
#19
Dec2011, 01:23 PM

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#20
Dec2011, 01:50 PM

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To picture the kinds of geometric patterns the apprentice used in placing the objects, just think things like Feynman diagrams, processes involving conservation laws, etc. Thus, when the consciousness is turned on and sent on a trip along the blue guy's world line, the only way the consciousness could acquire any comprehension of the continuous sequence of 3D crosssection views of the 4dimensional manifold with embedded objects would be to psychologically adjust his X1 crosssection view so as to be in sync with a Lorentz boosted view of the universe. That's just because the apprentice formed the patterns in just the right way to produce the unique invariances that are normally associated with Lorentz boosts. If the blue guy did not view the universe across a Lorentz boost view there would not be the kind of correlation in the sequence of events unfolding around him that could produce a comprehensible experience. It's kind of analogous to the difference between listening to random noise and music. If you are in an environment of a loud audible random noise, yet there is a lone violin playing a melody somewhere in the background, your brain has a way of filtering the violin melody so that you comprehend the sound in spite of the noise. The symmetry of geometric patterns present in the 4D spatial universe makes possible some kind of correlation within the brain that plays some kind of role in the ability of the consciousness to recognize and comrehend. For the blue guy it was a matter of psychologically adjusting his crosssection view to the proper Lorentz boost that makes for an intelligible continuous sequence. The cosmic buttonpusher initially did not realize what his apprentice had done, but he quickly discovered the benefit of switching to a new metric so he could recognize the invariances and appreciate the local physics that resulted. He was quick to realize why the blue guy's consciousness automatically began scanning 4D space in the slanted X1' direction. But, it is important to note that it was not necessary for the cosmic buttonpusher to change metrics for his bird's eye view of the universe. The cosmic buttonpusher could happily muse over the varied patterns of objects placed on the positive definite metric space. He just wouldn't see the physics that the apprentice had built into the patternswhich are only apparent if you change the metric so as to recognize the invariances associated with the Lorentz boost. Arranging objects does not change the intrinsic mathematical properties of the manifoldit's topology, etc. The whole point of the story is to try to explain in what sense you can start with a positive definite manifold, yet then orient objects in a way that leads to the selection of an indefinite metric to make the orientation of objects intelligible. The quality of the four spatial dimensions did not change at all in that process. And there was certainly no rationale for regarding the 4th dimension as "time." And yes, you're right; it's the differential geometry and associated mathematical machinery. However, all through the physics Master's and PhD curriculum, in all of the functional analysis, tensor analysis, group theory, set theory, QM courses, classical field theory, special relativity, general relativity, and cosmology courses, none of my professors ever discussed manifolds in this context. I tried only two or three times to discuss this with my doctoral relativity advisor, but he was quite annoyed that I would allow myself to get so distracted from doing real physics. And he was right in terms of how I should have spent my time in that phase of education. 



#21
Dec2011, 04:12 PM

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But that doesn't mean the subject can't be learned. I learned it mainly from Misner, Thorne, & Wheeler, which I also had the advantage of not having to learn in school. But that may not be the best up to date source. Others here at PF could give better advice than I on where to look. 



#22
Dec2011, 05:56 PM

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#23
Dec2011, 10:34 PM

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One property in particular that you can't describe without the metric is causality: without the metric there is no way to tell whether a given pair of events is timelike, null, or spacelike separated, so you don't know what causal relationships are possible or forbidden between them. This is one big reason why the standard viewpoint considers the metric to be "part of the character of the space itself". 



#24
Dec2111, 05:36 PM

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In spite of this situation, it is not correct (in the view of the initial post here) to say that the metric accounts for the X4 as being either physical time or physical space. But yes, the metric is intimately associated with the revelation of the physics manifest on the manifold. But: It is the physics first (the very special arrangement of the 4D objects) that prompts for a successful selection of a metric. The L4 metric has been revealed to us, but only because the 4D objects have been arranged in that very special way. 



#25
Dec2111, 08:05 PM

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A couple more comments: The metric does not place a preference on the quality of the dimensions. The traditional view among physicists is the one PeterDonis has been advocating. Although the mathematical system applied in desribing special relativity theory does not force this view, Minkowski himself embraced it. Most physicists embraced the idea of time as the 4th dimension. However, that did not mean that they did not embrace the idea of the block universe. Weyl wrote: "The world does not happen, it simply is." Einstein apparently subscribed to this view as well (everyone always references Einstein's letter to the wife of his close friend, Besso, at the time of Besso's passing).
Typical of sentiments in the early years of special relativity is the commentary from the writings of Sir James Jeans on SpaceTime unity (book "Physics and Philosophy): "The physical theory of relativity suggests, although without absolutely conclusive proof, that physical space and physical time have no separate and independent existences; they seem more likely to be abstractions or selections from something more complex, namely a blend of space and time which comprises both. This is exactly the view that I've tried to refute with the beginning post of this thread. So, the two versions of block time: 1) A fourdimensional universe all there at once with physical time as the 4th dimension and 2) A fourdimensional universe all there at once with the 4th dimension as just another "physically spatial" dimension (we use "physically spatial" to avoid the confusion of the meaning of the mathematical abstract space that implies particular metricsmetrics that really do not force X4 to be either physical time or physical space). This physically spatial 4th dimension could be accompanied either by a 3D consciousness moving along a world line at speed c, or it could be accompanied by a consciousness that exists simultaneously all along the world line. By the way, a point made by Einstein ("Albert Einstein  PhilosopherScientist", Library of Living Philosophers, Edited by Paul Schilpp": "First a remark concerning the relation of the theory to 'fourdimensional space.' It is a widespread error that the special theory of relativity is supposed to have, to a certain extent, first discovered, or at any rate, newly introduced, the fourdimensionality of the physical continuum. This, of course, is not the case. Classical mechanics, too, is based on the fourdimensional continuum of space and time. But in the fourdimensional continuum of classical physics the subspaces with constant time value have an absolute reality, independent of the choice of the reference system. Because of this [fact], the fourdimensional continuum falls naturally into a threedimensional and a onedimensional (time), so that the fourdimensional point of view does not force itself upon one as necessary. The special theory of relativity, on the other hand, creates a formal dependence between the way in which the spatial coordinates, on the one hand, and the temporal coordinates, on the other, have to enter into the natural laws." And again, he is expressing a concept we've tried to refute in this thread. He could just as easily regarded the 4th dimension of Newton as a physically spatial dimension with time as a parameter. Nevertheless, his view gives me an opening to make a point of rebuttle to the notion that the metric detemines the physical nature of a dimension. In Einstein's example here we have a positive definite metrice (the X4' and X1' axes are not slanted symmetrically in accordance with the Lorentz boost at all. If a positive metric is associated with physically spatial coordinates only, then Einstein could not have his "time" as a 4th dimension in the Newtonian Euclidean 4D space. 



#26
Dec2111, 08:24 PM

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#27
Dec2111, 08:51 PM

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#28
Dec2111, 09:02 PM

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If all you are saying is that spacetime diagrams help to visualize relativity problems, I have no objection to that. But spacetime diagrams are tools; you don't have to adopt any particular position on whether or not X4 is "spatial" in order to use spacetime diagrams. 



#29
Dec2111, 09:04 PM

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#30
Dec2111, 09:22 PM

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#31
Dec2111, 10:45 PM

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a) Time as the 4th dimension. b) What is meant by "a mixture of time and space?" We know what that means mathematically, and we can easily see it mathematically when it is represented on a spacetime diagramits just a mathematical crosssection of a mathematical spacetime. But please offer some kind of description that would give us a concept of the mixture of space and time for which we would have no trouble comprehending. Discuss the quality or character of that mixture. I can comprehend in some sense the quality of space based on experience with X1, X2, X3. I can comprehend the notion of time from my direct psychological experience with time passing. How do you mix those two concepts and come up with a comprehensible concept? They are so different. The mathematical description is useful and necessary, but that alone does not provide a comprehensible concept. If I had been deaf, blind and without the sense of touch all my life, I may not be capable of having a comprehensible concept of space. My care giver could tell me when he is moving me from place to place, but the concept of space should be much more difficult to comprehend for someone in that state. I could comprehend a notion of time in that state. But, the notion of a mixture of space and time would be hopeless, even if my care giver could teach me mathematics, i.e., differential geometry and special relativity. 



#32
Dec2211, 12:33 AM

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#33
Dec2211, 09:55 AM

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Here is the writeup from the back cover of the book: In 1942, the logician Kurt Godel and Albert Einstein became close friends; they walked to and from their offices every day, exchanging ideas about science, philosophy, politics, and the lost world of German science in which both men had grown up. By 1949, Godel had produced a remarkable proof: In any universe described by the Theory of Relativity, time cannot exist. Einstein endorsed this result reluctantly, but he could find no way refute it, and in the halfcentury since then, neither has anyone else. Even more remarkable was what happened afterward: nothing. Cosmologists and philosophers alike have proceeded as if this discovery was never made. In "A World Without Time," Palle Yourgrau sets out to restore Godel to his rightful place in history, telling the story of two magnificent minds put on the shelf by the scientific fashions of their day, and attempts to rescue from undeserved obscurity the brilliant work they did together. I have another reference, a paper written by Godel for inclusion in a book dedicated to Einstein, the one I mentioned earlier edited by Arthur Schilpp. Schilpp mangaged to collect writings from a number of physicists, mathematicians and philosophers, all dedicated to commenting on Einstein's life and work, which formed one volume in Schilpp's series on Living Philosophers. I'll have to dig it out again and locate some pertinent excerpts from which Godel is describing his solutions to Einstein's equations of general relativity in the context of "destroying time." 



#34
Dec2211, 04:48 PM

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Yourgrau manages to convey fairly clearly what exactly Gödel demonstrated in his short paper, taking Einstein's theory of relativity and focussing on the knotty timeissue, presenting a world model in which he could show "that t, the temporal component of spacetime, was in fact another spatial dimension". The implications and consequences are profound and farreaching... 



#35
Dec2211, 06:06 PM

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Edit: Ok, I see that the last paragraph of the post is supposed to be the excerpt. It would still be nice to have a link to the entire review. 



#36
Dec2211, 06:17 PM

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The wording of the description of Godel's "remarkable proof" is already a red flag: the author is being way too dogmatic about what Godel's proof "means". The author appears to be a philosopher, not a physicist, which IMO is another reason to take what he says about the application of such proofs to physics with a grain of salt. John Stachel, a physicist, reviewed the book, and his review is online here: http://www.ams.org/notices/200707/tx070700861p.pdf He appears to have similar misgivings about the author's flat assertions about what Godel actually proved. This review, btw, also has an interesting quote from Einstein: "Time and space are fused in one and the same continuum, but the continuum is not isotropic. The element of spatial distance and the element of duration remain distinct in nature…" 


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