguments one way or the other about the existence of subsets of the universe with no matter in them at one time or another. I think it is generally felt that the space still exists, since there are other objects in the universe which one can use to refer to the location of this matterless region, and the matter elsewhere still influences the matterless regions, i.e. we can define a perfectly good metric in the region. I know we can define a perfectly good minkowski metric to a totally matterless universe as well, but I guess it is a question of whether that has any physical significance if there is nothing in the universe to notice it. Its pretty confusing, I haven't quite got a good handle on the philosophy in question here.
I think you misunderstood my shell - it has mass; it is a massive sphere with a spherical void inside of it so that mass distribution is uniform. IE a shell. I do not see why the inside of such a shell would experience no gravity.. there should be some space time curvature close to the edges. Everywhere in our universe space time has some kind of curvature. Maybe space time without curvature does not exist.
No. This can be proved using Newtonian mechanics. Cut the sphere up into little pieces. Each little piece has the same mass. Now the little pieces closer to you attract you stronger, but there are more little pieces that are further away. When you integrate over the full sphere, the force due little pieces close to you is canceled by the larger number of little pieces further away from you. No.
Excellent thread kurros, and indeed an insight imo. I reckon that it shows Einstein's basic 'fabric' concept could be fatally flawed. An alternative answer that allows for space without mass, is the concept of gravitons as individual particles. Space could have existed just before (the possible build-up of mega-springs of matter) leading to the big bang.
No way. I've tried problems like this before and the conclusion is that the gravitational pull will be stronger near the edges of this void than at the center. The center will experience less gravitational pull than an object at the edges. In other words if you place a particle with mass inside the void, it will accelerate to the nearest wall rather than remain stationary. Well, in fact, space-time without curvature does not exist. Gravity exerts its influence into infinity and so everywhere in space there should be curvature. If perhaps you reach the point where space time curvature becomes noticeably quantized then you will maybe get to a place where curvature is = 0 but at this point the observer will be curving space time so it is impossible to have space time with no curvature IE it does not exist.
Be reasonable- you are saying that there will be no gravitation inside of a hollow chamber !! Don't the tenets of logical positivism state that the act of observation interferes with the experiment? you will not ever observe spacetime without curvature in it. Therefore it does not exist.
Strange as it may be, there is indeed no gravitational field inside a spherically symmetric shell of uniform mass. I can prove the Newtonian version for you if you like, though if it enough for you the wiki page on Newtonian gravity states this fact in the section on "Bodies with spatial extent" (http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation). It comes from the famed Newton's shell theorem which Gietkeisel hates so much for some reason. Hmm, actually the page on the shell theorem has some stuff you might like to read too (http://en.wikipedia.org/wiki/Shell_theorem).
