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



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. 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."

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

As an aside, I might mention that there is one important type of
measurement--finding the length of an object in a given reference
frame--that is meaningfully "instantaneous" and does involve finding
the spatial distance between two events. The events, however, are
necessarily NOT separated by a time interval in this case. One event
is described by the spatial coordinates of the "front" of the object
at some time t along with the time t and the other event is described
by the spatial coordinates of the "rear" of the object at the SAME
time t along with the time t.

(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.

--
A. John Mallinckrodt http://www.csupomona.edu/~ajm
Professor of Physics mailto:ajm@csupomona.edu
Physics Department voice:909-869-4054
Cal Poly Pomona fax:909-869-5090
Pomona, CA 91768-4031 office:Building 8, Room 223