Hitoshi Kitada (hitoshi@kitada.com)
Mon, 12 Apr 1999 21:55:27 +0900
Dear Stephen,
Just some notes on what I overlooked...
----- Original Message -----
From: Stephen P. King <stephenk1@home.com>
Cc: <time@kitada.com>
Sent: Monday, April 12, 1999 10:24 AM
Subject: [time 223] Re: [time 218] Re: How to define length using LSs
> 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...
Since the path is "stationary" with respect to the observer's LS, the length
of the path is the same as that measured by the internal scale of the LS. My
definition above thus is concerned with the paths moving relative to the LS.
This transformation of length was used in time_IV.tex, I.3.2 (page 19, where
Lorentz transformation suffices) to measure the length between two centers of
mass of sublocal systems of the observed LS.
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?
Observation may be a kind of pattern recognition. Once the observer fixes the
pattern of what he is observing, he would begin the measurement of physical
values, based on the patterns that are recognized as points (i.e. classical
particles).
Maybe the reason why we "see" the universe as what the present physics
describes as expanding would be a consequence of the property of humans'
pattern recognition.
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,
I am not sure what relation is between the "various possible geodesics" and
"Feynman integral."
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
>
I will get the copies tomorrow...
Best wishes,
Hitoshi
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