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

[Hurkyl]

I guess that's the difference between a mathematician's take on this and a physicisits's take.
The mathematician says :"the maths says that and that's the end of the story. No big deal."

The physicist says "the maths says that...What does it imply? ...

I've been focusing on what 'the math says' because the validity of the mathematical argument was the question raised in the opening post! (And when I made the post you quoted, I thought atyy was denying its validity)

IAnd what it is telling us is that there is no actual physical meaning to the actual points of the manifold (where there is no matter)!

Well, we already new this, didn't we? We had the same conclusion from the translation invariance of Newtonian mechanics!

If I haven't made a mistake (and am remembering knot theory correctly), what the hole argument is telling us is this:

Suppose for simplicity that we have omniescient knowledge of the outside of the hole. We probe the hole by sending a test particle through it. The only physically meaningful thing we can say is where how it entered and exited the hole. There is no other physically meaningful information!

And, I believe it's also true that if we send in lots of point particles, the only information to be gained is where and how they entered and exitted, and if any of them crossed paths. (And the # of path crossings, the sequence of crossings a test particle experiences, and that sort of thing) But that's in a fuzzier area of my knowledge of knot theory.

(The above assumes that we are considering a hole consisting of one coordinate chart -- i.e. it's diffeomorphic to R^4. Otherwise, we might get a little bit of homotopical information too)

Now, in lqg...

In fact, while thinking about this yesterday, I realized that the whole thing I just described setup vaguely resembles a spin network, and I'm now wondering if this was relevant at all in the motivation of LQG.


Incidentally, if we can send test 'strings' through the hole (or enough test particles arranged so that we can convince ourselves they approximate a string), I believe there are a handful of more interesting things to say, but I know very little about that.

 

[ecce.monkey]

I think that I finally convinced myself that I understood the solution.
You obviosuly agree that g(x) and g'(x') describe the same physics.
Do you agree that g(x) and g'(x) are both mathematical solutions to the (empty space) Einstein's equations?

No I don't agree here. They may be solutions to the same empty mathematical form of an equation, but not to some concrete physical field equation. It's trivial that g'(x) is going to solve an equation of the same form as the EFE, but it's a bit hopeful that replacing x' by x using a simple transliteration is going to solve _the_ EFE. General Covariance (as I understand it) says an equation that is GC takes the same form in different coordinate systems. This is not the same as saying that GC equations of the same form are equivalent. You must take into account the content, in this case the physics, when measuring the equivalency. Especially if you are going to draw conclusions about the physics.

I'll illustrate my belief with Norton's toy example (see the above link posted by yossell) and my toy example...

In Norton's case he simply shows that an empty differential equation has two different solutions. He just finds two different curves whose second derivatives are zero, whoopy do. There's no other content in these equations. Or if you like he ends up finding solutions to two _different_ field equations, f''=0 and F''=0. He should be showing how to get two different curves for one equation.

In my case the only way to say x=5 is a circle is to change the physics. In that example the physics is the distance. Hurkyl shows you can call x=5 a circle by redefining distance. But I didn't stipulate a circle is a curve of points that have a constant any old thing. If you want to retain the physics, or the reality of it, if it's going to have some sense, then x=5 is not a circle.

So as I said before Einstein seems to have forgotten that he didn't add the physics when he tried to show that GC is unphysical.

Then the next step is mathematically trivial but physically highly nontrivial. These two different metrics *have* to describe the same physics.

Now, obviously, if you draw a manifold and calculate, say, a geodesic using the two different metrics, the geodesics will be different! At this point we have two choices. Either give up the whole general covariance as a fundamental principle (which Einstein reluctantly did for a while if I understand correctly) or accept what this is telling us as physically correct. And what it is telling us is that there is no actual physical meaning to the actual points of the manifold (where there is no matter)! And I am not saying there is no meaning to the coordinates, I am saying there is no meaning to the actual points! This is very profound and is the issue people in loop quantum gravity for example keep mentionning and refer to as diffeomorphism invariance.

Given there are two different geodesics, but the experience of a particle remains the same, can you elaborate on how this leads to a philosophical statement about meaning of the manifold points? To me of course it means there's just something fundamentally wrong with the logic that lead to the multiple geodesics.

By the way, I think you should have been a bit more respectful to Hurkyl :shy:
He/she has been very patient. It's normal that people use different languages sometimes and both parties must be patient and respectful in trying to understand each other's point of view.

I will attempt harder to contain myself, I tried but it came out. This is a public forum however, so there are bound to be all shades of characters, including (I only suspect mind you...) ones of a never-wrong, full-of-spin nature. I think we both pointed out the same flaw in Hurkyl's hole argument (essentially that he as overloading the meaning of his apostrophes) and as far as I can see the defence was to spin the language up quite a few notches.

 

[Hurkyl]

No I don't agree here. They may be solutions to the same empty mathematical form of an equation, but not to some concrete physical field equation.

I think you misunderstand what it means to be the same form.

Here's an example of a criterion being of the same form.

Consider two tangent vectors v and w to some point P of the Euclidean plane. Let $[v]_{x,y} = (v_x, v_y)$ denote the coordinate representation of v with respect to some orthonormal rectilinear coordinate system (with P not the origin), and let $[v]_{r, \theta} = (v_r, v_\theta)$ be its coordinate representation under the corresponding polar coordinates.

The condition that v and w are orthogonal is a condition that has the same form in both coordinate systems. In the (x, y) basis, the criterion is

$$v_x w_x + v_y w_y = 0$$​

and in the $(r, \theta)$ basis, the criterion is

$$v_r w_r + v_\theta w_\theta = 0$$​

In both cases, the relation has the same form

f(a, b) = 0​

where f is the function defined by

$$f(\mathbf{a}, \mathbf{b}) = a_1 b_1 + a_2 b_2$$​

For example, in the (x,y) basis, this relation is

$$f([v]_{x,y}, [w]_{x,y}) = 0$$​

which simplifies to

$$v_x w_x + v_y w_y = 0$$​

and in the $(r, \theta)$ basis, this relation is

$$f([v]_{r,\theta}, [w]_{r,\theta}) = 0$$​

Note that the only difference between these two relations is the basis with respect to which we represent the vectors as 2-tuples.

Correspondingly, we can observe things like how the coordinate vectors (1,0) and (0,1) represent orthogonal vectors both if we consider them as representing vectors in the (x,y) basis and as representing vectors in the $(r, \theta)$ basis.


An example of something not being of the same form is the distance formula of Euclidean geometry. In the (x, y) coordinate system, the distance between two points P and Q is:

$$d(P, Q) = \sqrt{ \left( x(P) - x(Q) \right)^2 + \left( y(P) - y(Q) \right)^2 }$$​

whereas in the $(r, \theta)$ coordinate system, the distance is

$$d(P, Q) = \sqrt{ r(P)^2 + r(Q)^2 - 2 r(P) r(Q) \cos\left( \theta(P) - \theta(Q) \right) }$$​

Again, let $[P]_{x,y} = (x(P), y(P))$ be the coordinate representation of the point P in the (x, y) coordinate system. Now, we can see that does not exist a function f(a, b) satisfying both:

$f([P]_{x,y}, [Q]_{x,y})$ is the distance between P and Q​

$f([P]_{r,\theta}, [Q]_{r,\theta})$ is the distance between P and Q​

On the other hand, if we demote the distance function to just another piece of added structure, then we can also ask for $[d]_{x,y}$ -- the coordinate representation of the distance function in (x,y) coordinates. (and similarly, in the $(r, \theta)$ coordinates) Then, the distance formula is given by the form

f(g, a, b) = g(a, b)​

and we see that

$f\left( [d]_{x, y}, [P]_{x,y}, [Q]_{x,y} \right) =$ distance between P and Q​

$f\left( [d]_{r, \theta}, [P]_{r,\theta}, [Q]_{r,\theta} \right) =$ distance between P and Q​

If it helps to read the above, the three arguments to f are, in order:
. a function that takes two 2-tuples as input
. a 2-tuple
. a 2-tuple


In case you're curious just what the heck the coordinate representation of d might be, it's a function that takes two 2-tuples as arguments and is given by:

$$[d]_{x, y}(\mathbf{a}, \mathbf{b}) = \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2}$$​

$$[d]_{r, \theta}(\mathbf{a}, \mathbf{b}) = \sqrt{ a_1^2 + b_1^2 - 2 a_1 b_1 \cos\left( a_2 - b_2 \right) }$$​

Hurkyl shows you can call x=5 a circle by redefining distance.

I had originally assumed you were taking this latter point of view, because:
(1) You're trying to understand an argument involving GR, where the metric tensor is merely a field which really and truly does have a different coordinate representation in different coordinate charts -- the analogous situation in your counterargument would be a distance function whose coordinate representation is also different in different charts
(2) You invoked a coordinate change that does not leave Euclidean geometry invariant

If you really and truly meant to consider the distance function of Euclidean geometry as having an absolute coordinate form, then you needed to restrict yourself to orthonormal coordinates -- your switch to polar coordinates was illegal, which is why you arrived at a problem in your counter argument.



P.S. I am avoiding tensor notation specifically to oppose the common abuse of thought that confuses a vector with its coordinate representation as a tuple of real numbers. While it is a very practical abuse of thought, it can also be a huge obstacle when you don't understand something. In that vein, I really shouldn't have used the notation $v_x$ and $v_y$ for the components of the coordinate representation of v w.r.t. (x,y), but alas, no convenient alternate notation sprung to mind.

 

[atyy]

So as I said before Einstein seems to have forgotten that he didn't add the physics when he tried to show that GC is unphysical.

Subtleties aside, it looks like everyone agrees on the hole argument (ie. metrics that are isometric represent the same physical spacetime), and the only real disagreement is whether Einstein made a clever mistake or a dumb mistake. If you think it's a clever mistake, then you say there's something to learn from it. If you think he made a dumb mistake, then there's nothing to learn from it, since the right answer was already obvious to you. It's funny to me that we say 'add the physics', when the solution is so dependent on Minkowski's geometrical formulation of special relativity, and Klein and Hilbert, both mathematicians around Einstein's time were more than aware of 'geometrical objects' that don't change with coordinates. Poincare wrote clearly and correctly that whether space is curved or not depends on whether we define light to follow straight lines. Lots of great physics, ie. connecting physical objects with mathematical objects, and being able to distinguish between mathematical convention and physical reality, came from mathematicians. But maybe it's ok to insult mathematicians - is it faint praise when Roger Penrose says that Ed Witten may be a better mathematician than Witten admits to being?

 

[ecce.monkey]

I think you misunderstand what it means to be the same form.

Here's an example of a criterion being of the same form.
...

No that is what I had in mind for form. However I am curious as to why you chose a distance formula as an example of something not retaining form...that would seem to be confusing the issue. I'm wondering if that's what you think I was saying, that the formula for distance was generally covariant? Well if so, no I was saying the definition of a circle as I gave it was (an attempt at being anyway) generally covariant.

I had originally assumed you were taking this latter point of view, because:
(1) You're trying to understand an argument involving GR, where the metric tensor is merely a field which really and truly does have a different coordinate representation in different coordinate charts -- the analogous situation in your counterargument would be a distance function whose coordinate representation is also different in different charts
(2) You invoked a coordinate change that does not leave Euclidean geometry invariant

If you really and truly meant to consider the distance function of Euclidean geometry as having an absolute coordinate form, then you needed to restrict yourself to orthonormal coordinates -- your switch to polar coordinates was illegal, which is why you arrived at a problem in your counter argument.

What do you mean distance having an absolute coordinate form? Again do you think I was saying distance was GC? Oh dear, re-reading your reply (#6) I think this is indeed what you thought I meant! Maybe, but then again you seemed to know I meant the circle definition previously. Maybe when I said we should use the same distance formula for step 3/4, when we are back in x,y, you thought we should use it all the time? I don't know, I just don't know where we misunderstand each other...

 

[atyy]

What do you mean distance having an absolute coordinate form? Again do you think I was saying distance was GC? Oh dear, re-reading your reply (#6) I think this is indeed what you thought I meant! Maybe, but then again you seemed to know I meant the circle definition previously. Maybe when I said we should use the same distance formula for step 3/4, when we are back in x,y, you thought we should use it all the time? I don't know, I just don't know where we misunderstand each other...

Hmmm, almost everything Hurkyl says makes sense to me (and I can find a good sense for the minor things that are ambiguous). But ecce.monkey is not trying to find out what is logically and physically correct. He's trying to understand Norton's statement of the hole argument. What is missing is whether Norton believes his statement of the hole argument is logically and physically correct. If Norton thinks that his statement is logically and physically wrong, then of course, ecce and Hurkyl will never agree since Hurkyl is stating things that are logically and physically correct, but ecce is trying to correctly represent an argument which is logically and physically wrong (but presumably in a subtle way, and therefore interesting).

There doesn't seem to be general agreement about this. Smolin, Rovelli talk about the 'lesson' of the hole argument. Whereas Matthias Blau <http://www.unine.ch/phys/string/lecturesGR.pdf> says the argument is wrong and only of historical interest.

 

[atyy]

OK, I think I get it. The part where Hurkyl says to ecce "you forgot to transform your metric", is analogous to the part where ecce says to Einstein "you forgot to transform your physics". The part that ecce got wrong is that his definition of a circle as the locus of equidistant points is not a covariant definition of a circle, but a PHYSICAL definition of a circle - a real object like the One Ring, and distance is physically measured by the number of atoms along a taut, non-stretchable string. The covariant formula for the circle would be Hurkyl's f(g,a,b)=f(dxy,pxy,qxy)=f(duv,puv,quv)=constant. So step 2 would be u=5 and Hurkyl's new distance formula d=u^2+v^2-2*u*v*cos(u-v). Now instead of interpreting u,v as radial coordinates, we is interpret u,v as cartesian coordinates, drawing u,v as horizontal and vertical axes - this is analogous to converting g'(x') to g'(x). Now the analogous error is to say that when u,v are drawn as cartesian coordinates, u=5 looks nothing like the One Ring, so our covariant formula cannot represent the One Ring! But Hurkyl says the distance formula represents the physics of the taut, non-stretchable string, and that if you want to reinterpret u,v as cartesian coordinates, you must do so for u=5 AND d=u^2+v^2-2*u*v*cos(u-v), ie. if you want to distort the ring in cartesian coordinates, you must also distort the taut string. In other words, you forgot to add the physics since the distance formula represents the fact that you you started with a distanced physically defined using a taut non-stretchable string.

One difference is while we distorted the ring, but forgot to distort the string- Einstein distorted the string (metric), but forgot to distort the ring (points of coincidence of trajectories of pairs of test particles).

Maybe we can get an even closer analogy if we only change coordinates on part of the One Ring?

 

[ecce.monkey]

OK, I think I get it. The part where Hurkyl says to ecce "you forgot to transform your metric", is analogous to the part where ecce says to Einstein "you forgot to transform your physics".

Maybe you meant to quote me as "you forgot to add the physics", if so yes I think it is analogous. The difference being I guess Hurkyl makes the transformation a positive legitimate step to defend the hole argument, a step the hole argument doesn't actually take.

The part that ecce got wrong is that his definition of a circle as the locus of equidistant points is not a covariant definition of a circle, but a PHYSICAL definition of a circle - a real object like the One Ring, and distance is physically measured by the number of atoms along a taut, non-stretchable string.

I agree with the second part but why is this not covariant? How does the statement change with a change in coordinates? Not at all because points and distance have the same meaning in any system, though they are stated or calculated differently. Doesn't matter to the rest of your argument though...

The covariant formula for the circle would be Hurkyl's f(g,a,b)=f(dxy,pxy,qxy)=f(duv,puv,quv)=constant. So step 2 would be u=5 and Hurkyl's new distance formula d=u^2+v^2-2*u*v*cos(u-v). Now instead of interpreting u,v as radial coordinates, we is interpret u,v as cartesian coordinates, drawing u,v as horizontal and vertical axes - this is analogous to converting g'(x') to g'(x). Now the analogous error is to say that when u,v are drawn as cartesian coordinates, u=5 looks nothing like the One Ring, so our covariant formula cannot represent the One Ring! But Hurkyl says the distance formula represents the physics of the taut, non-stretchable string, and that if you want to reinterpret u,v as cartesian coordinates, you must do so for u=5 AND d=u^2+v^2-2*u*v*cos(u-v), ie. if you want to distort the ring in cartesian coordinates, you must also distort the taut string.

Exactly, but I hope you agree the hole argument makes no such step, and hence my issue with it. By distorting the taut string you have changed the original "field equation", you have changed what distance (the "physics") means in cartesian coordinates. You can no longer claim g'(x) is solving the original field equation but a distortion of it.

 

[ecce.monkey]

Hmmm, almost everything Hurkyl says makes sense to me (and I can find a good sense for the minor things that are ambiguous). But ecce.monkey is not trying to find out what is logically and physically correct. He's trying to understand Norton's statement of the hole argument. What is missing is whether Norton believes his statement of the hole argument is logically and physically correct. If Norton thinks that his statement is logically and physically wrong, then of course, ecce and Hurkyl will never agree since Hurkyl is stating things that are logically and physically correct, but ecce is trying to correctly represent an argument which is logically and physically wrong (but presumably in a subtle way, and therefore interesting).

There doesn't seem to be general agreement about this. Smolin, Rovelli talk about the 'lesson' of the hole argument. Whereas Matthias Blau <http://www.unine.ch/phys/string/lecturesGR.pdf> says the argument is wrong and only of historical interest.

I think your take on the argument is close to correct:)

 

[Hurkyl]

Maybe when I said we should use the same distance formula for step 3/4, when we are back in x,y, you thought we should use it all the time?

Yes, I did. My point was that if you're mimicking the hole argument, you can't¹ use the same distance formula in steps 3/4! All of the relevant objects have to be transformed -- your definition of the circle involved both an equation defining a curve and a notion of distance, so you cannot insist on using the same distance formula in both of the 'solutions' to the circle definition.

So when you complained that I used the new coordinate form of the distance function after making the transformation...

Exactly, but I hope you agree the hole argument makes no such step, and hence my issue with it.

I don't see what you think is missing. The only objects used in the EFE are derived from
. The metric tensor
. The stress-energy tensor
. Assorted real number constants

The hole argument accounts for all of these -- the stress-energy tensor is zero in the hole, and thus its coordinate representation is invariant. The real number constants are also invariant under coordinate changes. The metric tensor is appropriately transformed under the coordinate change. What additional step do you think is missing?



1: Unless you restrict yourself to Euclidean covariance rather than general covariance.

 

[ecce.monkey]

Yes, I did. My point was that if you're mimicking the hole argument, you can't¹ use the same distance formula in steps 3/4! All of the relevant objects have to be transformed

No, here is a link to the hole argument as described by Norton:
http://www.pitt.edu/~jdnorton/papers/decades.pdf
See p801 and 802. The transliteration is the last step, g'(x) is supposed to solve the same equation. There's no _transformation_, just _transliteration_. You've got to stop there when mimicking the hole argument.

your definition of the circle involved both an equation defining a curve and a notion of distance, so you cannot insist on using the same distance formula in both of the 'solutions' to the circle definition.

All I'm insisiting on is if we're back in the _same_ coordinate system that we used to start with, i.e. the cartesian x,y, we should use the _same_ formula for the physical distance.

I don't see what you think is missing. The only objects used in the EFE are derived from
. The metric tensor
. The stress-energy tensor
. Assorted real number constants

The hole argument accounts for all of these -- the stress-energy tensor is zero in the hole, and thus its coordinate representation is invariant. The real number constants are also invariant under coordinate changes. The metric tensor is appropriately transformed under the coordinate change. What additional step do you think is missing?

For one thing this is supposed to be a general argument about GC, not EFE in particular. For another thing you have slipped in "the metric tensor is appropriately transformed". I mean, come on! A bit of sleight of hand there! It's our same argument.

 

[atyy]

I agree with the second part but why is this not covariant? How does the statement change with a change in coordinates? Not at all because points and distance have the same meaning in any system, though they are stated or calculated differently. Doesn't matter to the rest of your argument though...

I guess I was confused (Hurkyl too, it seems) whether you mean the distance function to be given by

Eqn 1: d(x,y)=x^2+y^2.

or by Hurkyl's

Eqn 2: f(g,a,b)=f(x^2+y^2,O,(x=5-y^2,y=5-x^2))=f(x^2+y^2-2*x*y*cos(x-y),O,(x=5,y=y))=5

Exactly, but I hope you agree the hole argument makes no such step, and hence my issue with it. By distorting the taut string you have changed the original "field equation", you have changed what distance (the "physics") means in cartesian coordinates. You can no longer claim g'(x) is solving the original field equation but a distortion of it.

I do think the hole equation takes the step, and the field equation is NOT distorted. So if you take Eqn 1 and define that to be "covariant", then there is a distortion of the form of the equation, unlike the hole argument. If you take Eqn 2 and define that to be "covariant", then there is no distortion of the form of the equation, analogous to there being no distortion of the form of the field equation in the hole argument. However, if you plot x=5 as if x,y coordinates were Cartesian, rather than radial, then x=5 wouldn't look like the One Ring to you, ie. there is no distortion of the form of the Eqn 2, but the figure you plotted looks like a gross distortion of the One Ring. The wrong conclusion to draw at this point, and which was Einstein's original mistake, is to conclude that covariant Eqn 2 cannot represent the One Ring. The right conclusion is that covariant Eqn 2 can represent the one Ring, you must plot both the One Ring and the string in radial coordinates, or you must plot both the One Ring and the string in cartesian coordinates, or any other strange way of plotting (x,y). The thing about a "covariant" equation is that it distorts the string and the ring together in such a way that its form is not distorted, Einstein's original mistake was to think that it only distorted the string, not remembering that it also distorted the ring.

 

[ecce.monkey]

I guess I was confused (Hurkyl too, it seems) whether you mean the distance function to be given by

Eqn 1: d(x,y)=x^2+y^2.

or by Hurkyl's

Eqn 2: f(g,a,b)=f(x^2+y^2,O,(x=5-y^2,y=5-x^2))=f(x^2+y^2-2*x*y*cos(x-y),O,(x=5,y=y))=5

I didn't say distance function, I said distance as in a distance of 10 metres.

I do think the hole equation takes the step, and the field equation is NOT distorted. So if you take Eqn 1 and define that to be "covariant", then there is a distortion of the form of the equation, unlike the hole argument. If you take Eqn 2 and define that to be "covariant", then there is no distortion of the form of the equation, analogous to there being no distortion of the form of the field equation in the hole argument. However, if you plot x=5 as if x,y coordinates were Cartesian, rather than radial, then x=5 wouldn't look like the One Ring to you, ie. there is no distortion of the form of the Eqn 2, but the figure you plotted looks like a gross distortion of the One Ring. The wrong conclusion to draw at this point, and which was Einstein's original mistake, is to conclude that covariant Eqn 2 cannot represent the One Ring. The right conclusion is that covariant Eqn 2 can represent the one Ring, you must plot both the One Ring and the string in radial coordinates, or you must plot both the One Ring and the string in cartesian coordinates, or any other strange way of plotting (x,y). The thing about a "covariant" equation is that it distorts the string and the ring together in such a way that its form is not distorted, Einstein's original mistake was to think that it only distorted the string, not remembering that it also distorted the ring.

The hole argument _is_ Einstein's argument. You are defending Einstein's hole argument but then saying Einstein was wrong...? Of course the way you solve the hole argument, by always transforming rather than transliterating, by including the metric in the transliteration so that your original field equation is not distorted, is the trivial solution to the argument, but this is not how the argument is stated.

 

[atyy]

The hole argument _is_ Einstein's argument. You are defending Einstein's hole argument but then saying Einstein was wrong...? Of course the way you solve the hole argument, by always transforming rather than transliterating, by including the metric in the transliteration so that your original field equation is not distorted, is the trivial solution to the argument, but this is not how the argument is stated.

I think Einstein made the mistake one step later than you think, ie. he did transform the metric.

 

[atyy]

And then after he had transformed the metric, he forgot that transforming the metric means transforming the string.

 

[atyy]

Eqn 1: d(x,y)=x^2+y^2.

or by Hurkyl's

Eqn 2: f(g,a,b)=f(x^2+y^2,O,(x=5-y^2,y=5-x^2))=f(x^2+y^2-2*x*y*cos(x-y),O,(x=5,y=y))=5

Ooops, I copied Hurkyl's equation wrongly and that's full of typos. Look up the correct ones on Hurkyl's post above.

 

[Hurkyl]

All I'm insisiting on is if we're back in the _same_ coordinate system that we used to start with, i.e. the cartesian x,y, we should use the _same_ formula for the physical distance.

Allow me to paraphrase a bunch of stuff, and you can tell me if I get you.

. You have chosen a background geometry'(the Euclidean plane)
. You chose a solution to the circle criterion
. You applied the hole argument, using the Cartesian-to-polar coordinate transformation
. You observed that the result was not a circle according to your chosen background geometry.

If that's correct, then the problem I'm trying to point out is as follows:

The arithmetic of the hole argument works by invoking a symmetry of the background structure. General relativity assumes only a differentiable manifold, and is generally covariant, which is why any diffoemorphism can be used in the usual hole argument. However, the Euclidean plane is only Euclidean covariant -- in that context, the (analogy to the) hole argument is only expected to work with Euclidean motions. If you repeat your orignal argument, but use a translation or a rotation instead of the Cartesian-to-polar coordinate transformation, you'll find that everything works out 'correctly'. More succintly,

the hole argument : general covariance :: your circle argument : Euclidean covariance


Another problem is that one of the primary features of GR is its background independence (which is a consequence of general covariance). GR doesn't make any a priori assumptions of geometry¹; it's just another dynamical variable. Insisting upon a background geometry in your analogy is very much a violation of the 'spirit' of GR. I am under the vague impression that this extends to your understanding of GR: that there is some sort of meaning to things like 'distance' that is independent of the metric tensor field.



1: Technically speaking, it does assume a reasonable notion of differentiability, but that doesn't really count as geometry, since it doesn't give you any notion of lengths or angles or similar things.

 

[Hurkyl]

I think Einstein made the mistake one step later than you think, ie. he did transform the metric.

And then after he had transformed the metric, he forgot that transforming the metric means transforming the string.

He really did that? As far as I can tell, the method of argument is exactly the same as one that proves, among other things, that absolute position is physically meaningless in Newtonian mechanics¹; I would be quite surprised if Einstein made such a simple mistake in its application to the hole. But that is sort of irrelevant, because even if Einstein made a mistake, that doesn't mean we have to, and we can still apply the method of argument in a correct way to make inferences.


1. The argument I'm thinking of is:
. Transform the entire universe by a translation
. Show that the transformed universe is indistinguishable from the original universe (e.g. by showing the transformation is indistinguishable from a coordinate change)
. Note that position is not invariant under translation
. Conclude that absolute position is not physically meaningful

 

[ecce.monkey]

Allow me to paraphrase a bunch of stuff, and you can tell me if I get you.

. You have chosen a background geometry'(the Euclidean plane)
. You chose a solution to the circle criterion
. You applied the hole argument, using the Cartesian-to-polar coordinate transformation
. You observed that the result was not a circle according to your chosen background geometry.

If that's correct, then the problem I'm trying to point out is as follows:

The arithmetic of the hole argument works by invoking a symmetry of the background structure. General relativity assumes only a differentiable manifold, and is generally covariant, which is why any diffoemorphism can be used in the usual hole argument. However, the Euclidean plane is only Euclidean covariant -- in that context, the (analogy to the) hole argument is only expected to work with Euclidean motions. If you repeat your orignal argument, but use a translation or a rotation instead of the Cartesian-to-polar coordinate transformation, you'll find that everything works out 'correctly'. More succintly,

the hole argument : general covariance :: your circle argument : Euclidean covariance

OK, if this is all true and given my lack of grounding in diff geom, I give up and need to mull over it all and do more reading. I think it's worth it if a deeper understanding of GR and things like gauge invariance are of consequence here.

But if you care to answer them here are some probably dumb questions because I don't have time to think about it at this stage:

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

Thanks indeed for your patience.

 

[atyy]

He really did that? As far as I can tell, the method of argument is exactly the same as one that proves, among other things, that absolute position is physically meaningless in Newtonian mechanics¹; I would be quite surprised if Einstein made such a simple mistake in its application to the hole. But that is sort of irrelevant, because even if Einstein made a mistake, that doesn't mean we have to, and we can still apply the method of argument in a correct way to make inferences.

Yes, Einstein first used the hole argument, while searching for the correct equations for gravity, but before he found them, to argue that generally covariant equations could not describe gravity. Later, he discovered the generally covariant equations that provide what still remains our best description of gravity. He then realized that his earlier argument was wrong, and corrected it.

1. The argument I'm thinking of is:
. Transform the entire universe by a translation
. Show that the transformed universe is indistinguishable from the original universe (e.g. by showing the transformation is indistinguishable from a coordinate change)
. Note that position is not invariant under translation
. Conclude that absolute position is not physically meaningful

I have to confess that I have never understood this argument although it is absolutely standard, has nothing to do with special or general relativity, and can be made entirely on the basis of high school physics. Surely I have to be somewhere! Let's say I am on an infinitely large lattice of identical atoms. The particular atom I'm on is a distinct atom from its identical neighbour. I would rather say absolute position has a meaning, but the space I'm in has translational symmetry, and the experimental consequence is that I can translate all my experiments in space, and they will give the same result. 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.

 

[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.