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A Potential Problem with the Lorentz transformations



I think I may have found a problem with the way the spacetime interval
is measured in special relativity (SR). The accepted way to measure the
spacetime interval between two events, in SR, is to measure the distance
and time intervals between each event and the position of the origin of
each frame at the time of each event. My contention is that this is not
a valid method for measuring the components of the spacetime interval
between events (except for the trivial case v = 0), since vectors drawn
between the events in each frame would not coincide _in_spacetime_
(that's not to say they must coincide in each frame - in general, they
won't).

In order to clarify the difference between coinciding in spacetime and
coinciding in each frame, consider the two dimensional case of an
orthogonal rotation of two dimensional reference frames _in_space_
about a common origin. A vector between the origin and a point in space
will, in general, have different components in each frame, therefore,
the vectors will not _appear_ to coincide from the viewpoint of each
frame, but will coincide in space.

Now consider an inertial frame F' in uniform relative motion with speed
v > 0 in the x-direction relative to frame F. The first event E1,
occurs at the origins of F and F' when the origins coincide, thus, at
E1, x = x' = 0, y = y' = 0, z = z' = 0, t = t' = 0. Say the second
event E2 occurs at x = x, y = y, z = z and t = t in the unprimed frame
F. In SR, The Lorentz transformation equations from F to F' give
the coordinates of E2 in F' as

x' = gamma(x - vt)
y' = y
z' = z
t' = gamma(t - xv/c^2)

where gamma = 1/sqrt(1 - v^2/c^2). In this case, it is claimed that x',
y', z' and t' are also the components of the spacetime interval between
E1 and E2. I claim that they are _only_ the coordinates of E2 and _not_
the components of the spacetime interval between the two events, since
the measurements of the coordinates of E2 in F' are made from the
position of the origin of F' at t' = t', not the position of the origin
of F' at t' = 0. The heads of the two vectors between the events in the
two frames coincide in spacetime (notice that I didn't say they coincide
in each frame - in general, they don't), but the tails do not.
Therefore, the spacetime interval is described by two separate vectors
_in_spacetime_, not one. In my opinion, this makes the Lorentz
transformation equations geometrically incapable of describing the
components of the spacetime interval between E1 and E2.

Observers in the two frames must agree on the position and time of the
first event, since their clocks and meter sticks were 'synchronized'
at that event. That 'point' in spacetime is the same for both frames
and they must always agree that it will be common to both frames, in
order for the tails of the vectors to coincide in spacetime.

All other events must be measured from that common 'point'. In
non-moving frames, it's okay to use the origins of each frame, at later
times, as a reference point for events, but for moving frames, it
isn't. For moving frames, if the origins coincide at the first event,
they won't at later events. In order to be consistent, geometrically,
the second event must be measured from the same place in both frames,
that is, a place that observers in both frames agree on - the origin of
the unprimed frame in this case. They do not need to agree on the
locations and times of later events, for the vectors to coincide in
spacetime, since their measurement devices are no longer necessarily
synchronized.

Therefore, I say that the space and time coordinates of event E2 in F'
should be measured from the position of the origin of F' at t' = 0
(that is, the position of the origin of F at t' = t'), not from the
position of the origin of F' at t' = t'.

An analogy might be made using the Galilean transformations

x' = x - vt
y' = y
z' = z
t' = t

The measurement of x' in the primed frame is made from the position of
the origin of the primed frame at t' = t', not at t' = 0 (actually, the
measurement of t' is from the 'position' of the origin in time at t' =
0, but the measurements of x', y' and z' are made from the position of
the
origin in space at t' = t'). Therefore, vectors drawn between the
origins of the two frames and the points P=(x,y,z,t) and
P'=(x',y',z',t') do not coincide in space. The heads of the vectors
coincide in space (but they don't coincide in each frame), however,
the tails of the vectors do _not_ coincide in space.

I would like to hear your comments about the geometrical validity of
this argument. Thanks.

--
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