[time 225] Re: [time 224] Re: How to define length using LSs

Stephen P. King (stephenk1@home.com)
Mon, 12 Apr 1999 10:50:13 -0400

Dear Hitoshi,

Hitoshi Kitada wrote:
> Dear Stephen,
> Just some notes on what I overlooked...
> > > > 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.

        Could we elaborate more on this thinking of the Lorentz transformation?
There is something strange about the geometry of these paths that I
don't have words for right now. :( I need to talk to my friend Paul
again, he helps me clear up my thinking. It has something to to with how
these "stationary" paths related to simultaneity planes...

> > 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).

        I agree! :) But, the "fixing" is not necessarily a "one step" process.
There is the possibility that more than one "way of observing" the
physical values is available. This is analogous to the choice of
languages that one can use to communicate an idea and/or the choice
between metric or English standards of measurement. It is in the
communication between observers that a "process of elimination" takes
place. I am wondering if the communication between LSs, as classical,
has any statistical correlation with the subLSs of those LS. I know that
they are "independent", but maybe there is some similarity in their
properties that would allow for correlation's... (BTW, is this
"independence" an "orthogonality" property?)
> 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.

        I think so! This is why I am looking into models of pattern recognition
as a heuristic guide to how to think about this. :)
> > 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."

        I am thinking here of the Feynman summation of the paths. Is the
"equivalence class" idea applicable here? The way that topology treats
curves between two points look like this... My thinking here stems from
the fact that the 'usual' treatment of trajectories in spacetime assumes
properties that are absolute and a priori, e.g. infinitesimal
differentiation and integration, and we know this to be false, there is
only an "approximation" to such! :)

> I will get the copies tomorrow...
> Best wishes,
> Hitoshi



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