Derivation of Lorentz Transformation
Thomas Smid
Einstein's derivation of the Lorentz transformation (as given in his
book 'Relativity: The Special and General Theory').
It claims to be elegant and simple but appears to be mathematically
inconsistent to me (as far as my understanding of maths goes anyway).
It is only necessary here to examine the initial equations for this,
which describe the 'equations of motion of a light signal' in the
unprimed and primed reference frames, i.e.
(1) x-ct=0
(2) x'-ct'=0
where c is the speed of light (which obviously has to be a constant >0)
In the same way, for the propagation of a signal in the opposite
direction
(3) x+ct=0
(4) x'+ct'=0
>From equations (1)-(4) (my numbering), the Lorentz transformation is
then derived by some algebraic manipulations in the reference.
However, if one subtracts equation (1) from (3) above, one obtains
(5) 2ct=0
which means that for any time t>0
(6) c=0,
in contradiction to the requirement that c>0.
The fundamental equations on which the derivation is based are
therefore mathematically inconsistent.
Any comments on this?
Thomas
YBM
> I came across a webpage http://www.bartleby.com/173/a1.html showing
> Einstein's derivation of the Lorentz transformation (as given in his
> book 'Relativity: The Special and General Theory').
>
> It claims to be elegant and simple but appears to be mathematically
> inconsistent to me (as far as my understanding of maths goes anyway).
>
> It is only necessary here to examine the initial equations for this,
> which describe the 'equations of motion of a light signal' in the
> unprimed and primed reference frames, i.e.
>
> (1) x-ct=0
> (2) x'-ct'=0
> where c is the speed of light (which obviously has to be a constant >0)
what are x and x' here ?
> In the same way, for the propagation of a signal in the opposite
> direction
> (3) x+ct=0
> (4) x'+ct'=0
and here ?
> However, if one subtracts equation (1) from (3) above, one obtains
> (5) 2ct=0
if you take two equation out of the context of the meaning of
the meaning of the variables, could you combine them ?
if a dog run at speed v to my right, the equation of movement is :
1. x=vt
if another dog run at speed v to my left, the equation of movement is :
2. x=-vt
if one subtracts (1) from (2), one obtains :
0=2vt => v=0
dogs cannot run, not even move.
any comment on this ?
Harry
"Thomas Smid" <thoma...@gmail.com> wrote in message
news:1125436553....@o13g2000cwo.googlegroups.com...
Interesting! IMO, you showed that if a signal is assumed to propagate at
zero speed, than this would at the same time be compatible with propagation
to the right AND with propagation to the left - but infinitely slow of
course ...
Harald
Igor Khavkine
> I came across a webpage http://www.bartleby.com/173/a1.html showing
> Einstein's derivation of the Lorentz transformation (as given in his
> book 'Relativity: The Special and General Theory').
>
> It claims to be elegant and simple but appears to be mathematically
> inconsistent to me (as far as my understanding of maths goes anyway).
>
> It is only necessary here to examine the initial equations for this,
> which describe the 'equations of motion of a light signal' in the
> unprimed and primed reference frames, i.e.
>
> (1) x-ct=0
> (2) x'-ct'=0
> where c is the speed of light (which obviously has to be a constant >0)
>
> In the same way, for the propagation of a signal in the opposite
> direction
> (3) x+ct=0
> (4) x'+ct'=0
>
>>From equations (1)-(4) (my numbering), the Lorentz transformation is
> then derived by some algebraic manipulations in the reference.
>
> However, if one subtracts equation (1) from (3) above, one obtains
> (5) 2ct=0
> which means that for any time t>0
> (6) c=0,
> in contradiction to the requirement that c>0.
When you subtract equations (1) and (3), you assume both hold at the
same time. Equation (1) describes a 45 degree line extending to the
right of the origin, while equation (3) describes a 45 degree line
extending to the left of the origin. If both equations are assumed to
hold at the same time, they describe only the points of intersection of
these two lines. There is only one such point, it is the origin with
x = 0 and t = 0.
The equation (5) that you obtain implies t = 0, since c is taken to be a
non-zero constant. Which is exactly the t coordinate of the point of
intersection I described above. Moreover, strictly speaking equation (6)
cannot not follow because there does not exist any t > 0 that will
satisfy both (1) and (3).
Nothing to see here folks, move along.
Igor
the softrat
Yeah: learn about the difference between vectors and scalars. Learn
that equations (1) & (2) and (3) & (4) describe different physical
states. Note that the 'direction' *is* significant. Realize that it is
over your head to attempt to re-do the derivation of the Lorenz
Transformation.
HTH
the softrat
Sometimes I get so tired of the taste of my own toes.
mailto:sof...@pobox.com
--
Whoever said nothing is impossible never tried slamming a
revolving door.
juanrgo...@canonicalscience.com
See
http://canonicalscience.blogspot.com/2005/08/what-is-history-of-relativit=
y-theory.html
for additional details and references. I am preparing a new extended
corrected version with more references and further data. This interesting
debate -and people that has corrected and improved to me- will be properl=
y
acknowledged therein.
****************************
<thoma...@gmail.com> on 1 Sep 2005 22:25:52 +0000 (UTC) wrote:
I came across a webpage http://www.bartleby.com/173/a1.html showing
Einstein's derivation of the Lorentz transformation (as given in his
book 'Relativity: The Special and General Theory').
It claims to be elegant and simple but appears to be mathematically
inconsistent to me (as far as my understanding of maths goes anyway).
****************************
Einstein copied relativity theory from Lorentz and Poincar=E9. It present=
ed
like new his research from a reorganization of work of others and did not
cited. For example, instead of use the LT like a postulate and derive
constancy of c how Poincar=E9 did, Einstein elevated constancy of c alrea=
dy
present in Poincar=E9 writings to a new postulate (note that I say to "ne=
w
postulate" no "new" alone) and derived (so say) the LT from it. There is
not new physics.
However, the derivation is mathematically wrong and based in hidden
assumptions that after even Einstein recognized (at least used three
additional principles to two known of Einstein SR).
Also others derivations of Einstein are wrong, for example his derivation
of E=3Dmc^2. This indicates that Einstein knew previously the results tha=
t
supposedly he was "deriving from application of logical steps to his two
postulates".
-------
Juan R. Gonz=E1lez-=C1lvarez
Center for CANONICAL |SCIENCE)
Robert Low
> It is only necessary here to examine the initial equations for this,
> which describe the 'equations of motion of a light signal' in the
> unprimed and primed reference frames, i.e.
> (1) x-ct=0
> (2) x'-ct'=0
This pair of equations tells you that a particular straight
line through the origin has the same description in two
different coordinate systems. These relationships are
satisfied *only* by points on this line. The understanding
you should have is that a point in space-time has its x coordinate
equal to c times its t coordinate iff its x' coordinate
is c times its t' coordinate. This requirement restricts
the type of linear equations that can related (x,t) and
(x',t').
> (3) x+ct=0
> (4) x'+ct'=0
This tells you that an entirely different straight line
through the origin also has the same description in these
two coordinate systems. These relationships are satisfied
only by points on this other line. Likewise, we now see
that a point in space-time has its x coordinate plus
c times its t coordinate equal to 0 iff its x' coordinate
plus c times its t' coordinate is equal to 0. This
requirement also restricts the linear relationships
that can hold between (x,t) and (x',t').
Now, equations (1) and (3) are both simultaneously satisfied only
by points on both of those straight lines, and the only such point
is the one with x=t=0.
What you've done is (almost) to solve equations (1)
and (3) to find where the lines cross, but then insisted
that this work for t>0 (which it can't, unless c=0).
cma...@yahoo.com
Let's clean up this thing first. The primed reference frame plays no
role at all in this "paradox": if you only had equations (1) and (3),
you would still be able to reach the same conclusion. So, this reduces
to:
(1) x-ct=0, for the propagation of a signal in the positive direction
(3) x+ct=0, for the propagation of a signal in the negative direction
Now let's see what this really means. We have two distinct signals
here, but we are reusing the same variable x to represent their
position in time. It's usually informally acceptable, but only if we
are very careful about it. Formally, these two distinct entities must
be represented by two distinct variables, such as x1 and x2. So the
equations should be written as:
(1) x1-ct=0, for the propagation of a signal in the positive direction,
(3) x2+ct=0, for the propagation of a signal in the negative direction.
And now, of course, the paradox is gone.
By fusing x1 and x2 like you did, you basically added the following
hidden equation:
x1(t) = x2(t) = x(t), for all t.
The semantic of this equation is that both signals are always at the
same position. And obviously, this is only possible if c = 0.
Chris
Juan R.
Basically you are right.
Einstein begin with
x = ct (1)
and follow with
x' = ct' (2)
both equations are valid for arbitrary times t >= 0.
Next he says
"Those space-time points (events) which satisfy (1) must also satisfy
(2). Obviously this will be the case when the relation"
(x' - ct') = lambda (x - ct) (3).
Next he begins again doing for the same t
x = - ct
and
x' = - ct'
and derives
(x' + ct') = mu (x + ct) (4).
note the change of sign (because now Einstein is using x = -ct and x' =
- ct'). Then says "By adding (or subtracting) equations (3) and (4),
and introducing for convenience the constants a and b in place of the
constants lambda and mu... we obtain the equations"
x' = ax - bct
ct' = act - bx
denoted by (5). (5) is valid for arbitrary times and was derived by
adding or substrating (3) and (4). But (3) is derived from
x = ct and (4) is derived from x = - ct.
Basically Einstein is sayyng that at the same non-zero t.
x = ct and x = -ct.
Juan R.
Center for CANONICAL |SCIENCE)
Igor Khavkine
> Thomas Smid wrote:
> > I came across a webpage http://www.bartleby.com/173/a1.html showing
> > Einstein's derivation of the Lorentz transformation (as given in his
> > book 'Relativity: The Special and General Theory').
> >
> > It claims to be elegant and simple but appears to be mathematically
> > inconsistent to me (as far as my understanding of maths goes anyway).
> > Any comments on this?
> Basically you are right.
Nope.
> Einstein begin with
>
> x = ct (1)
>
> and follow with
>
> x' = ct' (2)
>
> both equations are valid for arbitrary times t >= 0.
>
> Next he says
>
> "Those space-time points (events) which satisfy (1) must also satisfy
> (2). Obviously this will be the case when the relation"
>
> (x' - ct') = lambda (x - ct) (3).
>
> Next he begins again doing for the same t
>
> x = - ct
>
> and
>
> x' = - ct'
>
> and derives
>
> (x' + ct') = mu (x + ct) (4).
This point may not be clear to every one and is the source of your
error in the claim that Einstein made a mistake.
The equation x = ct defines a line passing through the origin in the
(x,t) plane. This equation can also be written as x-ct=0, in which case
this line is the set on which the function f(x,t)=x-ct vanishes. Note
that f(x,t) is a linear function of the coordinates. From simple linear
algebra, any other linear function of the coordinates that vanishes on
the same set of points must be a scalar multiple of f(x,t). Together
with the assumption that g(x,t) = x'(x,t)-ct'(x,t) is a linear function
of the coordinates, this implies equation (3) and similarly equation
(4). These equations hold as equalities of functions defined on the
(x,t) plane. In other words, they hold for all x and t, not just the
ones satisfying x = ct and x = -ct.
I think the assumption that g(x,t) be a linear function of the
coordinates can be dispensed with, but it will make the derivation
slightly more complicated.
> note the change of sign (because now Einstein is using x = -ct and x' =
> - ct'). Then says "By adding (or subtracting) equations (3) and (4),
> and introducing for convenience the constants a and b in place of the
> constants lambda and mu... we obtain the equations"
>
> x' = ax - bct
> ct' = act - bx
>
> denoted by (5). (5) is valid for arbitrary times and was derived by
> adding or substrating (3) and (4). But (3) is derived from
> x = ct and (4) is derived from x = - ct.
The fact that (3) and (4) are derived from the equations x = ct and x =
-ct is irrelevant. As I've shown, they both hold for *all* x and t, not
just the point at the origin.
> Basically Einstein is sayyng that at the same non-zero t.
>
> x = ct and x = -ct.
I hope you can now see that it is not at all what he is saying.
Igor
Juan R.
> Juan R. wrote:
> > Thomas Smid wrote:
s
> > > book 'Relativity: The Special and General Theory').
> > >
> > > It claims to be elegant and simple but appears to be mathematically
).
> [...]
> > > Any comments on this?
>
> > Basically you are right.
>
> Nope.
Obviously, Smid is right in his original claim.
Smid is talking of the derivation of the LT by Einstein. I am also
talking of the derivation of the LT by Einstein. I am not saying that
the LT was correct or uncorrect, or best derived using other procedure.
We are talking about Einstein derivation of the LT.
> > Einstein begin with
> >
> > x =3D ct (1)
> >
> > and follow with
> >
> > x' =3D ct' (2)
> >
> > both equations are valid for arbitrary times t >=3D 0.
> >
> > Next he says
> >
> > "Those space-time points (events) which satisfy (1) must also satisfy
> > (2). Obviously this will be the case when the relation"
> >
> > (x' - ct') =3D lambda (x - ct) (3).
> >
> > Next he begins again doing for the same t
> >
> > x =3D - ct
> >
> > and
> >
> > x' =3D - ct'
> >
> > and derives
> >
> > (x' + ct') =3D mu (x + ct) (4).
>
> This point may not be clear to every one and is the source of your
> error in the claim that Einstein made a mistake.
It is clear that Einstein wrote!
> The equation x =3D ct defines a line passing through the origin in the
> (x,t) plane.
Einstein begin with equation (1) even is numered in his derivation!
It is true that mathematically the equation x =3D ct defines a line
passing through the origin in the (x,t) plane, also is true that
physically defines a light signal in own Einstein words.
> This equation can also be written as x-ct=3D0, in which case
> this line is the set on which the function f(x,t)=3Dx-ct vanishes. Note
> that f(x,t) is a linear function of the coordinates. From simple linear
> algebra, any other linear function of the coordinates that vanishes on
> the same set of points must be a scalar multiple of f(x,t).
But was not Einstein wrote. He begins by considering exclusively the
points of the light signal verifying x =3D ct and is obvious why uses
light signals, because after utilizes the postulate of constancy of
ligh velocity c for writing x' =3D ct' in the other frame. Without that
postulate, he only could use your mathematical arguments and wrote an
undetermined function h(x',t') in the other frame. But he obtained the
explicit expression for the other linear function Einstein wrote x' =3D
ct' or h =3D x' - ct'.
> with the assumption that g(x,t) =3D x'(x,t)-ct'(x,t) is a linear functi=
on
> of the coordinates, this implies equation (3) and similarly equation
> (4). These equations hold as equalities of functions defined on the
> (x,t) plane. In other words, they hold for all x and t, not just the
> ones satisfying x =3D ct and x =3D -ct.
This is you have add now when you already know what is the LT. This is
not Einstein wrote. Einstein said
"Those space-time points (events) which satisfy (1) must also satisfy
(2). Obviously this will be the case when the relation"
(x' - ct') =3D lambda (x - ct) (3)
is fulfilled in general, where lambda indicates a constant; for,
according to (3), the disappearance of (x - ct) involves the
disappearance of (x' - ct').
That is, Einstein is very clear after of introducing equation (3).
Einstein continues to uphold that equation (1) x =3D ct IS valid in (3).
In fact (3) was derived from (1) and (2) and (2) was derived from (1)
more the postulate of constancy of c in different frames!!
Next doing EXACTLY the same, Einstein derives equation (4). Which
obviously (in Einstein derivation) also satisfies x' =3D - ct'.
> (4). These equations hold as equalities of functions defined on the
> (x,t) plane. In other words, they hold for all x and t, not just the
> ones satisfying x =3D ct and x =3D -ct.
Here you are a bit ambiguous "they hold for all x and t, not just the
ones satisfying x =3D ct and x =3D -ct." I think that you are saying that
also are valid for x =3D ct and x =3D -ct.
In any case, that is you are saying a posteriori, it is not that
Einstein said, and this is the reason that he BEGAN with x =3D ct and
after BEGAN with x =3D -ct, because he applied the principle of constancy
of light for deriving the expression for the ligh signal in the OTHER
frame. You are reasoning with abstract functions like f(t,x) and modern
thinking, Einstein is reasoning with light signals and HIS formulation
of SR based in two principles.
> I think the assumption that g(x,t) be a linear function of the
> coordinates can be dispensed with, but it will make the derivation
> slightly more complicated.
We are not talking of what is the best derivation of the LT. We are
talking of Einstein derivation.
> > note the change of sign (because now Einstein is using x =3D -ct and =
x' =3D
> > - ct'). Then says "By adding (or subtracting) equations (3) and (4),
> > and introducing for convenience the constants a and b in place of the
> > constants lambda and mu... we obtain the equations"
> >
> > x' =3D ax - bct
> > ct' =3D act - bx
> >
> > denoted by (5). (5) is valid for arbitrary times and was derived by
> > adding or substrating (3) and (4). But (3) is derived from
> > x =3D ct and (4) is derived from x =3D - ct.
>
> The fact that (3) and (4) are derived from the equations x =3D ct and x=
=3D
> -ct is irrelevant. As I've shown, they both hold for *all* x and t, not
> just the point at the origin.
In your own words "ARE DERIVED FROM x =3D ct and x =3D - ct".
But that is NOT irrelevant!!!!!
We are not talking that the LT was correct or incorrect. We are talking
of Einstein derivation of the LT. As said in the forum of history of
relativity and here also. Einstein derivations are full of mistakes,
implicit asumptions, incorrect logic, etc.
> > Basically Einstein is sayyng that at the same non-zero t.
> >
> > x =3D ct and x =3D -ct.
>
> I hope you can now see that it is not at all what he is saying.
>
> Igor
Therefore, Einstein begins with asumption x =3D ct and obtains the
expresion x' =3D ct' applying the 2=BA postulate of his SR. Obtains
equation (3) that -he clearly states- verifies the initial (x =3D ct)
from was derived. Using the same argument but now (x =3D -ct) Einstein
derives (4) which ALSO verifies the initial equation of the bgining of
the derivation (now x =3D - ct) since is valid for light signals in the
negative axis. Next Einstein does
(3) + (4)
and
(3) - (4)
which implies that Einstein is asuming that x =3D ct and x =3D - ct both
are correct at the same time, since the final result contains only a
"x".
If Einstein had done xA =3D ct and xB =3D -ct, with xA =3D/=3D xB then bo=
th xA
and xB would arise after of adding (or subtracting) equations (3) and
(4). But then one does not obtain result Einstien obtained, this is the
reaosn Einstien works with the same "x" in (3) and (4)
I see no problem if he begins with x =3D ct and after begin with x =3D - =
ct
and uses both equation in DIFFERENT environments. But Einstein is using
x =3D ct in (3) and x =3D - ct in (4) at the SAME time.
Therefore, if (3) and (4) are valid at the SAME time in Einstein
DERIVATION. Einstein IS doing x =3D ct and x =3D -ct at the same t >=3D 0.
Just Smid said and i remarked.
Igor Khavkine
> Igor Khavkine wrote:
> > Juan R. wrote:
> > > Thomas Smid wrote:
> > > > I came across a webpage http://www.bartleby.com/173/a1.html showing
> > > > Einstein's derivation of the Lorentz transformation (as given in hi=
> s
> > > > book 'Relativity: The Special and General Theory').
> > > >
> > > > It claims to be elegant and simple but appears to be mathematically
> > > > inconsistent to me (as far as my understanding of maths goes anyway=
> ).
> > [...]
> > > > Any comments on this?
> >
> > > Basically you are right.
> >
> > Nope.
>
> Obviously, Smid is right in his original claim.
Again. No.
> Smid is talking of the derivation of the LT by Einstein. I am also
> talking of the derivation of the LT by Einstein. I am not saying that
> the LT was correct or uncorrect, or best derived using other procedure.
> We are talking about Einstein derivation of the LT.
So am I. Everything that Einstein says in the appendix to his book is
mathematically correct. In my last message I tried to explain exactly
why one of the less obvious steps is indeed correct.
It seems, however, that you still have objections. Unfortunately, I
simply do not understand them or cannot see a logical coherence in
these objections. From what you wrote, I can only conclude that you are
either not willing or not able to check the validity of Einstein's
derivation yourself, even with the aid of provided explanations.
In either case, I see little point in continuing this conversation.
Igor
Juan R.
>
> > Smid is talking of the derivation of the LT by Einstein. I am also
> > talking of the derivation of the LT by Einstein. I am not saying that
> > the LT was correct or uncorrect, or best derived using other procedure.
> > We are talking about Einstein derivation of the LT.
>
> So am I. Everything that Einstein says in the appendix to his book is
> mathematically correct. In my last message I tried to explain exactly
> why one of the less obvious steps is indeed correct.
My presentation of topic was clear, moreover i followed Einstein
'proof' closely. I simply cannot understand your reject of the
evidence.
In your OWN last words
> The fact that (3) and (4) are derived from the equations x =3D ct and x=
=3D
> -ct is irrelevant. As I've shown, they both hold for *all* x and t, not
> just the point at the origin.
That is, you agree that equations (3) and (4) IN Einstein's appendix
are derived FROM x = ct and x = -ct. but that is NOT irrelevant because
is that Smid said.
After you claim "As I've shown", but you are not Einstein and we talk
of Einstein REALLY did nor that a posteriori you interpret the
derivation adding things are not present in the original document.
Einstein exactly says:
"Those space-time points (events) which satisfy (1) must also satisfy
(2). Obviously this will be the case when the relation
(x' - ct') = lambda(x - ct). . . . . .(3)
is fulfilled in general, where lambda indicates a constant; for,
according to (3), the disappearance of (x - ct) involves the
disappearance of (x' - ct').
"If we apply quite similar considerations to light rays which are being
transmitted along the negative x-axis, we obtain the condition"
(x' + ct') = mu(x + ct). . . . . .(4)
"By adding (or subtracting) equations (3) and (4) [...]"
That resulting relation (5) may be valid for arbitrary x and times t is
just your *personal* addition.
Moreover, in my opinion, you continue to be ambiguous. For instance, in
your Sep 7, 8:23 am post, you replied to Smid:
"The equation (5) that you obtain implies t = 0, since c is taken to be
a
non-zero constant. Which is exactly the t coordinate of the point of
intersection I described above. Moreover, strictly speaking equation
(6)
cannot not follow because there does not exist any t > 0 that will
satisfy both (1) and (3)."
Let me take Einstein derived eq (5). If i understand to you, you are
claiming that equations x = ct and x = -ct cannot be applied to the
same t unless t = 0. Therefore, for t > 0 in your own words:
"Equation (1) describes a 45 degree line extending to the
right of the origin, while equation (3) describes a 45 degree line
extending to the left of the origin. If both equations are assumed to
hold at the same time, they describe only the points of intersection of
these two lines. There is only one such point, it is the origin with
x = 0 and t = 0."
one may either apply one or other newer both to the same time. Ok! Let
me ignore Einstein really did and following your advice.
I apply ONLY x = ct to final (5), i obtain by adding or subtracting
x' - ct' = 0
and
x' + ct' = 2(a - b) ct
substituting first in the second and using def of a and b
2 x' = 2 mu ct.
which is NOT the equation (2) of Einstein unless c = 0 or t' = t or
both.
Etc.
There are many other mathematical and physical errors in the
derivation. Of course, there is absolutely none logic in Einstein's
derivation. The mathematics of his original derivation are completely
wrong and unphysical. This is the reason that Poincare -great
mathematician- newer stated that constancy of c was a postulate of
relativity, just a theorem derived from the group property of the LT.
That the LT follows from the constancy of c and the PoR is another of
the myths around Einstein figure...
Daryl McCullough
>Igor Khavkine wrote:
>> So am I. Everything that Einstein says in the appendix to his book is
>> mathematically correct. In my last message I tried to explain exactly
>> why one of the less obvious steps is indeed correct.
>
>My presentation of topic was clear, moreover i followed Einstein
>'proof' closely. I simply cannot understand your reject of the
>evidence.
Look, your original post (and Thomas' as well) made a simple
error. What Einstein assumed was that there existed linear
functions
x'(x,t)
and
t'(x,t)
relating the coordinates (x,t) of any event in one frame to
the coordinates (x',t') in the second frame. Einstein didn't
make the functional dependence explicit, but I think it
clarifies things.
His constraint on these functions is this: Light has the invariant
speed c in both frames. What that implies for the functions x'
and t' is this:
For any t,
x'(ct,t) = c t'(ct,t)
x'(-ct,t) = - c t'(-ct,t)
Now, it is just a fact of algebra that any linear functions
x' and t' satisfying those constraints *also* satisfies the
constraint that there exists constants lambda and mu such that
for all x, for all t,
x'(x,t) - c t'(x,t) = lambda (x - vt)
x'(x,t) + c t'(x,t) = mu (x + vt)
Note: these equations hold for *all* x and t, not just
the special cases x=ct or x=-ct.
--
Daryl McCullough
Ithaca, NY
Thomas Smid
> The equation x = ct defines a line passing through the origin in the
> (x,t) plane. This equation can also be written as x-ct=0, in which case
> this line is the set on which the function f(x,t)=x-ct vanishes. Note
> that f(x,t) is a linear function of the coordinates. From simple linear
> algebra, any other linear function of the coordinates that vanishes on
> the same set of points must be a scalar multiple of f(x,t). Together
> with the assumption that g(x,t) = x'(x,t)-ct'(x,t) is a linear function
> of the coordinates, this implies equation (3) and similarly equation
> (4). These equations hold as equalities of functions defined on the
> (x,t) plane. In other words, they hold for all x and t, not just the
> ones satisfying x = ct and x = -ct.
If f(x,t)=x-ct vanishes, then clearly x=ct (and correspondingly for the
other equations), so I don't see how you can come to the conclusion
that the equations hold in other cases as well. This would obviously be
a contradiction in terms.
The equations
(x' - ct') = lambda (x - ct)
(x' + ct') = mu (x + ct)
were only set up bei Einstein *because* each side is identically zero
(which also means that lambda and mu must be arbitrary by the way).
In general one would not be entitled to write these equations in the
first place.
Thomas
Igor Khavkine
> Igor Khavkine wrote:
>
> > The equation x = ct defines a line passing through the origin in the
> > (x,t) plane. This equation can also be written as x-ct=0, in which case
> > this line is the set on which the function f(x,t)=x-ct vanishes. Note
> > that f(x,t) is a linear function of the coordinates. From simple linear
> > algebra, any other linear function of the coordinates that vanishes on
> > the same set of points must be a scalar multiple of f(x,t). Together
> > with the assumption that g(x,t) = x'(x,t)-ct'(x,t) is a linear function
> > of the coordinates, this implies equation (3) and similarly equation
> > (4). These equations hold as equalities of functions defined on the
> > (x,t) plane. In other words, they hold for all x and t, not just the
> > ones satisfying x = ct and x = -ct.
>
> If f(x,t)=x-ct vanishes, then clearly x=ct (and correspondingly for the
> other equations), so I don't see how you can come to the conclusion
> that the equations hold in other cases as well. This would obviously be
> a contradiction in terms.
The function f(x,t) is not identically zero. For instance f(1,0) = 1 !=
0. But it does take the zero value on the line x=ct. Picture the graph
of this function. Since it is linear, its graph, z=f(x,t) is just a
plane in the (x,t,z) space. This plane intersects the x-t plane
precisely along the line x=ct, which is precisely where it vanishes.
> The equations
> (x' - ct') = lambda (x - ct)
> (x' + ct') = mu (x + ct)
> were only set up bei Einstein *because* each side is identically zero
> (which also means that lambda and mu must be arbitrary by the way).
No, you still misunderstand why this equality exists. Here's an
exercise for you. Write down all *linear* functions of x and y which
vanish along the line x=y. For any two such functions h1(x,y) and
h2(x,y), can you find a constant k such that h1(x,y) = k h2(x,y)
everywhere on the x-y plane? If you can, then you've performed the same
derivation as Einstein. If you can't, then you have to go back and hit
your linear algebra textbooks.
Igor
Thomas Smid
> The function f(x,t) is not identically zero. For instance f(1,0) = 1 !=
> 0. But it does take the zero value on the line x=ct. Picture the graph
> of this function. Since it is linear, its graph, z=f(x,t) is just a
> plane in the (x,t,z) space. This plane intersects the x-t plane
> precisely along the line x=ct, which is precisely where it vanishes.
The line x=ct is not only the place where the function f(x,t)=x-ct
vanishes, it is also the only place where it is defined. This is in
fact easy to prove:
Our *condition* is
(1) x=ct .
We now *define* a function f(x,t) such that
(2) f(x,t)=x-ct.
We are now making the *statement* 'There exist values x,t for which
(3) f(x,t) != 0 '
Can this statement possibly be true? Let's see. Because of (2), (3) is
identical to
(4) x-ct != 0
which however would mean that
(5) x != ct
in contradiction to our condition (1).
This proves that the statement is false and that f(x,t) must thus be
identically zero.
Thomas
Thomas Smid
> His constraint on these functions is this: Light has the invariant
> speed c in both frames. What that implies for the functions x'
> and t' is this:
>
> For any t,
>
> x'(ct,t) = c t'(ct,t)
> x'(-ct,t) = - c t'(-ct,t)
>
> Now, it is just a fact of algebra that any linear functions
> x' and t' satisfying those constraints *also* satisfies the
> constraint that there exists constants lambda and mu such that
>
> for all x, for all t,
>
> x'(x,t) - c t'(x,t) = lambda (x - vt)
> x'(x,t) + c t'(x,t) = mu (x + vt)
>
> Note: these equations hold for *all* x and t, not just
> the special cases x=ct or x=-ct.
The equations
(1) (x' - ct') = lambda*(x - ct)
(2) (x' + ct') = mu*(x + ct)
can only hold for all values of x and t if lambda=mu.
Proof:
Assume (1) holds for all x and t. It holds then also in particular for
those x,t for which x=ct (and x'=ct'). If assumed to hold for negative
x as well, it would then also hold in particular for all x=-ct (and
x'=-ct') i.e. we would have
(3) -2ct' = -lambda*2ct
or
(4) 2ct' = lambda*2ct .
However from (2) we have for the particular case x=ct (and x'=ct')
(5) 2ct' = mu*2ct
and by comparison with (4) it follows that the equations can only hold
both if
(6) lambda=mu .
On the other hand, if (1) is only restricted to positive x and (2) to
negative, one could not add or subtract the equations as for any given
x only one of the equations could be applied.
Thomas
Juan R.
> Look, your original post (and Thomas' as well) made a simple
> error. What Einstein assumed was that there existed linear
> functions
>
> x'(x,t)
> and
> t'(x,t)
>
> relating the coordinates (x,t) of any event in one frame to
> the coordinates (x',t') in the second frame. Einstein didn't
> make the functional dependence explicit, but I think it
> clarifies things.
>
> speed c in both frames. What that implies for the functions x'
> and t' is this:
>
> For any t,
>
> x'(ct,t) = c t'(ct,t)
> x'(-ct,t) = - c t'(-ct,t)
>
> Now, it is just a fact of algebra that any linear functions
> x' and t' satisfying those constraints *also* satisfies the
> constraint that there exists constants lambda and mu such that
>
> for all x, for all t,
>
> x'(x,t) - c t'(x,t) = lambda (x - vt)
> x'(x,t) + c t'(x,t) = mu (x + vt)
>
> Note: these equations hold for *all* x and t, not just
> the special cases x=ct or x=-ct.
First, i may note that 'v' above is a typo (as confirmed by Daryl
personal communication). It is 'c'.
Are we talking of Einstein did, we interpret now he did or of the
derivation of the LT from two postulates?
*If* we talk of x = ct or -ct and the appealing to the PoR, then one is
/not/ deriving nothing, since math is flawed. This is and continue to
be Smid point. I agree that *if one begins from equations for light
signals the derivation is mathematically incorrect*.
*If* we talk of derivation of LT from two postulates of SR. There is no
such one /derivation/ since SR does not follow from those two
postulates alone.
In mathpages [http://www.mathpages.com/rr/s8-08/8-08.htm]:
"The two famous principles of Einstein's 1905 paper are /not/
sufficient to uniquely identify special relativity, as Einstein himself
later acknowledged. One must also stipulate, at the very least,
homogeneity, memorylessness, and isotropy."
If we talk of that for light signals For any t,
x'(ct,t) = c t'(ct,t)
x'(-ct,t) = - c t'(-ct,t)
||system A||
I agree. If not are talking that for all x, for all t,
x'(x,t) - c t'(x,t) = lambda (x - vt)
x'(x,t) + c t'(x,t) = mu (x + vt)
||system B||
But i do not agree that the general case -system B- follows -is derived
from- the special case -system A-.
In the logical step from system A to system B, one needs to introduce
ADITTIONAL POSTULATES.
I see two cases:
I) Daryl and Igor Case: x in system B is /any/ x.
Then it does not follow -is not derived- from special case x=ct. We are
*postulating* that there is a linear relationship valid for any event
with lambda = lambda(v) and mu = mu(v). Of course, the special case
-system A- does /not/ uniquely identify system B.
II) Smid Case: we only know PoR and the constancy of light (Einstein's
postulates of SR), therefore we begin from x = ct or x=-ct.
In that case, as explained several times, the derivation is just
imposible.
In both cases I and II, the LT does not follow from the Einstein
postulates.
mark...@yahoo.com
> [Einstein's derivation] claims to be elegant and simple
> but appear to be mathematically inconsistent
To (correctly) rephrase your (1) and (2):
If x', t' are related as functions x' = A(x,t), t' = B(x,t) then the
functions satisfy the identity:
A(ct,t) = c B(ct,t)
and (3) and (4):
A(-ct,t) = -c B(-ct,t).
You said
> However, if one subtracts equation (1) from (3) above [etc.]
and there's nothing to subtract; you simply read everything wrong; and
I supplied the correct and more explicit reading above.
More generally, one can prove that if the functions
A(x,y,z,t), B(x,y,z,t), C(x,y,z,t), D(x,y,z,t)
satisfy the identity
A(ctuw,ctux,ctv,t)^2 + B(ctuw,ctux,ctv,t)^2
+ C(ctuw,ctux,ctv,t)^2 = c^2 D(ctuw,ctux,ctv,t)^2
where
u, v stand respectively for cos(theta), sin(theta)
w, x stand respectively for cos(phi), sin(phi)
-- the identity holding for all theta, phi and t; then the coordinate
transformation described by
x' = A(x,y,z,t), y' = B(x,y,z,t)
z' = C(x,y,z,t), t' = D(x,y,z,t)
must be a combination of the following:
(1) Lorentz transformation on the x and t coordinates
(2) Euclidean rotation on the x, y, z coordinates
(3) A spatial translation x -> x + h
(4) A temporal translation t -> t + h
(5) A space-time rescaling (x,y,z,t) -> (kx,ky,kz,kt), k > 0
(6) A parity reversal x -> -x
(7) A time reversal t -> -t
Einstein, I believe, assumed in his 1905 paper that the functions had
to be continuous and differentiable. However, that's not necessary to
assume. It FOLLOWS solely from the identity above: both continuity and
differentiability.
That, in turn, has to do with the fact that all of Minkowski geometry
(including definitions of lengths, angles, time intervals, congruence,
etc.) can be constructed from nothing more than the relation
A -- B <==> events A and B can be joined by a light signal
The complete structure of both the spatial geometry and temporal logic
of Minkowski geometry are completely derivable from this, and nothing
but this, subject to a finite set of axioms. That was (in part) proven
by A.A. Robb in 1914; and (in part) proven in the 1960's.
Dirk Van de moortel
> Daryl McCullough wrote:
>
> > His constraint on these functions is this: Light has the invariant
> > speed c in both frames. What that implies for the functions x'
> > and t' is this:
> >
> > For any t,
> >
> > x'(ct,t) = c t'(ct,t)
> > x'(-ct,t) = - c t'(-ct,t)
> >
> > Now, it is just a fact of algebra that any linear functions
> > x' and t' satisfying those constraints *also* satisfies the
> > constraint that there exists constants lambda and mu such that
> >
> > for all x, for all t,
> >
> > x'(x,t) - c t'(x,t) = lambda (x - vt)
> > x'(x,t) + c t'(x,t) = mu (x + vt)
> >
> > Note: these equations hold for *all* x and t, not just
> > the special cases x=ct or x=-ct.
>
> The equations
>
> (1) (x' - ct') = lambda*(x - ct)
> (2) (x' + ct') = mu*(x + ct)
>
> can only hold for all values of x and t if lambda=mu.
>
> Proof:
>
> Assume (1) holds for all x and t. It holds then also in particular for
> those x,t for which x=ct (and x'=ct').
Yes, then the equation reduces to
0 = 0
which is correct.
> If assumed to hold for negative
> x as well, it would then also hold in particular for all x=-ct (and
> x'=-ct') i.e. we would have
>
> (3) -2ct' = -lambda*2ct
> or
> (4) 2ct' = lambda*2ct .
So these equations (3) and (4) are valid for all events that satisfy
x = - c t
>
> However from (2) we have for the particular case x=ct (and x'=ct')
>
> (5) 2ct' = mu*2ct
This equation only valid for all events that satisfy
x = c t
>
> and by comparison with (4) it follows that the equations can only hold
> both if
>
> (6) lambda=mu .
No, we just have seen that you can only take the equations
(4) and (5) together for events that satisfy
x = c t
and
x = - c t
together, in other words for events that satisfy
{ x = 0
{ t = 0
and thus also
{ x' = 0
{ t' = 0
So when you take the equations
{ (4) 2ct' = lambda*2ct
{ (5) 2ct' = mu*2ct
you can only do that for t = t' = 0, so you actually have
{ (4) 0 = lambda * 0
{ (5) 0 = mu * 0
From these equations you cannot deduce that
lambda = mu.
This is very elementary lower high school level algebra.
>
> On the other hand, if (1) is only restricted to positive x and (2) to
> negative, one could not add or subtract the equations as for any given
> x only one of the equations could be applied.
When we write
x = c t,
we actually mean
{ (x,t) | x = c t }
and we read it:
the set of events for wich the coordinates satisfy
the relation x = c t.
When we write
x = - c t,
we actually mean
{ (x,t) | x = - c t }
and we read it:
the set of events for wich the coordinates satisfy
the relation x = -c t.
In lower high school we learn how to find the intersection
of sets for which the equations are given. We find
{ (x,t) | x = c t } Intersect { (x,t) | x = - c t }
= { (0,0) }
Did you already cover this at school?
Dirk Vdm
Daryl McCullough
about algebra. Thomas and Juan are accusing Einstein
of making simple algebraic mistakes. They are wrong
about this, but the disagreement is really about simple
mathematics, not about physics.
Thomas Smid says...
>The equations
>
>(1) (x' - ct') = lambda*(x - ct)
>(2) (x' + ct') = mu*(x + ct)
>
>can only hold for all values of x and t if lambda=mu.
>
>Proof:
>
>Assume (1) holds for all x and t. It holds then also in particular for
>those x,t for which x=ct (and x'=ct'). If assumed to hold for negative
>x as well, it would then also hold in particular for all x=-ct (and
>x'=-ct') i.e. we would have
>
>(3) -2ct' = -lambda*2ct
>or
>(4) 2ct' = lambda*2ct.
You are getting confused because you are leaving
out the functional dependence. x' and t' are functions
of x and t. So the correct equations, including the
functional dependence, is
(1) x'(x,t) - c t'(x,t) = lambda*(x - ct)
(2) x'(x,t) + c t'(x,t) = mu*(x+ct)
Now, you want to look at (1) for the special case in which
x = -ct. Okay, in that case we have:
x'(-ct,t) - c t'(-ct,t) = - lambda * 2ct
Under our assumptions that light has speed c in both
frames, we get x'(-ct,t) = -c t'(-ct,t), so we have
(3) -2c t'(-ct,t) = - lambda * 2ct
If you look at equation (2) in the special case in which
x=ct, you get something similar:
(4) 2 c t'(ct,t) = mu * 2ct
Now, what you want to do is to add the two equations (3) and
(4). But note that t' is a *function*, not a number. In equation
(3), that function is evaluated at the arguments -ct,t. In equation
(4), that function is evaluated at the arguments +ct,t. So when
you add (3) and (4) what you get is:
2 c (t'(ct,t) - t'(-ct,t)) = 2ct (mu - lambda)
The left side is *not* equal to zero, in general. You
were misled into thinking it was zero because you were
ignoring the functional dependence of t' on the arguments
x and t. In general t'(ct,t) will not be equal to t'(-ct,t)
(except in the special case t=0).
Juan R.
> This discussion is not actually about physics, it is
> about algebra. Thomas and Juan are accusing Einstein
> of making simple algebraic mistakes. They are wrong
> about this, but the disagreement is really about simple
> mathematics, not about physics.
It is true that Smid is wrong in his last post. I wrote a reply on why
last Smid conclusion, mu =3D lambda, is incorrect, but was rejected due
to non-ascii format. Now it appears unnecesary i submit again seing two
above replies.
However, i continue to claim that the Lorentz transformation cannot be
derived from the principle of relativity more the constancy of c in
different -inertial frames- as said already in Sep 7:
"However, the derivation is mathematically wrong and based in hidden
assumptions that after even Einstein recognized (at least used three
additional principles to two known of Einstein SR)."
I and many other authors are claiming this, including respected
relativists like Fock or even the own Einstein in last years. See for
example http://www.mathpages.com/rr/s8-08/8-08.htm for the same
criticism:
"The two famous principles of Einstein's 1905 paper are /not/
sufficient to uniquely identify special relativity, as Einstein himself
later acknowledged."
Einstein later acknowledged in the so-called 'Morgan document' -written
in 1921- that he had tacitly invoked in that paper additional and
important assumptions.
Any attemp to prove that Einstein *derivation* is correct is flawed. It
is completely imposible to derive -i.e. logically obtain without
additional premises- the LT from two Einstein postulates. This was the
reason Poincar=E9 -great mathematician- newer take constancy of c like a
postulate.
It would remain clear after of many posts but i will remark again that
the Lorentz transformation does not follow from Einstein (1), (3) and
corresponding equations for light rays which are being transmitted
along the negative x-axis. Einstein did not explicit them in the book,
but I call them (1') and (3')
(1) and (3) more (1') and (3') DOES NOT IMPLY (3) and (4)
That is, there is not *derivation*. The correct logic of a real
*derivation* is
(1) and (3) more (1') and (3') *MORE adittional assumptions* IMPLIES
(3)
and (4)
But those *adittional assumptions* are not the constancy of c nor the
PoR.
If -CASE I- x, x', t, and t' in Einstein (3) and (4) are related to
light signals, then i agree with Smid in that derivation is flawed. If
-CASE II- x, x', t, and t' in Einstein (3) and (4) are for *ANY* event,
then they do not follow from constancy of c nor from the PoR. Are *new*
equations invoked by Einstein like a new 'postulate'
In both cases -I and II- the Lorentz transformation is not being
*derived*.
It is really interesting how Daryl continues to sustain that Einstein
*derived* the LT, when he own Einstein recognized that he did not that
FROM PoR and the constancy of c, that are the two only *physical*
premises explicitely cited in his INITIAL 'derivation'.
Einstein acknowledged in 1921 that he had /tacitly invoked in that
paper additional and important assumptions/.
Curiously my point -that LT cannot be *derived* from two Einstein
postulates alone- was also maintained by recognized relativists like
Fock, and is also supported by mathematicians in mathpages i cited
above.
The PoR + constancy of c -the two Einstein principles- do not uniquely
identify SR.
Thomas Smid
> More generally, one can prove that if the functions
> A(x,y,z,t), B(x,y,z,t), C(x,y,z,t), D(x,y,z,t)
> satisfy the identity
> A(ctuw,ctux,ctv,t)^2 + B(ctuw,ctux,ctv,t)^2
> + C(ctuw,ctux,ctv,t)^2 = c^2 D(ctuw,ctux,ctv,t)^2
> where
> u, v stand respectively for cos(theta), sin(theta)
> w, x stand respectively for cos(phi), sin(phi)
> -- the identity holding for all theta, phi and t; then the coordinate
> transformation described by
> x' = A(x,y,z,t), y' = B(x,y,z,t)
> z' = C(x,y,z,t), t' = D(x,y,z,t)
> must be a combination of the following:
> (1) Lorentz transformation on the x and t coordinates
> (2) Euclidean rotation on the x, y, z coordinates
> (3) A spatial translation x -> x + h
> (4) A temporal translation t -> t + h
> (5) A space-time rescaling (x,y,z,t) -> (kx,ky,kz,kt), k > 0
> (6) A parity reversal x -> -x
> (7) A time reversal t -> -t
I presume you are referring to the procedure that derives the Lorentz
Transformation from the invariance of a spherical wave propagation i.e.
if you have the equation
(1) x^2 +y^2 +z^2 = c^2*t^2
then
(2) x'^2 +y'^2 +z'^2 =c^2*t'^2 .
Now this procedure is actually also mathematically incorrect in my
opinion.
You can see this already by inserting the formal Lorentz Transformation
(as given by Einstein)
(3) x' = ax -bct
(4) ct' = act -bx
into (2), which yields (taking the y and z - components as 0)
(a^2-b^2)*x^2 = (a^2-b^2)*c^2*t^2 .
The quadratic form of the light propagation equations is therefore
fulfilled for all values of the coefficients 'a' and 'b' in the Lorentz
Transformation (3),(4), and not just for the specific well known
values.
The reason for this is that if you have the more general transformation
(5) x' =Ax -Bct
(6) ct'=Dct -Ex
and insert this into (2), one has (again assuming the y and z
components as 0)
(7) (Ax -Bct)^2 = (Dct- Ex)^2
and after evaluating the expressions on both sides
(8) (A^2-E^2)*x^2 -2xct*(DE-AB) = (D^2-B^2)*c^2*t^2 .
Now the argument usually used in this derivation is that in oder to be
consistent with Eq.(1), one has to demand that
(9) A^2-E^2 =1
(10) DE-AB = 0
(11) D^2-B^2 =1 .
(see for instance http://www.mathpages.com/rr/s1-06/1-06.htm (towards
the bottom of the page) (note that the definition of the coefficients
is slightly different from mine)
Now this is clearly a mathematically incorrect conclusion. It may be
justified for the linear term, but in general if one has the equations
x^2 = y^2
a*x^2 =b*y^2 ,
one can not conclude that a=1 and b=1, because by inserting the first
into the second equation one has merely
a*x^2 = b*x^2
i.e.
a=b.
So applied to (8), one could actually only conclude
(12) A^2-E^2 = D^2-B^2
which obviously would not be enough to determine the constants (even if
one takes B as known by associating it with the velocity v).
So it is quite obvious that only by making the incorrect conclusions
(9)-(11), could one arrive at a specific solution for the
coefficients.
Thomas
Thomas Smid
> "Thomas Smid" <thoma...@gmail.com> wrote in message
> news:1128786130.9...@g47g2000cwa.googlegroups.com...
>
> > Daryl McCullough wrote:
> >
> > > His constraint on these functions is this: Light has the invariant
> > > speed c in both frames. What that implies for the functions x'
> > > and t' is this:
> > >
> > > For any t,
> > >
> > > x'(ct,t) = c t'(ct,t)
> > > x'(-ct,t) = - c t'(-ct,t)
> > >
> > > Now, it is just a fact of algebra that any linear functions
> > > x' and t' satisfying those constraints *also* satisfies the
> > > constraint that there exists constants lambda and mu such that
> > >
> > > for all x, for all t,
> > >
> > > x'(x,t) - c t'(x,t) = lambda (x - vt)
> > > x'(x,t) + c t'(x,t) = mu (x + vt)
> > >
> > > Note: these equations hold for *all* x and t, not just
> > > the special cases x=ct or x=-ct.
> >
> >
> > (1) (x' - ct') = lambda*(x - ct)
> > (2) (x' + ct') = mu*(x + ct)
> >
> > can only hold for all values of x and t if lambda=mu.
> >
> > Proof:
> >
> > Assume (1) holds for all x and t. It holds then also in particular for
> > those x,t for which x=ct (and x'=ct').
>
> 0 = 0
> which is correct.
>
> > x as well, it would then also hold in particular for all x=-ct (and
> > x'=-ct') i.e. we would have
> >
> > (3) -2ct' = -lambda*2ct
> > or
>
> So these equations (3) and (4) are valid for all events that satisfy
> x = - c t
>
> >
> > However from (2) we have for the particular case x=ct (and x'=ct')
> >
> > (5) 2ct' = mu*2ct
>
> This equation only valid for all events that satisfy
> x = c t
That's what I said. It was other people, not me, who claimed that both
equations would hold for all values of x and t. I merely showed that
*if* you assume that both equations hold for all values of x and t (as
you would have to do if you want to add or subtract the equations)
then you would deduce lambda=mu. Nobody questions that mathematically
you can generalize the condition 'if x=ct then x'=c't' to the equation
x'-ct'=lambda*(x-ct), but you are dealing not just with one condition
here but with two, and you can not generalize both without getting into
mathematical conflict.
Besides, some people seem to forget that this is a problem of physics
and not just of mathematics: the point is that the condition ' if x=ct
then x'=ct' 'only applies to events associated with the propagation of
a light signal. If instead you have a number of independent events in
the unprimed frame that by accident (or by some sort of arrangement)
follow the equation x=ct, you can not conclude that x'=ct' (the same
applies to x=-ct, x'=-ct'). You can therefore neither make the
generalizations x'-ct'=lambda*(x-ct) and x'+ct'=mu*(x+ct) unless
actually x=ct and x=-ct respectively and the events are associated with
the propagation of a light signal (after all there are no other linear
functions of x and t that describe the propagation of light apart from
x=ct and x=-ct).
Thomas
Dirk Van de moortel
It was you who made the following claim in
http://groups.google.com/group/sci.physics.research/msg/f1829512c28e5622
| The equations
|
| (1) (x' - ct') = lambda*(x - ct)
| (2) (x' + ct') = mu*(x + ct)
|
| can only hold for all values of x and t if lambda=mu.
And I have just shown that you gave a faulty proof for this.
Now I will give a simple proof of the fact that both equations (1)
and (2) hold for all pairs (x,t) and their corresponding (x',t').
Given this:
(G1) The transformation between coordinates (x,t) and (x',t') as used
in two inertial frames is assumed to be linear.
(G2) If all the events on a light signal are described by a world line in
one coordinate system with the equation x = c t, then the equation
transforms to x' = c t' in the other coordinate system and vice versa.
(G3) If all the events on a light signal are described by a world line in
one coordinate system with the equation x = -c t, then the equation
transforms to x' = -c t' in the other coordinate system and vice versa.
We prove this:
There are numbers Lambda and Mu such that the following equations
hold for all events that have coordinates (x,t) in one frame and the
corresponding coordinates (x',t') in the other frame:
( x' - c t' ) = Lambda * ( x - c t )
( x' + c t' ) = Mu * ( x + c t )
Proof:
Linearity of the transformation (given as G1) means that there are
some numbers P, Q, R, S such that the transformation can be written
as
{ x' = P x + Q t
{ t' = R x + S t
If you understand what a coordinate transformation is, then you
know that both equations are valid together for all events having
coordinate pair (x,t) in one frame and the corresponding pair (x',t')
in the other frame.
---------
We now use given (G2).
The transformation of the world line with equation
x = c t
can be found by inserting this in the transformation and eliminating
x and t. We find:
{ x' = P c t + Q t
{ t' = R c t + S t
resulting in
{ x' = (c P + Q) t
{ t' = (c R + S) t
which gives after eliminating t:
x' / t' = (c P + Q) / (c R + S)
or equivalently
x' = (c P + Q) / (c R + S) t'
In other words: given an event with coordinates (x,t) on the
light signal with equation
x = c t ,
we know that the corresponding coordinates (x',t') of that
same event satisfy the equation
x' = (c P + Q) / (c R + S) t'
Knowing with (G2) that these coordinates (x',t') satisfy the equation
x' = c t' ,
this implies that
(c P + Q) / (c R + S) = c
and thus that
(c P + Q) = c (c R + S)
and, for later use in (*):
Q - c S = - c (P - c R)
So we have found a relation between the coefficients P, Q, R
and S of the transformation. This relation is clearly independent
of the values of (x,t) and (x',t'), so it does not depend on the
chosen event. It is a constraint on the coefficients imposed by
the given (G2).
So now we take the transformation equations,
{ x' = P x + Q t
{ t' = R x + S t ,
which are valid for all events, and find that they satisfy
x' - c t' = P x + Q t - c (R x + S t)
= (P - c R) x + (Q - c S) t
= (P - c R) x - c (P - c R) t (*)
= (P - c R) (x - c t)
= Lambda ( x - c t ) ,
provided we take Lambda as
Lambda = P - c R.
---------
We now use given (G3).
The transformation of the world line with equation
x = - c t
can be found by inserting this in the transformation and eliminating
x and t. We find:
{ x' = - P c t + Q t
{ t' = - R c t + S t
resulting in
{ x' = (- c P + Q) t
{ t' = (- c R + S) t
which gives after eliminating t:
x' / t' = (Q - c P ) / (S - c R)
or equivalently
x' = - (Q - c P) / (c R - S) t'
In other words: given an event with coordinates (x,t) on the
light signal with equation
x = - c t ,
we know that the corresponding coordinates (x',t') of that
same event satisfy the equation
x' = - (Q - c P) / (c R - S) t'
Knowing with (G3) that these coordinates (x',t') satisfy the equation
x' = - c t' ,
this implies that
(Q - c P) / (c R - S) = c
and thus that
(Q - c P) = c (c R - S)
and, for later use in (**):
Q + c S = c (P + c R)
So we have found another relation between the coefficients P, Q, R
and S of the transformation. This relation is clearly independent
of the values of (x,t) and (x',t'), so it does not depend on the
chosen event. It is a constraint on the coefficients imposed by
the given (G3).
So now we take the transformation equations,
{ x' = P x + Q t
{ t' = R x + S t ,
which are valid for all events, and find that they satisfy
x' + c t' = P x + Q t + c (R x + S t)
= (P + c R) x + (Q + c S) t
= (P + c R) x + c (P + c R) t (**)
= (P + c R) (x + c t)
= Mu ( x + c t ) ,
provided we take Mu as
Mu = P + c R.
---------
So we have proven that given the transformation
{ x' = P x + Q t
{ t' = R x + S t ,
there are numbers Lambda and Mu such that the following
equations hold for all events that have coordinates (x,t) in
one frame and the corresponding coordinates (x',t') in the
other frame:
( x' - c t' ) = Lambda * ( x - c t )
( x' + c t' ) = Mu * ( x + c t ) ,
namely
Lambda = P - c R
Mu = P + c R
QED
---------
If you haven't seen any of the above at school yet, you have the
opportunity to learn this before your friends do. If there is something
you don't understand about it, I recommend that you ask on
sci.math. This is pure mathematics [- and actually I'm wondering
what all this is doing in sci.physics.research -]
Dirk Vdm
Dirk Van de moortel
So what?
The linear transformation
{ x' = a x - b c t
{ c t' = a c t -b x
is more general than the one with the specific well known values.
The fact that a more general form preserves a property does not
invalidate the fact that the more specific form would preserve
that same property.
If you take a still more general form, like you do below, you
will need other specific arguments and assumptions to force the
sphericality to be invariant under the transformation.
You have the logic in the wrong direction.
One does not *conclude* from (8) that
A^2-E^2 = D^2-B^2 =1 .
One does *satisfy* (8) if one takes -for example-
A^2-E^2 = D^2-B^2 =1
and D E = A B
>
> which obviously would not be enough to determine the constants (even if
> one takes B as known by associating it with the velocity v).
Of course not.
You need other assumptions and arguments to choose the
values 1.
>
> So it is quite obvious that only by making the incorrect conclusions
> (9)-(11), could one arrive at a specific solution for the
> coefficients.
Again, you have your logic wrong.
(9)-(11) are not conclusions. They are sufficient conditions.
What you are doing now, is attacking your own misconception
of the logic of the articles you try to understand.
Hopefully you will soon learn the difference between "sufficient"
and "necessary conditions", and between implications and
equivalences at school. Everything will become clearer then.
Try sci.math if you have problems with this. I'm sure that you
will get help there.
Dirk Vdm