Re: Derivation of Lorentz Transformation

[markwh04@yahoo.com]

>From Thomas Smid:
> [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.

[Thomas Smid]

markwh04@yahoo.com wrote:

> 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]

markwh04@yahoo.com wrote:

> 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]

markwh04@yahoo.com wrote:

> 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]

markwh04@yahoo.com wrote:

> 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]

markwh04@yahoo.com wrote:

> 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]

markwh04@yahoo.com wrote:

> 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]

markwh04@yahoo.com wrote:

> 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]

markwh04@yahoo.com wrote:

> 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]

markwh04@yahoo.com wrote:

> 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

[Dirk Van de moortel]

"Thomas Smid" <thomas.smid@gmail.com> wrote in message
news:1129136222.474476.97160@f14g2000cwb.googlegro ups.com...

> markwh04@yahoo.com wrote:
>
> > 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.

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.

>
> 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

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

[Dirk Van de moortel]

"Thomas Smid" <thomas.smid@gmail.com> wrote in message
news:1129136222.474476.97160@f14g2000cwb.googlegro ups.com...

> markwh04@yahoo.com wrote:
>
> > 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.

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.

>
> 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

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

[Dirk Van de moortel]

"Thomas Smid" <thomas.smid@gmail.com> wrote in message
news:1129136222.474476.97160@f14g2000cwb.googlegro ups.com...

> markwh04@yahoo.com wrote:
>
> > 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.

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.

>
> 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

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

[Dirk Van de moortel]

"Thomas Smid" <thomas.smid@gmail.com> wrote in message
news:1129136222.474476.97160@f14g2000cwb.googlegro ups.com...

> markwh04@yahoo.com wrote:
>
> > 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.

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.

>
> 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

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

[Dirk Van de moortel]

"Thomas Smid" <thomas.smid@gmail.com> wrote in message
news:1129136222.474476.97160@f14g2000cwb.googlegro ups.com...

> markwh04@yahoo.com wrote:
>
> > 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.

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.

>
> 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

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

[Dirk Van de moortel]

"Thomas Smid" <thomas.smid@gmail.com> wrote in message
news:1129136222.474476.97160@f14g2000cwb.googlegro ups.com...

> markwh04@yahoo.com wrote:
>
> > 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.

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.

>
> 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

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

[Dirk Van de moortel]

"Thomas Smid" <thomas.smid@gmail.com> wrote in message
news:1129136222.474476.97160@f14g2000cwb.googlegro ups.com...

> markwh04@yahoo.com wrote:
>
> > 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.

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.

>
> 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

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

[Dirk Van de moortel]

"Thomas Smid" <thomas.smid@gmail.com> wrote in message
news:1129136222.474476.97160@f14g2000cwb.googlegro ups.com...

> markwh04@yahoo.com wrote:
>
> > 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.

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.

>
> 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

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

[Dirk Van de moortel]

"Thomas Smid" <thomas.smid@gmail.com> wrote in message
news:1129136222.474476.97160@f14g2000cwb.googlegro ups.com...

> markwh04@yahoo.com wrote:
>
> > 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.

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.

>
> 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

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