Defying Intrinsic Geometry

[bji]

Hi all. The following was initially posted as a response in this thread, but I believe that it may be more appropriate in this forum since it deals with some of the mathematical underpinnings of relativity. So I would like to discuss this topic here instead of in the other thread. Thanks!

 

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.

 

For cripes' sake! How do I get this forum software to display my posts as separate posts instead of merging them??? I don't want them to be merged! They are separate posts and I want them to be displayed that way!!! That being said, since I can't seem to separate my posts, what I wrote later will have to appear merged here:

 

Posted a few days later by bji:

 

Well no one else is responding to my questions so I guess I'll have to

 

I've been thinking about this a bit more and I think that perhaps fundamentally my "problem" in understanding intrinsic geometry is that I 'assume' that all spaces in which other spaces may be embedded, are themselves Euclidean. Every 'illustration' that one is going to be able to come up with for the concept of intrinsic geometry is going to be embedded in a 3d space. We 'think' in 3d space so it's impossible for us to visualize geometries except in relation to Euclidean 3d space (even when we imagine a 2d space such as a plane, we are imagining it as observers looking "down" on the plane from some place in 3d space).

 

So I guess it should come as no surprise that if I assume that all spaces are Euclidean, that I can't come up with a way to visualize intrinsic geometry.

 

Please don't take this conclusion as meaning that I refute anything that I said in my previous post though; I still believe all of it, and if there are errors in my logic, I would appreciate having them pointed out to me.

 

What it all seems to come down to, and I think my major complaint about the concept of intrinsic geometries is - you cannot visualize them. If you try to visualize them, you must do so from the perspective of an embedding Euclidean 3d space; and with the assumption that the embedding space is Euclidean, one will always draw the conclusion that I drew earlier (that to observers in the embedded space, geometry will always look Euclidean). So perhaps what I am really saying here is that I have a theory:

 

For any space embedded within a Euclidean space, observers within the embedded space will always perceive geometry as being Euclidean, and be unable to measure any 'curvature' of their space.

 

I really think that this theory is true, although I don't have the mathematical rigor to back it up. It's probably already been proven geometrically by somebody else 300 years ago; so if someone could point me to that proof, I would be very appreciative!

 

So the only remaining issue is: can we 'assume' that there is such a thing as intrinsic geometry? If the only way to formulate an intrinsic geometry is by describing it as a curved space embedded within a higher-order space, is there any reason to believe that it makes any 'sense' to then simply 'remove' the higher-order embedding space, leaving 'only' the 'curved' space, and then saying that this space has an 'intrinsic geometry'? If so, what is the justification for that, except as a 'leap of faith' that one has to make without any logical justification? If no, then how can we talk about the intrinsic curvature of spacetime of GR without having to appeal to higher dimensions in which that curvature exists?

[Martin]

 

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.

...

 

You are right about the effect of "zooming in" and considering smaller and smaller trianges.

In a curved space the total interior angle depends on the size of the triangle.

 

for example, uniform positive curvature means that you would notice as you make the triangles larger and larger the total angle increases.

 

and just as you imagined in your post, an inhabitant of the curved space would notice that as he makes the triangles smaller and smaller the total angle goes down closer and closer to 180 degrees---the flat limit.

 

On the other hand, if I understand what you are contending in those two earlier paragraphs, then you are not right. An inhabitant of a sphere surface would notice that if he makes his triangle large enough the sum of the interior angles can be 270 degrees.

By changing the size he can make it vary from 180 up to 270, and actually even more.

[Country Boy]

Hi all. The following was initially posted as a response in this thread, but I believe that it may be more appropriate in this forum since it deals with some of the mathematical underpinnings of relativity. So I would like to discuss this topic here instead of in the other thread. Thanks!

 

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.

 

 

 

Extrinsic geometry makes perfect sense to me.

 

 

 

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?

Well, that depends upon how you define 'physical curvature' doesn't it? You could certainly define it as "giving a specific result on this experiment".

 

 

 

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'.

Are you saying that you reject the very concept of measuring angles? If you have two straight lines intersecting at a point, the "measure of the angle between them" has very specific definition which works as well for angles on a sphere as on a plane. (Oh, and "straight line" can be given a specific definition that would include great circles on a sphere.) Now, we come to your: a being in the surface of the sphere would not see any difference between his "lines" and your "lines"- and would measure angles in exactly the same way. The being would however find that angle sum in a triangle depends upon the size of the triangle. Whether he would then come to the conclusion that his space was "curved" I can answer to- but certainly we can distinguish between one "curvature" and another through intrinsic measurements. That was a theorem of differential geometry due to Gauss.

 

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.

On what basis do you contend that? Indeed, what definitions are you giving to "angle", "line", etc.? Those are all carefully defined in differential geometry- as well as the notion of "intrinsic measurement".

 

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.

No, we wouldn't observe anything of the sort- and you can't contend that we would because you haven't done the experiement! The "experiment" can be done, mathematically, in differential geometry and gives the results that you disagree with.

 

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.

And you don't recognise from that that they would have to agree that there are trangles in which the angle sum is not 180 degrees? Choose a point on this line which returns to its starting point, A, and move at right angles to it. Since the first line meets itself, this second line will have to meet it in a second point, B- it's not to difficult to show that it must also be at right angles. Go back to any third point, C, on the new line, turn at right angles and go until you meet the first line, which must happen, at D. You now have two triangles in which two of the angles are right angles- and so the angle sum must be greater than 180 degrees.

 

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.

 

For cripes' sake! How do I get this forum software to display my posts as separate posts instead of merging them??? I don't want them to be merged! They are separate posts and I want them to be displayed that way!!! That being said, since I can't seem to separate my posts, what I wrote later will have to appear merged here:

 

Posted a few days later by bji:

 

Well no one else is responding to my questions so I guess I'll have to

 

I've been thinking about this a bit more and I think that perhaps fundamentally my "problem" in understanding intrinsic geometry is that I 'assume' that all spaces in which other spaces may be embedded, are themselves Euclidean. Every 'illustration' that one is going to be able to come up with for the concept of intrinsic geometry is going to be embedded in a 3d space. We 'think' in 3d space so it's impossible for us to visualize geometries except in relation to Euclidean 3d space (even when we imagine a 2d space such as a plane, we are imagining it as observers looking "down" on the plane from some place in 3d space).

 

So I guess it should come as no surprise that if I assume that all spaces are Euclidean, that I can't come up with a way to visualize intrinsic geometry.

 

Please don't take this conclusion as meaning that I refute anything that I said in my previous post though; I still believe all of it, and if there are errors in my logic, I would appreciate having them pointed out to me.

 

What it all seems to come down to, and I think my major complaint about the concept of intrinsic geometries is - you cannot visualize them. If you try to visualize them, you must do so from the perspective of an embedding Euclidean 3d space; and with the assumption that the embedding space is Euclidean, one will always draw the conclusion that I drew earlier (that to observers in the embedded space, geometry will always look Euclidean). So perhaps what I am really saying here is that I have a theory:

 

For any space embedded within a Euclidean space, observers within the embedded space will always perceive geometry as being Euclidean, and be unable to measure any 'curvature' of their space.

 

I really think that this theory is true, although I don't have the mathematical rigor to back it up. It's probably already been proven geometrically by somebody else 300 years ago; so if someone could point me to that proof, I would be very appreciative!

 

So the only remaining issue is: can we 'assume' that there is such a thing as intrinsic geometry? If the only way to formulate an intrinsic geometry is by describing it as a curved space embedded within a higher-order space, is there any reason to believe that it makes any 'sense' to then simply 'remove' the higher-order embedding space, leaving 'only' the 'curved' space, and then saying that this space has an 'intrinsic geometry'? If so, what is the justification for that, except as a 'leap of faith' that one has to make without any logical justification? If no, then how can we talk about the intrinsic curvature of spacetime of GR without having to appeal to higher dimensions in which that curvature exists?

 

Sorry I was late in reading this. You might want to look for a basic book on "Differential Geometry". You should find that much of it is devoted to "intrinsic properties" and especially the "Gaussian" or "intrinsic" curvature which is defined through "intrinsic" measurements.

[bji]

You are right about the effect of "zooming in" and considering smaller and smaller trianges.

In a curved space the total interior angle depends on the size of the triangle.

 

I think you misread me; the 'zooming in' I spoke of is how I visually tried to describe one way to mentally model "going" from 3d space to 2d space. What is really happening when one 'moves' from being an observer in the 3d space to an observer in the 2d space' date=' is that the 'third dimension' smoothly (in my visualization) 'shrinks', which is the same thing as saying that the perceived curvature of the 2d sphere 'straightens out'; at each time t in my visualization, the '3rd dimension' in which we have modelled the 2d space, is slightly 'smaller' than the '3rd dimension' at time t - 1. At each time t, the 2d sphere would look less 'curved' than it did at time t - 1, and as a result, all lines on the sphere would look straighter and all angles would look smaller. And eventually, at some time t', the 3rd dimension will have been eliminated completely (from our model), leaving only two spatial dimensions, those of the surface of the 2d sphere that we were modelling in 3d space. And at that point, lines would be straight and what were 90 degree angles, would now be 60 degree angles, when observed by an observer existing strictly within the confines of the surface of the sphere (and *not* within the 3d space in which the surface of the sphere is modelled).

 

Since the model of the 2d sphere in 3d space is just a mental model, I felt like it was OK to take the liberty of imagining the 3d component of the space disappearing gradually, and it seems logical to me to describe what one would 'observe' during this process as the straightening of the lines and shrinking of the angles; the net result being a Euclidean plane, that we as observers now are 'on', having lost our 'height' away from the plane when the 3rd dimension was lost.

 

Of course, at this point, we are 2d observers in a 2d world, which is impossible for humans to visualize, so we do so by imagining once again that we are not 'in' the plane, but sitting in some privileged 3d position 'above' the plane looking down on it and its triangle.

 

for example, uniform positive curvature means that you would notice as you make the triangles larger and larger the total angle increases.

 

I contend that such an observations could only be made if one were "outside" of the 2d space defined by the surface of the sphere, and thus your definition of 'uniform positive curvature' only has meaning if the surface is embedded in a Euclidean 3d space

 

and just as you imagined in your post' date=' an inhabitant of the curved space would notice that as he makes the triangles smaller and smaller the total angle goes down closer and closer to 180 degrees---the flat limit.

[/quote']

 

Since it is not possible to ever reach the limit, the triangle will never have angles that sum to 180 degrees; it will always be fractionally more, although the fraction may be very small if the triangle is very small. Thus, for any such 'curved space', the sum of the angles is greater than 180 degrees, which means that the observer who is measuring this curvature must either a) exist in the 3d space defining the curve, and thus we must require a 3d Euclidean space for this observer to be in, or b) exist in the 2d space defined by the curve, in which case the observer would never observe triangles with angles summing to more than 180 degrees in the first place.

 

On the other hand' date=' if I understand what you are contending in those two earlier paragraphs, then you are not right. An inhabitant of a sphere surface would notice that if he makes his triangle large enough the sum of the interior angles can be 270 degrees.

By changing the size he can make it vary from 180 up to 270, and actually even more.[/quote']

 

What exactly is the formula for calculating the angles of a triangle in this 'curved space' then? I contend that however you write it, it must contain more than just two parameters. My point being that if it has more than two parameters, then it should be considered as having more than two dimensions, since each parameter describes a degree of freedom in a spatial dimension. You could try to say that only two of those three parameters describes a degree of freedom in a spatial dimension, and the third parameter is just a parameter defined 'continuously over the 2d space', but why is that any more sensible than just calling all three 'dimensions'? Why can't we just call that third parameterized value 'altitude'?

 

I think that this is directly relevant to the question of whether or not an observer on the surface of the sphere could ever detect its curvature, as in:

 

If any parameter in the equation defining an angle on the surface of a sphere includes a component which corresponds to 'altitude', then an observer entirely within the surface of that sphere would always see that component as equalling zero, since that observer, existing entirely within the surface of the sphere, cannot 'see' altitude. Thus any equation for computing an angle on the surface of a sphere, when considered from the perspective of an observer on the surface of the sphere, does not include any altitude-dependent parameter; and when the altitude-dependent parameter is taken out, we are left with the normal Euclidean equation. And thus any angle on the surface of a sphere measured as 90 degrees using the 3d equation for measuring angles, will be measured as 60 degrees when measured using the equation as it would be known to observers on the 2d surface of the sphere.

[Martin]

Fortunately HallsofIvy is here answering your questions.

 

I think you and I may be talking at cross purposes---not really making contact.

I know for sure that someone in a curved space can measure triangles with total angle T more than pi.

 

(are you OK with radians? pi = 180 degrees?)

 

Back almost 200 years ago Carl Gauss realized this. It is not so new. Carl tried to get money from the Hanover government to actually try measuring some large triangles defined by lightbeams, to see if space itself was curved. He was also doing some surveying work, so it would have fitted in with the practical job he was doing. If he could get the idea in 1820, then you and I can grasp it today.

 

And now we actually DO this, in effect, with astronomical observations of the Omega parameter!

 

Even if our space is not embedded in a higher dimension space it can still have a RADIUS OF CURVATURE. If we aren't in a higher dim, then this radius not a physically realized distance----it is just one possible way to measure how curved our space is.

 

 

Try this, with the simple example of a 2-sphere---ordinary sphere surface.

If A is the area of a triangle (drawn with geodesics, the straightest possible line living in the surface)

and if T is the total interior angle

then can you find R, the radius of curvature, by

 

[math]R^2 = \frac{A}{T - \pi}[/math] ?

 

If that is true it means that if the area A is measured in square lightyears, that the formula is going to give you the radius of curvature R expressed in lightyears.

[bji]

Well, that depends upon how you define 'physical curvature' doesn't it? You could certainly define it as "giving a specific result on this experiment".

 

I guess what I'm trying to get is an 'intuitive' understanding of what 'curvature of space' means for spaces which are not embedded in other spaces. Because there is this word - 'curve' - which in my vocabulary specifically requires that the curvature of a surface of dimension N requires a dimension of degree N + 1 in which to 'express' that curvature. Also, there is the fact that mental models used as examples in talking about intrinsic geometry always seem to be making this point for me by showing an example of intrinsic geometry as a curved space embedded within a higher dimensional space, and I have issues with these models, which are glossed over in every example I have ever seen, which I am trying to express here in order to understand why the model works.

 

Are you saying that you reject the very concept of measuring angles? If you have two straight lines intersecting at a point' date=' the "measure of the angle between them" has very specific definition which works as well for angles on a sphere as on a plane. (Oh, and "straight line" can be given a specific definition that would include great circles on a sphere.) Now, we come to [b']your[/b]: a being in the surface of the sphere would not see any difference between his "lines" and your "lines"- and would measure angles in exactly the same way. The being would however find that angle sum in a triangle depends upon the size of the triangle. Whether he would then come to the conclusion that his space was "curved" I can answer to- but certainly we can distinguish between one "curvature" and another through intrinsic measurements. That was a theorem of differential geometry due to Gauss.

 

What I'm saying is that I reject the notion that measuring angles in 2d space is the same thing as measuring angles in 3d space. By whatever metric you try to come up with, it seems to me that there are fundamental differences in how one would measure an angle in 2d space as one would measure it in 3d space. In 3d space, there is a *third dimension* to take into account, so you necessarily must alter your mechanism for measuring the angle to account for it. If you do not - then you have just projected the angle onto a 2d plane and are measuring it according to the same rules that you would measure the angle in 2d. Which I think defeats the purpose of trying to call the measurement process the 'same'.

 

Yes, I know, there is mathematical notation that would allow you to compute these angles, but does that really mean that computing an angle in 2d is the 'same' as computing an angle in 3d (of a vertex made by curved lines), except 'by definition' - and why should I ascribe to that definition instead of others, especially if that definition suffers from the logical contradictions which I am trying to describe in my examples?

 

On what basis do you contend that? Indeed' date=' what definitions are you giving to "angle", "line", etc.? Those are all carefully defined in differential geometry- as well as the notion of "intrinsic measurement".

[/quote']

 

I have no doubt that the mathematical language of differential geometry allows these things to be defined in a self-consistent way. However, what I want is to be satisfied that the fundamental assumptions on which this mathematical language is based - that it 'makes sense' to talk about 'curvature' of a space that is not defined in terms of a higher dimension.

 

No' date=' we wouldn't observe anything of the sort- and you can't contend that we would because you haven't done the experiement! The "experiment" [b']can[/b] be done, mathematically, in differential geometry and gives the results that you disagree with.

 

I will try to read up on differential geometry, then, and see if I can become satisfied with the logic involved.

 

And you don't recognise from that that they would have to agree that there are trangles in which the angle sum is not 180 degrees? Choose a point on this line which returns to its starting point' date=' A, and move at right angles to it. Since the first line meets itself, this second line will have to meet it in a second point, B- it's not to difficult to show that it must also be at right angles. Go back to any third point, C, on the new line, turn at right angles and go until you meet the first line, which must happen, at D. You now have two triangles in which [b']two[/b] of the angles are right angles- and so the angle sum must be greater than 180 degrees.

 

This is an awesome argument and is exactly the kind of thing that I can visualize and does seem at first blush to present a problem for my arguments. But I realize after some thought that your argument is flawed.

 

The problem with your argument is in the "rotate 90 degrees" step. In the same way that, a 90 degree angle on the surface of the sphere when observed by a 3d observer outside the sphere, looks like a 60 degree angle to a 2d observer on the surface of the sphere, a 90 degree angle to an observer bound by the 2d space of the surface of the sphere, would look like a 135 degree angle to a 3d observer from outside the sphere. We could see as the 3d observer that the resulting lines would not cross; the 2d observer would have to make four 135 degree angles before he would intersect with his original line. In the 2d view, he would have made four 90 degree angles and thus a square; in the 3d view, he would have made four 135 degree angles and also thus a square on the surface of the sphere.

 

I think you and I may be talking at cross purposes---not really making contact.

I know for sure that someone in a curved space can measure triangles with total angle T more than pi.

 

I think I understand what you've been saying, but I am not sure you are visualizing what I am saying in a sufficient manner to see the points I am trying to make. The thing I am trying to point out is very subtle and it requires that you do not just assume that the definition of intrinsic geometry is valid. You have to first assume that it may not be valid; and so at every point where terminology is used in defining intrinsic geometry, or an example of intrinsic geometry is invoked, you have to question that terminology and how what it describes fits into the model at hand.

 

For example, in HallsofIvy's response, he just assumed that a 90 degree rotation to an observer on the surface of the sphere would be the same thing as a 90 degree rotation within the plane tangent to the surface of the sphere at that point. However, this begs the question, how does rotation along a plane external to the sphere in question, whose definition requires an external 3d space in which to define it, constitute 'intrinsic geometry'? My assertion is that it does not - if you want to define rotation in this manner, then you have to accept that you're not really describing 'intrinsic geometry' - you're describing a geometry which implicitly requires a higher dimension in which to define the operations constituting that geometry. (My apologies to HallsofIvy if I have misconstrued his argument - I assume that the way that he wanted to define 'rotation' was within the plane tangent to the surface of the sphere at that point; but maybe he meant something else - if so, I'd like to try to understand what it was)

 

Alternately, if you do not want to appeal to a higher dimension in defining the operations of the geometry, then you have to accept that the geometry as perceived by 2d observers on the surface of the sphere would look Euclidean, and their '90 degree angles' would look like 135 degree angles to 3d observers, and result in squares in both frames of reference. Similarly for 60 degree angles in the 2d space, which look like 90 degree angles in the 3d space and in both frames of reference, make equilateral triangles according to the Euclidean definition thereof.

 

Back almost 200 years ago Carl Gauss realized this. It is not so new. Carl tried to get money from the Hanover government to actually try measuring some large triangles defined by lightbeams' date=' to see if space itself was curved. He was also doing some surveying work, so it would have fitted in with the practical job he was doing. If he could get the idea in 1820, then you and I can grasp it today.

 

And now we actually DO this, in effect, with astronomical observations of the Omega parameter!

[/quote']

 

I have no doubt that we can do this. I also don't doubt that there are definitions of geometry which describe things like the surface of a sphere, and are perfectly self-consistent; but I still believe, according to the arguments that I have given, that this implies and requires a higher space in which the curved space must be embedded. There must be extrinsic geometry involved; you can call 'extrinsic geometry in which you ignore the embedding space even though it's there' intrinsic geometry if you want to, but if you ever try to describe the space of our physical reality in this way, please don't choose to call it intrinsic - please accept that by defining curvature of the space, you are implicitly requiring that there be a higher dimension in which the space is embedded!

 

In which case, GR would require higher-order spatial dimensions.

 

Even if our space is not embedded in a higher dimension space it can still have a RADIUS OF CURVATURE. If we aren't in a higher dim' date=' then this radius not a physically realized distance----it is just one possible way to measure how curved our space is.

[/quote']

 

My point is that the radius of curvature can only be meaningfully defined within a higher dimension. You can say this higher dimension is not really a physical dimension like the dimensions in which you are defining your intrinsic geometry - but I have to wonder, what is it then? If the higher dimensions are used mathematically in exactly the same way as the dimensions of the intrinsic geometry, then why wouldn't it be just as physical? How do you justify saying that objects can move in the physical dimensions, which produces, in the higher-dimensional equations of movement, a corresponding change in coordinates in the higher dimension, but that only those changes of coordinates in the actual physical dimensions count as movement, and that the changes of coordinates in the higher dimensions do not count as movement, but instead are some 'imaginary' thing?

 

Try this' date=' with the simple example of a 2-sphere---ordinary sphere surface.

If A is the area of a triangle (drawn with geodesics, the straightest possible line living in the surface)

and if T is the total interior angle

then can you find R, the radius of curvature, by

 

[math']R^2 = \frac{A}{T - \pi}[/math] ?

 

If that is true it means that if the area A is measured in square lightyears, that the formula is going to give you the radius of curvature R expressed in lightyears.

 

I believe that your definition of "geodesics, the straightest possible line in the surface" requires the use of higher dimensions in its definition, as does your definition of 'total interior angle' does as well. If they do not, please give me their definitions without appealing to any higher dimension or parameter within their formula which is unaccounted for when all positional parameters of the 2d space are taken into consideration. If they do, then I think you are just begging the question at hand - how can you call the space 'intrinsic', completely independent and not requiring an embedding space, when the definition of the geometric operations for the 'intrinsic' space are defined only in terms of an embedding space?

 

All that I'm saying is, you can define geometries this way for sure, but you have to accept that their definitions require higher dimensions.

 

Please don't get frustrated by my responses to your posts. I really do appreciate your taking the time to try to explain these things to me. I'm not trying to be belligerent or willfully misunderstand what you are trying to say. If you have the patience, I would very much appreciate your thoughts and the thoughts of others on my continued responses to this topic. If you do not have the patience to continue, then please accept my thanks for taking as much time as you have to try to convince me of the validity of your (and Gauss') arguments.

[Martin]

I believe that your definition of "geodesics, the straightest possible line in the surface" requires the use of higher dimensions in its definition, as does your definition of 'total interior angle' does as well. If they do not, please give me their definitions without appealing to any higher dimension...

 

You just have a mistaken idea. when you take a course in differential geometry you learn definitions of stuff that don't appeal to higher dimension or presuppose any embedding. We are just talking about wellknown nuts and bolts here. You say the nuts and bolts do not exist. I've been there and seen that they do.

 

If you want me to give definitions in an informal context like this, I just go the way of popularizers like in the Flatland book. I never read these popularizations but I think I know how they explain intuitively. It is pretty obvious.

 

Imagine you actually were a 2D creature living in a 2D surface and you had a tape measure or a piece of string to measure with, then you could imagine measuring the shortest distance between two points.

 

Imagine you and a buddy stretch the string. That basically defines a geodesic. It is a shortest path or is made up of segments which are shortest paths which you and a buddy can check with string. At no point does an outside space have to exist----you don't need to go there or refer to it.

 

Lets shift to the 3D case because it is easier to imagine. You and a buddy live in a curved 3D world. You have a piece of string. You stretch it and find the shortest path between two points. that is a geodesic. It's simple. You didnt have to refer to any higherdimension space that your world is embedded. It could well NOT be embedded.

 

Same with angles. To measure angles we don't have to refer to any higher dimension world. You just have a place where two geodesics intersect and that defines sectors (pie slices) and you compare the pie slices.

One is the same as the other, or bigger, or smaller, or the two slices add to make a straight line (a geodesic) and so on.

[bji]

You just have a mistaken idea. when you take a course in differential geometry you learn definitions of stuff that don't appeal to higher dimension or presuppose any embedding. We are just talking about wellknown nuts and bolts here. You say the nuts and bolts do not exist. I've been there and seen that they do.

 

I am glad to hear that you have understood differential geometry to this extent. It is your expertise in the subject that I am appealing to here. You've seen the nuts and bolts - can you show them to me' date=' show me how they are defined? Because so far, in my understanding, you've only given me definitions of nuts which require the assumption of bolts, and definitions of bolts that require assumption of nuts. I have not yet seen how the definition of intrinsic geometry is not circular.

 

Lets shift to the 3D case because it is easier to imagine. You and a buddy live in a curved 3D world. You have a piece of string. You stretch it and find the shortest path between two points. that is a geodesic. It's simple. You didnt have to refer to any higherdimension space that your world is embedded. It could well NOT be embedded.

 

Can we take the example you gave before, of how we could detect that our curved 2d world in the shape of the surface of a sphere, is curved, by noticing that the angles composing a triangle get larger as we make the triangle larger?

 

Can you please define the formula for computing the sum of the angles for an equilateral triangle on the surface of a sphere?

 

Can you do it appealing only to two parameters in the equation?

 

If not, how can you say that the surface of the sphere could ever be divorced from the space in which the sphere is defined, if the geometry you would use to describe objects on the surface of the sphere, such as the equation giving the sum of the angles of an equilateral triangle, cannot be expressed except with three parameters?

 

My belief is that you cannot write an equation giving the sum of the angles of an equilateral triangle on the surface of the sphere without using enough parameters to sufficiently account for all three dimensions in which that triangle actually exist in the 3d space defining the sphere in which the triangle is embedded. You can only remove parameters if you 'hide' them in the definition of other parameters; for example, maybe you would try to use some kind of 3d polar coordinates or something, of which you only need two to define the lines making up the triangle, and thus end up with an equation of only two parameters, which are expressed in terms of 3d polar coordinates. But in this case, you have still made my point - because the definition of 3d polar coordinates requires a 3d space in which to define them, so you are still requiring a 3d space in which to express the equations, even if you have produced a formula with only two parameters.

 

Rather than give visual examples such as stretched strings, can we focus on this mathematical example? Because there are too many loaded terms when we talk about visual examples; to you, your example with the stretched string demonstrates that the observer in the space can measure the curvature of the space. But to me, you haven't done anything other than restate what you said previously; because you haven't explained how they will ever be able to perceive or measure any aspect of the string that isn't strictly contained within the two dimensions of their perception (saying that they can perceive the curvature because there is curvature, and that there is curvature because they can perceive it, is to me a circular definition).

 

So I think it's better to stick to the mathematical question I have given; there will be less room for misinterpretation of terms to add to the confusion.

 

Thanks again, and best wishes!

[Martin]

...Can you please define the formula for computing the sum of the angles for an equilateral triangle on the surface of a sphere?

..

 

maybe we are discovering the source of your confusion. I'm not sure but maybe.

 

one way people living in a curved space can find out the average curvature of the space around them is to measure triangles experimentally

 

and see how the excess or deficit angle changes with size

 

you don't need a FORMULA for what the sum of angles SHOULD be

you actually measure the angles and add them up

 

it us very empirical and practical handson

 

you don't want to assume ahead of time that you are on a spherical surface with uniform positive curvature or some other pure abstract geometry

 

reality is warty and dimply and bumpy.

 

Actually that was Gauss idea of what to do around 1820 and 1830 I think .

 

Now astronomers have a different method of measuring, involving the CMB, and galaxy counts.

 

It also does not involve assuming a higherdimension embedding.

 

You look and see how fast the volume of a sphere grows as you increase the radius. If it grows slower than the cube of the radius then you have positive curvature (and the angles of a very large triangle would total less than 180 degrees)

 

BTW the "radius of curvature" concept is just a abstract mathematics way of quantifying curvature. You don't have to use it. You could also use 2pi time it and call it the "circumference of curvature". Or some other thing. It doesnt presume a higherdimensional embedding exists or some outside space exists. It is just a way to assign a number to curvature so you can say more or less curved and quantify it.

[bji]

maybe we are discovering the source of your confusion. I'm not sure but maybe.

 

one way people living in a curved space can find out the average curvature of the space around them is to measure triangles experimentally

 

and see how the excess or deficit angle changes with size

 

you don't need a FORMULA for what the sum of angles SHOULD be

you actually measure the angles and add them up

 

Thanks again for your responses, they are much appreciated.

 

But wouldn't one need geometry just to make sense of the measurements that one is making? And doesn't that then beg the question of, how is the geometry defined, such that the results that we are measuring can be reasoned about, and conclusions drawn from them? And isn't this exactly the question that we are talking about - whether or not it makes sense to define intrinsic geometry of spaces?

 

If I assumed Euclidean geometry, and then I measured the angle of some triangle floating in the 3d space that I live in and can perceive, and the result was that the angles did not sum to 180 degrees, I would have a couple of choices:

 

1) I can invent a geometry to describe what I have measured, calling it intrinsic geometry, in which I simply allow for triangles of greater than 180 degrees, and 'parallel' 'straight' lines which eventually touch, by defining them that way. This doesn't actually explain anything, it just gives a mathematical basis for talking about what I've already perceived.

 

2) I can use the existing Euclidean geometry that I already know, but extend it to four dimensions, and assume that the explanation for the measurements I have made is that the space I can perceive, and in which I am measuring the lines and angles, is only a part of the whole space of reality, and that there are extra spatial dimensions in which I am moving and the apparatus which I am using to make the measurements are moving, and that the movements I am making in this 4 (or higher) dimensional space are what are causing my measurements to give me the results that I am getting, even though I can only directly perceive three of the dimensions involved. Maybe it is my mind that is the limitation, being able to model the four dimensional world in which I am living as three dimensional, so that information which is coming into my four-dimensional eyes is being 'lost' when collapsed to the three dimensions in which I think; and so lines which are really curved in my 4 (or higher) dimensional world, are being perceived to me as straight.

 

3) I can assume that there is something wrong with the way I have made the measurements, and try to find my mistake, believing implicitly that I live in a Euclidean 3d space and so any measurement that defies this must be an incorrect measurement

 

I think that (3) is clearly bogus. I must accept the observations I make, and not question them based on preconditions that I place on the results that they must give. So I wouldn't conclude that.

 

I think that (1) is bogus too. I'm just 'pretending' that myself and all of the apparatus which I use to make measurements, are subject to some parameterizations on space and how it affects my measurements, that I give the name 'intrinisic geometry' to, but that is logically inconsistent, unless I also allow that there can be paramaterizations on the lengths and angles that will be measured in my space, that are somehow independent of spatial dimensions. I don't know how I can justify saying that my measurements are being affected by some third parameter, that changes the way lengths and angles are measured, but that is not simply the manifestation of a higher dimension. I think that I must conclude that there is a higher dimension in which this parameterization can be defined.

 

And so I am left with (2), which I think actually makes some sense. Not that I am trying to draw conclusions about how my perceptive organs or how my mind works or anything like that; that is just speculation for me. The basic principle of (2) that I am trying to espouse is that I must conclude that what I perceive is an artifact

of my lower-order perception of events occuring in a higher-order dimension.

[fredrik]

Bji, I think you'll get some ideas if you look up a differential geometry book. I bet there are online pdf's too.

 

It's clear that it can in a local sense at least, make sense to "imagine" the space embedded in a higher dimensional space, but but point is that this embedding is ambigous and not unique. That is, given the constraint of your "lower dimensional perception" as you put it, there are several ways to imagine this beeing embedded in higher spaces, and this embedding simply can't be determinied from withing your constrained perception, that is the main point. What can be determined (in principle, in the classical domain) is the intrinsic part of the geometry.

 

What is beyond that is not uniquely determinable by something "living under the constraint" the surface means.

 

It's not that it doesn't make sense to imagine a higher dimensional embedding, it's just that it's not unique, and thus should be "shaved off" the description, and to get rid of arbitrary embeddings and keep only the things that are non-ambigous.

 

This is all in line with trying to purify our theories, to contain relations between real observables only, and to do away with constructs that while useful, are ambigous. So that we only have left the physical degrees of freedom so to speak.

 

Still I'm not sure if you take issue with this from a physical point, or a mathematical point. After all differential geometry is mathematics and there are a bunch of axioms that build geometry, like the concept of straight and parallell lines etc. If you wonder what the mapping of these things to reality is, then there are more understandable objections, but then that's what I called the second issue in the other thread. And I don't think that is your question?

 

/Fredrik