Understand Einstein Hole Argument: Norton's Expln & General Covariance

[atyy]

What are the criteria (only symmetry of background structure, whatever that is?) such that I can legitimately transliterate g'(x') to g'(x) and still call it a solution to a GC equation?

OK, maybe Norton's notation is confusing, with the use of dummy-like variables.

Maybe take a look at eqn 5, 6 in:
http://arxiv.org/abs/gr-qc/0603087

Basically, by definition of general covariance (eqn 5 in that paper), you are allowed to transform the metric as Hurkyl did.

That paper agrees with your complaint that it is a trivial sleight of hand (section 2.2.1): "It seems clear that any equation that has been written down in a special coordinate system ... can also be written in a ... covariant way by introducing the coordinate system – or parts of it – as background geometric structure."

Nonetheless, it is correct (indeed, how could it be wrong, since you can always do it by sleight of hand), so that's not where Einstein made his mistake.

 

[Hurkyl]

1) In what way have I restricted myself to a Euclidean plane?

By insisting, in the (x,y) coordinates, that we must use the distance^2 = dx^2 + dy^2 formula for distance.

2) What are the criteria (only symmetry of background structure, whatever that is?) such that I can legitimately transliterate g'(x') to g'(x) and still call it a solution to a GC equation?

If it's an equation involving only g, you can always do it; x' and x are both differentiable coordinate charts in the same atlas, and so the change-of-coordinate transformation is a diffeomorphism. The truth of a generally covariant equation is invariant under diffeomorphisms, and so one field is a solution iff the other one is.

If the equation involves other variables, the above remains true if you transform everything involved.

That x' and x are related by a diffeomorphism is essential; if you used a nondifferentiable coordinate chart (or even some notion of a discontinuous one), then things will break down. (because those aren't symmetries implied by GC)

If the coordinate charts aren't global, then this is just working locally to the coordinate charts. There may or may not be issues passing to your entire manifold. (The hole argument deals with this by leaving the outside of the hole unchanged, and insisting that the two coordinate charts must agree outside of the hole)

3) Dropping the circle analogy altogether, is there another simple analogy I can use to convince myself that I can go from g'(x') to g'(x)?

For this purpose, the circle one is good when handled properly, and has the added feature that it's easier to see the importance of restricting your attention to symmetries. (It's easier to imagine nonEuclidean transformations than nondifferentiable transformations, especially given how important differential analysis is to current physics) Like I've said, if you repeat the circle analogy, but build the new coordinates by translating/reflecting/rotating the old coordinates (rather than making a Cartesian-to-polar change). And if in doing so you think the coordinates are superfluous (i.e. you think "why not just translate everything on the plane"), then that is correct; the use of coordinates is just an intermediate step for constructing an active transformation, and are otherwise unimportant.

Of course, the fact that coordinate symmetry implies "active" symmetry is interesting.

 

[Hurkyl]

If absolute position has no meaning, how does one even define translation? OK, I'm a goon, you don't have to reply to this (unless you'd like to be amused by a pointless debate), since I've heard the standard argument for years and still don't understand it.

Absolute position does have meaning -- but its only relative to additional non-physical choices we've made. (i.e. relative to a gauge choice)

Let E be the 3-dimensional Euclidean group. The situation can be described mathematically by working exclusively with E-sets, E-spaces, et cetera. That is, sets, spaces, etc. that come equipped with a notion of how they change when you apply elements of E. The relevant fact is that space doesn't have any "E-points": that is, no elements fixed by E. But if we break the symmetry, we can talk about ordinary points.

Actually, there is a way to talk about points without breaking the symmetry. There is a notion of a 'generalized E-point', which is roughly equivalent to the idea of an indeterminate variable. This let's us reason in many of the ways we're used to doing -- just with the caveat that it is impossible to plug in an actual value for the variable.

 

[ecce.monkey]

By insisting, in the (x,y) coordinates, that we must use the distance^2 = dx^2 + dy^2 formula for distance.

Only because that's how distance is defined in cartesian coords. But see below...

For this purpose, the circle one is good when handled properly, and has the added feature that it's easier to see the importance of restricting your attention to symmetries. (It's easier to imagine nonEuclidean transformations than nondifferentiable transformations, especially given how important differential analysis is to current physics) Like I've said, if you repeat the circle analogy, but build the new coordinates by translating/reflecting/rotating the old coordinates (rather than making a Cartesian-to-polar change). And if in doing so you think the coordinates are superfluous (i.e. you think "why not just translate everything on the plane"), then that is correct; the use of coordinates is just an intermediate step for constructing an active transformation, and are otherwise unimportant.

But I just find that sort of transformation unconvincing because it is just simple translations within the same coord system, and so too trivial.

Now I'm thinking that the circle example is an unfortunate choice, because I was trying to use distance as an independent physical "constant", which I wanted to be independent of and ancillary to the real problem at hand. I was thinking of it as mass or electric charge say, that should not be redefined. Maybe my problem is I am trying to understand the g'(x') =>g'(x) problem in too much of a general way, that "g" is just a general mathematical functional solution to some GenCov equation. But maybe the argument that you can go from g'(x') to g'(x) _relies_ on the fact that g is a metric and so is defining distance, maybe even to say defining relative coordinates. That I can live with I think (even without being completely convinced by a proof). It's at that point (for me) the circle analogy breaks down. The solutions x^2+y^2=25 or r=5 do not define distance themselves and so can't stand for g. If this is true I then have no problem with it being a genuine problem as opposed to a blunder, then the point cooincidence argument makes sense in the context, and I understand how this leads one to talk about gauge invariance. Maybe for now anyway.

 

[Hurkyl]

But I just find that sort of transformation unconvincing because it is just simple translations within the same coord system, and so too trivial.

I agree, it is fairly trivial. But in that regard, the analogy is apt; the technical content of the hole argument really is trivial, and wholly unremarkable! As far as I know, the only reason the hole argument is a big deal is for conceptual reasons -- that people are used to extremely rigid structures like Euclidean and Minkowski space, so they get shocked by a demonstration of just how little structure a generally covariant theory has.

So, for the purposes of understanding the technical side of it, I think the circle analogy is fine.

I was thinking of it as mass or electric charge say, that should not be redefined.

Hrm... now that you bring it up... I note there is a relevant symmetry here too. For example, there would be no detectable difference if you replaced the universe with another one in which you doubled all masses and forces, halved the gravitational constant and the permittivity of free space, and so forth. Again, the same reasoning is in play: we promote a coordinate change (changing our units of mass) into an actual transformation which produces a new universe that is indistinguishable from the original universe.

But this lesson has already been learned, I suppose: we already know mass is relative. When we stating something is '1 kg', we really don't mean to indicate an absolute quantity, but instead only to relate its mass to the mass of other objects. (such as a standardized object) I just never thought of it in this way before.

 

[atyy]

Actually, there is a way to talk about points without breaking the symmetry. There is a notion of a 'generalized E-point', which is roughly equivalent to the idea of an indeterminate variable. This let's us reason in many of the ways we're used to doing -- just with the caveat that it is impossible to plug in an actual value for the variable.

I didn't understand any of that, but that sounds interesting. Any references you recommend?

 

[Hurkyl]

I didn't understand any of that, but that sounds interesting. Any references you recommend?

Well, my knowledge of this sort of language mainly comes from topos theory (a subfield of category theory). While it provides an incredibly useful language for expressing things like this as well as other useful things (e.g. making precise the notation used for doing calculations with fields), I don't think it has really filtered down to the masses yet, so that's probably not useful unless you're predisposed to that sort of thing. (I've mainly learned topos theory from this text)


However, the specific mathematics I was referring to is that of group actions -- groups acting on sets, manifolds, vector spaces, and that sort of thing. The relevant cases here are Lie groups, which (to my knowledge) are useful to physics in many ways, so probably worth studying. Alas, I can't recommend any specific references.

 

[atyy]

Well, my knowledge of this sort of language mainly comes from topos theory (a subfield of category theory). While it provides an incredibly useful language for expressing things like this as well as other useful things (e.g. making precise the notation used for doing calculations with fields), I don't think it has really filtered down to the masses yet, so that's probably not useful unless you're predisposed to that sort of thing. (I've mainly learned topos theory from this text)


However, the specific mathematics I was referring to is that of group actions -- groups acting on sets, manifolds, vector spaces, and that sort of thing. The relevant cases here are Lie groups, which (to my knowledge) are useful to physics in many ways, so probably worth studying. Alas, I can't recommend any specific references.

Let's see. The way I think about it is that in Newtonian physics, the metric is a system of rigid rulers, and you can translate experiments without affecting the rulers (translation makes sense), and furthermore, without affecting the experimental outcome (translational symmetry). In GR, the metric is also a system of rigid rulers, but you cannot even translate an experiment (matter) without affecting the rulers, because matter and metric interact. In both theories, if you move everything, then everything stays the same (ie. equivalent to an arbitrary coordinate change). So I would say in both theories, within one model universe, absolute position makes sense. But in both theories, absolute model universes don't make sense, because if you change model universes (ie. move everything), then nothing changes.