Re: Did Einstein Repeal Newton's Second Law?
From: Eric Baird (eric_baird_at_compuserve.com)
Date: 10/06/04
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Date: Wed, 6 Oct 2004 03:28:55 +0000 (UTC)
On Mon, 27 Sep 2004 19:12:24 GMT, "Androcles"
<androc1es@nospamblueyonder.co.uk> wrote:
>
>"Tom Roberts" <tjroberts@lucent.com> wrote in message
>news:cj9m8q$cin@netnews.proxy.lucent.com...
>| Bjoern Feuerbacher wrote:
>| > you [...] talk only about Special Relativity, as far as
>| > I can see. But what about General Relativity? It changes Newton's
>| > first law, doesn't it? So, shouldn't the second law also be changed?
>|
>| The changes required by GR are the same as everything else -- these laws
>| apply only locally. In GR, global properties are probelematical,
>
>LOL!
>Eric Baird:
> How far Einstein's general theory actually implements this principle
> still seems to be a matter of dispute in some parts.
>Roberts:
> Not among people knowledgable about GR.
>
>
>There is no dispute, people don't actually disagree, but it is
>"problematical".
Since my name was mentioned ...
I think that these are two different issues: the "matter of dispute"
(now possibly agreed upon) was whether or not GR actually implemented
the principle of relativity in its purest form, making all motion
purely relative.
My understanding is that people originally expected that it would, but
that there was some controversy (1950's? 1960's?) when it was realised
that perhaps "Einstein's GR" was still not fully "Machian", and
supported genuine "rotating universe" solutions. If Tom is saying that
nowadays most everyone "in the know" realises that GR is /not/ fully
Machian, then that suit sme fine, I agree with Tom and with that
collective interpretation, I think that's why the current
implementation of GR sucks. ;D
Some folks here argue that of course GR does implement things like the
equivalence principle fully ... I didn't want to get into arguments
about what the correct form of "mainstream GR" /ought/ to be and how
it /ought/ to handle these problems, so I limited myself to saying
that it was still disputed "in some parts".
I personally think that a "general" theory /should/ be fully Machian,
but that the condition of SR-compliance has ended up trashing GR's
principles and implementation so that it's a much more disappointing
theory than it could have been. So I seem to be agreeing with Tom
about the basic facts, but the two of us are obviously reaching very
different conclusions about what those facts seem to imply.
-------------------
Now, on issue #2 -- global properties becoming problematical under GR
...
I'm afraid I'm going to have to agree with Tom again:
If we are saying the the "motion" of an object has energy, and that
energy produces gravitation (so that, eg, a "rotating" star appears to
have more mass, because of the kinetic energy locked up in its
rotation), then it really isn't straightforward deciding exactly what
official "mass value" we ought to write in for the star when doing our
global accounting, because the energy bound up in its rotation is now
really more a function of the energy bound up in the star's relative
motion within a /system/ of masses.
We start to put less emphasis on the isolated KE of an individual
mass, and more emphasis on the recoverable KE of a system, and the
overall energy of the system might be different to the result that we
get by calculating all the components individually in a simple way and
then adding them up. Things can get complicated.
For instance, suppose that we want to find the KE of a rotating star:
We might be very clever and model the region without the star, place a
gyroscope at the star's location to come up with an agreed definition
of what counts in practice as a /physical/ "nonrotating frame" there,
then put the star in position, divide it into concentric cylindrcal
shells, calculate the KE for each shell from its rest mass and motion
wrt that identified "non-rotating" reference, and then just sum all
the individual energies to get the total kinetic energy of the star.
But unfortunately, since the rotation of the star is supposed to drag
light, the gyro readings at the location would be expected to be
different once the rotating star has been placed in position, and our
initial reference for what counts as a "non-rotating frame" at the
star's location is now wrong.
The shape that we calculate for the metric due to "background"
considerations gets changed by the existence of the star, and also by
the star's motion relative to the rest of the stuff in our model.
It's also no longer obviously valid to calculate the rotational
energies of the different layers independently and sum, because it
seems likely that each layer may find it "easier" to rotate thanks to
the dragging effects of the adjacent layers, so the total kinetic
energy of the star might be less than we originally calculated.
If we redo the calculation but this time take the /new/ rotating-frame
reference that is slightly dragged around by the star's total mass, we
get a lower nominal value for the star's rotation speed (wrt the new
reference frame), and consequently a lower calculated rotational KE,
which /might/ be that same earlier "reduced" value, or it might not
(dunno). Then we have to decide whether or not we need to factor in
the way that the choice of dragging frame might change within the star
depending on the change in density through the star, and whether or
not any of the new values that we calculate then need to be fed back
into the start of the calculations as a successive approximation
method (or not). Ug.
Since the strength of the dragging effect doesn't then just depend on
radius, and mass, and relative rotation and background field strength
but also on the density of the star's gravitational field (wrt
background), and since we seem to need to correct our initial "naive"
KE calculations for the exterior field strength of the star, then if
we have two identical spinning stars, and squish them together into
the space originally occuped by just one, then /perhaps/ the KE of the
resulting double-density star would be less then the combined KE's of
the two originals ... we'd no longer be able to guarantee that the
predicted KEs from "normal" calculations sum correctly -- perhaps two
identical spinning stars end-to end with a common spin and spin axis
have less total KE than if they were widely separated, and perhaps if
they then politely coalesced into a single-star volume we might expect
a "spin-up" to have to happen in order to conserve total angular
momentum. Dunno.
So /then/ it looks as if perhaps it's not just the mass and density
and relative speeds of a collection of stars that gives the total KE
(and momentum, etc) of the group (which should then show up as part of
the "total gravitation" of the group), but that perhaps the
/distribution/ of those smaller masses also affects the total mass.
And at this point one probably has a strong urge to be somewhere else,
working on a different problem!
<grin>
-------------------
Even if many of the casual suggestions in the above paragraphs turn
out to be quite wrong and misguided, these are the sorts of issues one
runs into, and /proving/ that they are wrong and misguided is probably
not easy.
How would you work out the contribution of a very large galaxy's
rotation to it's mass, when you aren't sure how rotational KE sums in
a system that is so far removed from other systems that it's starting
to become an island universe, and so that perhaps its own rotational
state is starting to become a reasonable definiton of a /non/ rotating
frame, for physics inside the galaxy? Exactly how far does the
background gravitational field drop off between two adjacent distant
galaxies, and how far does this weakening translate into a reduced
energy requirement for rotation-related distortion to appear in the
region (it should perhaps take less energy to distort a "rarefied"
region than a dense one)? Maybe in that rarefied intergalactic region
where a lot of the the "relative galactic rotation" distortion would
seem to be happening, the higher local rate of timeflow there (due to
the lower background field strength) results in the rotating galaxy
being assigned a lower nominal rotation speed (as seen from the
rarefied distorted region) than /we'd/ give it, making the relative
rotation of widely-separated galaxies easier than we'd expect, with a
lower energy bill?
There's loads of these sorts of issues, and they ain't easy.
So, like Tom says, mass isn't always a well-defined property under GR
... but IMO this can't be blamed on special relativity (aw shucks) or
on other specific aspects of GR's implementation of "general"
relativistic arguments, I think that most of the same basic
difficulties probably show up in any other advanced gravitational
model that agrees that "localised" energy is associated with gravity
and is at least approximately Machian.
How thoroughly do we define "localised"? We normally say that light
doesn't have mass -- there's a qualification that "captive" light does
seem to contribute mass to its "prison" ... but if we zoom out so far
on a galaxy that light whizzing across the galaxy now seems to be
crawling across it incredibly slowly compared to the scale of the
overall structure, and slower than in the intergalactic regions,
should we now say that the "free" light is now at least partially
"restricted" within the galaxy and /does/ contribute to the galaxy's
external gravitational field?
Or what if a cosmological region had enough total mass, over a large
enough scale, to bend light right around, would the region then have a
reduced exterior gravitational field because its geometry was partly
"turned away" from the rest of the outside universe, or would it have
an increased gravitational field because it was genuinely "hoarding"
light whose energy would then contribute to its effective contained
mass?
Tricky.
In fairness to the GR folks, they do seem to be being very upfront and
honest about this particular issue (there's probably no real reason
for them not to be, since competing theories would seem to suffer from
the same difficulties).
I rather liked Matt Visser's quickie "briefing notes" on the subject
in "Science " ;)
: Science, Vol 282, Issue 5387, 249 , 9 October 1998 ::
::
:: " Many of the standard concepts of Newtonian physics,
:: such as energy, momentum, and angular momentum
:: (the momentum carried by rotating objects) [HN1], are
:: surprisingly elusive in Einstein's general relativity. The
:: problem is that the geometry of space and time [HN2]
:: is itself dynamic, so we do not know exactly what the
:: static foundations should be. Archimedes said "give me
:: a place to stand and I will move the Earth." It is the lack
:: of a definite place to stand that makes definitions of mass
:: and momentum so tricky in general relativity. "
:: ...
And there are other immediate difficulties, such as the nasty way that
if you have a satellite orbiting your rotating star and use the
orbital characteristics of the satellte to measure the star's mass,
the result that you get would seem to depend on the direction that
your satellite orbits in ... so we might measure different "clockwise"
and "anticlockwise" values for the mass of the star and its
gravitational pull. <!>
Again, not particularly GR's fault IMO.
Although I do personally think that we'd have more tools at our
disposal for tackling /some/ of the problem gravitational issues if we
changed GR by dropping the the idea of a "clean" reduction to flat
spacetime ... even then, the list looks pretty scary, and I think that
"total mass" would probably still not be a "well defined" concept in
the above sense.
=Erk= (Eric Baird)
: <walks onto lecture hall stage>
: " Here is a blank blackboard, and here is a piece of chalk ...
: And from these starting assumptions, the rest of the theory inevitably follows.
: Thank you. "
: <exits stage>
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