I've been reading about the Einstein hole argument, and i fail to understand what makes active diffeomorphisms "special" compared to passive diffeomorphismsm also known as good old coordinate transformations.

Supposedly this article is a good overview of the problem. I've read this, but i feel that i must be missing something, since i can't feel surprised by the fact that two different coordinate atlas that are identical up to some global time slice and differ afterwards (in the article they call it coordinate transformations that are the same before a global time slice and differ later) will imply (through the field equations) evolutions of the metric that are identical but they will also differ later. Yes, i keep saying to myself while reading this, of course they must differ when the coordinates differ and be equal when the coordinates are equal, that is what happens when you change coordinates!

Now, some other interpretation of the hole argument states that a solution to a GR coordinated system with matter will have a natural mapping to another solution to a completely different system with matter that so happens to have the same coordinate system. Not sure what that does even mean, a manifold is sufficiently defined with a collection of charts and coordinate transformations in the overlap of the charts, also known as a manifold atlas. (yes there are subleties for these assumptions when one is involved with exotic spheres or four dimensional spaces, where there are non-equivalent differentiable structures, but let's put a pin on that for now)

Some other interpretation is that you need matter (and actual observers) to actual pinpoint invariants that are preserved by these active diffeomorphisms, since all observables are done in relation with these. That is a fair statement, but seems to me pretty much devoid of any actual content, since it is natural that one will always make statements relative to, say, the earth, the sun, the milky way, even if the statements are written in manifestly covariant expressions, one will always take a collection of preferred charts to write fields in the given coordinates, which the measurement apparatus are calibrated and gauged against.

Is there something am i missing? i feel like i'm going nuts over the silliness i perceive in this argument

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    $\begingroup$ On and off I've been trying to see what's really behind the active/passive difference too. The best explanation I've found so far is in Gaul and Rovelli's article (section 4). Still not totally happy I get the significance though... $\endgroup$ – twistor59 Jun 25 '12 at 6:31
  • $\begingroup$ I just found this new paper, where they discuss some of the issues with this active/passive distinction which we're struggling with. $\endgroup$ – twistor59 Jun 25 '12 at 7:24
  • $\begingroup$ It's a silly argument. The passive/active distinction is useless, and it is only confusing things to make the distinction. There is no distinction. $\endgroup$ – Ron Maimon Jun 25 '12 at 8:40
  • $\begingroup$ @twistor59 i'm starting to read that reference, but looks pretty balanced view, i'll try to comment more after i've read at least half of it. Thanks! $\endgroup$ – lurscher Jun 25 '12 at 18:07
  • $\begingroup$ Minor comment to the post (v1): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$ – Qmechanic Feb 13 '14 at 6:28

It's a silly argument, as Einstein admitted in 1915, when he sorted out all the gauge invariance business. The reason it was confusing to him is because he wanted deterministic equations, and he had a gauge freedom in the equations, so it was obvious that he couldn't get deterministic equations without fixing a gauge. The hole argument is just saying this: that the future can't be determined from the past, because you can make a coordinate transformation which is only nontrivial in the future.

So determinism doesn't work in the stupid way of determining the future metric uniquely, but it works in the only way it is supposed to work, by determining the metric up to diffeomorphisms. You must remember that Einstein was working before gauge invariance was formulated as a concept (he formulated this idea in the resolution of the hole argument), so he wasn't comfortale with the idea that determinism is only up to a relabeling of the variables.

The passive/active distinction

The passive/active distinction is something people make up to try to distinguish between what they consider two different things: moving a ball two units to the left, or moving the coordinate system two units to the right. The argument that these people give is that I can actually put my hands on the ball and move it two units to the right, but if I change coordinates, the ball stays put and I move the coordinates by doing something mental. So the two things are different.

This is a wrong and confusing point to make, because if you make a consistent picture, either purely mathematical or purely physical, the distinction vanishes. If you want a physical positivistic definition of a coordinate frame, it comes with metersticks and clocks in places in space that tell you how to measure the positions of objects. Moving the frame to the left is like moving all those metersticks and clocks, and it is equivalent to physically moving the ball to the right. Alternatively, if you think of the ball as a mathematical model ball, with a position in a mathematical model universe inside a computer, moving the ball to the right in the mathematical model world is exactly the same (and just as much a mental game) as moving the coordinates to the right.

The distinction between the two activities comes when you compare moving the ball in the model (which is always mental) to moving the ball in the world (which is always physical). The mental/physical map is being invoked here, and instead of saying "oh yes, this is the map between the model and the world", the active/passive afficianado insists that there are really two different kinds of mental transformations in the model: moving the ball or moving the frame, and only one of them is physical motion, and the other is a frame relabeling. This distinction is just a philosophical error mixing up the model/object divide (which is conceptually present) with the moving the object one way/moving the frame the other way distinction, which is completely bogus. There is no distinction really, and the active/passive business is just an annoying thing that gets in the way of learning symmetry.

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  • $\begingroup$ ok, what about statements such as "all physical theories are invariant under passive diffeomorphisms, but only GR is invariant under active diffeomorphisms, which QED or any other theory in a fixed background are not" (continued) $\endgroup$ – lurscher Jun 25 '12 at 13:30
  • $\begingroup$ for a moment i thought this was trying to say that different physical solutions can be mapped with a transformation similar in embodiment to a coordinate transformation, but that maps two actually different geometries, similarly to how a conformal mapping in the complex plane would relate different solutions to the harmonic electrostatic potentials $\endgroup$ – lurscher Jun 25 '12 at 13:32
  • $\begingroup$ what about this other statement? is it true in some sense? $\endgroup$ – lurscher Jun 25 '12 at 13:33
  • $\begingroup$ @lurscher: I'm thinking about it. I know what you mean, so I wouldn't object if you said it, but it's not the way I state it. Maybe this is an example of where there is an active/passive distinction (but I don't think so). I prefer to say it as "the metric is nondynamical", but this is not dealing with it completely. I'll think about it more. $\endgroup$ – Ron Maimon Jun 25 '12 at 18:43
  • $\begingroup$ i know, when i said "i thought this was trying to say.." i didn't mean your answer, i meant that whatever gist of relevance to the whole argument probably might have to do with such mapping of different solutions. Such thing if true would be really interesting, but as you said, if it is true, is not clearly stated in the literature as such $\endgroup$ – lurscher Jun 25 '12 at 18:46

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