Hi Ng
this is a reply to Tom Roberts on the subject of my critique of SRT. The
discussion started in thread about LIGO, but I wanted my text (which I
have written last night and which follows now) to be found more easily
than in the 'backyard' of an old thread about LIGO.
This text is actually the rewritten version from what I have written
earlier in German and covers now only a certain part (first part of §3)
of 'On the Electrodynamics of Moving Bodies' by A. Einstein from 1905,
translated into English in this version in 1923:
https://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf
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§ 3. Theory of the Transformation of Co-ordinates and Times from a
Stationary System to another System in Uniform Motion of Translation
Relatively to the Former
1) Let us in “stationary” space take two systems of co-ordinates, i.e.
two systems, each of three rigid material lines, perpendicular to one
another, and issuing from a point. Let the axes of X of the two systems
coincide, and their axes of Y and Z respectively be parallel. Let each
system be provided with a rigid measuring-rod and a number of clocks,
and let the two measuring-rods, and likewise all the clocks of the two
systems, be in all respects alike.
2) Now to the origin of one of the two systems (k) let a constant
velocity v be imparted in the direction of the increasing x of the other
stationary system (K), and let this velocity be communicated to the axes
of the co-ordinates, the relevant measuring-rod, and the clocks. To any
time of the stationary system K there then will correspond a definit
position of the axes of the moving system, and from reasons of symmetry
we are entitled to assume that the motion of k may be such that the axes
of the moving system are at the time t (this “t” always denotes a time
of the stationary system) parallel to the axes of the stationary system.
3) We now imagine space to be measured from the stationary system K by
means of the stationary measuring-rod, and also from the moving system k
by means of the measuring-rod moving with it; and that we thus obtain
the co-ordinates x, y, z, and ξ, η, ζ respectively.
4) Further, let the time t of the stationary system be determined for
all points thereof at which there are clocks by means of light signals
in the manner indicated in § 1; similarly let the time τ of the moving
system be determined for all points of the moving system at which there
are clocks at rest relatively to that system by applying the method,
given in § 1, of light signals between the points at which the latter
clocks are located.
5) To any system of values x, y, z, t, which completely defines the
place and time of an event in the stationary system, there belongs a
system of values ξ, η, ζ, τ, determining that event relatively to the
system k, and our task is now to find the system of equations connecting
these quantities.
6) In the first place it is clear that the equations must be linear on
account of the properties of homogeneity which we attribute to space and
time. If we place x’= x − vt, it is clear that a point at rest in the
system k must have a system of values x’, y, z, independent of time.
7) We first define τ as a function of x’, y, z, and t. To do this we have
to express in equations that τ is nothing else than the summary of the
data of clocks at rest in system k, which have been synchronized
according to the rule given in § 1.
8) From the origin of system k let a ray be emitted at the time τ0 along
the X-axis to x’, and at the time τ1 be reflected thence to the origin of
the coordinates, arriving there at the time τ2 ; we then must have
1/2(τ0 + τ2 ) = τ1 ,
or, by inserting the arguments of the function τ and applying the
principle of the constancy of the velocity of light in the stationary
system:
½[τ(0, 0, 0, t) + τ(0, 0, 0, t +x’/(c – v)+x’/(c + v)]= τ(x’, 0, 0, t
+x’/(c – v)).
remarks
Ad 1) lines are NOT material. The concept of one-dimensional lines is a
human brain child, since in nature we do not have anything material with
only one dimension.
Material objects are three-dimensional. A ‘rigid line’ is a product of
thought, hence not material.
Space cannot be measured by rigid rods. Only material objects in close
proximity and at rest in respect to the measuring device could be
measured that way.
The length of the ‘rigid rod’ is certainly not well defined, if we don’t
find other means than the rod itself.
Length of material objects is also subject to change by certain influences.
A frame of reference should consist of coordinates and a definition of
time, which is valid throughout all of that frame of reference (I call
that ‘f-o-r’ in the future).
Einstein used a different concept, where time is a function of the
location in a certain f-o-r.
This is possible, but very strange. I personally would reject this concept.
Ad 2) “and let this velocity be communicated to the axes of the
co-ordinates..” is quite funny.
I have no idea, what Einstein actually meant, but certainly he didn’t
want to communicate with lines.
Ad 3) You cannot measure space with rods.
So imagine you had a long rod and start to measure space. You could
eventually carry that around and hold it in all directions possible, but
that would be a measurement of anything.
Ad 4) This is the main conceptual error of SRT:
Light is quite fast, but has finite velocity. So it takes time for light
to pass from A to B. Now, where do you put that time of transition of
the signal?
Einstein used a method, where that transit time is not taken into
consideration, but the time ‘seen’ at remote clocks is taken for time at
that position.
I personally would reject this concept entirely, since the real time
required for light signals to travel would then vanish and apparent time
seen at remote clocks is taken as real instead.
That is conceptually wrong.
Ad 5) Seems to be ok.
Ad 6) I had a lot of trouble with “If we place x’= x – vt”.
The small Latin letters are used as coordinates in K. So all of them
belong to f-o-r K.
‘x’ seems to belong to a certain spot in K. But Einstein was writing
about a certain fixed spot in k. This spot has small Greek letters as
coordinates names.
The f-o-rs K and k are related in this way: xsi= x+ v*t.
So what relation does x’= x – vt mean?
I use the orientation of the axis, that higher values of x or xsi pint
to the right (as we usually do on paper). So that equation would
describe a movement to the left, while the entire f-o-r k would move to
the right with velocity v in respect to K.
The equation would in effect nullify the movement of k in respect to K,
since its own movement gets subtracted.
If ‘x’ is actually wrong and xsi is meant instead, then this would
cause, that moving f-o-r and non-moving f-o-r change their position,
since now K is moving to the left in respect to k.
That is certainly correct and actually a good idea, but does not relate
to the problem in consideration, since a point at rest in k is not
independent of time t (from K). It is independent of time tau in k and
has coordinates named with small Greek letters.
Measured in coordinates of K we had to add something for the movement of
k and that would be: xsi(t)= x+ vt, while eta and zeta are not affected.
Ad 7) I actually do not understand, what quantity is meant with x’.
So how is the correct interpretation of x’?
Ad 8) Since k is moving in respect to K, a ray emitted from the origin
of k and reflected in K would not have equal length to travel on the way
to the mirror and on the way back. Actually the way back is longer,
since the emitter moves in respect to the mirror ‘to the left’ (away).
If we flip the relation and say, that k is at rest and K moving, than
the length to travel for the signal is influenced by the movement, but
the time to travel back and forth would be equal.
(This apparent contradiction is quite interesting, but was not covered
in SRT.)
The equation
½[τ(0, 0, 0, t) + τ(0, 0, 0, t +x’/(c – v)+x’/(c + v)]= τ(x’, 0, 0, t
+x’/(c – v))
is in my opinion wrong, since tau is a linear function, that has four
vectors as arguments.
‘Linear function’ allows to multiply the argument with a constant
factor. So we could multiply the factor ½ inside the function. Then we
could multiply ½ to all of the components of the vector.
This would make all of the first three components zero. Another rule
for linear functions is
f(a) + f(b) = f(a+b).
So the right side of the equation could not have any non-zero value in
the first position of the vector.
Thomas Heger