The following text is from Appendix I in Einstein's 1920 "Relativity":

_____________________________
A light-signal, which is proceeding along the positive axis of x, is
transmitted according to the equation
x = ct
or
x - ct = 0...........(1)

Since the same light-signal has to be transmitted relative to k' with
the velocity c, the propagation relative to the system k' will be
represented by the analogous formula

x' - ct' = 0...................................(2)

Those space-time points (events) which satisfy (1) must also satisfy
(2). Obviously this will be the case when the relation

(x' - c't) = L(x - ct)........................(3)

is fulfilled in general, where L indicates a constant; for, according
to (3), the disappearance of (x - ct) involves the disappearance of
(x' - ct').
__________________________________________________ ______



Here Einstein obtains (3) from (1) and (2) which can be presented as

(1)and(2) -- (3)..............................(A)

In the text however he says the opposite: WHEN (3), THEN (1)--(2),
where -- is the sign of the biconditional (-- means that when (1),
then (2) and vice versa). This can be expressed as

(3) -- [(1)--(2)]...........................(B)

From a mathematical point of view, (B) is correct and (A) is not, so
Einstein is correct in stating (B) in the text. However (B) is useless
- the purpose of Einstein is to OBTAIN (3) and not to advance (3) as a
premise. So he is forced to carry out the wrong procedure (A) leading
to (3), but to "support" it in the accompanying text by stating,
essentially, the correct inference (B).

Pentcho Valev

"Pentcho Valev" wrote in message
om...
The following text is from Appendix I in Einstein's 1920 "Relativity":

_____________________________
A light-signal, which is proceeding along the positive axis of x, is
transmitted according to the equation
x = ct
or
x - ct = 0...........(1)

Since the same light-signal has to be transmitted relative to k' with
the velocity c, the propagation relative to the system k' will be
represented by the analogous formula

x' - ct' = 0...................................(2)

Those space-time points (events) which satisfy (1) must also satisfy
(2). Obviously this will be the case when the relation

(x' - c't) = L(x - ct)........................(3)

is fulfilled in general, where L indicates a constant; for, according
to (3), the disappearance of (x - ct) involves the disappearance of
(x' - ct').
__________________________________________________ ______



Here Einstein obtains (3) from (1) and (2) which can be presented as

(1)and(2) -- (3)..............................(A)

In the text however he says the opposite: WHEN (3), THEN (1)--(2),
where -- is the sign of the biconditional (-- means that when (1),
then (2) and vice versa). This can be expressed as

(3) -- [(1)--(2)]...........................(B)

From a mathematical point of view, (B) is correct and (A) is not, so
Einstein is correct in stating (B) in the text.

From a mathematical point of view Einstein is making assumptions, notably
linearity. Such popularizations are not meant for the mathematically
sophisticated. If however you are not mathematically sophisticated (ie
having studied university level mathematics) then I suggest you not really
in a position to judge.

However (B) is useless
- the purpose of Einstein is to OBTAIN (3) and not to advance (3) as a
premise. So he is forced to carry out the wrong procedure (A) leading
to (3), but to "support" it in the accompanying text by stating,
essentially, the correct inference (B).


Look are you interested in understanding relativity or in sorting out
misconceptions in Einstein's popular writings? The two goals are not the
same. Einstein is not the last word on relativity. I strongly advice you
to examine modern writings on the subject not pitched at a popular audience
such as the link I gave previously (http://arxiv.org/abs/physics/0110076).

Come on - what is your agenda? People around here will help you if you want
to understand relativity - they will not help you if all you want to do is
relativity bash by examining Einstein's popularizations.

Thanks
Bill

"Bill Hobba" wrote in message ...

"Pentcho Valev" wrote in message
om...

[snip]

From a mathematical point of view, (B) is correct and (A) is not, so
Einstein is correct in stating (B) in the text.

From a mathematical point of view Einstein is making assumptions, notably
linearity. Such popularizations are not meant for the mathematically
sophisticated. If however you are not mathematically sophisticated (ie
having studied university level mathematics) then I suggest you not really
in a position to judge.

F.w.i.w.
net-ops.be
and in the same thread I liked Alfred's contribution:
http://groups.google.com/groups?&thr... ng.google.com

Dirk Vdm


However (B) is useless
- the purpose of Einstein is to OBTAIN (3) and not to advance (3) as a
premise. So he is forced to carry out the wrong procedure (A) leading
to (3), but to "support" it in the accompanying text by stating,
essentially, the correct inference (B).


Look are you interested in understanding relativity or in sorting out
misconceptions in Einstein's popular writings? The two goals are not the
same. Einstein is not the last word on relativity. I strongly advice you
to examine modern writings on the subject not pitched at a popular audience
such as the link I gave previously (http://arxiv.org/abs/physics/0110076).

Come on - what is your agenda? People around here will help you if you want
to understand relativity - they will not help you if all you want to do is
relativity bash by examining Einstein's popularizations.

Thanks
Bill

"Dirk Van de moortel" wrote
in message ...

"Bill Hobba" wrote in message
...

"Pentcho Valev" wrote in message
om...

[snip]

From a mathematical point of view, (B) is correct and (A) is not, so
Einstein is correct in stating (B) in the text.

From a mathematical point of view Einstein is making assumptions,
notably
linearity. Such popularizations are not meant for the mathematically
sophisticated. If however you are not mathematically sophisticated (ie
having studied university level mathematics) then I suggest you not
really
in a position to judge.

F.w.i.w.

net-ops.be
and in the same thread I liked Alfred's contribution:

http://groups.google.com/groups?&thr... ng.google.com

Thanks for the links Dirk. Always interesting to see how others approach
these things. My take was this:

Since x - ct and x' - ct' is only zero when considering event on a light ray
then we can legitimately write x - ct/x' - ct' and not have divide by zero
problems. Now x' and t' can be considered functions of x and t thus x -
ct/x' - ct' = L(x, t). Now when x - ct becomes zero so does x' - ct' which
mans we have 0/0 which is undefined but most importantly is not the dreaded
infinity you can get when dividing by zero. Thus we can define it as the
limit of L(x,t). Thus it is defined for all x, t. Hence we have x' - ct' =
L(x,t) (x - ct) (1) Similarly we can do the same for x + ct to get x' + ct'
= U(x,t)(x + ct) (2) So far all ok. The thing is of course Einstein left
out these steps (no problem because it is not too hard to see) but then goes
no to say L (x, t) must be a constant. Now this of course is true - but is
in fact logically equivalent to saying that the transformations connecting
x', t' and x, t are linear (as you correctly point out in you link). Just
to recap the detail - from (1) and (2) we have

x' = (L + U)/2 x - (L -U)/2 ct and t' = -(L - U)/2c x + (L + V)/2 t

Now since the transformation is assumed linear we can solve for L and U in
terms of the linear constants of the transformation thus justifying the fact
that L must itself be a constant. To me this is the important, true, but
crucial step Einstein did not make explicit - hence my comment about
linearity. This does not make Einstein wrong or anything like that - simply
something a more mathematically sophisticated reader would spot - and should
do further reading to see what is happening or if he was smart enough figure
out himself (I was not - when I read that passage, oh so many years ago now,
I scratched my head over it and just said - ok my math is not up to it at
this stage but I have no doubt it is true).

You mention in the link above that it must be linear for all sorts of
reasons. One obvious one is that since free particles move with constant
velocity then they trace out straight world lines in both frames and so for
those familiar with linear algebra it must be linear. But as people who
read my stuff know I am symmetry guy and prefer symmetry arguments. Now if
we shift the 0 on the x axis or change the time of the clocks by a constant
then free particles still move at constant velocity so we still have an
inertial frame. Now the POR says the law of physics are the same in all
inertial frames - thus they are the same if we shift the 0 on the x axis or
change the zero on the clocks. This is the definition of homogeneous in
space and homogeneous in time. Now from the calculus we know that for a
small increment in x, delta x and a small increment in t delta t (delta x',
delta t') = A(x,t) (delta x, delta t). But homogeneity implies the 2x2
matrix A must not depend on x and t (or you will violate that the laws of
physics must be the same regardless of x and t). Hence (delta x', delta t')
= A (delta x, delta t). Thus if we break (x, t) into the sum of a large
number of small changes in x and t we have (x',t') = A (x,t) + C or if we
ensure the origins coincide and the clocks are both zero there (x',t') = A
(x,t) i.e. is linear.

Again I want to make it clear to the original poster that what he is quoting
from is a popularization written by Einstein. Such by their very nature
leave out detail and it is not difficult to find problems with it. They can
be rectified, but a better understanding of what is going on can be obtained
from other sources such as the links I gave to a proper derivation of the
Lorentz transformation that clearly sates all assumptions being made and the
rigorous mathematical development of those assumptions. All the original
poster seems to want to is to try and prove Einstein wrong by a superficial
analysis of what, in a sense (since it was meant as a popularization) is a
superficial analysis. Such a course is not the way of genuine intellectual
debate. It is the way of those caught up in the 'Unskilled and Unaware of
It: How Difficulties in Recognizing One's Own Incompetence Lead to Inflated
Self-Assessments' (see http://www.apa.org/journals/psp/psp7761121.html)
phenomena that you have provided a link to on your site. What they do is
when they come across something they do not understand instead of saying
what I did all those years ago (I do not see it at this stage but I have no
doubt it is true) they say - I am smarter than Einstein and all these
supposed smart physics dudes - I can see a problem they do not. It is a
clearly a problem they have and makes understanding relativity even more
difficult because at first sight it is counter intuitive. It is only after
some acquaintance with it and an 'internalization' of its fundamental
tenants you understand why it must be so.

Thanks again for the links
Bill


Dirk Vdm


However (B) is useless
- the purpose of Einstein is to OBTAIN (3) and not to advance (3) as a
premise. So he is forced to carry out the wrong procedure (A) leading
to (3), but to "support" it in the accompanying text by stating,
essentially, the correct inference (B).


Look are you interested in understanding relativity or in sorting out
misconceptions in Einstein's popular writings? The two goals are not
the
same. Einstein is not the last word on relativity. I strongly advice
you
to examine modern writings on the subject not pitched at a popular
audience
such as the link I gave previously
(http://arxiv.org/abs/physics/0110076).

Come on - what is your agenda? People around here will help you if you
want
to understand relativity - they will not help you if all you want to do
is
relativity bash by examining Einstein's popularizations.

Thanks
Bill

"Bill Hobba" wrote in message ...

"Dirk Van de moortel" wrote
in message ...

"Bill Hobba" wrote in message
...

"Pentcho Valev" wrote in message
om...

[snip]

From a mathematical point of view, (B) is correct and (A) is not, so
Einstein is correct in stating (B) in the text.

From a mathematical point of view Einstein is making assumptions,
notably
linearity. Such popularizations are not meant for the mathematically
sophisticated. If however you are not mathematically sophisticated (ie
having studied university level mathematics) then I suggest you not
really
in a position to judge.

F.w.i.w.

net-ops.be
and in the same thread I liked Alfred's contribution:

http://groups.google.com/groups?&thr... ng.google.com

Thanks for the links Dirk. Always interesting to see how others approach
these things. My take was this:

Since x - ct and x' - ct' is only zero when considering event on a light ray
then we can legitimately write x - ct/x' - ct'

Just a notation nitpick: you better write it as
(x - ct) / (x' - ct')
[sorry, hope you don't mind]

and not have divide by zero
problems. Now x' and t' can be considered functions of x and t thus x -
ct/x' - ct' = L(x, t). Now when x - ct becomes zero so does x' - ct' which
mans we have 0/0 which is undefined but most importantly is not the dreaded
infinity you can get when dividing by zero. Thus we can define it as the
limit of L(x,t). Thus it is defined for all x, t. Hence we have x' - ct' =
L(x,t) (x - ct) (1) Similarly we can do the same for x + ct to get x' + ct'
= U(x,t)(x + ct) (2) So far all ok. The thing is of course Einstein left
out these steps (no problem because it is not too hard to see) but then goes
no to say L (x, t) must be a constant. Now this of course is true - but is
in fact logically equivalent to saying that the transformations connecting
x', t' and x, t are linear (as you correctly point out in you link). Just
to recap the detail - from (1) and (2) we have

x' = (L + U)/2 x - (L -U)/2 ct and t' = -(L - U)/2c x + (L + V)/2 t

Now since the transformation is assumed linear we can solve for L and U in
terms of the linear constants of the transformation thus justifying the fact
that L must itself be a constant. To me this is the important, true, but
crucial step Einstein did not make explicit - hence my comment about
linearity. This does not make Einstein wrong or anything like that - simply
something a more mathematically sophisticated reader would spot - and should
do further reading to see what is happening or if he was smart enough figure
out himself (I was not - when I read that passage, oh so many years ago now,
I scratched my head over it and just said - ok my math is not up to it at
this stage but I have no doubt it is true).

Yes, I have scratched mine pretty often as well :-)
Those are i.m.o. in fact the best and most productive moments.



You mention in the link above that it must be linear for all sorts of
reasons. One obvious one is that since free particles move with constant
velocity then they trace out straight world lines in both frames and so for
those familiar with linear algebra it must be linear. But as people who
read my stuff know I am symmetry guy and prefer symmetry arguments. Now if
we shift the 0 on the x axis or change the time of the clocks by a constant
then free particles still move at constant velocity so we still have an
inertial frame. Now the POR says the law of physics are the same in all
inertial frames - thus they are the same if we shift the 0 on the x axis or
change the zero on the clocks. This is the definition of homogeneous in
space and homogeneous in time. Now from the calculus we know that for a
small increment in x, delta x and a small increment in t delta t (delta x',
delta t') = A(x,t) (delta x, delta t). But homogeneity implies the 2x2
matrix A must not depend on x and t (or you will violate that the laws of
physics must be the same regardless of x and t). Hence (delta x', delta t')
= A (delta x, delta t). Thus if we break (x, t) into the sum of a large
number of small changes in x and t we have (x',t') = A (x,t) + C or if we
ensure the origins coincide and the clocks are both zero there (x',t') = A
(x,t) i.e. is linear.

That is a nice intuitive way of looking at it as well...


Again I want to make it clear to the original poster that what he is quoting
from is a popularization written by Einstein. Such by their very nature
leave out detail and it is not difficult to find problems with it. They can
be rectified, but a better understanding of what is going on can be obtained
from other sources such as the links I gave to a proper derivation of the
Lorentz transformation that clearly sates all assumptions being made and the
rigorous mathematical development of those assumptions. All the original
poster seems to want to is to try and prove Einstein wrong by a superficial
analysis of what, in a sense (since it was meant as a popularization) is a
superficial analysis. Such a course is not the way of genuine intellectual
debate. It is the way of those caught up in the 'Unskilled and Unaware of
It: How Difficulties in Recognizing One's Own Incompetence Lead to Inflated
Self-Assessments' (see http://www.apa.org/journals/psp/psp7761121.html)
phenomena that you have provided a link to on your site. What they do is
when they come across something they do not understand instead of saying
what I did all those years ago (I do not see it at this stage but I have no
doubt it is true) they say - I am smarter than Einstein and all these
supposed smart physics dudes - I can see a problem they do not. It is a
clearly a problem they have and makes understanding relativity even more
difficult because at first sight it is counter intuitive. It is only after
some acquaintance with it and an 'internalization' of its fundamental
tenants you understand why it must be so.

Thanks again for the links
Bill

No problem - say thanks too google :-)
Cheers,
Dirk Vdm