Stephen P. King (stephenk1@home.com)
Sun, 11 Apr 1999 21:24:37 -0400
Dear Hitoshi,
I will lay out the first part of my thinking about Weyl's theory... I
will be very busy in the next couple days so I hope that you will think
about this slowly... :)
Hitoshi Kitada wrote:
snip
> Even for the outside of an LS, the time coordinate of the LS can be used to
> define the mesh (coordinate system) for seeing other LS's. Just metric looks
> different as the definition of length outside the LS is different from the
> inside as below.
If the LSs have independent clocks and lengths, would they not act, if
we could see only the Riemannian manifold X, as a clock and length
attached at each point of it. We would not have to assume that an object
moved around on X would have to carry a "persistent" memory of duration
and length scales with it. Does this make sense? This is very similar to
a point that Eddington makes in his explanation of Weyl's theory.
pg.168-170
"There is an arbitrary assumption in our geometry up to this point,
which it is desirable now to point out. We have based everything on the
"interval," which, it has been said, is something which all observers,
whatever their motion or whatever their mesh-system, can measure
absolutely, agreeing on the result. This assumes that they are provided
with identical standards of measurement - scales and clocks. But if A is
in motion relative to be and wishes to hand his standards to B to check
his measures, he must stop their motion; this means in practice that he
must bombard his standards with material molecules until they come to
rest. Is it fair to assume that no alteration of the standards is caused
by this process? Or if A measures time by the vibrations of a hydrogen
atom, and space by the wavelength of the vibration, still is it
necessary to stop the atom by a collision in which electrical forces are
involved.
The standard of length in physics is the length in the year 1799 of a
bar deposited at Paris. Obviously no interval is ever compared directly
with that length; there must be a continuous chain of intermediate steps
extending like a geodetic triangulation through space and time, first
along the past history of the scale actually used, then through
intermediate standards, and finally along with the history of the Paris
meter itself. It may be that these intermediate steps are of no
importance - that the same result is reached by whatever route we
approach the standard; but clearly we ought not to make the assumption
without due consideration. We ought to construct out geometry in such a
way as to show that there are intermediate steps, and that the
comparison of the interval with the ultimate standard is not a kind of
action at a distance.
To compare intervals in different directions at a point in space and
time [here I think of this as a point in X. SPK] does not require this
comparison with a distant standard. The physicist's method of describing
phenomena near a point P is to lay down some form for comparison (1) a
mesh-system, (2) a unit of length (some kind of material standard),
which can also be used for measuring time [with LSs we start with a way
of measuring time, but this is immaterial. SPK], the velocity of light
being unity. [this assumption of "the velocity of light being unity" I
find problematic! SPK] With this system of reference he can measure in
terms of his unit small intervals PP' running in any direction from P,
summarizing the results in the fundamental formula
ds^2 = g_11 dx_1^2 + g_22 dx_2^2 + ... + 2g_12 dx_1 dx_2 + ...
If now he wishes to measure intervals near a distant point Q, he must
lay down a mesh-system and a unit of measure there. He naturally tries
to simplify matters by using what he would call the *same* unit of
measure at P and Q, either by transporting a material rod or some
equivalent device. If it is immaterial by what route the unit is carried
from P to Q, and replicas of the unit carried by different routes all
agree on arrival at Q, this method is at any rate explicit. The question
whether the unit at Q defined in this way is *really* the same as that
at P is mere metaphysics. But if the units carried by different routes
disagree, there is no unambiguous means of identifying a unit at Q with
the unit at P. [The alteration of a rod by the transportation via
different routes is well illustrated by the twins paradox! rods and/or
clocks that are subject to de/accelerations along the way, will differ
when they are brought together again. SPK] ... If there is this
ambiguity the only possible course is to lay down (1) a mesh-system
filling all the space and time considered [within LS theory, we think of
this mesh-system as the co-ordinate system specific to that LS observer.
SPK] , (2) a definite unit of interval, or gauge, *at every point in
space and time*. [I think of this as if the observer LS fibers each
point in its co-ordinate system of reference with a unit clock and
length equal to its internal unit clock and length, thus different
observers can have differing fibrations if their unit clocks and lengths
differ. SPK] The geometry of the world referred to such a system will
complicated than that of Riemann hitherto used; and we shall see that it
is necessary to specify not only the 10 g's, but four other functions of
position, which will be found to have an important physical meaning.
The observer will naturally simplify things by making the units of
gauge at distant points as nearly as possible equal, judged by ordinary
comparisons. But the fact remains that, when the comparison depends on
the route taken, exact equality is not definable [perhaps we could use
"class equivalence" closure to denote the spatial synchronization of LS
frames of reference? SPK]; and we have therefore to admit that the
*exact* standards are laid down at every point independently.
It is the same problem over again as occurs in regard to mesh-systems.
We lay down particular rectangular axes near point P; presently we make
some observations near a distant point Q. To what co-ordinates shall the
latter be referred? The natural answer is that we must use the same
coordinates as we were using at P. But, except in the particular case of
flat space, there is no means of defining exactly what coordinates at Q
are the *same* as those at P. In many cases the ambiguity may be too
trifling to trouble us; but in exact work the only course is to to lay
down a definite mesh-system extending throughout space, the precise
route of the partitions being necessarily arbitrary. We now find that we
have to add to this by placing in each mesh a gauge whose precise length
is arbitrary. [we would generalize this by your explanation of
uncertainty! SPK] Having done this the next step is to make measurements
of intervals (using our gauges) This connects the absolute properties
[weakened by our understanding of uncertainty! SPK] of the world with
our arbitrarily drawn mesh-system and gauge-system. And so by
measurement we determine the g's and the new additional quantities,
which determine the geometry of our chosen system of reference, and at
the same time contain within themselves the absolute geometry of the [we
should not use this term "the", since there are many - an nonenumerable
amount of "worlds" possible given that they are defined by the possible
interactions of LSs! SPK] world - the kind of sp-ace-time which exists
in the field of our experiments."
> > Again I am thinking in external terms. How is the "speed c" defined for
> > a group of LSs communicating to each other? Do we assume an absolute
> > interval for all? How do we get around the need of a invariant interval
> > such as is assumed in GR? An LS defines a local clock, do they also
> > define a local "length"?
>
> Light speed c in vacuum and the time of an LS are used to measure the length
> of the path (outside the LS) through which the light passes from the begining
> to the end, under the situation that the path is stationary with respect to
> the LS. Thus c (speed of ligh in vacuum) is assumed as an absolute constant.
> The length of a path moving with respect to the LS is defined by relativistic
> change of coordinates (e.g. Lorentz transformation).
I don't think of this "stationary path" as a priori existing, it is
dependent on the decomposition of the LSs, among other things... I think
of it as a construction. But this is a difficult idea to discuss. :(
Most physicists just assume that a light cone structure exists a priori,
but this is not the case in Local Times theory. If we think of light
rays as the extremal geodesic curves expressing the causal connections
between events, which we might think the "centers of mass" are.
When an observer makes an observation, is it a selection process? Could
we think of the various possible geodesics as competing against each
other? I know that this idea is usually explained using the Feyman sum
of paths, but such is not a computable or constructable idea. I am only
trying to explore ideas here, I hope that we can hone in on the
mathematical details later... :)
> > > > 4) would massive particles follow such a Minkowskian structure if
> > > > gravity is very weak?
> > >
> > > I am not sure enough, but at least by my working hypothesis any particles do
> > > not follow Minkowskian structure if considered inside an LS.
> >
> > I am speaking here of the outside situation...
>
> Outside an LS, 4) is true.
My ideas of how this structure is constructed follows Eddington...
Later,
Stephen
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