weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) (03/24/86)
I'd like to apologize to Jim Giles for getting nasty with him. I guess I just wasn't making myself clear at first. Also, I ended up bringing my copy of MTW in from home to the office, which for me is a two mile walk these days. Strange and inaccurate versions of general relativity seem to be rather common: once it got its myth of incomprehensibility, comprehension was no longer a requirement for speaking about it. A good way to clear out the educational system's brainwashing efforts here is to read Albert Einstein's original works on the subject. They shine with a crystal clearness with no one's muddy simplifications to hide the true beauty of general relativity. (And Pais _Subtle is the Lord_ is a fine guidebook to these originals.) From "Die Grundlage der allgemeinen Relativitaetstheorie", _Annalen der Physik_, 49, 1916 (from the Dover paperback translation): Of all imaginable spaces ... in any kind of motion relatively to one another, there is none which we may look upon as privileged a priori.... THE LAWS OF PHYSICS MUST BE OF SUCH A NATURE THAT THEY APPLY TO SYSTEMS OF REFERENCE IN ANY KIND OF MOTION. [p 113] This view of space and time [that space and time coordinates have intrinsic physical meaning] has alway been in the minds of physicists.... But we shall now show that we must put it aside and replace it by a more general view, in order to be able to carry through the postulate of general relativity.... In a space which is free of gravitational fields we introduce a Galilean system of reference K(x,y,z,t) and also a system of co-ordinates K'(x',y',z',t') in uniform rotation relatively to K. Let the origins of both systems, as well as their axes of Z, permanently coincide. We shall show that for a space-time measurement in the system K' the above definition of the physical meaning of length and times cannot be maintained. [Because of special relativity, the circumference is measured by contracted rods and so seems longer, so the spatial geometry is not Euclidean. Clocks too are our position dependent in K'. [Editorial comment here: I earlier posted a remark that fake gravity induced by accelerations cannot change the geometry. The point is that while only mass-energy can alter the space-time geometry, different coordinate choices CAN alter your spatial geometry. It is the frame chooser's responsibility to make sure he knows what he is doing, not Einstein's.] We therefore reach this result:--In the general theory of relativity, space and time cannot be defined in such a way that differences of the spatial co-ordinates can be directly measured by the unit measuring-rod, or differences in the time co-ordinate by a standard clock. The method hitherto employed for laying co-ordinates ... breaks down.... So there is nothing for it but to regard all imaginable systems of co-ordinates, on principle, as equally suitable for the description of nature. This comes to requiring that:-- THE GENERAL LAWS OF NATURE ARE TO BE EXPRESSED BY EQUATIONS WHICH HOLD GOOD FOR ALL SYSTEMS OF CO-ORDINATES, THAT IS, ARE CO-VARIANT WITH RESPECT TO ANY SUBSTITUTIONS WHATEVER (GENERALLY CO-VARIANT). [pp 115-117] [In choosing among co-ordinate systems], there is no immediate reason for preferring certain systems of co-ordinates to others. [p 117] From _The Meaning of Relativity_, fifth edition: The case that we have been considering [K and K' in relative rotation] is analogous to that which is presented in the two-dimensional treatment of surfaces. It is impossible in the latter case also, to introduce co-ord- inates on a surface (eg the surface of an ellipsoid) which have a simple metrical significance, while on a plane the Cartesion co-ordinates, x1, x2, signify directly lengths measured by a unit measuring rod. Gauss overcame this difficulty, in his theory of surfaces, by introducing curvilinear co-ordinates which, apart from satisfying conditions of continuity, were wholly arbitrary, and only afterwards these co-ordinates were related to the metrical properties of the surface. In an analogous way we shall introduce int he general theory of relativity arbitrary co-ordinates, x1, x2, x3, x4, which shall number uniquely the space-time points, so that neighbouring events are associated with neighbouring values of the co-ord- inates; otherwise the choice of co-ordinates is arbitrary. We shall be true to the principle of relativity in its broadest sense if we give such a form to the laws that they are valid in every such four-dimensional system of co-ordinates, that is, if the equations expressing the laws are co-variant with respect to arbitrary transformations. [p 61] ----------------------------------------------------------------------- As a final point, the assertion that most frames in general relativity are local Lorentz frames is just plain false. Off the top of my head there are Schwarzschild, Reissner-Nordstrom, Kerr, Boyer-Linquist, Kruskal-Szekeres, Taub-NUT, plane wave, de Sitter, anti-de Sitter, Roberston-Walker, Godel, and mixmaster coordinate systems, and not one of them is a local Lorentz frame! It is true that the theoretical justification of certain points is easier to do with calculations in a local Lorentz frame. So keep reading MTW Jim, you'll get to the good stuff eventually. ucbvax!brahms!weemba Matthew P Wiener/UCB Math Dept/Berkeley CA 94720
jlg@lanl.ARPA (Jim Giles) (03/24/86)
In article <12603@ucbvax.BERKELEY.EDU> weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) writes: >Of all imaginable spaces ... in any kind of motion relatively to one >another, there is none which we may look upon as privileged a priori.... >THE LAWS OF PHYSICS MUST BE OF SUCH A NATURE THAT THEY APPLY TO SYSTEMS >OF REFERENCE IN ANY KIND OF MOTION. [p 113] > ... >In a space which is free of gravitational fields we introduce a Galilean >system of reference K(x,y,z,t) and also a system of co-ordinates >K'(x',y',z',t') in uniform rotation relatively to K. Let the origins of >both systems, as well as their axes of Z, permanently coincide. We >shall show that for a space-time measurement in the system K' the above >definition of the physical meaning of length and times cannot be maintained. >[Because of special relativity, the circumference is measured by contracted >rods and so seems longer, so the spatial geometry is not Euclidean. Clocks >too are our position dependent in K'. [etc.] All this is but to say that given sufficient mathematical sophistication you can correctly work out what's going on in either coordinate system. I never doubted this. But it still doesn't answer my original objection to the Russell quote: given the frames K and K' - AT MOST one of them will be able to spin up a gyroscope and observe NO precession. In addition, the frame which measures no precession on its gyroscopes will be fixed relative to the distant stars to many significant figures of measurement (in the neighborhood of Earth: within 0.1 arc seconds per year (MTW pp. 1119- 1121)). To this extent, AT LEAST, there seems to be a prefered frame with respect to rotation. Now, I guess you are arguing that the coincident event of zero precession locally and being fixed relative to the distant stars is 'merely a matter of convenience'. Well, I'm not buying it. And neither did Einstein, who spent considerable effort to determine why this might be so. This is why I brought Mach's principle into this discussion - which not only predicts that there would be a 'prefered' frame, but gives a relativistic explanation for the effect (it also predicts that any local variation from the 'prefered' frame would be insignificant). While it is certainly convenient to do one's calculations in a non-rotating coordinate system, there also seems to be something physically significant about such systems as well. J. Giles Los Alamos
kort@hounx.UUCP (B.KORT) (03/24/86)
Matthew Wiener treats us to some nuggets from Einstein: >Of all imaginable spaces ... in any kind of motion relatively to one >another, there is none which we may look upon as privileged a priori.... >THE LAWS OF PHYSICS MUST BE OF SUCH A NATURE THAT THEY APPLY TO SYSTEMS >OF REFERENCE IN ANY KIND OF MOTION. [p 113] > >THE GENERAL LAWS OF NATURE ARE TO BE EXPRESSED BY EQUATIONS WHICH HOLD >GOOD FOR ALL SYSTEMS OF CO-ORDINATES, THAT IS, ARE CO-VARIANT WITH RESPECT >TO ANY SUBSTITUTIONS WHATEVER (GENERALLY CO-VARIANT). [pp 115-117] > >[In choosing among co-ordinate systems], there is no immediate reason for >preferring certain systems of co-ordinates to others. [p 117] OK. Let's see what we can do with these gems. Let's substitute "discussions" for "spaces", "emotion" for "motion" "SENTIENT BEINGS" for "PHYSICS" "BELIEF" for "REFERENCE" or "CO-ORDINATES" We get... Of all imaginable discussions ... in any kind of emotion relatively to one another, there is none which we may look upon as privileged a priori.... THE LAWS OF SENTIENT BEINGS MUST BE OF SUCH A NATURE THAT THEY APPLY TO SYSTEMS OF BELIEF IN ANY KIND OF EMOTION. [meta-p 113] THE GENERAL LAWS OF NATURE ARE TO BE EXPRESSED BY EQUATIONS WHICH HOLD GOOD FOR ALL SYSTEMS OF BELIEF, THAT IS, ARE CO-VARIANT WITH RESPECT TO ANY SUBSTITUTIONS WHATEVER (GENERALLY CO-VARIANT). [meta-pp 115-117] [In choosing among belief systems], there is no immediate reason for preferring certain systems of belief to others. [meta-p 117] I don't know if this exercise leads anywhere, but the above example of the "copy-with-substitute" procedure does seem to preserve a certain sensibility to it. I wonder if anyone else has noticed the deep isomorphism between the laws of physics and the laws of human nature. --Barry Kort ...ihnp4!hounx!kort
gwyn@brl-smoke.ARPA (Doug Gwyn ) (03/24/86)
In article <12603@ucbvax.BERKELEY.EDU> weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) writes: >It is true that the theoretical justification of certain points is easier >to do with calculations in a local Lorentz frame. From the point of view of the more general theory, it is important to realize that a general metric cannot be completely diagonalized by real (differentiable) coordinate transformations, so a local Lorentz frame is not always possible. A theory in which it IS always possible is one that neglects other possible metric-related physical effects besides gravitation. In answer to the fellow who wanted to know what the difference is between space and space-time: When one diagonalizes the symmetric part of the metric (the only part considered in normal general relativity) using real coordinate mappings, one of the four diagonal elements will have different sign than the other three. In the transformed (local Lorentz) frame, that coordinate will be associated with the direction of time while the other three are spatial.
weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) (03/26/86)
In article <875@lanl.ARPA> jlg@a.UUCP (Jim Giles) writes: >All this is but to say that given sufficient mathematical sophistication >you can correctly work out what's going on in either coordinate system. I >never doubted this. But it still doesn't answer my original objection to >the Russell quote: given the frames K and K' - AT MOST one of them will be >able to spin up a gyroscope and observe NO precession. And when did Russell say that was false? 'precession' is a frame-dependent phenomena, as you have just shown, and is not covariant. Russell said the difference between the two was 'mathematical convenience'. Perhaps you would have been happier if he went on to say the mathematically simpler was the one that gets the standard physics terminology. > In addition, the >frame which measures no precession on its gyroscopes will be fixed relative >to the distant stars to many significant figures of measurement (in the >neighborhood of Earth: within 0.1 arc seconds per year (MTW pp. 1119- >1121)). To this extent, AT LEAST, there seems to be a prefered frame with >respect to rotation. Preferred if you like your calculations to be simple and your terminology to be standard. Not preferred if you want to understand the geometry of space-time itself. In either K or K', the curvature will come out zero. That is geometry. That is where the real physics lies. Does the above paragraph sound like it is from MTW? It should. >Now, I guess you are arguing that the coincident event of zero precession >locally and being fixed relative to the distant stars is 'merely a matter >of convenience'. Well, I'm not buying it. And neither did Einstein, who >spent considerable effort to determine why this might be so. This is why >I brought Mach's principle into this discussion - which not only predicts >that there would be a 'prefered' frame, but gives a relativistic >explanation for the effect (it also predicts that any local variation >from the 'prefered' frame would be insignificant). But physics today rejects the notion of a preferred frame. That is the meaning of general covariance. Given a frame, it is the individual framer's responsibility to correlate his coordinate measurements with physical reality. There is no apriori meaning. Thus in studying black holes, a lot of effort must go into figuring out what 'r' and 't' refer to physically. If you hold the earth as fixed in your frame, and then calculate that Alpha Centauri is going 9490 times faster than light, you have abdicated your responsibility of relating coordinate numbers to physical notions. The only possible meaning of Mach's principle in general relativity today is as Doug Gwyn pointed out, is in the notion of boundary conditions. >While it is certainly convenient to do one's calculations in a non-rotating >coordinate system, there also seems to be something physically significant >about such systems as well. Yes. There is physical significance to inertial frames. ucbvax!brahms!weemba Matthew P Wiener/UCB Math Dept/Berkeley CA 94720
weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) (03/27/86)
In article <2049@brl-smoke.ARPA> gwyn@brl.ARPA writes: >In answer to the fellow who wanted to know what the difference is >between space and space-time: When one diagonalizes the symmetric >part of the metric (the only part considered in normal general >relativity) using real coordinate mappings, one of the four >diagonal elements will have different sign than the other three. >In the transformed (local Lorentz) frame, that coordinate will be >associated with the direction of time while the other three are >spatial. I don't believe this. The fellow asked for laymen's terms only. ucbvax!brahms!weemba Matthew P Wiener/UCB Math Dept/Berkeley CA 94720
gwyn@brl-smoke.UUCP (03/28/86)
In article <751@hounx.UUCP> kort@hounx.UUCP (B.KORT) writes: >[In choosing among belief systems], there is no immediate reason for >preferring certain systems of belief to others. [meta-p 117] > >I don't know if this exercise leads anywhere, but the above example >of the "copy-with-substitute" procedure does seem to preserve a certain >sensibility to it. I wonder if anyone else has noticed the deep >isomorphism between the laws of physics and the laws of human nature. I take strong issue with considering the substituted quotation to be a "law of human nature". Unlike coordinate systems, which are an artifical contrivance, human beings are existing concrete entities with specific characteristics, so one very well WOULD expect there to be preferred belief systems, moral codes, etc. derived from the actual nature of human beings. I do not grant equal credence to a criminal's belief in his right to kill me and in my belief that I have the right to live. This isn't really a physics topic, but that's where you posted.
gwyn@brl-smoke.UUCP (03/29/86)
In article <12699@ucbvax.BERKELEY.EDU> weemba@brahms.UUCP (Matthew P. Wiener) writes: >In article <2049@brl-smoke.ARPA> gwyn@brl.ARPA writes: >>In answer to the fellow who wanted to know what the difference is >>between space and space-time: When one diagonalizes the symmetric >>part of the metric (the only part considered in normal general >>relativity) using real coordinate mappings, one of the four >>diagonal elements will have different sign than the other three. >>In the transformed (local Lorentz) frame, that coordinate will be >>associated with the direction of time while the other three are >>spatial. > >I don't believe this. The fellow asked for laymen's terms only. So? I would like a layman's explanation of how the universe works in 25 words or less, but I don't think it's very likely. The problem with simpler, inaccurate explanations is that one then has to work even harder to correct all the misperceptions that the oversimplified explanation produces. Since nobody had answered the question about "the difference between space and spacetime" (see! already a misperception) satisfactorily, I decided to supply an accurate answer as an aside to another posting. I really don't think this topic could be explained to a layman without many pages of discussion and examples, in any event; without resorting to something like my principal axes of the metric tensor explanation, a layman's explanation would sound something like: "At any place in the universe, at any time, there are three directions of space, which have no preferred alignment, and one direction of time. Time is that which a clock measures, and space can be measured by yardsticks." I don't think that would be particularly enlightening. Worse yet, a clever layman is likely to ask embarrassing questions such as: "Is there any place in the universe with only two space directions? Two space and two time? etc.?" Then one would have to explain more accurately anyway. If you think you can answer the layman's question better, please do so, and more power to you. "We know for certain, for instance, that for some reason, for some time in the beginning, there were hot lumps." -- Firesign Theater, "I Think We're All Bozos On This Bus"