cpf@batcomputer.UUCP (03/25/86)
After several very bad articles on this subject, there have recently
been some very good ones, especially the one that noted that the Earth
does rotate, and that this is unambiguous because the geometry near the
Earth is a Kerr geometry. However, I feel that the lack of significance of
coordinate frames should be stated forcefully.
Coordinates are NOT fundamental in general relativity. What is fundamental
the geometry of spacetime that is described by the metric
ds**2 = gAB(p)dxAdxB, where g, the metric, is 10 functions of the point
p in space time. All the physics is here. We can now make any (almost)
coordinate transformation, xA = xA(xAold), and get exactly the same physics.
(Although it is nice if the transformation is invertible and sufficiently
differentiable.)
An example of a legal transformation is:
x0 -> x0, x1 -> x1, x2 -> x2 , x3 -> x3 - v * x0, where v is a
constant. Pick v = 10. We now have a coordinate system in which every
"velocity" is faster than light! What does that mean? It means that
a dx3/dT for a time-like particle is greater than 1 (c = 1). In fact,
in this coordinate system, the velocity of light is greater than the
"velocity of light"! So what? A light-like world line still has interval
zero (ds**2 = 0). In this coordinate frame, the metric is somewhat strange
looking, but the physics is normal. If I want to make a coordinate
system in which the earth is motionless (i.e dx{1,2,3}/dT = 0 for
points on the earth's surface, I can do that (in an infinite number of ways).
If I pick a simple way of doing that, Alpha Centauri's velocity will
have some strange looking numbers in it, but no physics will be changed
by my changing coordinate systems.
To restate this, coordinate systems have no reality -- we are free
to choose any one we want to. Normally we choose on the basis of simplicity,
and this conditions us to expect certain things of a coordinate system,
but nothing stops us from choosing a perverse one. They all give the
same physics.
I can do anything to my coordinates that I want to:
x0 -> x0**2, x1 -> exp (x2) - x3, x2 -> gamma(x1**2), x3 -> arctanh(x3).
I don't even want to thing about what my metric would look like
(even for flat space), but there is nothing that stops me from doing
it.
As a final example of the meaninglessness of coordinates, consider
two point masses orbiting each other. This problem is not exactly
soluble, but I can make an approximate solution: Newtonian gravity.
I can now make a coordinate transformation so that both particles are
"stationary"; that is x1, x2 and x3 are constant for all x0. The
metric will look odd, with trigonometric functions of x0 in strange
places, but it will be a correct description of the physics (up to the
Newtonian approximation). It is not a particularly useful description,
but is a correct one.
To sum up, the evolution of the coordinates of a particle are completely
meaningless until one looks at the metric (that is, the geometry) and
figures out what the particle is actually doing!
[P.S. For the semi-expert. If you don't understand what I am talking
about in the next paragraph, don't worry about it.]
An example of a coordinate system with some perverse features is
ordinary Schwarzschild geometry. It seems like no particle will every
reach 2M, since it takes infinite (coordinate) time to reach 2M.
Actually, of course, a particle does reach 2M in finite proper time,
and in due course achieves it demise at r = 0. One can remove this
singularity by going to another coordinate system, e.g Kruskal-Szekeres.
(This coordinate singularity is, in its way, more disturbing than the
"faster than light" velocities of a rotating coordinate system, and
has produced considerable confusion among science fiction writers.)
--
--------------------------------------------------------------------------------
Courtenay Footman ARPA: cpf@lnsvax.tn.cornell.edu
Lab. of Nuclear Studies Usenet: {decvax,ihnp4,vax135}!cornell!lnsvax!cpf
Cornell University Bitnet: cpf%lnsvax.tn.cornell.edu@WISCVM.BITNETjlg@lanl.ARPA (Jim Giles) (03/26/86)
In article <438@batcomputer.TN.CORNELL.EDU> cpf@batcomputer.UUCP (Courtenay Footman) writes: >After several very bad articles on this subject, there have recently >been some very good ones, especially the one that noted that the Earth >does rotate, and that this is unambiguous because the geometry near the >Earth is a Kerr geometry. However, I feel that the lack of significance of >coordinate frames should be stated forcefully. Bravo!! I have been trying to get this point across for some time now. I have confined my discussion to coordinate systems because it is easier to explain that way. But the bottom line is still - 'The Earth DOES rotate, and this is UNAMBIGUOUS'. There are some on the net who will take an overzealous remark by Bertrand Russell at face value over the whole of GR - and then accuse anyone who disagrees with them of ignorance! J. Giles Los Alamos
jlg@lanl.ARPA (Jim Giles) (03/28/86)
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.
To reintroduce the subject: Matt Weiner 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, Weiner 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. 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. Matt Weiner from Berkeley
says it doesn't :-).
Last week, Matt Weiner 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.'
(The Lorentz geometry is being introduced in analogy to Euclidean
geometry, hence the term 'corresponding'.)
Mr. Weiner, 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. 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.
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, Matt Weiner 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.
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 Matt Weiner is.
"But, Matt Weiner can't be wrong," he thinks, "he's from Berkeley!"
He now proceeds down a trail of increasingly confusing, incorrect,
and inconsistent reasoning from which there is no escape.
III. Experimenter number two also goes through the same reasoning.
But, when he finds his results are inconsistent, he thinks: "Maybe
this Weiner 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."
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. I noticed the errors in Mr.
Weiner's submission when I first read it. But, since he included with
it a lot of ad-hominem abuse directed at me, 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. 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.
J. Giles
Los Alamosgwyn@brl-smoke.UUCP (03/28/86)
In article <1005@lanl.ARPA> jlg@a.UUCP (Jim Giles) writes: >In article <438@batcomputer.TN.CORNELL.EDU> cpf@batcomputer.UUCP (Courtenay Footman) writes: >>After several very bad articles on this subject, there have recently >>been some very good ones, especially the one that noted that the Earth >>does rotate, and that this is unambiguous because the geometry near the >>Earth is a Kerr geometry. However, I feel that the lack of significance of >>coordinate frames should be stated forcefully. > >Bravo!! I have been trying to get this point across for some time now. I >have confined my discussion to coordinate systems because it is easier to >explain that way. But the bottom line is still - 'The Earth DOES rotate, >and this is UNAMBIGUOUS'. There are some on the net who will take an >overzealous remark by Bertrand Russell at face value over the whole of GR - >and then accuse anyone who disagrees with them of ignorance! The statement "the Earth rotates" is meaningless until you specify with respect to what. Nobody is arguing that the Earth is not rotating with respect to the observed universe-environment. The question is, is that different from the universe-environment rotating with respect to the Earth? (Is the latter question even well-posed?) You haven't answered that. (Or if you have, claiming that there is a difference, then either you don't fully understand the depth of the question or else you claim something that is not actually known at present.) I do agree that people should be less insulting in technical discussions. This posting is not intended as an insult..
weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) (03/29/86)
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
gwyn@brl-smoke.UUCP (03/29/86)
I finally dug out my copy of MTW to see what was in Box 1.3. If a region of space-time is indeed inertial, then the coordinate-free representation of special relativistic physics is correct. The introduction of coordinates can be done in many ways; if done carefully so that a local Lorentz frame results, then the simplified form of special relativistic coordinate-based equations can be used, since the metric tensor is diagonal is such a frame; if a more general frame, such as a rotating one, is introduced instead, then one has to treat the metric tensor correctly, in addition to being more careful in formulating the coordinate-based equations of special relativity. Despite this, one does not have to resort to general relativity to treat physics in an inertial space, even if one chooses to use rotating coordinate systems; however, some of the tools commonly employed in general relativity are needed. On the other hand, to relate events at non-neighboring points of the space-time continuum to each other when space-time is noninertial (e.g. in the presence of gravitating bodies), it is essential to use general relativity rather than special relativity. This can be traced to the existence of a non-trivial gauge field in such a region. The question of rotation of the whole universe about a stationary Earth definitely is the non- local type of question that requires general relativity. One final point: Empty space-time is NOT necessarily flat, even in conventional general relativity. In the generalized theory of Eddington/Schr"odinger, in fact, space-time cannot be flat anywhere.