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From: jlg@lanl.ARPA (Jim Giles)
Newsgroups: net.physics
Subject: Re: Coordinate frames in GR
Message-ID: <1035@lanl.ARPA>
Date: Thu, 27-Mar-86 19:43:47 EST
Article-I.D.: lanl.1035
Posted: Thu Mar 27 19:43:47 1986
Date-Received: Sat, 29-Mar-86 06:13:34 EST
References: <438@batcomputer.TN.CORNELL.EDU> <1005@lanl.ARPA>
Reply-To: jlg@a.UUCP (Jim Giles)
Organization: Los Alamos National Laboratory
Lines: 116

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 Alamos