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Re: A Geometrical Proof of the Non-invariance of the Spacetime Interval



John Mallinckrodt wrote:

Like Hugh, I don't understand the point of this thread and I
certainly don't see that it has anything to do with relativity, let
alone the invariance of the spacetime interval.

As I said very early on, I'm not referring to the invariance of the
_magnitude_ of the spacetime interval. I'm referring to the lines drawn
between the coordinates of the events in each frame. I claim that they
don't coincide in spacetime, therefore, they don't describe an invariant
entity (the spacetime four-vector between the events).

Except for a couple
of ineffective early posts, I have tried to stay out of the
discussion in an attempt not to prolong it. Neverthless, DR's most
recent questions seem to indicate a willingness to listen to the
responses. Accordingly, I throw caution to the wind and write ...

David Rutherford wrote:

In your opinion,

(1) Is it possible for observers in a reference frame to claim that the
spatial location of an event(events) is(are) fixed in that frame, at all
times? That is, can they always claim that it(they) occurred at a given
set of spatial coordinates, in that frame?

The first question suggests a fundamental misunderstanding of the
definition of the word "event" as it is used in physics. An event
takes place at a specific location and time. The description of that
location and time will be in terms of frame-dependent coordinates,
but, once assigned in any given frame, those coordinates simply ARE
the coordinates of the event in that frame. Thus, to wonder whether
the coordinates or the location of the event they represent are
"fixed at all times" is simply not to understand the meaning of the
word event. Events do not in any way "occur in" or "belong to"
reference frames. Thus, it is meaningless to ask "what happens" to
the coordinates of the event in some reference frame "as time passes."

Unless you have a way to measure the spatial distance between events at
different times, you have to find some way to determine the location of
the occurrence of the earlier event at the time of the later event, in
order to measure the distance between them, at that time. How do you do
that if asking what the coordinates of the earlier event are "as time
passes" is meaningless?

The awkwardness of the second question suggests the same fundamental
misunderstanding, but I suppose that it does have a simple answer:
Yes.

(2) Can two reference frames, in uniform motion with respect to each
other, both claim that the spatial locations of the events are fixed in
their frame, at all times.

See the answers to questions 1.

(3) Where and when do you measure the spatial distance between two
events that are separated by a time interval, in a frame? Is it an
instantaneous measurement?

If for some reason you want to know the spatial distance between the
locations at which two events occur, you use their spatial
coordinates and the Pythagorean theorem in the usual way.

I might want to find the components of the spacetime interval between
events, in the two frames, or I might want to see if the Lorentz
transformations give the results they claim to.

Clearly then, the spatial distance between two events is a frame-dependent
quantity.

I never claimed that it wasn't, in case you thought I did (except in the
Galilean case).

In general I wouldn't call this a "measurement" and it
certainly isn't in any meaningful sense "instantaneous." You can
perform the calculation at your leisure at any time.

Please tell me exactly how you would do the measurements of the distance
between breakfast and lunch in each frame. That is, where do you measure
the events from and at what time do you make the measurements?

(4) If you had breakfast in Los Angeles and lunch in San Diego, would
you really tell _anybody_ that you had breakfast and lunch in the
same place?

This seems to be a *practical* question so I will first give a
practical answer: I would most likely not say that I had breakfast
in the same place. This is because we have all adopted an *implicit*
convention of referring all measurements to observers who are
stationary with respect to the surface of the Earth. This
well-accepted, if usually unconscious convention leads us to say
otherwise utterly ridiculous things like, "I was going 80 mph!" with
full confidence that other people will understand what we MEAN.

On the other hand, as Hugh has suggested, if I simply stayed at a
dining table in the restaurant facility of a very large, uniformly
moving habitat of some kind, had breakfast as Los Angeles whipped by,
and had lunch as San Diego whipped by, I can more easily imagine
saying that I had both meals in the same place. Lower the drapes and
eliminate the engine rumble and it gets really easy.

Note that everything I have said above applies equally well to both
classical and special relativistic physics.

Please see my answer to Hugh regarding this question.

--
Dave Rutherford
"New Transformation Equations and the Electric Field Four-vector"
http://www.softcom.net/users/der555/newtransform.pdf

Applications:
"4/3 Problem Resolution"
http://www.softcom.net/users/der555/elecmass.pdf
"Action-reaction Paradox Resolution"
http://www.softcom.net/users/der555/actreact.pdf
"Energy Density Correction"
http://www.softcom.net/users/der555/enerdens.pdf
"Proposed Quantum Mechanical Connection"
http://www.softcom.net/users/der555/quantum.pdf