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From: harald....@epfl.ch (Harry)
Newsgroups: sci.physics.relativity
Subject: "length contraction" according to Lorentz and Poincare - An example
Date: 19 Dec 2004 11:24:17 -0800
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In a recent posting I saw the claim:
"Under SR, A thinks that B's ruler is shorter. B thinks that A's
ruler is shorter.
Under LET, A thinks that B's ruler is shorter. B thinks that A's
ruler is *longer*."
"LET" referred to the mathematical physics as found in Lorentz' 1904
paper, and "SR" referred to the mathematical physics of as found in
Einstein's 1905 paper.
I don't think that that is correct; in any case, Lorentz obviously
didn't feel that he should be stuck to some little errors and
incomplete understanding that he had when writing a certain paper, and
neither did Einstein. After 1905 Lorentz taught relativity and he did
not propose a separate "LET" as there is no difference as far as
mathematical or experimental physics is concerned.
That reciprocity of observations directly follows from Lorentz'length
contraction and clock slowdown is not immediately obvious.
Recently I wrote for another newsgroup an example of the
Lorentz-Poincare interpretation of "length contraction" and "time
dilation" without making use of the Lorentz transformations.
I copied it below as it may help some with intuitive understanding
("feeling").
_____________________________________________________________________________
EXAMPLE OF APPARENT MUTUAL LENGTH CONTRACTION:
A MOVING RULER THAT PASSES A STATIONARY ROD
Text books show that the inverse Lorentz transformation follows from
the Lorentz transformation. However, abstract mathematical evidence
does not fully satisfy everyone, and the philosophical interpretations
are open for debate. It may appear impossible that with a shortened
ruler one could measure other objects to be shortened.
It is helpful to understand the possible mechanism of the phenomena. I
will try to transmit here the understanding thanks to the explanation
according to Lorentz, by only using his "length contraction"
(contraction of objects) and "local time" (clock slowdown), in
combination with Poincare's calibration procedure.
It should be kept in mind that Lorentz combined the "absolute space"
concept of Newton's physics (used for observer-independent inertia)
with the "ether" concept of Maxwell (used as a carrier for waves);
what follows should not be confused with Einstein's interpretation of
relativity or the Einstein-Minkowski's Space-Time concept of
relativity.
THE EXAMPLE:
If a steadily moving system has two clocks, each at one end of a
ruler, and the velocity relative to the stationary ether is v, then,
according to Lorentz-Poincare's theory,
its length L = L0 / gamma,
with gamma = 1/(SQRT(1-v^2/c^2), L the length of the ruler in
movement, L0 its length in rest. Clock rates are expected to reduce by
the same factor.
We will assume that this can be measured with a system of coordinates
that is itself resting in the ether. (In practice only effects on
clocks have been directly measured, although MMX was explained with
length contraction.) Only in the ether frame is light speed truly
isotropic, i.e. c in all directions.
For simplicity, for this example we will use a system of reference for
defining durations and lengths that is resting in the ether or
"absolute space", so that v is the "absolute speed" (= relative to
"absolute space"). we'll also only consider one dimension. Thus by
definition only in this frame will duration and length measurements
provide the true values.
Let's take v=0.8c, so that gamma = 5/3. Thus according to the theory,
the ruler will be contracted to 3/5 of its rest length: for this
example we'll assume a ruler that is 6 m in rest, so it will be 3.6 m
long. Similarly, the clocks will tick at 3/5 of the rate in rest,
making one tick 0.6 s of duration. Because this is determined relative
to the stationary ether or absolute space, the effects correspond to
physical reality.
Now, what length will be measured with the moving system, when it
passes a 6 m long rod that is in rest?
According to the theory, the measured relative speed is the same: the
velocity v'= -v in standard notation, or, what I prefer for
mathematical symmetry: defining approaching speeds as positive, v'=v.
(Primed symbols are the values as determined with the moving reference
system.)
That the perceived speeds are equal can be demonstrated as follows.
V'=V:
If the clocks have been synchronized with light or radio pulses, the
synchronization proceeded for example as follows, with a signal sent
from the middle towards the clocks:
Light signal to both clocks: assumed by convention to take 3[m]/c[m/s]
or 3/3E8 = 10 ns.
Note the convention: those observers, ignorant of the speed of their
system, use as working hypothesis that light speed is isotropically c
relative to their chosen reference system.
However, in reality and as noticed from the rest system,
Light signal to front clock:
1.8 [m] / {(1-0.8)*c} [m/s] = 9/c s
Light signal to rear clock:
1.8 [m] / {(1+0.8)*c} [m/s] = 1/c s
------ +
Synchronization error : 8/c s = 26.7 ns
Thus the number of ticks that C2 is ahead of C1 is:
t2'_offset = 3/5 * 8/c = 4.8/c ticks = +16E-9 ticks.
If we define t'=0 at the moment that clock C1 meets the rear end of
the rod, that will be indisputably so for C1 : t1'=0.
C2 C1
ruler ---------------------- --> v
rod______________________________________
However, from the system in rest it will be noted that at that moment
C2 already indicates 16 ns.
C2 will meet the front of the rod when the 3.6 m ruler has reached it,
moving at 0.8c, thus after 3.6/(0.8*c) = 4.5/c s. C2, ticking slow,
will then have advanced by:
delta_t'= 3/5 * 4.5/c = 2.7/c ticks or 9E-9 ticks.
Thus in total:
t2'= t2'_offset + delta_t' = 4.8/c + 2.7/c = 7.5/c ticks or 25E-9
tick.
Therefore, according to speed determination in the moving system, its
apparently 6 m long ruler was passed in 25 apparent ns, or at 0.8c, so
that the speed measurement with the erroneous instruments still
provides the correct result.
THE LENGTH MEASUREMENT:
Now we are ready to determine the length of the stationary rod
according to the moving system:
1. If the length is determined with a clock, based on the measured
speed, obviously the slower tick rate causes the same underestimation
of the rod's length: it counts only 3/5*25 = 15 nanoticks for passing
the rod, so that the rod's length seems to be:
0.8c [m'/s'] * 15E-9 [s'] = 3.6 m'.
2. More complicated is it if we compare lengths. For that we need to
determine at what point of the ruler t'=0 when the rod's left end
passes it.
C2 C1
-------------------- --> v
rod______________________________________
We established that t2'=0 at a time 8/c= 26.7 ns earlier, thus when
the clock is 8/c * 0.8c = 6.4 m to the left. Funny enough, C1 has then
not yet met up with the rod, but the time as determined from C1 is
then -26.7 ns so that isn't relevant. What matters is that the front
end of the rod has not met with C2 when the clock time is 0, so that
the rod seems shorter than the ruler. In principle the mechanism is
thus explained.
For a precise determination one needs to either send a light signal
from C2 to the rod and back at t2=0, or (for the calculation less
complicated!) one needs to have clocks everywhere and then the clock
that reaches t'=0 when the front end of the rod passes it will
establish the apparent length.
In theory that should be at point: x'= 3/5*6 [m] = 3.6 marks (a mark
is an apparent meter). The clock at that point would have an offset as
follows: 3/5* 16E-9 ticks = +9.6 nanoticks.
That point is 3/5*x'= 2.16 m to the left of C1. Thus the ruler has
then 6-2.16 m= 3.84 m to go before C1 meets up, or 3.84/(0.8c)= 16 ns
before t=0. That corresponds to -9.6 nanoticks. As a result, the clock
at that point will then indeed indicate 0 - as it should be.
In the general case that both reference systems are in motion, all
measured lengths and durations will be distorted and not be conform
reality.
The Lorentz transformations (Poincare, 1905) follow directly from the
discussed model and were also used by Lorentz.
Harald van Lintel, Dec. 2004.