Re: Can the Special Theory of Relativity be "explained"

On Jun 25, 9:50 am, andy everett <vze2q...@xxxxxxxxxxx> wrote:
Is there a possible visualization of our Universe from which Laws of
Nature such as Einstein's Special Theory of Relativity might be implied.

Thank you for any thoughts.

Yes. There are several derivations of the Lorentz transformation,
based on the following assumptions.
Consider two "inertial" frames of reference, with frame F' moving at
velocity v wrt F, along their common x-axes. Assume
(1) The quantities (x',y',z',t') in frame F and (x,y,z,t) in frame F
are related linearly.
(2) y' = y and z' = z (i.e., rulers perpendicular to the motion remain
unchanged in length; this is almost required by the fact that a moving
ruler and a stationary ruler can everywhere coincide at an instant if
they are oriented perpendicular to their motion.)
(3) There is no preferred direction.
(4)IIf clocks beat faster or slower in F' than in F when F' moves to
the right at speed v, the same rate of beating applies to clocks in F"
moving to the left with speed v.
(5) If F' moves to the right wrt to F at speed v, there exists an
intermediate frame F0 that sees F' moving to the right with some speed
v0 and sees F moving to the left at speed v0.
(6) If F1, F2, F3, ..., Fn are successive frames that move at speed v
to the right wrt each other (F1 moves with velocity v as seen by F, F2
moves with velocity v as seen by F1, F3 moves with velocity v as seen
by F2, etc.) then Fn moves to the right as seen from F, for any
integer n >= 1 and any v > 0.

Various books and papers show the following (as a THEOREM, not a hand-
waving conjecture):
There exists a *universal* constant k >= 0 (k has dimensions 1/
velocity^2) such that x' = g*(x - v*t), y' = y, z' = z and t' = g(t-
k*v*x), where g = 1/sqrt(1 - k*v^2). Note: universality of k means
that it is the same for all pairs of inertial frames. Assumptions (1)-
(5) lead to the transformation as above; assumption (6) prevents us
from having k < 0. Note that either k = 0 (0 is a universal constant),
giving x' = x - v*t, t' = t (Galilean) or else k > 0. In the latter
case we can write k = 1/c^2, where c > 0 is some universal constant
having dimensions of velocity. This case gives the Lorentz
transformation.

Note: nowhere have we mentioned light, speed of light, or anything
like it. At this point we can say that since Galilean transformations
seem good for small speeds at least, either Galilean is exact or else
c must be large. Whether or not Galilean is exact or the more general
for is required is, presumably a matter of experiment and observation.
In the Lorentz case, whether there is any physical object that moves
at speed c is another matter, presumably again a matter of experiment.
Note, though, that that in the Lorentz case speed c is invariant
between inertial frames, so if there is something having speed c in
one inertial frame, it has speed c in all inertial frames. Also, c
would be a "speed limit". Experiment suggests that there is deviation
from Galilean and that there is, indeed, such a thing that moves at
speed c in physical reality, namely, light. However, even if that were
not true---even if light speed varied from one frame to another---
special relativity could /still be true/, but the constant "c" in the
Lorentz transformation would just be some "parameter" that is at least
as large as the largest observed speed of light or anything else
known.

Some of the arguments showing the above can be found in the book
"Essential Relativity", by Wolfgang Rindler, Springer-Verlag (1977). I
think there are also freely downloadable pdf files dealing with this
issue, but I don't recall the references.

R.G. Vickson
.

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