Conceptual connection between intrinsic geometry and relational information

[fredrik]

The threads on expansion of the universe caused me to take the opportunity to express an interesting reflection that relates to the still open question of the conceptual unification of QM and GR.

 

There two conceptual areas that commonly are discussed, from both philosophical and conceptual viewpoints:

 

1) Relativity, both special and general, which relate to the concept of geometry and in particular intrinsic geometry. In the general picture of a curved surface, there are two kinds of curvature - extrinsic and intrinsic.

 

The extrinsic curvature relates to how the surface "curves" relative to the higher dimensional embedding space - for example a 2D-surface curved in 3D embedding.

 

The physical curvatures in GR are the intrinsic ones. Although one can mathematicall imagine it embedded in a higher dimensional space, this embedding is not unique, and more important non-physical.

 

So the physical curvature is what can be deduced from an observer that lives on the surface, not in the embedding space.

 

Martin wrote in http://www.scienceforums.net/forum/showthread.php?t=30787

 

the key thing being able to describe growth from the inside, without visualizing the outside

 

to be a cell inside the trunk of a tree and know that the tree is growing even if you can't look from the outside and see the bark of the tree. even if you don't know if the bark exists.

 

or to be a cell in somebody's leg-bone and aware that it is growing without having any idea of the outward shape

 

the person who invented the methods for INTERNAL GEOMETRY was Riemann in 1850. Using Riemann tools you can study the shape of something that has no exterior. Riemann geometry enabled Einstein to take the 1915 GR step.

 

Riemann tools enable one to explore curvature internally even if the outside does not exist---even if there is no outside.

 

2) Quantum mechanics, which at least ín some interpretations, and according to Bohr, shoud dela with the observers information about the system. Ie. QM, according to Boht speaks about what we can say about nature, not what nature is. In essencse, I like to think of this as intrinsic information vs extrinsic information.

 

Like the analogy of internal vs external geometry, the observer constrained to the surface, can not necessarily say that there can not exists in some sense an embedding. The point is that, this embedding is unverifiable for him.

 

This is conceptually related to the idea that, there can always "in some sense" exists information that the observer doesn't have, but then as in the geometry example, information truly hidden from the observer, has no impact on his expected life. At the same instant it does, it's not hidden anymore.

 

Similarly, if we consider an observer embedded in an environment, one comes to a situation that is strikingly similar to the geometry. An observer, is embedded in an environment. And anything the observer "knows" about the environment, must be encoded in the part of the system that we call the observer. So one can consider two kinds of information, intrinsic to the observer, and extrinsic as stored in the environment.

 

In geometry as in GR, the intrinsic one is the physical one, any embeddings are non-physical, redundant and ambigous.

 

One may be tempted to draw the same parallell, that in QM, the physical correspondence is the intrinsic information, and the externally "imagined" information encoded in the environment are non-physical are judged relative the observer. This is really the same argument as in the GR case.

 

But this leads to a completely subjective view of things - which is sort an "issue". It would seem to totally wreck objectivity, since each observer would have it's own physical picture, and it seems alot of people doesn't see any sensible way to do science. The usual way to resolve this, is to violate the intrinsic information an instead IMO, pick an embedding that many seem to agree upon.

 

But here is IMO things missing. I think the parallell here is conceptually interesting, and comparaing different problems with each other may open for new ideas.

 

Maybe the missing understanding is how, objectivity can emerget, out of the seemingly only sensible non-ambigous, but subjective, physics?

 

I just wanted to pose this reflection. Maybe someone might add their own reflection to this. It might be interesting.

 

/Fredrik

[ajb]

What you say about curvature is indeed true. Also, any manifold (space-time) can be embedded in [math]\mathbb{R}^{n}[/math] for some [math]n[/math]. However, it is not known if space-time is embedded inside a larger dimensional manifold. String theory etc. suggest that it is, but really we just don't know.

 

What I will say is that something like 99% of physics can be explained in terms of geometry. For me geometry is the language of physics.

[fredrik]

Ajb, what's your opinion and interpretation on QM in this context?

 

Should we expect that the geometries must be constructible relative to the observer in the sense that there exists in the observers memory or internal microstate, and "image" of the environment, that may in general disagree with other observers. The analogy here is that the subjectively physical geometry so to speak is the one that can be "in principle" be mapped out by an observer living there - ie the intrinsic geometry.

 

intrinsic geometry ~ intrinsic information, but when you add the information thinking, then the geometry itself is of course part of the information - so it seems that the truly "physical geometry" - adding the QM stuff - is subjective to the observer living on the manifold, making it very strange.

 

Classically one can imagine that an observer living on the manifold can "in principle", with clocks and rules find out the global geometry. But for several reasons, this doesn't make sense. One is that how do we expect a finite observer living on the manifold, to be able to relate to such a massive map? (information capacity issue) information wise, it makes no sense. The second issues is the time and information "ages" and changes (time stamping issue).

 

I'm curious if you have a different view of this, in your geometrized thinking so to speak?

 

I'm not trying to "solve" these big issues in this thread, but I'm curious to hear alternative views and angles, crazy or not, doesn't matter to me. Some madness might even help.

 

/Fredrik

[foodchain]

intrinsic to the observer, and extrinsic as stored in the environment.

/Fredrik

 

I don’t understand how any particular object can be separated from the environment as is. For instance if for some strange reason plants decided to change and stop giving off air the amount of change that would bring on would surely be noticeable. Yet the entire "web" of it all seems to be just that, the entirety of the environment. That’s one reason I always questioned the use of that word in such a context, to me the line of environment and individual "thing" is not defined to a point that I can digest really. Its easy to see this in many fields of study scientifically speaking I think. Such as with geology, the earth gets treated as a system which is dynamic in relation to its various components, but really its that concert that is deriving the environment so to speak. As far as QM goes I don’t understand fully the scope of environment or why anything is anything to be honest, why history can exist, or why stars seem to be evident in the observable universe. Which I think again leads to the context of an environmental variable at the least.

 

For instance vision has quantum properties, normal human vision. So am I viewing a constant superposition defined to some particular state at a constant, why, and for the reason that the universe or the earth is far older then the human species what defines an observation or a measurement at a quantum scale? Personally I think its just physical interaction overall, which I think would yield an environment regardless. Yet within the quantum domain that inherent nature of a probability makes me question how to define nature of things quantum truly that do not change, ever, such as the uncertainty principal.

 

*On edit.

 

I have read some papers I guess on QM that seem pretty far out to me from where I sit. To actually ponder the many worlds hypothesis is actually something I don’t think any human could really do successfully outside of pure math. For instance a take on that, which is a supplement if not challenge to doctrine or Copenhagen, would that make our observable universe just a quantum blip in an infinite quantum blip that basically rehashes circuits of some sorts(what state of matter would that be)? I fear even uttering(or thinking on) such thoughts simply because they do appeal to me of holding a certain state of madness or at least do not happen to be rational in any sense of modern or standard, yet no one can exclusively deny really what could be an endless array of interpretations I think.

[fredrik]

I don’t understand how any particular object can be separated from the environment as is.

 

In a certain sense I agree completely. This is also I think another important angle of the problem.

 

While the problem of drawing a line between two parts is real, the fact that certain self-organisation takes place that is still more or less identifiable as a stable sub-system makes the question relevant.

 

In my thinking, I imagine that for consistency, ANY subsystem could be interpreted as an observer. However, not all observers are "stable" so to speak. So I think the particular subsystems that become "observers" distinguish themselves this way.

 

For example, put a biological organism into the "wrong" environment. You can do this without problem, and it's still fairly clear what the distinction is between the organisms and it's environment, however the organism will probably die. But in some cases, the organism may be so strongly dependent on the environment, that it's even hard to define transporting it to another environment, since it would probably imply immediate destruction of the system.

 

So the same way that say a particular environment in biology favours appearance of certain organisms, in physics a particular environment might make favour certain particles. And in the abstraction I was trying to make a particle qualifies as an observer as much as anything else.

 

Also I imagine that an observer can increase it's information capacity by learning how to predict and make use of the environment. Humans have a lot of tools, like books and computers. But it still takes a certain level of sophistication to do that.

 

To describe how an observer can increase it's mass, and therfore intertial stability spontaneously is something that begs for a better understanding. And I think the environment plays a key here since spontanesous or not must depend on the environment. This is why one would expect a different "zoo" for each environment, but in each case there should be an emergent logic.

 

/Fredrik

[ajb]

Ajb' date=' what's your opinion and interpretation on QM in this context?

[/quote']

 

I tend not to try to interpret quantum mechanics. Far greater people than I have tries and any interpretation rarely adds to the calculations.

 

 

 

Should we expect that the geometries must be constructible relative to the observer in the sense that there exists in the observers memory or internal microstate' date=' and "image" of the environment, that may in general disagree with other observers. The analogy here is that the subjectively [b']physical geometry[/b] so to speak is the one that can be "in principle" be mapped out by an observer living there - ie the intrinsic geometry.

 

intrinsic geometry ~ intrinsic information, but when you add the information thinking, then the geometry itself is of course part of the information - so it seems that the truly "physical geometry" - adding the QM stuff - is subjective to the observer living on the manifold, making it very strange.

 

I think you will have to be very careful about what are the observables in the theory. The curvature tensors are not observables as it is not gauge invariant. The Ricci scalar could be.

 

As photons travel along geodesics it is possible in principle to use photons are tracers and map the geometry that way.

 

Classically, the geometry is independent of observers. In a full quantum theory I don't know. The role of the observer is fundamental in quantum theory. How do you "quantise" the observer ?

 

Classically one can imagine that an observer living on the manifold can "in principle"' date=' with clocks and rules find out the global geometry. But for several reasons, this doesn't make sense. One is that how do we expect a finite observer living on the manifold, to be able to relate to such a massive map? ([b']information capacity issue[/b]) information wise, it makes no sense. The second issues is the time and information "ages" and changes (time stamping issue).

 

I'm curious if you have a different view of this, in your geometrized thinking so to speak?

Certainly an observer can map out the local geometry by studying the motion of test particles or light rays. Global issues (topology) are more difficult.

 

 

I'm not trying to "solve" these big issues in this thread' date=' but I'm curious to hear alternative views and angles, crazy or not, doesn't matter to me. Some madness might even help.

/Fredrik[/quote']

 

People do use non-commutative geometry to unify quantum mechanics and relativity. It is not something I have any specialist knowledge of, but what I will say is that "God ever geometrizes"

[fredrik]

I think you will have to be very careful about what are the observables in the theory.

 

In the way things are, and from a mathematical point of view, I agree completely.

 

But one might ask - and I do - does non-observables has a place in a physical theory. It's to me, a sign of mathematical redundancy - by the same reasoning that the non-physical embeddings are.

 

If these non-observable things, can not be removed, or chosen arbitrarily, without destroying the physics, then it suggest that we are making use of an ambigous embedding or background which is disturbing, isn't it?

 

Classically, the geometry is independent of observers. In a full quantum theory I don't know. The role of the observer is fundamental in quantum theory. How do you "quantise" the observer ?

 

The quantization procedure is a bit of a vague notion itself, usually when starting with a classical theory, and then taking it into the quantum domain. I'm not sure that's the way to go.

 

I'm not sure but I try to start with the notion of distinguishability. Either one can consider this as an elementa of discretensss, or one can consider it as a cutoff in continuum models, where you cut off anything with sufficiently low measure, and then either the measure is "0" or it's not. This is probably a weak point in the reasoning though. This notion of distinguishability keeps existing throughout the complexity.

 

I expect that once we see the right way of reasoning, quantization and gravity phenomenan are not in contradiction, and one shouldn't have to "quantize" gravity, I'd expect it to come hand in hand.

 

/Fredrik

[ajb]

Non-observables do indeed play a role in physics. See gauge theories and gauge symmetries.

 

About embeddings, if space-time is embedded in some higher manifold there would (presumably) be a gauge (and maybe a conformal) symmetry here. That is all embedding diffeomorphic to each other are equivalent. This is simply the statement that physics is independent of the coordinates chosen.

 

It would be nice if one can start from a fully quantised theory. But our understanding of quantum field theory is not fully developed and as such it is usually necessary to start from a classical theory and quantise it. (whatever that exactly means).

[foodchain]

In a certain sense I agree completely. This is also I think another important angle of the problem./Fredrik

 

Do you think quantum mechanics could be applied to geologic processes for study into origins of life questions? Such as trying to equate possible quantum mechanical basis for energy in a system to take the form of early life similar to bacteria(metabolism really for my idea) for instance? I cant help but to view trophic phenomena as simply more geologic process in regards to matter and energy interactions. I want to know how to study such a question in a quantum perspective? I am thinking of taking linear algebra, would it also be good to take differential equations to supplement? I want to apply a consistent histories thinking to it all along with someway to describe the environment as a nature of the quantum world, for this reason I am interested in einselection for instance. I would also of course like to use such as a basis for natural selection and move natural selection out of a definition to only biological phenomena and more or less natural phenomena in general?

 

Its just a weird idea I have interest in as hobby. I would like to learn the best way to get to a point you can apply quantum theory to say such a question if possible in a format to derive some math. I was wondering if you could describe what math skills are really needed to work with quantum theory?

[Martin]

Fredrik, thanks for the quote in post #1.

It is a nice short essay. It's clear and it draws an interesting parallel: QM forces on us a kind of self-restraint about what we can say about nature, as Bohr indicated. Indeed there might be deterministic hidden variables or if not there might be some fundamental structure underlying QM, but it seems as if we cannot see it. Analogous to knowing our space from the inside and not being able to speak meaningfully about wherein it may or may not be embedded.

 

Cool.

[fredrik]

Non-observables do indeed play a role in physics. See gauge theories and gauge symmetries.

 

Thanks for your comments Ajb, as far how current physics works, there are alof of non-observables. But I was trying to ask what to expect from the next revolution to unify both GR relational ideas with the observer relational ideas of QM.

 

I think that somehow, we haven't yet incorporated the beauty of both QM and GR into our conceptual thinking. I am trying to probe that. It seems you very much like the initiated roads, I feel like that's taking me to the wrong place so that's why I try to find my way through the bush.

 

In the gauge thinking, I can't help making a loose reflective association between the gauge degrees of freedom and the degrees of freedom that the identity of the observers means. Since in the global geometrical thinking, once seems to describe the world the observers lives in, but not from the observers view, but from the birds view - this is a bit unsatisfactory. In th birds view the local observers seems arbitrary and lacking physical reality, instead the connections between them are objective - gauges choices are arbitrary but gauge connections are not - ie. The birds view can not explain the subjective choice, but it can certainly explain their connections. I guess this is as close to your thinking I can stretch, I'm not sure what you think of this.

 

Also the use of symmetries are too wild IMO. It's more to me like a kind of tool in reasoning, that _enforcing_ various symmetries locally and globally we can make conclusions and find new phenomenology. But from the observational point of view, which IMO is sort of the only physical view - albeit arbitrary from the point of view of "god" or the "birds view", I want to understand the dynamics of how this symmetries arise as a resuly of local information processing and retention.

 

I kind of suspect that we may need a new formalism, that respects all current ideals from the first construct.

 

/Fredrik

 

Do you think quantum mechanics could be applied to geologic processes for study into origins of life questions?

...

I would also of course like to use such as a basis for natural selection and move natural selection out of a definition to only biological phenomena and more or less natural phenomena in general?

 

If we are first talking about the normal standard QM, I don't think has much to add to those question.

 

However the associations I do, are more in the sense of a pictures development of quantum mechanics, and it's interpretations. But I suspect that to make sense of these at this point fuzzy things, a normal standard appreciation of the basiscs is needed. My strange reflections is not standard QM.

 

Its just a weird idea I have interest in as hobby. I would like to learn the best way to get to a point you can apply quantum theory to say such a question if possible in a format to derive some math. I was wondering if you could describe what math skills are really needed to work with quantum theory?

 

For the basics, i'd say linear algebra and ordinary differential equations.

 

Made simple, the "states" in quantum mechanics make a vector space. Measurements correspond to linear operators. And the operators have eigenvalues. The eigenvectors or (eigenfunctions) are found by eigenvalue problems. If you have a basis already, this is an algebraic problem. If not, one usually solves an eigenvalue problem which is a differential equation to find the eigenfunctions. For example the orbitals in the hydrogen atoms correspond to eigenfunctions. In QM the concept of innerproduct is also important as it is used to computer expectation values and probabilities. All the terms are from linear algebra. Technically though, QM is usually dealing with infinite linear vector spaces, or spaces or functions, studied in more details in functional analysis. But for a first QM course, I think most people have not studied advanced functional analysis. Alot of it is generalisations from normal linear algebra, but conceptually it's similar.

 

One can also study the operators in QM from an group algebraic perspective. But this is all mathematical classifications. I'd say for a basic appreciation of the very basics, linear algebra and basic differential equations an eigenvalue problems is about what you need. But then that's not where it ends, probably where it starts

 

The axioms in QM is what attaches the formalism to reality, analyse this carefully. Some basic familiarity with probability theory and basic statistics might help a little too, when the frequentists interpretation is discussed.

 

/Fredrik

 

It's clear and it draws an interesting parallel: QM forces on us a kind of self-restraint about what we can say about nature, as Bohr indicated. Indeed there might be deterministic hidden variables or if not there might be some fundamental structure underlying QM, but it seems as if we cannot see it. Analogous to knowing our space from the inside and not being able to speak meaningfully about wherein it may or may not be embedded.

 

I'm glad you could read out what I tried to say. This in my thinking, strongly relates also to the issue of regularization in an abstract sense, maybe we can call it self-regularization - which is an important part of action formulations. I'm not if that makes sense, or adds any clarity to my previous reflections on "geometry counting" - like, from what point of view is the counting constructed?

 

(I'm still waiting for Rovelli's book btw)

 

/Fredrik

[foodchain]

However the associations I do, are more in the sense of a pictures development of quantum mechanics, and it's interpretations. But I suspect that to make sense of these at this point fuzzy things, a normal standard appreciation of the basiscs is needed. My strange reflections is not standard QM./Fredrik

 

I was thinking of trying to model environmental chemical behavior from a theoretical point of view using QM. Such as looking for environment which could favor or support various microbial metabolisms. I was thinking on how to link autotroph behavior to early or primordial stages of evolution in a sort of process produced by a biogeochemical function or stage for instance.

 

I was thinking maybe that the use of thermodynamics as proposed by einselection really in modeling possible chemical ecologies that could produce early autotroph behavior in possibly the earths crust or other geologic features like underwater volcanic vents or chimneys.

 

Thank you also for your reply on the math question, that cleared up a lot of things for me.

[fredrik]

Depending on exactly what your goal is, it sounds like a good plan is to acquire a broad knowledge both in molecular biology, quantum chemistry and also physics. The fields called quantum chemistry and quantum biology tries to study more complex chemical and biological systems in terms of QM.

 

I suspect, based on my own development, that as you learn more, you will also find the right ways. Before I studied any QM at all, long time ago, I had a very naive view of reality, relative to now. I would not be able to chose the right way back then, so I like to think that the focus is always on the next step.

 

"Always in motion is the future" like Master Yoda said

 

/Fredrik

 

To get back to the idea in the OP, in the general quest of unification and trying to find the common denominator between QM and GR that naturally allows a new formalism, that respects the best of two worlds so to speak, I like drawing parallells to Martins explanations on the GR stuff in the other thread.

 

I get the impression that Martin seems to prefer the GR side of things, and try to add QM ontop of that, rather than the other way around.

 

I'm try to reflect and perhaps find new angles, and I particulary find this statement interesting.

 

4. relativity teaches us not to expect distances to be constant (except when anchored to something material like rock or metal or part of a bound system where forces operate to prevent change)

 

In a certain sense the notion of a distance measure between two points, can be considered as the distance between two states (as in possible locations), and the shortest distance between any two given states is thus sort of like a measure of a potential a priori transition between the two states. This in line with Ariel Catichas ideas to associate "distance in space" as a kind of probability for these two "states", "locations" of simply "pieces of information" to be mixed up! So distance is a kind of parametrisation of locality itself, and that space is arranged as per locality "ordering", so that states that are less likely to be mixed up are "farther apart".

 

How can one interpret "relativity teaches us not to expect distances to be constant" so as to get closer to measurement and information ideals (that I personally think is (or rather should be) the core of QM, in line with Bohr's ideal)?

 

Then if we consider a distance to be a sort of an expected probability measure that two distinguishable states are mixed up, or alternatively as a measure of ther distinguishability, then the bold statement suggest that we should not expect that expected measures are constant, or in effect that we should not expect that our expectations are certain.

 

This is, in my thinking, right in line with the analogies of intrinsic vs extrinsic geometry, and information.

 

That is, from he incomplete, inside frog-view, it is not possible to rate our own ultimate expectations. This directly connects to non-unitarity.

 

Unitarity, means that we in our theories, take our expectations to be laws! This makes no sense, in the suggested philosophical analysis I'm trying to do in what I consider to be the natural continuation of Bohr's philosophy.

 

The missing part in QM, to which a clue can be find in the GR analogies IMO, is that QM considers "measurements" and relations between measurements. But of course, there is no such thing as a measurement without an observer.

 

And also in line with the GR concept of "parallell transport" of say tangentspace vector, information must be transported between observers, and due to the fact that not all observers have the same capacity to store information (including expectations, and rating systems), it seems very strange to a priori expect these transformations to be unitary.

 

If you think about this one can even to a cetain extent draw loose parallells to the popular concept of regularization and renormalization, if you consider scaling over complexity of the observer. This seems to suggest a natural regularization scheme that is easily pictures to have a physical meaning. The cutoff is at the complexity scale, with of course one would loosely associate considering the holographic ideas, to related to the observers "entropy/energy/mass".

 

If we picture that the actions calculations, is actually implemented in the observers "microstructure", then it's entirely obvious that the naturaly cutoff is the complexity of the observers - there is physical a limit to what the observer can simply fit in his smallness. And this regularization is physical, by letteint the observational resolution go to infinity, it corresponds physically to letting the observer grow in information capacity and thus we would expect - mass!

 

/Fredrik

[bji]

I am a 'layman', once trained to some degree in math (I have a B.S. in Math/Computer Science from CMU), but having long since forgotten most of it. I have been reading a bit recently about 'intrinsic geometry', and am especially interested in it because it is required by general relativity. And I have some problems with the concept of 'intrinsic geometry' that I hope someone can help me out with.

 

1) Relativity' date=' both special and general, which relate to the concept of geometry and in particular intrinsic geometry. In the general picture of a curved surface, there are two kinds of curvature - extrinsic and intrinsic.

 

The extrinsic curvature relates to how the surface "curves" relative to the higher dimensional embedding space - for example a 2D-surface curved in 3D embedding.

[/quote']

 

Extrinsic geometry makes perfect sense to me.

 

The physical curvatures in GR are the intrinsic ones. Although one can mathematicall imagine it embedded in a higher dimensional space' date=' this embedding is not unique, and more important non-physical.

[/quote']

 

I don't understand this part. If one does not imagine it mathematically embedded in a higher dimensional space, then how does one define the 'physical curvature' of the space?

 

So the physical curvature is what can be deduced from an observer that lives on the surface' date=' not in the embedding space.

[/quote']

 

I contend that an observer living on the surface can never deduce any physical curvature in the surface. Because they cannot perceive the 'dimension(s)' in which the curvature is occurring (this begs my previous question above - I am assuming here that there is an embedding in a higher dimension involved), and presumably any device they use to measure within their space cannot 'perceive' the higher-order dimension(s) either, then they will never make a measurement or perceive anything that suggests any higher-order dimension(s).

 

The classic example used to illustrate non-Euclidean and intrinsic geometry is the space consisting of the surface of a sphere. It is claimed that in this space, the 'angles of a triangle sum to more than 180 degrees'. First - I dispute that the concept of 'angle' and 'tringle' and 'degree' mean the same thing in this space as they do in a Euclidean plane, so I don't think it really means anything to say that in the one case the sum is more than 180 and in the other case it is always 180 (although everything I have ever read on this topic always suggests these two facts as if they indicate something profound about geometry), because the 'it' in question is a different 'it'.

 

I contend that an observer on the surface of the sphere would always a) perceive that they are on a flat plane, and b) perceive the geometry of objects in their world as identical to those of a Euclidean geometry, including that the sum of the angles of a triangle is 180 degrees.

 

In other words, they would never be able to detect any 'physical curvature' of their space. And I think this same principle holds for any space; my belief is that to any observer in any space, geometry looks Euclidean and physical curvature of the space cannot be detected.

 

Let's take the example of an observer on the surface of the sphere trying to measure the angles of what we, sitting outside of the sphere in the 3d space in which the sphere is embedded, can see is a triangle composed entirely of right-angles when viewed from 3d space.

 

How can we visualize what they would perceive? Well we can imagine "warping" from our 3d view of the system, to their view as observers on the plane. This "warping" is the same thing has "flattening the surface of the sphere into a plane" - we'd be looking at the surface of the sphere, and seeing it as a 3d object with all of the lines drawn on it as curves in 3d space (curves which coincide with the 3d space which is the surface of the sphere). As we "zoomed in" on the surface of the sphere, we'd observe the lines "straightening out" as the third dimension in which we exist and are observing shrinks. In addition, the angles between the lines would shrink. So as the 3rd dimension shrinks away, we see the curved lines of the triangle becoming straight and the 90 degree angles of its vertices narrowing to 60 degree angles. When our transformation from a 3d world to the 2d world of the surface of the sphere is complete, we'd see that we are on a flat 2d plane, looking at a triangle that has straight lines for its edges and 60 degree angles at its vertices.

 

And all of the non-Euclidean geometry that we had been perceiving when we were 3d observers, would have been reduced to 2d geometry now that we are confined to the 2d world of the surface of the sphere.

 

We would, while we exist within the 2d world of the surface of the sphere, have no way of detecting that we are anywhere other than in a perfectly flat plane governed by Euclidean geometry.

 

I think that by extension, the only way to perceive the curvature of any space, is to exist as an observer in a higher dimension in which that space exists.

 

And so, I reject the notion of intrinsic geometry. There is only extrinsic geometry, and if one is to have a curved space, then one must have higher order dimensions in which that curved space is embedded. And furthermore, if one is to perceive the curvature of that space, one must exist as an observer in the higher-order dimension.

 

As a result, I don't believe in the intrinsic curvature described by general relativity. I think that if there is curvature involved, it is because our 3d space exists as a 3d curved space embedded in real physical higher dimensions.

 

The question then is, why can't we observe these higher dimensions, and why can't we move in them? If we can see space "curving" indirectly by observing light rays being bent by gravity, then we must be observing from the perspective of at least 4 dimensions (and probably more, given that we need to allow for 3 degrees of freedom in which the light can bend, not just one), and yet, we don't perceive anything "else" outside of the curved 3d space that lets us know that we are looking through N-dimensional (with N >= 4) space.

 

Oh - one interesting thing that I forgot to mention. Although an observer confined within the 2d space of the surface of the sphere, and having no way to perceive any extra dimensions outside of this surface, would perceive their reality as that of a flat plane with Euclidean geometry, they would notice the peculiar fact that if they travel far enough in one direction, they will end up back at the same point that they started at.

 

This would be the only way for them to detect that they were in a non-Euclidean space. That being said, they would not be able to draw any conclusions about the "shape" of that space, except to say that it is a closed space. Well maybe the fact that they ended up at the same place in the same orientation would tell them something as well. Also, heading off in a different direction and measuring how far it takes, relative to their first trip around the universe, to get back to the same spot, and what orientation they are in when they get there, might give them some more information.

 

I am not sure exactly how much they could deduce about the shape of their universe in this way, but at the very least, they could tell that they were in a closed space.

 

Hi. I believe that my question was best posed in a thread in the "Relativity" forums, so I have moved my post there and would appreciate any responses to be made in that thread. Thank you!

 

The link to the thread is here.

[fredrik]

You got some comments in the other thread, but here is some more.

 

To me this issues can be decomposed into several subissues.

 

- The first issues is what intrinsic vs extrinsic curvature means in classical (no quantum stuff) geometry.

 

- another issue is how to properly define the degrees of freedom that defines spacetime and it's curvature to be realistic observables and to try to figure out the relation between geometry and information, and what information content there is in the geometry itself, and to where _this_ information relates.

 

My impression is that your questions regards the first issue, so I think my post was not making sense because it aims to deal mainly with the second issue and my line of reasoning was to appeal to intuition that if you understand the first issue, the second issue isn't far away by an abstraction.

 

The 2D surface of a 3D space is just a simple example, picked because it can be visualized to illustrate a deeper concept that doesn't necessarily depend on anything "visual". In the sphere in 3D space there are both intrinsic curvature in the 2D surface, and there is the extrinsic curvature how the 2D surface curves in the third dimension so to speak. This curvature in the third dimension does not yield an intrinsic curvature.

 

To appreciate the difference: compare a sphere with a cylinder. Both are obviously curved relative to the embeddeding, right? But the cylinder has curvature only into the third dimension, so an ant walking on a part of a cylinder cylinder would see it indistinguishable from a flat plane, because a cylinder is nothing but a flat planned "wrapped up", but the wrapping is done without _deforming the plane_ - this is the key.

 

As you know from playing with paper, you can fold a paper into a cylinder without stretching the sheet. But you can not simply make a paper sphere out of a plane piece of paper without stretching the paper fibres or

making wrinkles.

 

The difference between a cylinder and a sphere should give a simple visualisation of the difference between two things that are both obviously curved in the 3D space, but where only one has an intrinsic curvate (stretching). Intrinsic curvature happens when the paper or surface is deformed differently in different locations.

 

Does this make sense, or am I missing your point?

 

/Fredrik

[bji]

To me this issues can be decomposed into several subissues.

 

- The first issues is what intrinsic vs extrinsic curvature means in classical (no quantum stuff) geometry.

 

Yes, this is a question that is interesting to me; I rephrase the question slightly as 'is there any representation of intrinsic geometry that does not require an external dimension in which to define that geometry? And if not, then does it make sense to call such a geometry intrinsic'?

 

- another issue is how to properly define the degrees of freedom that defines spacetime and it's curvature to be realistic observables and to try to figure out the relation between geometry and information' date=' and what information content there is in the geometry itself, and to where _this_ information relates.

[/quote']

 

This is a very interesting question as well, and unfortunately I don't understand the mechanisms of quantization of time and space well enough to know how this question is materially different from the first question; so for the moment, I will leave it aside as you suggest.

 

My impression is that your questions regards the first issue' date=' so I think my post was not making sense because it aims to deal mainly with the second issue and my line of reasoning was to appeal to intuition that if you understand the first issue, the second issue isn't far away by an abstraction.

[/quote']

 

Well actually, I have to admit that I chose your post simply because it dealt with a question that I wanted to ask, and not so much because the topic you were dealing with specifically was confusing to me. I actually did a google search on 'intrinsic geometry' to try to find a discussion I could join so that I could ask my question. My apology if to some degree this 'hijacks' your discussion; but it seemed to me to be at least somewhat related to the discussion at hand. If this is not true, then feel free to ignore my post - I put it in a separate thread where it is being discussed on its own merits anyway ...

 

The 2D surface of a 3D space is just a simple example' date=' picked because it can be visualized to illustrate a deeper concept that doesn't necessarily depend on anything "visual". In the sphere in 3D space there are both intrinsic curvature in the 2D surface, and there is the extrinsic curvature how the 2D surface curves in the third dimension so to speak. This curvature in the third dimension does not yield an intrinsic curvature.

[/quote']

 

It is the intrinsic curvature in the 2d surface that I do not understand. From a 'visualization' standpoint, it is impossible to imagine the curvature of a space which has no higher dimension to 'curve in', so it makes me wonder how one can ever divorce an intrinsic space from a higher dimension. Also, the fact that geometric operations on intrinsic spaces seem to always be defined in terms of higher dimensions, makes me wonder how they can ever be separated. And finally, the fact that (and I am assuming here because I don't know the math well enough, but I can't imagine how it can work any other way) the 'intrinsic geometry' of a space such as a 2d sphere would require a third 'parameter' seems to suggest to me that there must be a third 'dimension' implicit in the concept. You can say that the third 'parameter' does not represent a dimension; but I can't quite understand how this can be justified when a simpler explanation would be that the third parameter represents a third dimension.

 

For example, here is a definition that I have read for the angle between two lines in spherical geometry:

 

The angle between two lines in spherical geometry is the angle between the planes of the corresponding great circles, and a spherical triangle is defined by its three angles.

 

The only way to describe the planes of the corresponding great circles is to using all degrees of freedom in a 3d space. One way to define a plane is via a normal vector; and a normal vector in 3d space requires parameterization in all three coordinates (x, y, and z). So defining an angle in spherical geometry requires all three dimensions in the space embedding the sphere to be used. And so the mathematical formulae for defining the angle between two lines in a spherical geometry must have three parameters, one for the x, y, and z dimensions. Because the formulae implicitly require all three dimensions, how can one say that the geometry of the sphere is 'intrinsic' and independent of one of those dimensions?

 

I have read a different definition for the 'dimensions' of a spherical space; the 'x' dimension would be 'latitude' and the 'y' dimension, longitude. In this way, only two parameters would only need to be used to locate a point in the space. However, the concept of latitude and longitude themselves are implicitly 3d concepts, so I don't see how one is escaping having to deal with a third dimension by using parameters like this to define the geometry of a sphere.

 

To appreciate the difference: compare a sphere with a cylinder. Both are obviously curved relative to the embeddeding' date=' right? But the cylinder has [b']curvature only into the third dimension[/b], so an ant walking on a part of a cylinder cylinder would see it indistinguishable from a flat plane, because a cylinder is nothing but a flat planned "wrapped up", but the wrapping is done without _deforming the plane_ - this is the key.

 

But I don't see how this challenges my point; the ant on the cylinder would perceive its geometry as being Euclidean. That was my point exactly, except that I extend this to also be true for an ant walking on the surface of a sphere. And in the cylinder case or the sphere case, geometry is defined implicitly by requiring a third dimension in which to frame the geometry, so I don't think there is any way to separate the cylinder from the space it's embedded in, and thus, no way to have an 'implicit' geometry.

 

As you know from playing with paper' date=' you can fold a paper into a cylinder without stretching the sheet. But you can not simply make a paper sphere out of a plane piece of paper without stretching the paper fibres or

making wrinkles.

 

The difference between a cylinder and a sphere should give a simple visualisation of the difference between two things that are both obviously curved in the 3D space, but where only one has an intrinsic curvate (stretching). Intrinsic curvature happens when the paper or surface is deformed differently in different locations.

[/quote']

 

I can see that there are geometric differences between these two shapes, for sure. But from the point of view of the ant on either surface, both would look Euclidean. The big difference between the cylinder and the sphere is that with the sphere, angles would be measured differently by the ant than by a 3d observer, whereas for the cylinder, they would not. This is analogous to the sphere requiring 'stretching' in order to be smoothed out, whereas the cylinder would not. The net result is the same for the ant: it perceives Euclidean space.

 

And when you say "Intrinsic curvature happens when the paper or surface is deformed differently in different locations" - that is the root of the question at hand. How can a surface intrinsically have curvature that is different in different locations, without that curvature being expressed as a quantity in a higher dimension in which the curved space is embedded? What exactly do you use to define the degree of curvature at any point in the space, except as a parameterization, this parameterization being most logically represented as a higher dimension?

 

Does this make sense' date=' or am I missing your point?

[/quote']

 

If my responses to your post make sense, then your post made sense to me, as I was able to produce a meaningful response. If my responses make no sense, then I must not have understood you

 

Thanks so very much for replying. All I really want is to be able to say that GR requires an external spatial dimension in which to frame its spacetime curvature; and I think this follows from my arguments that a non-Euclidean lower-dimension space cannot be separated from its containing space and still have any meaningful geometry. But I know that just about every mathematician and physicist in the world disagrees with me, and it's much more likely that I just have a lack of understanding, than that everyone else is wrong.

 

/Fredrik

[fredrik]

And when you say "Intrinsic curvature happens when the paper or surface is deformed differently in different locations" - that is the root of the question at hand. How can a surface intrinsically have curvature that is different in different locations, without that curvature being expressed as a quantity in a higher dimension in which the curved space is embedded? What exactly do you use to define the degree of curvature at any point in the space, except as a parameterization, this parameterization being most logically represented as a higher dimension?

 

I think there are different, and other ways to understand intrinsic geometry besides visualisations. It's probably a matter of personal preference. It seems your desired to visualize everything litteraly is part of the confusion?

 

One of my favourites is to consider the geometry of information. But I think this is more abstract than the geometry, and I doubt this will help but here goes briefly. This is an interesting analogy that connects statistics to geometry. Consider an abstract state space, basically think of it as an indexation of distinguishable states.

 

Now consider that as you "live" and participate in interactions, or "sample this state space" you acquire experience and a buildup of a historical frequencies (~probabilities) occurs. Now clearly it may be that some states prove - from history - to be more frequently populated, this means that to these states you kind og assign a higher a priori probability.

 

Now in this statistical world, what is the closest thing we can think of as a "geodesic"? It's the most likely path a random walker would take constrained to take himself between two points (or states). Clearly the state space which have a a priori higher probability so to speak will "attract" the random walker and kind of curve the state space. The "regions" or high a priori probability is also the places where any random walker is most likely to be found.

 

So intrinsically speaking, one can consider a geodesic to be something like the path with the higest a priori probability to be taken by a random walker. This way, one can imagine various geometrical concepts without the typical geometry analogies so to speak.

 

This way, a geodesic is really nothing but the "most expected path" between two states, but while fully allowing variation around this expectation. but as the interactions continue, expectations are updated, and so is to "geometry".

 

/Fredrik

 

Also note how I in my personal "imaging" considers geometry to be accumulated somehow, as a condensation of history. Clearly this means I consider geometry kind of to be information, and thus can be rated. This suggest that there is to be expect a sort of natural intertia of geometry itself, from the information poitn of view. And this may hint, how the relation between "dynamics in spacetime" (ie. random walking) and the "dynamics OF spacetime" is connected, because this connection is conceptually as I think of it, the updating of expectations in response to revised history. And of course the key here is that ALL history can not be retained. Only a condensation of the supposed most significant history. The amount of retained history in my thinking relates to the information capacity of the random walker. Because strange as it seems, the random walker is the only valid observer.

 

In GR, the energy and mass distribution controls how the spacetime itself evolves, via Einsteins field equations. Similarly of course, the dynamics in spacetime is given as usual. This non-linear feedback is IMO quite analogous to the above ramblings.

 

/Fredrik

[fredrik]

I decided to analyse Rovelli's LQG ideas so some week ago I got his book "Quantum Gravity", and last night in bed I skimmed parts of the Quantum mechanics chapter, to see his view of things.

 

And I was delighted to see that he in sections 5.6.3 (Information) and 5.6.4 (Spacetime relationism versus quantum relationism) expresses a eflection that is strongly related to the topic of this thread.

 

Rovelli writes on p.221 in QG

 

"Thus, locality ties together very strictly the spacetime relationism of GR with the relationism underlying QM. It is tempting to try to develop a general conceptual scheme based on this observation. This could be a conceptual scheme in which contiguity is nothing else than manifestation, or can be identified with, the existence of a quantum interaction. The spatiotemporal structure of the world would then be directly determined by who is interacting with whom. This is, of course, very vague, and might lead nowhere, but I find the idea intriguing."

 

This is a different view of expressing something that I like. I definitely share his idea that the idea is intriguing.

 

I also share Rovelli's relational view of QM. But on one significant point, I get the feeling from my incomplete reading of this book that he (at least conceptually) simplifies a crucial point:

 

I several places he talks about information the observer "have" about the system, and says something that it doesn't need to be stored. I have to read on in his book, but if he is somehow not accounting for that information capacity of the observer that constrains his possilbe observations and "possession of relations", then I disagree, because this constrain is in my eyes a key factor. This is exactly the factor that ordinary QM, disrespects, and IMO quite probably connected to many divergence issues.

 

But perhaps Rovelli's view in this chapter is to explain his view of _standard QM_, THEN I almost completely agree. And it will be interesting to read the rest.

 

/Fredrik

[fredrik]

In the goal of this thread, to develop more conceptual intutition of QM and GR and their similarities rather than differences, as a guide to learn more about this, here is another angle of reflection along the same lines.

 

Anyone having more reflections on my relfections from a different viewpoint is appreciated.

 

In short General relativity the dynamics is described at two levels.

 

(I) The dynamics in - or relative to - spacetime.

 

This is usually expressed so that a particle subject to no non-gravitational forces, follow a geodesic in spacetime.

 

(II) The dynamics of spacetime itself.

 

This is usually expressed by einsteins field equations, which is a relation between the geometry of spacetime and the matter and energy distribution in the stress energy tensor.

 

If the mass of the test particle is small enough to not distort spacetime, the dynamics is effectively that of a particle moving in a fixed, but curved background. This background is determined by the energy and mass distribution of the environment.

 

for example when a stellar dust particle circles the earth in space.

 

But the nontrivial things happen when the particle is massive enough to significantly distort it's own environment. That means that for each infinitesimal change of position of this particle, there is an infinitesimal change of the entire geometry of spacetime!

 

This means that, as the sytem evolves, the "geodesics" keep changing too.

this is for example when several extremely massive neutron stars occur in many body problems, all of the participans make massive contributions to curving the spacetime.

 

If we let that be a simple idea of classical GR. How can we now rethink, or reinterpret, those principles in terms of something that is easier to merge with the information nature of QM?

 

I propose the following conceptual analogy as an alternative to "visualisations".

 

In short, dynamics is described at two levels.

 

(I') The dynamics in - or relative to - prior expectations.

This can be trivially expressed so that a particle subject to no unexpected feedback, evolves as expected.

 

Put this way (I') appears almost trivial.

 

(II') The dynamics of the expectations themselves.

The dynamics of expectations, can be decomposed into two parts.

 

a) expected dynamics (as induced by constraints)

b) unexpected feedback

 

Obviously the unexpected dynamics is inherently unpredictable. So our best bet is to go base decisions one expected dynamics, but leave the door open for unexpected events, because given insight of our incompletness, the unexpected is still somehow expected.

 

For me at least, this gives a fairly clear conceptual vision, suggesting some deeply interesting parallells between QM's information perspective and GR's relative views.

 

Some key conceptual issues I personally see is:

 

- The association of movement along geodesic, with the "expectated change". And if we from a pure informational view, can induce such an "expected change", then this defines a geometry of our information.

 

- The idenficitation of dynamical geometry, with evolving expectations, because as changes occur, our set of knowledge changes, and this updates our expectations.

 

Questions:

 

- In describing the theory general relativity, there is a birds view present, which can also IMO be seen as a background expectation that isn't induced from a real observer. It's somehow an external observer or god. This is the sense in which GR is deterministic, and I also think that a superobserver does qualify as a kind of background.

 

- In a sense one might be tempted to say that unexpected feedback is related to when we have "open systems". But IMO, closed vs open systems is not possibly a valid intial condition. Because how would you know if the system is closed or open in advance? It must clearly by an idealisation.

 

So is this a "problem"? It seems so, but I think that this problem may also be part of the solution to the problem of time. Because at first it seems we are just drifting and drifting, it's not possible to come to a certain conclusion! Frustrating! But maybe this is nothing but the drive for time?

 

I think this is extremely interesting.

 

- What can one say, about the structure formation in such open crazy world? Is it possible to make any generic predictions about probable structures, in the case of some minimal assumptions? It may seem thay anything is possible, but that isn't necessarily a problem at all as long as everything isn't equally "probable". To make gigantic jumps from this chaos to everyday life probably isn't possible, because it would be a logical jump. Perhas the first ad simlpest thing to elaborate is the microstructure of reality. What are the simplest, nontrivial structures that we would expect in such crazy world?

 

We talk about information, but where is this information encoded? How does even the concept of probability make sense at this point? At the same time it gets crazy and unpredictiable, it seems to get _simpler_. Because the complex things doesn't exists, so the "chaos" seems self-restraining.

 

Ie. how is this very theory relating to a hypotetical simplistic prehistoric observer who might be nothing but a flip flop device? If you are a flip-flop device, how can you improve, and evolve?

 

Einstein fought with trying to establish the relation between spacetime geometry and the matter and density coupling. His answer was Einstein field equations.

 

Now we are at a similar situation, to establish the relation between the subjective expectations (defining expected "generalised geodesics") and the intrinsic information of an observer.

 

IE. How does an observer, with incomplete information compute the optimum strategy for his actions? This contains many subquestions indeed, and one question is to what extent Einsteins field equations can be exploited to directly solve this, or do we simply have to invent a new fundamental relation that would be the generalisation of einsteins field equations, where we take the step from the mechanistic spacetime view, to a generic information space?

 

/Fredrik