Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.3 4.3bsd-beta 6/6/85; site ucbvax.BERKELEY.EDU Path: utzoo!watmath!clyde!burl!ulysses!ucbvax!brahms!weemba From: weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) Newsgroups: net.physics Subject: Re: Coordinate frames in GR Message-ID: <12762@ucbvax.BERKELEY.EDU> Date: Sat, 29-Mar-86 02:01:03 EST Article-I.D.: ucbvax.12762 Posted: Sat Mar 29 02:01:03 1986 Date-Received: Sun, 30-Mar-86 02:31:54 EST References: <438@batcomputer.TN.CORNELL.EDU> <1005@lanl.ARPA> <1035@lanl.ARPA> Sender: usenet@ucbvax.BERKELEY.EDU Reply-To: weemba@brahms.UUCP (Matthew P. Wiener) Organization: University of California, Berkeley Lines: 298 Jim Giles writes: >People who have been following this discussion are aware that I have, >so far, tried to keep the discussion general enough for the layman to >follow (at least, roughly). I have avoided using terms like 'Kerr- >Newman geometry' and 'Minkowski spaces' in order not to lose the >average reader in a cloud of opaque terminology. I continue this >effort, in spite of attempts by others on the net to obscure what >should be a simple point. I too have posted laymen summaries, which were apparently too obscure for JG here. However, JG is dead wrong, and does NOT know what he is talking about. Anyone who claims that in a rotating frame Alpha Centauri is going 9490 times faster than light is speaking gibber. >To reintroduce the subject: Matthew Wiener of Berkeley has been defending >the statement by Bertrand Russell that rotation of the Earth vs. >rotation of the universe around the Earth is 'merely' a matter of >convenience. I think Russell was just being a little over-zealous in >his support of relativism here, Wiener is being more than over-zealous - >he is being obstinate. Part I. below gives a summary of my position >with no complex terminology and no coordinate systems. I am correct and will continue to be correct (I hope). Do not, people, try to understand what JG is saying. He is INCORRECT, and is following standard laymen's errors. >I. Consider an isolated region of space distant from any large masses. > The geometry of space-time is flat in this region for most practical > purposes. Vibrating test particles (ie. pendula) appear to co-rotate > with respect to the distant stars. Spinning test particles (ie. > gyroscopes) appear to have a fixed axis with respect to the distant > stars. > > Consider, now, another isolated region which is a few Giga-parsecs from > the first. Again, the same observations of test particles give the > same observations. In addition, any observer who can see BOTH sets of > experiments will notice that the test particles co-rotate with each > other as well as the stars distant from each. That is, pendula and > gyroscopes precess identically - even if seperated by large distances. > > But, this can't have any significance. Matthew Wiener from Berkeley > says it doesn't :-). I have never said any such nonsense. I have said that the significance is a frame dependent effect in previous postings, and I will continue to say such. General relativity is not interested in frame dependent effects. I have said that and I will continue to say that. >Last week, Matthew Wiener posted an article which quoted a passage from >'Gravitation' by Mizner, Thorne, and Wheeler. To make my point clear, >I will reproduce the passage in question here: > > 'In spacetime the intervals ("proper distance," "proper time") > between event and event satisfy the corresponding theorems of > Lorentz-Minkowski geometry (Box 1.3). These theorems lend > themselves to empirical test in the appropriate, very special > coordinate systems: [...] (local Lorentz coordinates; local > inertial frame) in the local Lorentz geometry of physics. > However, these theorems rise above all coordinate systems in > their content. They refer to intervals and distances.' Notice that MTW refer to local Lorentz frames as VERY SPECIAL. Not as ALL. They are very special because the mathematics gets very simple. >(The Lorentz geometry is being introduced in analogy to Euclidean >geometry, hence the term 'corresponding'.) > >Mr. Wiener, in his commentary on this passage, made the claim that the >theorems in question (those of the Lorentz-Minkowski geometry) applied >equally to ALL coordinate systems. They said ALL, they meant ALL. I speak English like everybody else on this net. ALL is ALL, and not 'local Lorentz' or 'very special'. > This is not true (particularly for >the equations in box 1.3, which apply correctly only to Lorentz frames - >that is, local inertial reference frames when orthonormal coordinates >are used). To make this clear, consider sections II. and III. below. Sections II,III of the box are divided into part A and part B. Part A is coordinate-free. The formulas in part A thus apply to ALL coordinate systems. Part B refers to these same formulas in Lorentz frames. Thus, any formulas put here hold only in Lorentz frames. Notice that MTW always say that part B only refers to Lorentz frames. That is how I figured out that part B only refers to Lorentz frames and not to ALL frames. I'm pretty good at reading. >II. Consider the same sort of isolated region. Along come two > experimenters. They decide (for their own reasons) that they need > a coordinate system fo physical measurements of their experiments. > Since the local region of space-time is flat (ie. Lorentzian) they > both decide to use a Lorentz frame as their coordinate system. ^^^^^^^ > The first experimenter has been reading this discussion on galaxy-net, > and decides that, since rotation is merely a matter of convenience, he > will fix his 'frame' to his rotating spacecraft (it's rotating to > prevent his equipment from floating about the lab in an inconvenient > way :-). Now, Matthew Wiener has assured him that the formulae in Box > 1.3 of 'Gravitation' apply to ANY coordinate system - including > rotating ones. These are the equations of Special Relativity. The equations in part A refer to any coordinate system. The ones in part B refer to Lorentz frames only. This is what MTW say, this is what I say. If the experimenter is in a local Lorentz frame, they he can use the formulas in part B. If he is not, he cannot, and must use the computationally less convenient formulas in part A. > This first experimenter then proceeds to make his measurements and > finds, to his surprise, that they are inconsistent! His conclusion > is that either Special Relativity is wrong, or Matthew Wiener is. > "But, Matthew Wiener can't be wrong," he thinks, "he's from Berkeley!" His conclusion is that we both are correct. He reads part B, and realizes he can apply these formulas if and only if he is in a Lorentz frame. He reads part A, and realizes he can apply those formulas no matter what frame he is in. > He now proceeds down a trail of increasingly confusing, incorrect, > and inconsistent reasoning from which there is no escape. No, the experimenter rereads box 1.3, part B and discovers those formulas apply only to Lorentz frames. When MTW say Lorentz, they mean Lorentz. When they mean ALL, they say ALL. And so do I and every experimenter in the world I know of. >III. Experimenter number two also goes through the same reasoning. > But, when he finds his results are inconsistent, he thinks: "Maybe > this Wiener guy has a screw loose (from spending too much time > getting dizzy in rotating coordinate systems no doubt :-). Maybe > there IS a significant difference between rotating and non- > rotating coordinate systems." This guy also rereads box 1.3. He too notices the difference between ALL and Lorentz, because he too knows English. (English is the international language of physics, by the way, so I'm not being chauvinistic here.) :-) The only significance that Lorentz frames have is that they are computationally the simplest to write down and study. > The second experimenter then repeats his observations in a > Lorentzian frame which co-rotates with the distant stars (and his > local pendula, etc.). Now he finds that his results are consistent > with predictions of Special Relativity. Now he thinks: "Gee, > rotation was not a matter of 'mere' convenience after all. There > is a significant effect here." >I owe an apology to those readers of the net who have had to wade >through this discussion for an extra week. The only apologize you owe is for the incorrect bull you've been feeding. > I noticed the errors in Mr. >Wiener's submission when I first read it. But, since he included with >it a lot of ad-hominem abuse directed at me, The abuse came later when JG kept repeating his incorrect assertions. > I thought it would be fun >to watch him try to extract his foot from his mouth. In the interest of >this, I gave a number of blatant clues that his submission was >incorrect: but he seems to be unaware that the tough bony thing he's >chewing is his foot. >For anyone who has been struggling though this debate (especially those >who obtained a copy of 'Gravitation'), I would like to assure you that >the equations of Special Relativity are obeyed only within local >inertial reference frames. Which version of the equations? The ones in box 1.3 part A hold in all coordinate frames. The ones in box 1.3 part B hold in Lorentz frames. I have known for years that part A holds in ALL frames, and that part B holds in Lorentz frames only. I have known this because I can read what is written in front of me. > And, especially, the equations in Box 1.3 of >'Gravitation' are obeyed only within Lorentz frames (ie. local frames >with orthonormal coordinates). To apply these equations to any other >coordinate systems would require mathematical tools which are quite >beyond the scope of Special Relativity (though, well within the grasp of >General Relativity). This is not to say that these mathematical tools >are too complex to be handled in Special Relativity: Newton would probably >have understood how to transform from a rotating coordinate system to >a non-rotating one. It's just that these techniques are not part of >the relevant subject matter of Special Relativity. Stop changing the subject. Bertrand Russell *was* talking about General Relativity when he said the difference between rotating and non-rotating frames was mathematical convenience. Bertrand Russell was correct as you now seem to be admitting here. To quote from Einstein again, in "Die Grundlage der allgemeinen Relativaetstheorie," Annalen der Physik, 49, 1916, the landmark paper where Einstein first published the field equations: The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion. If JG wants to tell me that Einstein did not know relativity, go ahead. Does the net want to take a vote on whether to believe JG or Einstein? I have decided to include a proof that Alpha Centauri does NOT go faster than light in a rotating frame. JG has asserted that it goes 9490 times faster. The discussion now is technical. ---------------------------------------------------------------------- Let K(t,r,h,z) be a frame in polar coordinates. (I use h for theta.) Let K'(t,r,H,z) be rotating at angular velocity w with respect to K. So H=h+wt. The metric in K is 2 2 2 2 2 2 2 ds = - c dt + dr + r dh + dz The metric in K' is 2 2 2 2 2 2 2 2 2 2 ds = (r w - c ) dt + dr + r dH + dz - 2r w dtdH If an object (say Alpha Centauri) is at rest with respect to K, it moves along (t,R,0,0) from t=0 to T. The proper separation along its path is gotten by integrating ds. We get icT imaginary. Thus, the separation is time-like. That is, the object is moving slower than light. What happens in K'? We integrate along (t,R,tw,0) from t=0 to T. We get icT imaginary. Again, the separation is time-like. Again, the object is moving slower than light. The separation between two events in general relativity is a feature of the geometry, and not of the coordinate system. That is why the value came out the same. --------------------------------------------------------------------------- I shall go even further. In both K and K' I shall show the path of the object is a geodesic. Again, a geodesic is a geometric notion, not a frame dependent notion. As both K and K' are frames, I expect to get the same answer in both coordinate systems. i The geodesic equation is that a path x (s) satisfies: 2 i ----- i j d x \ i dx dx --- + > Gamma -- -- = 0 2 / jk ds ds ds ----- j,k where Gamma are the Christoffel symbols of the second kind. In the metric K we have that the only nonzero ones are: r h h Gamma = -r, Gamma = Gamma = 1/r. hh rh hr i t In our path the only nonzero x was x (s)=s. Thus all derivatives come out zero, except for the first derivative of the time coordinate of the path. As the corresponding Christoffel symbols are zero when one of the indices is t, and the geodesic equation is satisfied. Passing to K', we find the nonzero Christoffel symbols are: r r r r 2 Gamma = -r, Gamma = Gamma = wr, Gamma = -w r, HH Ht tH tt H H H H Gamma = Gamma = w/r, Gamma = Gamma = 1/r. tr rt Hr rH i t H The only nonzero x in the K' version of the path is x (s)=s and x (s)=ws. Their s derivatives are 1 and w respectively. All second derivatives are zero. Thus the geodesic equations reduce to i i 2 i Gamma + 2w Gamma + w Gamma = 0. tt tH HH The only nonzero Gamma here are when i=r and i=H. When i=r we get 2 2 (-w r) + 2w (wr) + w (-r) = 0. And when i=H we get 2 (0) + 2w (0) + w (0) = 0. As you can see, the main difference between the two frames is that the computation is a bit more complicated in K' then in K. But the physics is the same. --------------------------------------------------------------------------- As MTW says on page 19-23, and JG quoted without understanding English: In spacetime the intervals between event and event satisfy the corresponding theorems of Lorentz-Minkowski geometry. These theorems LEND themselves to EMPIRICAL TEST in the appropriate VERY SPECIAL coordinate systems: ... the local Lorentz geometry of physics. However, the theorems RISE ABOVE ALL coordinate systems in their content. They refer to intervals or distances. Those distances [do not] call on coordinates for their definition. [emphasis mine] ucbvax!brahms!weemba Matthew P Wiener/UCB Math Dept/Berkeley CA 94720