bji

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.

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. 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!

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'? 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. 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.

Doesn't 'the sum of the perturbative forces of all other planets' part cancel out the 'center of mass of the universe' part, such that you only need to use one or the other? In other words, doesn't taking into consideration those perturbative forces eliminate the need to add them into the equation calculating the center of mass of the galaxy? Thus the only body left in calculating the center of mass of the galaxy would be the sun, leaving the center of mass of the sun being the correct center of mass to use in the calculation of the precession of Mercury. I don't have enough of an educational background in either mathematics or physics to describe what I am saying with formulae; but perhaps my comments give you a starting point for deducing the correct answer yourself? I hope so

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

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. 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

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.

But doesn't GR only enter into the equation if the clocks in question are each at a significantly different part of the gravity well? In that case, wouldn't the effects of GR not matter to synchronized clocks on the Earth's surface, no matter how far apart on the surface the clocks were? True, but I specifically discounted that possibility when I framed my question. What I'm interested in is, what aspects of modern technology require explanation of the observed phenomena that GR explains? If there were no explanation, and no way to predict these effects, what technologies would not work? So far the only answer has been GPS, because it requires precise synchronization of clocks that are separated by a significant distance in Earth's gravity well. There hasn't been any other technology suggested yet. Wouldn't it be possible to measure the discrepancy between clocks at different altitudes, and come up with a formula that would predict the discrepancy for arbitrary altitudes? If so, then even without a theory explaining why the discrepancy exists, it might have been possible to account for it anyway, at least to a certain degree of precision, which may have been enough for GPS satellites. It's all speculation though. I do find it interesting personally. My main motivation for asking this is that, to some degree, in absence of a formal education in physics, I make some decisions about the 'quality' of theories based on how much practical knowledge they have enabled. For example, the laws of thermodynamics, aside from all of the logical and observational evidence which supports them, have enabled technologies that we could never have had without them, because just knowing the "rules of the game" allows one to with so much more precision formulate ideas about practical applications. Without laws such as these, one would have to use random trial-and-error in places that the laws allow knowing ahead of time which approach is going to work, or at the very least, significantly diminishes the size of the space of possible designs that need to be searched through to find the one 'that works'. Another analogy would be architecture; without Euclid's geometry, it would have been very difficult to have build some of the fantastic structures that mankind has invented. Instead of being able to predict which configuration of support structures would be necessary in building a tall building, we'd have to use trial and error, which would be prohibitively slow and expensive. I personally question some of the fundamental assumptions of GR (which is not to say that I reject the theory - I just find that it's unsatisfactorily explained to me, almost certainly because of my limited education in math and physics - but the parts that I do understand, I have some unanswered questions; see my other thread about "Denying Intrinsic Geometry" for an example); and so I'm curious to know how much "practical application" of GR there has been, to help me feel more confidence that the theory's potential validity has been borne out by enabling practical applications. Whereas well-trained scientists evaluate the quality of a theory by examining whether or not it can predict effects that previously were unexplained or, even better, undetected (thus demonstrating that the theory wasn't made just to fit the bounds of existing knowledge, but has explainative powers that extend beyond existing knowledge, thus enhancing existing knowledge, which is the true purpose of science), I like to also evaluate the quality of a theory by examining the technologies that it has enabled; if I can hold something in my hand that does something that would have been impossible without the theory (for example, a computer; or a cell phone; or the steering wheel of an automobile), then I gain confidence in the theory in the same way that scientists gain confidence in a theory by evaluating its pure predictive powers. Sorry to bring religion into this, but this is one reason that I don't put any faith in the Christian Bible; because it hasn't enabled any technologies, it hasn't proven itself (to me) that any of the supposed 'facts' in it (about the origins of the universe, or anything else it tries to explain) have any merit. As a historical document describing events which occurred in the past, it may have some merit; but as an explanation of how our universe works and how it was created, for the reasons I detailed above, it holds no value at all for me.

Thank you for your answer, it was very informative. It prompted me to do a google search for "gravitational time dilation effect on gps", the results of which confirm what you have said. In particular, the Wikipedia article on the issue was very comprehensive and enlightening. I wonder if scientists wouldn't have been able to find a way to measure and predict the discrepancy of atomic clocks at medium earth orbit, even without GR? Total speculation of course, but I'm guessing that someone clever would have found a way to periodically re-synchronize the clocks, so that the error could be bounded by the frequency of re-synchronization. Combined with known measured quantities for the drift (which the wikipedia page says is 45 microseconds per day), which could have been incorporated into the system even without knowing what caused the drift, maybe GPS could have been implemented (perhaps with less accuracy than we have now) without GR (or even SR?). Once again I appreciate the response. Certainly GPS seems to be the best example of a technology that as it is currently implemented requires the theory of general relativity to function properly. Any others? Oh - and as a follow-up - this part I don't think I believe. I don't think that cell phones or the internet require GPS. Even without any satellite communication at all, we could still have the internet (undersea cables to most places, radio to places that would be uneconomical to reach by undersea cable) - consider that the internet existed before GPS, so clearly it was possible, at least at the limited scope of the internet in the 1970's. And probably we could still have cell phone service too (at the very least, by using the aforementioned internet to carry calls beyond one's local cell tower). But GPS, for sure we couldn't have that.

Please forgive my choosing such a provocative title for this thread. I specifically shortened the question I am trying to answer to this phrase because I believed that it would be the most thought-provoking. The question I am really trying to ask is (and I realize that all answers to this question will be speculation, I'm not looking for the 'right' answer, just people's opinions on the answer, backed as much as possible by expertise in theoretical and applied physics): if Einstein had never been able to come up with the general theory of relativity, and no subsequent scientist had been able to, and there were still to this day no satisfactory theories to explain some of the aberrations of cosmologic observation that general relativity explains, then how would our world be different? The fundamental aspect of this question that I am most concerned about is, what aspects of modern technology would be impossible without general relativity? And, are any of these technologies in any domain other than cosmology? I ask this because my gut feeling, which you are welcome to tell me is entirely wrong because it is just a guess, and not even an educated one, is that the only things that general relativity is required for is explanations of phenomenon that we observe with telescopes and other instruments which measure phenomenon on a cosmologic scale. For example, I know that general relativity can explain the precession of Mercury (which is the fact that we see light from Mercury appearing in a slightly different place than we expect, right?), and also can explain "gravity lenses" and other anomolies of astrophysical observation. But what else would we be unable to explain? Are there fundamental aspects of, say, silicon transistors, or nuclear reactors, or space travel, or high-energy physics, or lasers, or chemistry, or anything else, that would prevent these things from advancing to the state that they have today? My motivation for the question comes from knowing (or at least, feeling pretty sure that I've read in the past, although I forget the details) that special relativity has important implications for many modern technologies, such that if we didn't have the theory of special relativity, we would have been unable to account for aspects of forces and particles that would have made some technologies impossible (for example, never being able to send a spacecraft to the moon because our trajectory calculations would miss relativistic effects and thus be incorrect). However, I am not sure if the same is true for general relativity - especially considering that its sole advancement in human knowledge (in my very limited understanding) is an explanation of how gravity's effect on matter can be modelled; but since none of our modern technologies (that I know of) depend on relativistic effects of gravity, I wonder if we would or wouldn't be in the same place we are today from a technology standpoint if Einstein had never come up with general relativity at all? Thank you!

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.