Rosé and A-M: Geometrization of Quantum Mechanics

[Careful]

? Surely the inner product in a Riemannian geometry is tensorial? So the definition of conjugacy is too? And therefore the statements remain true even though the elements in them change with diffeomorphism equivalent frames. And when you lift these covariant statements to the tangent spaces in the two frames you get cvorrsponding geometric statemnts: two vectors and a projection are correct in both cases though the vectors are different. Not so?

HUH ? Did I somewhere claim that the definition of conjugacy is non tensorial ? What I said is that in the general case outlined in dg-ga/9702017, the definition of D depends upon the notion of conjugacy (although this does not apply to the special case Torsten uses) and not only upon D_E, D_F and the bundle map \alpha !

 

[Kea]

Careful

If you type [ itex ]\alpha[ /itex ] for $\alpha$ and [ tex ]\alpha[ /tex ] for $$\alpha$$ (but WITHOUT the spaces in the brackets) your posts would be much easier for people to read.

Cheers

 

[Careful]

Careful

If you type [ itex ]\alpha[ /itex ] for $\alpha$ and [ tex ]\alpha[ /tex ] for $$\alpha$$ (but WITHOUT the spaces in the brackets) your posts would be much easier for people to read.

Cheers

I am sure you can figure it out It took me already more than one hour to type that in.

 

[selfAdjoint]

HUH ? Did I somewhere claim that the definition of conjugacy is non tensorial ? What I said is that in the general case outlined in dg-ga/9702017, the definition of D depends upon the notion of conjugacy (although this does not apply to the special case Torsten uses) and not only upon D_E, D_F and the bundle map \alpha !

Well, I think this case in Nair is handled by well-known theorems about complex bundles over manifolds, say the holomorphic theorem. But I don't want to keep jumping around to different papers. Right now I am getting Harvey and Lawson's characteristic currents (math.DG/9407216) under my belt (this paper is a little gem!), and then I'll come forward to its consequent, the Nair paper.

I agree that Torsten's 1996 paper is expressed too tersely, perhaps because writing in English was then a labor for him. He might not have wanted to write more paragraphs for the same reason that you don't want to put [ tex ] [ /tex ] around your LaTeX constructions .

 

[Careful]

Well, I think this case in Nair is handled by well-known theorems about complex bundles over manifolds, say the holomorphic theorem. But I don't want to keep jumping around to different papers. Right now I am getting Harvey and Lawson's characteristic currents (math.DG/9407216) under my belt (this paper is a little gem!), and then I'll come forward to its consequent, the Nair paper.
I agree that Torsten's 1996 paper is expressed too tersely, perhaps because writing in English was then a labor for him. He might not have wanted to write more paragraphs for the same reason that you don't want to put [ tex ] [ /tex ] around your LaTeX constructions .

I never said that the Torsten paper was too condense, I merely stated that some claim he made is incorrect (notice that I never got any answer to this; neither did I get any answer to my complaints about ``the´´ example in his recent paper). I also do not dispute the content of the dg-ga/9702017 and math.DG/9407216 papers and you do *not* need to read these to understand what Torsten tries to say. These papers are actually dealing with a mathematically *different* situation from the one adressed by Torsten (as I explained). Perhaps you should better try to figure out what might be wrong in my *proof* that there is no curvature added in Torsten's case. It is rather obvious that in the generic situation of the above mentioned papers curvature is going to be added (I do not have to read the papers to know that) since the extra subtlety which kills it off in Torsten's case is not present there.

 

[Kea]

Right now I am getting Harvey and Lawson's characteristic currents (math.DG/9407216) under my belt...

Hey, guys, if you can easily follow Harvey and Lawson, it would be much appreciated if you could clarify some points (any of them!).

For instance, they mention a similarity with the Quillen formalism that lies behind the localisation theorems of Witten et al in TFTs. See page 7 where they mention several options for the approximation mode \chi, such as \chi \equiv 1 for t> 1 giving approximations supported near the singular set.

 

[selfAdjoint]

Hey, guys, if you can easily follow Harvey and Lawson, it would be much appreciated if you could clarify some points (any of them!).

For instance, they mention a similarity with the Quillen formalism that lies behind the localisation theorems of Witten et al in TFTs. See page 7 where they mention several options for the approximation mode \chi, such as \chi \equiv 1 for t> 1 giving approximations supported near the singular set.

Since I don't know anything about the Quillen formalism I don't think I can help you. The definitions of \chi seem clear enough and I can just follow their transgression arguments. Is there anything there you need?

 

[torsten]

I never said that the Torsten paper was too condense, I merely stated that some claim he made is incorrect (notice that I never got any answer to this; neither did I get any answer to my complaints about ``the´´ example in his recent paper). I also do not dispute the content of the dg-ga/9702017 and math.DG/9407216 papers and you do *not* need to read these to understand what Torsten tries to say. These papers are actually dealing with a mathematically *different* situation from the one adressed by Torsten (as I explained). Perhaps you should better try to figure out what might be wrong in my *proof* that there is no curvature added in Torsten's case. It is rather obvious that in the generic situation of the above mentioned papers curvature is going to be added (I do not have to read the papers to know that) since the extra subtlety which kills it off in Torsten's case is not present there.

At first some reaction on your last proof:
I know the difficulties in comparing the metrics of non-diffeomorphic manifolds. But in dimension 4 the situation is more friendly then in other dimensions: It is known (mostly the work of Quinn) that two homeomorphic 4-manifolds are diffeomorphic apart from a (possible collection of ) contractable 4-dimensional submanifold having the boundary of a homology 3-sphere. That can be considered as a kind of localisation. Thus a map f:M\to N is a diffeomorphism apart from that contractable piece. But that means we have to understand the special structure of the tangent bundle.
Now to your example: We consider a special class of 4-manifolds known as elliptic surfaces (=complex surfaces) as classified by Kodaira. Such 4-manifolds are fibrations over a Riemannian surface S where the fibers are tori except for a finite number of cases. All possibe cases for these exceptional fibers were classified by Kodaira. A logarithmic transformation is the local modification of N(T^2)=D^2\times T^2 in the 4-manifold M by using a cluing map \partial N(T^2)\to M-\partial N(T^2) which is a pair of maps (\phi,T) with \phi:S^1\to S^1 given by z\mapsto z^p. The tangent bundle over D^2 is a complex line bundle. Now we are in the sitation of Harvey and Lawson: the map \phi induces a map between the complex line bundles which is singular in z=0. Then we obtain a singular connection associated to this map. That modifies the trivial fibration N(T^2)=D^2\times T^2 to a non-trivial one with an exceptional fiber in z=0. That creates a cohomology class which agrees with the class of the exceptional fiber. (see the work of Gompf on the nucleus of elliptic surfaces)
In principle one can take the two coordinates z,z^*. But you can also take the one form dz/z and construct the other components by using the complex structure, i.e. z=x+iy. What is wrong with that approach?
In my 1996 gr-qc paper I discuss the exotic S^7 case but in the Class. Quant. Grav. paper I ommit it because of the known problems.
Secondly I don't understand why in your proof the connection in point (ii) (i.e. on \Sigma) vanishes. Maybe I'm to stupid to see it. Can you illuminate me?
Some more general words about the additional curvature. The work of LeBrun (based on Seiberg-Witten theory) showed that on an exotic 4-manifold there is NO metric of strictly positive scalar curvature. Thus the exotic structure has to change something on the manifold which modifies the curvature. That was the original motivation for our work.

 

[Careful]

**At first some reaction on your last proof:
I know the difficulties in comparing the metrics of non-diffeomorphic manifolds. But in dimension 4 the situation is more friendly then in other dimensions: It is known (mostly the work of Quinn) that two homeomorphic 4-manifolds are diffeomorphic apart from a (possible collection of ) contractable 4-dimensional submanifold having the boundary of a homology 3-sphere. That can be considered as a kind of localisation. **

?? You merely outline that you can ``clump´´ the ``non diffeomorphic´´ properties of both *manifolds* in four dimensions. This is *not* the issue I was reffering to (for my part: just start with two metric tensors on one and the same manifold). As you should know, manifolds by themselves are entirely uninteresting for gravitational physics: the only thing which matters are the causality and curvature properties of the metric. It is a very old issue how to compare two *different* metrics.


**Thus a map f:M\to N is a diffeomorphism apart from that contractable piece. But that means we have to understand the special structure of the tangent bundle. In principle one can take the two coordinates z,z^*. But you can also take the one form dz/z and construct the other components by using the complex structure, i.e. z=x+iy. What is wrong with that approach?**
**

Ah, but in this example you are assuming an identification between M and N has been made. Let me explain what the difference is between this example and what you said before. In your theory, you consider a map f between two different manifolds M and N and you try to define a difference D between the covariant derivative on N and that on M. This is not easy since M and N are two different manifolds, so on what bundle does D have to live? Now, you want to consider an expression of the form:

(df) D_{M} (df)^{-1} on M - \Sigma
D_{N} on \Sigma

Normally, (in the dg-ga/9702017) paper you would have a mixed expression (suppose df is not equal to zero on \Sigma) :

D = df D_{M} \beta + D_{N} (1 - df \beta) (++)

where \beta does now depend upon the chosen Riemannian structures. We want to have an expression of the form (D(V))(Z)(x) so it seems appropriate to me to put V in TM, Z in f*(TN) and define:

(D(V))(Z)(x) = df D_{M}(V(x)) \beta (Z(f(x))) + [D_{N}( df(V(x)) ) ] (1 -
df \beta ) Z(f(x))

This leads to the trouble I mentioned (now first let me answer your following question and then come back to the example).

** Secondly I don't understand why in your proof the connection in point (ii) (i.e. on \Sigma) vanishes. Maybe I'm to stupid to see it. Can you illuminate me? **

On \Sigma : (D(V))(Z)(x) = [D_{N} ( df(V)(x) )] (W(f(x)) where V is a (smooth) section of M in TM and W is a (smooth) section of M in f*(TN). The point is that df = 0 on \Sigma therefore df(V)(x) = 0 hence (D(V))(Z)(x) = 0. The difference with the references you quote is that there both bundles live on one manifold (and you can limit yourself to the bundle map). Here you cannot, and the only way to make sense of this is to pull back the cotangent bundle T*N to M. But the latter is a trivial operation on \Sigma (since you assume df to be null).


Now, let me go on with your example where you regard the four dimensional manifold as a fibre bundle (with 2-D fibers, the tori T^2) over a Riemann surface. The map $\phi$ leaves the fibres T^2 invariant and can be undone on D^2 so I do not see how it ``induces´´ a different differentiable structure (neither do I see how an active diffeomorphism $\phi$ on D^2 \subset Riemann surface can be reduced to a bundle map of a complex line bundle over the Riemann surface). But anyway, you seem to be saying that one should compare different BUNDLE connections over thîs particular Riemann surface. This means you collapse the four dimensional diffeomorphism group to the subgroup which leaves the particular fibration invariant (in either acts only on the base = Riemann surface). Moreover, a spacetime connection on the four manifold has *nothing* to do with a bundle connection over the Riemann surface, so this approach would be obviously flawed. It is easy to see that there are less degrees of freedom in the bundle connection and moreover both connections live on different structures and obey different transformation laws!

For your information, there is an an approach to 2+1 quantum gravity ('t Hooft, Deser, Jackiw et al) based upon classical solutions to the field equations with a distributional energy momentum tensor source (corresponding to spinning particles) which are everywhere locally Minkowski (except where the particle is - there you have a conical singularity). The singularity in the metric is *not* generated by applying a singular coordinate transformation,but is made visible by it (sorry for first mentioning otherwise). When you simply apply a singular coordinate transformation, you have nothing : no conical singularity and no tidal effects.

 

[Careful]

To wrap up this discussion, I shall give a very simple physical reason why the claim that a change of differentiable structure introduces matter is false. The obvious reason it that the Torsten-Helge ``construction´´ does not produce tidal effects outside the ``material body´´ (in either: no gravitational waves) . Consequently there is no gravitational force (and even no volume effect due to Ricci curvature - since no conical singularity is produced). I was hoping that I did not have to state it that explicitly, but this ``correction of technicalities´´ game has cost enough time.

Cheers,

Careful

 

[torsten]

Well, in my opinion the H-constraint problem is unsolvable; something which is well known to be true in the old geometrodynamics approach and I fail to see how introducing new variables can lead to any substantial progress (since the problem is really of a geometrical nature).
Let's assume for sake of the argument that a change of DS gives new physics (I am not convinced yet). I repeat that I fail to see how differential equations; which live on one differentiable structure can give rise to a *change* of the latter.
Cheers,
Careful

I have a question about the Hamilton constraint:
In a paper of Kodama, he showed that the exponential of the Chern-Simons action solves the Hamilton constraint and he had a problem with the momentum constraint (or diffeomorphism constraint). Why does this apporach fails? Chern-Simons theory has a lot to do with knot theory or spin networks.

 

[Careful]

I have a question about the Hamilton constraint:
In a paper of Kodama, he showed that the exponential of the Chern-Simons action solves the Hamilton constraint and he had a problem with the momentum constraint (or diffeomorphism constraint). Why does this apporach fails? Chern-Simons theory has a lot to do with knot theory or spin networks.

I thought that the Kodama state is not normalizable with respect the physical inner product. I am not going to go into detail to this subject here, neither am I interested in topological field theory approaches to quantum gravity for obvious reasons. As I said, the game on this tread is over; there is nothing physical about changing differentiable structures. If you want to discuss the Hamiltonian constraint, you are free to open another thread.

 

[selfAdjoint]

Careful, I have been reviewing this thread, and I have a question about what you posted in #89: here it is (with tex tags around your codes):

I understand what is written above (actually this theorem of Harvey and Lawson is pretty easy to see) but let me treat some stuff in detail. In the paper dg-ga/9702017, E and F are (let's restrict to real) vector bundles of the same rank over one differentiable manifold X, \alpha being a bundle map. It is assumed that the bundle map is singular upon a submanifold \Sigma and that there is a Riemannian metric on each bundle which allows for the definition of the conjugate \alpha^*. I shall first comment upon these issues and then apply it to your paper.

D is defined as D = \alpha D_E \beta + D_F (1 - \alpha \beta) where \beta = (\alpha^* \alpha)^{-1} \alpha^*. In case both Riemannian metrics are the same (in the obvious sense), \alpha \beta is the orthogonal projection operator on the image of \alpha and therefore does depend upon the choice of Riemannian metric (unless im(\alpha) = 0); actually the entire expression D does. So, it is not fair to say that this holds for fibre bundles E and F; since there is lot's of more structure involved

Why pull in the Riemannian metric here? \alpha is a bundle map; a product is defined on the fibers (which are vector spaces, thus converted into algebras) to enable the adjoint \alpha^* to be defined fiberwise, and therefore also the bundle map \beta. The projection you describe seems to me to be defined in the fibers. None of this requires recourse to any Riemanian metric frame, or indeed any particular basis in the fibers. Neither does the connection or curvature which can all be defined at the bundle level. That these bundle-geometric definitions project down onto something that is expressible in Riemannian geometry is of course trivially true, but that does not constrain the bundle maps, etc. It is "downstream" from them.

But I am probably misunderstanding your meaning, so could you enlighten me?

 

[Careful]

**Careful, I have been reviewing this thread, and I have a question about what you posted in #89: here it is (with tex tags around your codes):
Why pull in the Riemannian metric here? \alpha is a bundle map; a product is defined on the fibers (which are vector spaces, thus converted into algebras) to enable the adjoint alpha^* to be defined fiberwise, and therefore also the bundle map \beta. **

The vector spaces over x in E (call it V), and F (call it W) are different, \alpha_x : V -> W. So, how do you define the adjoint of a linear transformation ? You choose a basis in V, one in W, write out \alpha_x as a matrix and take the Hermitian conjugate of the associated matrix. Choosing the bases is equivalent (up to the respective unitary transformations) to introducing Riemannian metrics in V and W. Now, it seems logical to me that such choice of base is made smoothly, otherwise \beta would look even nastier on \Sigma (\beta is already not smooth there). Mathematically speaking, the Riemannian metric is a smooth section in the bundle of frames defined by the respective vector bundles (at least one which covers \Sigma which is all we need for our purposes).


**
The projection you describe seems to me to be defined in the fibers. **


sure


**None of this requires recourse to any Riemanian metric frame, or indeed any particular basis in the fibers. **


Sure it does ! There does not exist something like a ``canonical´´ projection operator of a space on a subspace.


**Neither does the connection or curvature which can all be defined at the bundle level. **


That is true, but that has nothing to do with projections...

 

[selfAdjoint]

So, how do you define the adjoint of a linear transformation ? You choose a basis in V, one in W, write out \alpha_x as a matrix and take the Hermitian conjugate of the associated matrix. Choosing the bases is equivalent (up to the respective unitary transformations) to introducing Riemannian metrics in V and W.

Once you have the inner product on V, say <,> you know there is a unique map \alpha^* satisfying \alpha^*(v) = \langle v,\alpha(v) \rangle. This is independent of any basis. Of course you can exhibit it in any basis but that is not part of its definition. It is as smooth as \alpha but clearly no smoother; if \alpha is not injective at some point \sigma \in V then \langle \sigma,\alpha(\sigma)\rangle is clearly undefined. Smoothness of \beta is obtained by the \chi approximation.

 

[Careful]

Once you have the inner product on V, say <,> you know there is a unique map \alpha^* satisfying \alpha^*(v) = \langle v,\alpha(v) \rangle. This is independent of any basis. Of course you can exhibit it in any basis but that is not part of its definition. It is as smooth as \alpha but clearly no smoother; if \alpha is not injective at some point \sigma \in V then \langle \sigma,\alpha(\sigma)\rangle is clearly undefined. Smoothness of \beta is obtained by the \chi approximation.

Are you pretending now that you never heard of a correspondence between a certain class of frames and a Riemannian metric ?? BTW selfadjoint, the map \beta is NOT smooth at all - if you would care to look at it you will see that there is an infinity replaced by a zero.

**if \alpha is not injective at some point \sigma \in V then \langle \sigma,\alpha(\sigma)\rangle is clearly undefined. **

Utterly false: if we assume V=W then this expression is perfectly well defined for all \alpha.

I promise you that when I make a stupid mistake I shall notify you so that you may scold upon me. But please, stop this ridiculous game about first grade algebra.


Cheers,

Careful

 

[Mike2]

To wrap up this discussion, I shall give a very simple physical reason why the claim that a change of differentiable structure introduces matter is false. The obvious reason it that the Torsten-Helge ``construction´´ does not produce tidal effects outside the ``material body´´ (in either: no gravitational waves) . Consequently there is no gravitational force (and even no volume effect due to Ricci curvature - since no conical singularity is produced). I was hoping that I did not have to state it that explicitly, but this ``correction of technicalities´´ game has cost enough time.
Cheers,
Careful

Force is a change of energy/mass/(differential structure?) with distance, and he has not come up with how things are changing with time (as well as with distance, I suppose). So I don't see how you can conclued no gravitational waves, etc, when the dynamics has not been derived yet. Have you jumped ahead of Torsten to prove that no dynamics is possible in this programme?

I read half of Frankel, The Geometry of Pysics, and half of Nakahara, Geometry, Topology, and Physics. But what area of study would you say would be a prerequisite for this stuff of Torsten's. And what books would you recommend. Thanks.

 

[Careful]

**Force is a change of energy/mass/(differential structure?) with distance, and he has not come up with how things are changing with time (as well as with distance, I suppose). So I don't see how you can conclued no gravitational waves, etc, when the dynamics has not been derived yet. Have you jumped ahead of Torsten to prove that no dynamics is possible in this programme? **

Torsten claims that a change of differentiable structure introduces a source term in the Einstein equations. This, he does by adding a singular part to an originaly smooth connection through a change of differentiable structure which *lives* on a three dimensional manifold (that is what he claims - it is not true of course). Now, start out with the flat connection and perform a coordinate transformation which is singular on \Sigma, then Torsten claims that the resulting connection (combined with a suitable transformation of the metric) is a solution to the Einstein field eqn's with a distributional source of matter. But this means, that his matter does not produce tidal effects (no gravitational waves/force), moreover he does not obtain any ricci curvature effect (volume contration). Hence, his ``matter´´ does not influence the gravitational field and vice versa; contrary to Einsteins theory (in 2+1 dimensions you can do that for non spinning particles, but ok, there we have no gravitational force anyway).

I shall present the argument differently : how do we know matter is present ? (a) We observe tidal effects (on light) (b) we observe that matter induces focal points (volume effect). Neither of these things are present (since the flat solution remains unatered outside the object), therefore we observe nothing. One simply seems to have forgotten that adding matter induces a GLOBAL operation on the physical quantities of interest.


**I read half of Frankel, The Geometry of Pysics, and half of Nakahara, Geometry, Topology, and Physics. But what area of study would you say would be a prerequisite for this stuff of Torsten's. And what books would you recommend. **

As I said, there is no Torsten stuff; at least not in the field of QG.

Cheers,

Careful

 

[Mike2]

I shall present the argument differently : how do we know matter is present ? (a) We observe tidal effects (on light) (b) we observe that matter induces focal points (volume effect). Neither of these things are present (since the flat solution remains unatered outside the object), therefore we observe nothing. One simply seems to have forgotten that adding matter induces a GLOBAL operation on the physical quantities of interest.

What is "outside the object" when matter, according to Torsten, are singularity points, or Delta functions, whose support is 3D, IIRC? I think all that needs to be shown is that the curvature is greater when particles are closer together? Then this would be the "force" you are referring to, right?

 

[Careful]

What is "outside the object" when matter, according to Torsten, are singularity points, or Delta functions, whose support is 3D, IIRC? I think all that needs to be shown is that the curvature is greater when particles are closer together? Then this would be the "force" you are referring to, right?

But his 3-D delta functions are supposed to represent matter ! The whole point he wants to make is that a change of diff structure induces singularities (on a 3-D support) in a (flat) background metric (or connection). But outside this 3-D support nothing happens with the metric (connection), so there is no propagation, no gravitational waves!
Moreover, I have PROVEN that nothing happens (even on this 3-D surface!) and have shown where his examples are flawed. I do not know what I can do more.

 

[marcus]

...
Moreover, I have PROVEN that nothing happens (even on this 3-D surface!) and have shown where his examples are flawed. I do not know what I can do more.

Careful, I'd really value some of your critical comments about the Charles Wang and Johan Noldus papers mentioned in this "List" thread
https://www.physicsforums.com/showthread.php?t=102147

especially the Noldus, which is new for me (I just noticed it)
I give a link in the last post on that thread
https://www.physicsforums.com/showthread.php?p=852574#post852574

 

[Mike2]

But his 3-D delta functions are supposed to represent matter ! The whole point he wants to make is that a change of diff structure induces singularities (on a 3-D support) in a (flat) background metric (or connection). But outside this 3-D support nothing happens with the metric (connection), so there is no propagation, no gravitational waves!
Moreover, I have PROVEN that nothing happens (even on this 3-D surface!) and have shown where his examples are flawed. I do not know what I can do more.

I thought the whole point with the delta function is that it integrates to the same value no matter what the size of the volume (support?) you integrate over. So if his conclusion is valid for one size support, then it is valid for all sizes of support so that in effect it is valid for all space, since any point in space might just as easily be used in some support. But now I wonder what the delta function is a function of. What physical field is the integrand inside the support integral?

 

[Careful]

**I thought the whole point with the delta function is that it integrates to the same value no matter what the size of the volume (support?) you integrate over. So if his conclusion is valid for one size support, then it is valid for all sizes of support**

True (I never claimed otherwise), but the support needs to be compact (moreover, I wonder what the volume measure is he wants to use since the metric would change). But I repeat, this is all NOT happening. I do not think this conversation is useful given the fact that there is really no such effect.

** so that in effect it is valid for all space, since any point in space might just as easily be used in some support. But now I wonder what the delta function is a function of. **

You cannot reasonably claim that the universe should consist of one particle.

 

[Mike2]

**I thought the whole point with the delta function is that it integrates to the same value no matter what the size of the volume (support?) you integrate over. So if his conclusion is valid for one size support, then it is valid for all sizes of support**

*True (I never claimed otherwise), but the support needs to be compact (moreover, I wonder what the volume measure is he wants to use since the metric would change). But I repeat, this is all NOT happening. I do not think this conversation is useful given the fact that there is really no such effect. *

"no such effect" as what? You've kind of lost me here.

If the delta integrates to the same value no matter what "size" the support is, then isn't this just another way of saying that this value is invariant wrt the metric, at least as far as scale is concerned, right?

** so that in effect it is valid for all space, since any point in space might just as easily be used in some support. But now I wonder what the delta function is a function of. **

*You cannot reasonably claim that the universe should consist of one particle.*

How is this a "claim" that the universe consists of one particle? If you've already covered this area, please indicate what post number it was covered in, and maybe I can read it anew in this light. Thanks.

 

[Careful]

**"no such effect" as what? You've kind of lost me here.**

There is NOTHING happening at a change of differentiable structure, please read my posts which adress this at a technical level.

**If the delta integrates to the same value no matter what "size" the support is, then isn't this just another way of saying that this value is invariant wrt the metric, at least as far as scale is concerned, right? **

You have missed my remark about WHICH VOLUME MEASURE to use (the metric would also transform on \Sigma).

Please, I have discussed these matters in details already (read it up)
Again, if you drop a compact form of matter in spacetime according to the authors prescription, the gravitational field outside this region is NOT changing in the procedure. I do not understand what your problem is ??

 

[Mike2]

***"no such effect" as what? You've kind of lost me here.***

**There is NOTHING happening at a change of differentiable structure, please read my posts which adress this at a technical level. **

Please don't take my comments as a challenge. I'm sure you are a better mathematician than I am. And I am asking for help. I only have a brief acquaintence with the math concepts that you discuss. And I'd like to know more. But I'm not sure what areas of math to study. Perhaps you could help me with that.

If you could be just a little more generous, I might actually begin to understand your objections. I'm not sure which of your previous posts would address the issues I raise. If you could at least give the post number (located in the upper right of each post), I might gain insight to your objections. Otherwise, I be lost in a sea of symbols as to which post you are referring to.

But as to your remarks above. I don't know what you mean by "NOTHING happening". I suppose the only issue at hand is whether the change in "differential structure" produces the change in curvature and is equivalent to adding another singularity as claimed.

***If the delta integrates to the same value no matter what "size" the support is, then isn't this just another way of saying that this value is invariant wrt the metric, at least as far as scale is concerned, right? ***

**You have missed my remark about WHICH VOLUME MEASURE to use (the metric would also transform on \Sigma).
Please, I have discussed these matters in details already (read it up)
Again, if you drop a compact form of matter in spacetime according to the authors prescription, the gravitational field outside this region is NOT changing in the procedure. I do not understand what your problem is ??**

It should be an easy matter to scroll through the post and refer to the post that best describes your argument. Otherwise, I'm afraid I'd be lost.

But it would sound as if you are saying that the form on the 3D support responsible for the curvature in the 4D manifold does not map points outside the 3D support. But isn't it true that the support is of arbitrary size? So if the form is defined only inside the support, but the support can be of any size, then doesn't this mean it is applicable for all space that could just as easily be included in some support or another.

 

[Careful]

Hi Mike,

To understand what I say, you just need to have some good grasp on the basis of differential geometry (and GR). I don't know if you are a math or physics oriented person, but a good physics book is the one by Misner, Thorne and Wheeler; a bit more mathematical Nakahara (but I guess you were doing that) Wald or Hawking and Ellis (my preferred one).

I think posts 99 and 100 are good starters. The claim at your third paragraph is correct with negative answers to all.

I briefly mention the volume measure in post 113 (the ``problem´´ is that in the spirit of Torsten and Helge this measure should also transform on \Sigma which -again- it does not.).

I hope this clarifies things

 

[selfAdjoint]

I read half of Frankel, The Geometry of Pysics, and half of Nakahara, Geometry, Topology, and Physics. But what area of study would you say would be a prerequisite for this stuff of Torsten's. And what books would you recommend. Thanks.

Mike you can find a lot of background material on Careful's objections in Nakahara's Chapter 7, Riemannian Geometry, and Chapters 9 and 10, Fibre Bundles and Connections on Fibre Bundles. In Chapter 7 he introduces you to the Levi-Civita Connection and its use in defining curvature, which is at the root of the disagreement between what we might calll the Brans-Nair-Asselmeyer program and Careful. In Chapter 9 Nakahara defines principle bundles and their features and in Chapter 10 he shows how to define connection and curvature through strictly bundle operations in principle bundles. He then goes on to show that in the case that the principle bundle is actually the tangent bundle of a Riemannian manifold, the bundle connection coincides with the Levi-Civita connection, and you recover all the Riemannian results, including even the Bianchi Identities. So there is no conflict between bundle construction and traditional Riemannian methods.

If I may suggest, I believe Careful's criticisms rest on two major assertions. First that the mapping between bundles that is key to the Asselmeyer 1996 paper is not well defined, and then that the connections have to be supported at the manifold level by smooth (C^{\infty} ) diffeomorphisms, which would destroy the paper's claim that there is actually a failure of diffeomorphism, a lack of injectivity, on a "set of measure zero", which can however be approximated to any degree by smooth constructions and therefore meaningfully integrated into Einstein's field equations. I am still researching these issues for my own peace of mind, in the absence of any response from Torsten or Helge on these points.

 

[Careful]

If I may suggest, I believe Careful's criticisms rest on two major assertions. First that the mapping between bundles that is key to the Asselmeyer 1996 paper is not well defined, and then that the connections have to be supported at the manifold level by smooth (C^{\infty} ) diffeomorphisms, which would destroy the paper's claim that there is actually a failure of diffeomorphism, a lack of injectivity, on a "set of measure zero", which can however be approximated to any degree by smooth constructions and therefore meaningfully integrated into Einstein's field equations. I am still researching these issues for my own peace of mind, in the absence of any response from Torsten or Helge on these points.

To name a few mathematical criticisms:
(a) the bundle in Torsten's example is *not* the tangent bundle to the four manifold (so the connections are not spacetime connections)
(b) If you want to define the singular connection rigourously, the construction becomes trivial and there is no curvature effect at all (as I computed explicitely)
(c) There is nothing happening to the geometry outside \Sigma, so an observer outside the ``singular region´´ will not detect anything at all.

But, it is even much easier to see that nothing happens if you think about how to complete M - \Sigma.

 

[marcus]

... the Charles Wang and Johan Noldus papers mentioned in this "List" thread
https://www.physicsforums.com/showthread.php?t=102147

especially the Noldus, which is new for me (I just noticed it)
I give a link in the last post on that thread
https://www.physicsforums.com/showthread.php?p=852574#post852574

Careful has already taken the trouble to comment on the Noldus paper, which I mentioned in the "List" thread. Thanks Careful. It was just a brief comment, and may not be his last word on it.

I am thinking that even though we always need to be focused, and this thread has been very intensively focused on the Torsten Helge idea, perhaps we should start a thread about another approach----not to spend ALL the critical talent on one thread.

Maybe i will start a thread on this Noldus idea, and see if there is any comment. It might be a good idea, and also might possibly be no good at all---I certainly cannot tell at first sight, without some help from others here.

One thing I can say is that Noldus is willing to take risk. he does not merely play it safe. i think this at least is a clear plus. he says that he will try to rebuild QM, put it on a new footing, to make it compatible with GR. the newly founded QM is supposed to APPROXIMATE the old one, but new experiments could distinguish and falsify it, if it is wrong.

So as not to conflict with the topic of THIS thread, namely the Torsten Helge paper, I will make a Noldus thread (not a list of rebel QG approaches, but a thread focused just on the Noldus paper)

Here it is:
https://www.physicsforums.com/showthread.php?t=103750