Lorentz transformations - a derivation
Timo Nieminen
transformations that seem to be making slow progress, I thought this
might be a worthwhile contribution. Feel free to copy and inflict on
correspondents in such threads!
This is a first draft only, and so could have a nice assortment of errors.
0. Introduction
The aim is to provide a simple and general derivation of the
homogeneous Lorentz transformations, without assuming that axes are
parallel, or that motion is along the x-axis. (Perhaps a revision
to avoid the use of Cartesian coordinates might be useful?)
Keeping in mind that the Lorentz transformations relate coordinates in
two inertial reference frames, we will restrict our attention to
such reference frames. At first, we will simply assume that all the
reference frames are in uniform relative motion (ie unaccelerating
and no rotational motion), and later, when some physics is introduced,
we will introduce the inertiality.
Permission is given to use the content of this post, including
publishing online, re-posting, etc.
Comments and corrections welcome.
0.1 Notation
r denotes a position vector; a number can be appended to distinguish
between two different position vectors, eg r1, r2. The components of
the position vector will, in general, differ between reference frames.
t denotes a time as measured in a given reference frame.
d_ij is the Kronecker delta.
Coordinates are specified by x, y, z or x1, y1, z1 etc when necessary.
The reference frame in which position vectors and times will
be specified when necessary by a "subscript" letter eg r_a, r1_a,
t_a, or (t,r)_a. Coordinates are x_a, y_a, z_a.
The scalar product of two vectors a and b is denoted by a.b
The product of two scalars, or of two matrices, is denoted by a b
The transpose of a matrix a is written as aT
Vectors are written as matrices with a single column when
used in matrix expressions; ie a is a column vector,
aT is a row vector.
Where a matrix is written in terms of its elements, the notation
[ a b c; d e f; g h i ] will be used to avoid problems with
non-fixed-width fonts. Here, a b c are the elements of the first row,
d e f the elements of the 2nd row etc.
Periods are left off ends of sentences where they could cause
confusion with mathematical notation (see above).
1. Rotations in 3D space
Consider a 3D Euclidean space with a Cartesian coordinate system such
that the distance between two points r1 and r2 is
ds = sqrt( (r1 - r2).(r1 - r2) )
Note that the scalar product is, in terms of coordinates,
r1.r2 = g_11 x1 x2 + g_22 y1 y2 + g_33 z1 z2
where g_11, g_22, g_33 are the diagonal elements of the metric tensor g.
For a Cartesian coordinate system, we have g = d_ij
Note that we can write this as a matrix product:
r1.r2 = r1T g r2
which, in a Cartesian coordinate system, is r1.r2 = r1T r2
If we consider two Cartesian coordinate systems with coincident origins,
we can ask what linear transformations of coordinates result in
distances being invariant.
Such a transformation must be of the form:
x_b = a_11 x_a + a_12 y_a + a_13 z_a + c_1
y_b = a_21 x_a + a_22 y_a + a_23 z_a + c_2
z_b = a_31 x_a + a_32 y_a + a_33 z_a + c_3
or, more compactly, we can write this as a matrix equation
r_b = A r_a + C
Since we have specified that the origins are coincident, we have
C = (0,0,0); the transformation must be homogeneous.
If we have r_a = r1_a - r2_a, the distance between the points specified
by positions vectors r1_a and r2_a must be the same in both coordinate
systems. Therefore
ds^2 = ds_a^2 = ds_b^2
= r_b.r_b
= (A r_a).(A r_a)
= (A r_a)T (A r_a)
= rT_a AT A r_a
which, since this must also equal r_a.r_a, means that
AT A = I
ie the matrices are orthogonal, and
inv(A) = AT
Therefore, the square of the determinant of A is
|A|^2 = 1
We can further note that 3x3 matrices with |A|^2 = 1 form a group under
matrix multiplication, termed O(3) - the three-dimensional orthogonal
group.
We can identify two distinct classes of transformations in O(3):
|A| = +1, which are pure rotations, and |A| = -1, which are rotations
combined with a reflection.
That these transformations form a group means that:
1. The result of one rotation/reflection followed by another
rotation/reflection can be obtained by a single rotation/reflection.
2. If we replace pairs of rotation/reflection transformations by
equivalent single transformations, the order in which we do so does
not matter. (Note that this is associativity, not commutativity!)
3. There is a rotation/reflection which leaves the coordinates unchanged.
4. For any rotation/reflection, there is an inverse transform that
restores things to the original state.
If we exclude reflections (ie we restrict ourselves to pure rotations
with |A| = +1, which we will call proper rotations), these conditions
are still satisfied, so proper rotations also form a group, denoted
SO(3). Since all proper (ie reflection-free) rotations must form a
continuous group containing the identity transformation, this provides
a general way of identifying the subgroup we are interested in - it
must contain I. Euler's theorem states that all 3D orthogonal
transformations with |A| = +1 are rotations.
1.1 Rotations in n-dimensional space
We will make a diversion into n-dimensional rotations, to see how we can
parameterise rotations, and actually write down the elements of a
rotation matrix.
Note that the considerations in the above section apply equally to
dimensions other than 3 - SO(1), SO(2), SO(4) etc are the groups of
proper rotations in 1, 2, and 4 dimensions.
1.1.1 1D
Since in 1D, we have |A| = A_11, the only 1D rotation matrix is [1].
1.1.2 2D
The transformation A has 4 matrix elements, but the orthogonality
relations provide 3 equations relating these, so only one free
parameter is required to describe a rotation. Therefore, we can give
a single element of SO(2), and generate all other elements by raising
it to a power. That is, given G, an element of SO(2), G^a is also an
element. We can proceed by choosing an "infinitesimal generator" S such
G = exp(-S)
Thus, we have
G^a = exp( - a S )
Noting that |A| = exp(Tr(S)), the requirement that |A| = 1 means that
Tr(S) = 0. Since inv(A) = exp(S), and inv(A) = AT, we must have
ST = -S, so S is antisymmetric. Since this requires all diagonal
elements to be zero, we also have Tr(S) = 0
The matrix
S = [ 0 -1; 1 0 ]
is a suitable infinitesimal generator, since any 2x2 antisymmetric
matrix can be written as the product a S
S has an interesting property:
S^2 = [ -1 0; 0 -1], S^3 = [ 0 1; -1 0 ] = -S, S^4 = -S^2 = I
Therefore, if we write the series expansion for exp(-aS), all of the
higher powers of S can be reduced to S and S^2. Using this, we find
exp(-aS) = - sin(aS) - cos(a S^2)
Since S^2 = -I, we can write any 2D rotation matrix as
R = [ cos(a) sin(a); -sin(a) cos(a) ]
in which we can immediately recognise our (originally abstract)
parameter a as the angle of rotation.
1.1.3 3+D
The same considerations apply. We need only write a set of infinitesimal
generators which are a basis set in terms of which any antisymmetric
matrix can be written. A suitable basis is:
S_1 = [ 0 -1 0; 1 0 0; 0 0 0 ]
S_2 = [ 0 0 1; 0 0 0; -1 0 0 ]
S_3 = [ 0 0 0; 0 0 -1; 0 1 0 ]
and we can write any antisymmetric matrix as
S = a_1 S_1 + a_2 S_2 + a_3 S_3
We can proceed as for 2D (with somewhat more difficulty!) and write
down the 3D rotation matrix in terms of the 3 parameters a_i (left
as an exercise for the reader!)
The astute reader might note that the top left 2x2 block of S_1 is
exactly the same as our 2D S, and must behave in the same way, so
S_1^3 = -S_1, S_1^4 = -S_1^2 etc. The same also applies for S_2 and
S_3. In the simple case where two of the three parameters a_i are
zero, we obtain transformations which we can easily recognise as
rotations about the x, y, and z axes, with the non-zero parameter
being the angle of rotation.
The extension to dimensions higher than 3 is elementary, although
writing down the elements of R explicitly in terms of a_i becomes
progressively more painful.
2. The Lorentz transformations
The mathematics of rotations gives us a simple mechanism to derive
the Lorentz transformations.
Consider a 4D coordinate system with metric tensor
g_00 = -1, g_11 = 1, g_22 = 1, g_33 = 1
A length interval is then
ds = sqrt( rT g r )
Homogenous linear transformations which leave this invariant must
satisfy AT g A = g, and since |g| is non-zero, we must have |A|^2 = 1
Restricting ourselves to proper rotations, we have |A| = 1
Since we have a metric tensor not equal to I, we must explicitly
include it when writing down our generator and infinitesimal generators.
We now require (g S) to be antisymmetric (we actually required this
for rotations in Cartesian systems, but since (g S) = (I S) = S, we
didn't write it down.
Thus, a suitable basis set for the infinitesimal generators is:
S_1 = [ 0 1 0 0; 1 0 0 0; 0 0 0 0; 0 0 0 0 ]
S_2 = [ 0 0 1 0; 0 0 0 0; 1 0 0 0; 0 0 0 0 ]
S_3 = [ 0 0 0 1; 0 0 0 0; 0 0 0 0; 1 0 0 0 ]
S_4 = [ 0 0 0 0; 0 0 -1 0; 0 1 0 0; 0 0 0 0 ]
S_5 = [ 0 0 0 0; 0 0 0 1; 0 0 0 0; 0 -1 0 0 ]
S_6 = [ 0 0 0 0; 0 0 0 0; 0 0 0 -1; 0 0 1 0 ]
Clearly, if we have a_1 = a_2 = a_3 = 0, our transformations are 3D
rotations of the last 3 coordinates, leaving the first coordinate
unchanged.
Since we now have S_1^3 = S_1 and S_1^4 = S_1^2, if we have only a_1
non-zero, we obtain
R = [ cosh(a_1) -sinh(a_1) 0 0; -sinh(a_1) cosh(a_1) 0 0; 0 0 0 0; 0 0 0 0 ]
and similarly for having only a_2 or a_3 non-zero.
We now have the Lorentz transformations and a general recipe for
writing any Lorentz transformation in terms of 6 parameters, of
which 3 specify a 3D rotation of the last 3 coordinates. Now it
is time to intoduce some physics.
3. Lorentz transformations in physics
To make use of the above mathemachinery, we note that we can specify
an event - a combination of a position vector and a time - as a 4D
vector (at,r) = (ar,x,y,z) where a is a scale factor so that ar and
x (and y and z) have the same units. Since x has units of length, and
t has units of time, the scale factor a has units of velocity.
We adopt the postulate that the laws of physics are the same in all
inertial reference frames (the Principle of Relativity).
This requires us to specify what is meant by
an inertial reference frame: a reference frame in which an object acted
on by zero force is either stationary or moves in a straight line at
constant speed. This means that dr/dt is independent of time in all
reference frames, where r(t) is the position of the force-free object.
If the object is inertial in any single reference frame, it will be
inertial in any reference frame related to the first by a linear
transformation. Therefore, the Lorentz transformations relate
inertial reference frames.
We adopt a further postulate: that the Maxwell equations correctly
describe the propagation of electromagnetic waves in free space in
all inertial reference frames. Directly from this, we see that the
speed of light in free space, c, must be the same in all in inertial
reference frames.
Therefore, c is a good choice of scale factor, since it must be the
same in all inertial reference frames, so we write our 4-coordinates
as (ct,r). It is worth noting that if we postulate instead that
either (a) we can use the same scale factor in all inertial reference
frames or (b) that there is a speed that is the same in all inertial
reference frames, we reach the same point, but without having identified
our scale factor as the speed of light in free space. In that way,
we could obtain a result that would be undisturbed by falsification of
the Maxwell equations (eg by measurement of a non-zero photon mass).
However, we will be content to use the historical postulate.
If we consider two event: the launching of a pulse of light, with
4-coordinates (ct1,r1), and its reception (ct2,r2), if the speed of
light is to be the same in all inertial reference frames, we must
have sqrt((r2 - r1).(r2 - r1))/(t2 - t1) = c in all frames. Therefore,
sqrt((r2-r1).(r2-r1)) = ct2 - ct1
(r2-r1).(r2-r1) = (ct2 - ct1)^2
-(ct2 - ct1)^2 + (r2-r1).(r2-r1) = 0
If we write (ct,r) = (ct2,r2) - (ct1,r1), the left hand side of the
above expression is
(ct,r).(ct,r) = (ct,r)T g (ct,r)
Therefore, a linear transformation under which the scalar product
invariant under a metric g_00 = -1, g_11 = g_22 = g_33 = 1 is
invariant results in the speed of light being the same in all
inertial reference frames.
The Lorentz transformations obtained in section 2 are the
transformations which meet these requirements, and therefore must
be the correct transformations relating coordinates (ct,r) in
different reference frames, if the Principle of Relativity is valid,
and the Maxwell equations are correct.
The parameters (a_4,a_5,a_6) are those required to specify a spatial
rotation. What are the other three parameters (a_1,a_2,a_3)?
Since the space origins (r = 0) of different reference frames only
need to coincide at t = 0, clearly the reference frames can be
in relative motion.
As measured in frame a, the origin of frame b moves at a constant
velocity B = dr_a/d(ct_a). Since B is constant, and the 4-origins are
coincident, B = r_a/(ct_a), where (ct_a,r_a) = Lba (ct_b,0,0,0)
Noting the Lorentz transformation resulting from only a_1 being
non-zero, the velocity in such a case would be (-tanh(a_1),0,0),
and (0,-tanh(a_2),0) and (0,0,-tanh(a_3)) when a_2 and a_3 are
the only non-zero parameters, we must have
(a_1,a_2,a_3) = B atanh(|B|) / |B|
for the transformation from a to b (the transformation above was
from b to a) and we are done!
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
AaronB
Timo Nieminen wrote:
> Given the rather long threads on derivations of the Lorentz
> transformations that seem to be making slow progress, I thought this
> might be a worthwhile contribution. Feel free to copy and inflict on
> correspondents in such threads!
[snip]
I'm just curious, why did you bother to go through all the effort of
this? The transformations were derived successfully about 100 years
ago. I don't think that mathematics has changes so extensively in that
time that the derivations done then are no longer applicable now.
A.
Edward Green
There is much I could learn from this, but of course, that's going to
be a lot of work, so I'll just offer a two penny comment instead:
> The aim is to provide a simple and general derivation of the
> homogeneous Lorentz transformations, without assuming that axes are
> parallel, or that motion is along the x-axis. (Perhaps a revision
> to avoid the use of Cartesian coordinates might be useful?)
The entire idea of "Lorentz transformation" in fact insists we will
only consider coordinate systems which are, in their own internal way,
Cartesian. The group of Lorentz transformations define precisely the
maximum generality of coordinate systems (up to change of scale) in
which physics looks particularly simple. To consider more general
coordinate systems we have to explicitly include the metric in our
physical laws, which maintains the appearance of coordinate equality,
but conceals the fact that some coordinate systems are more equal than
others.
Eugene Shubert
derivation is counterproductive to good science.
I believe physics is more interesting when special relativity
is derived from the weakest assumptions possible.
http://www.everythingimportant.org/relativity/special.pdf
Eli Botkin
I haven't gone through your derivation and so can't comment.
Just want to tell you that W. Pauli presents the general form of the
transformation equations as a footnote (pages 10 and 11) in his book "Theory
of Relativity", Pergamon Press, 1958.
Eli Botkin
"Timo Nieminen" <ti...@physics.uq.edu.au> wrote in message
news:Pine.LNX.4.50.0501061128280.13596-100000@localhost...
mme...@cars3.uchicago.edu
>Given the rather long threads on derivations of the Lorentz
>transformations that seem to be making slow progress, I thought this
>might be a worthwhile contribution. Feel free to copy and inflict on
>correspondents in such threads!
>
>This is a first draft only, and so could have a nice assortment of errors.
>
>
>S has an interesting property:
>
>S^2 = [ -1 0; 0 -1], S^3 = [ 0 1; -1 0 ] = -S, S^4 = -S^2 = I
>
>Therefore, if we write the series expansion for exp(-aS), all of the
>higher powers of S can be reduced to S and S^2. Using this, we find
>
>exp(-aS) = - sin(aS) - cos(a S^2)
I think that you meant to write
exp(-aS) = -sin(a)S - cos(a)S^2
>
...
>
>Therefore, c is a good choice of scale factor, since it must be the
>same in all inertial reference frames, so we write our 4-coordinates
>as (ct,r). It is worth noting that if we postulate instead that
>either (a) we can use the same scale factor in all inertial reference
>frames or (b) that there is a speed that is the same in all inertial
>reference frames, we reach the same point, but without having identified
>our scale factor as the speed of light in free space. In that way,
>we could obtain a result that would be undisturbed by falsification of
>the Maxwell equations (eg by measurement of a non-zero photon mass).
>However, we will be content to use the historical postulate.
>
This is good for the historical connection, but does at time produce
the (misguided) perception that relativity is electrodynamic in
origin. So, there is something to say for the route of a) there may
be an invariant speed (but only one, Landau deals with this), then b)
we do observe that c is invariant. In other words to convey the point
that the proper statement is not "c is the speed of light" but "the
speed of light is c".
Nice job, saved it for future reference. Just don't expect it to
penetrate the crackpots thick skulls:-)
Mati Meron | "When you argue with a fool,
me...@cars.uchicago.edu | chances are he is doing just the same"
Timo Nieminen
> Timo Nieminen <ti...@physics.uq.edu.au> writes:
> >
> >S has an interesting property:
> >
> >S^2 = [ -1 0; 0 -1], S^3 = [ 0 1; -1 0 ] = -S, S^4 = -S^2 = I
> >
> >Therefore, if we write the series expansion for exp(-aS), all of the
> >higher powers of S can be reduced to S and S^2. Using this, we find
> >
> >exp(-aS) = - sin(aS) - cos(a S^2)
>
> I think that you meant to write
>
> exp(-aS) = -sin(a)S - cos(a)S^2
Er, yes.
> >Therefore, c is a good choice of scale factor, since it must be the
> >same in all inertial reference frames, so we write our 4-coordinates
> >as (ct,r). It is worth noting that if we postulate instead that
> >either (a) we can use the same scale factor in all inertial reference
> >frames or (b) that there is a speed that is the same in all inertial
> >reference frames, we reach the same point, but without having identified
> >our scale factor as the speed of light in free space. In that way,
> >we could obtain a result that would be undisturbed by falsification of
> >the Maxwell equations (eg by measurement of a non-zero photon mass).
> >However, we will be content to use the historical postulate.
> >
> This is good for the historical connection, but does at time produce
> the (misguided) perception that relativity is electrodynamic in
> origin. So, there is something to say for the route of a) there may
> be an invariant speed (but only one, Landau deals with this), then b)
> we do observe that c is invariant. In other words to convey the point
> that the proper statement is not "c is the speed of light" but "the
> speed of light is c".
I think that, ideally, if one is using the modern postulate, rather than
the historical postulate, then it's best to spend a little time on
mechanics. Show that a massless object with any energy travels at the
invariant speed c, from which it follows that it travels at c in all
frames. Then, given the observation that photons are massless, it's pretty
clear what the speed of light has to be.
The historical postulate avoids this (but however assumes the Maxwell
equations - this piece had its origins in notes for an EM course where
that's already on deck), so I used it. I did feel that a note on the
alternative modern postulate was needed.
> Nice job, saved it for future reference. Just don't expect it to
> penetrate the crackpots thick skulls:-)
It's one way to get them to (supposedly) killfile me.
Bill Hobba
"Edward Green" <spamsp...@netzero.com> wrote in message
news:1104979077....@f14g2000cwb.googlegroups.com...
> Timo Nieminen wrote:
>
> There is much I could learn from this, but of course, that's going to
> be a lot of work, so I'll just offer a two penny comment instead:
>
> > The aim is to provide a simple and general derivation of the
> > homogeneous Lorentz transformations, without assuming that axes are
> > parallel, or that motion is along the x-axis. (Perhaps a revision
> > to avoid the use of Cartesian coordinates might be useful?)
>
> The entire idea of "Lorentz transformation" in fact insists we will
> only consider coordinate systems which are, in their own internal way,
> Cartesian.
No - it considers inertial coordinate systems. Inertial coordinates is a
different concept than Cartesian coordinates.
> The group of Lorentz transformations define precisely the
> maximum generality of coordinate systems (up to change of scale) in
> which physics looks particularly simple.
The Lorentz transformations apply to inertial frames. And yes the laws of
physics are often simpler in inertial frames. So?
> To consider more general
> coordinate systems we have to explicitly include the metric in our
> physical laws,
So Nuv does not appear in SR?
> which maintains the appearance of coordinate equality,
> but conceals the fact that some coordinate systems are more equal than
> others.
Some coordinate systems are more equal than others?????? Please describe to
me what these are and why they are more equal. The principle of general
invariance says otherwise. Note this does not say we can not tell say
inertial coordinate systems from non inertial ones - what is says is they
are equivalent ie no coordinate system is 'more equal' than others. There
are all equal in the formulation of the laws of physics.
Bill
Bill Hobba
<mme...@cars3.uchicago.edu> wrote in message
news:ix3Dd.22$25....@news.uchicago.edu...
Just adding my voice to what Mati said - nice job Timo. I too will keep it
for future reference. Thanks for the interesting post. It is good to see a
derivation direct from group theory.
Thanks
Bill
mme...@cars3.uchicago.edu
>On Thu, 6 Jan 2005 mme...@cars3.uchicago.edu wrote:
>
>> >
>> >
>> This is good for the historical connection, but does at time produce
>> the (misguided) perception that relativity is electrodynamic in
>> origin. So, there is something to say for the route of a) there may
>> be an invariant speed (but only one, Landau deals with this), then b)
>> we do observe that c is invariant. In other words to convey the point
>> that the proper statement is not "c is the speed of light" but "the
>> speed of light is c".
>
>I think that, ideally, if one is using the modern postulate, rather than
>the historical postulate, then it's best to spend a little time on
>mechanics. Show that a massless object with any energy travels at the
>invariant speed c, from which it follows that it travels at c in all
>frames. Then, given the observation that photons are massless, it's pretty
>clear what the speed of light has to be.
>
>The historical postulate avoids this (but however assumes the Maxwell
>equations - this piece had its origins in notes for an EM course where
>that's already on deck), so I used it. I did feel that a note on the
>alternative modern postulate was needed.
>
starting with a concrete case (Maxwell's equations) the way you do, and
only then turn around and show that the result is much more general.
>> Nice job, saved it for future reference. Just don't expect it to
>> penetrate the crackpots thick skulls:-)
>
>It's one way to get them to (supposedly) killfile me.
>
Wish you luck with this:-)
Timo Nieminen
> Timo Nieminen <ti...@physics.uq.edu.au> writes:
> >On Thu, 6 Jan 2005 mme...@cars3.uchicago.edu wrote:
> >
> >> This is good for the historical connection, but does at time produce
> >> the (misguided) perception that relativity is electrodynamic in
> >> origin. So, there is something to say for the route of a) there may
> >> be an invariant speed (but only one, Landau deals with this), then b)
> >> we do observe that c is invariant. In other words to convey the point
> >> that the proper statement is not "c is the speed of light" but "the
> >> speed of light is c".
> >
> >I think that, ideally, if one is using the modern postulate, rather than
> >the historical postulate, then it's best to spend a little time on
> >mechanics. Show that a massless object with any energy travels at the
> >invariant speed c, from which it follows that it travels at c in all
> >frames. Then, given the observation that photons are massless, it's pretty
> >clear what the speed of light has to be.
> >
> >The historical postulate avoids this (but however assumes the Maxwell
> >equations - this piece had its origins in notes for an EM course where
> >that's already on deck), so I used it. I did feel that a note on the
> >alternative modern postulate was needed.
> >
> Yes. Considering again, perhaps it is better to derive it first time
> starting with a concrete case (Maxwell's equations) the way you do, and
> only then turn around and show that the result is much more general.
Depends on whether it's part of a stand-alone relativity course, part of
an electromag course (my students will be on the receiving end of this in
2 months), or part of a mechanics course.
The latter should avoid tying it to Maxwell's equations or the speed of
light. Do the maths for Lorentz transformations and Galilei
transformations just as transformation groups sans physics, point out that
the Galilei transformations were assumed to hold and appeared to be OK.
Explore the physics entailed by the Lorentz transformations, and then
spring experimental results for time dilation on them.
The historical route really needs a more thorough coverage of the
experimental disproof of Galilei-symmetric ethers. Since there's a
plethora of such ethers (eg stationary, fully-dragged, partially-dragged -
and that's just one parameter) one needs to consider a whole constellation
of experiments. That doesn't belong in a mechanics course, even if it only
takes under an hour to cover briefly.
And while discussing motivations etc, I'll add (but more for the benefit
of the others who are reading than yours) that I prefer this group theory
approach over the usual rocketships and light signals approach since the
latter are usually restricted to a pure boost along the x-axis, and I
don't see that they make uniqueness obvious. The group theory approach
OTOH makes the uniqueness explicit.
> >> Nice job, saved it for future reference. Just don't expect it to
> >> penetrate the crackpots thick skulls:-)
> >
> >It's one way to get them to (supposedly) killfile me.
> >
> Wish you luck with this:-)
Heh! Androcles did before I could even get started on the maths.
Unfortunately it didn't stop hime from replying to my posts.
--
Timo
Timo Nieminen
> "Edward Green" <spamsp...@netzero.com> wrote:
> > Timo Nieminen wrote:
> >
> > > The aim is to provide a simple and general derivation of the
> > > homogeneous Lorentz transformations, without assuming that axes are
> > > parallel, or that motion is along the x-axis. (Perhaps a revision
> > > to avoid the use of Cartesian coordinates might be useful?)
> >
> > The entire idea of "Lorentz transformation" in fact insists we will
> > only consider coordinate systems which are, in their own internal way,
> > Cartesian.
>
> No - it considers inertial coordinate systems. Inertial coordinates is a
> different concept than Cartesian coordinates.
(mostly piggybacking, since I didn't see the original reply)
Well, the space part of the coordinates can be a Cartesian system, but
it's only a Cartesian coordinate system if it's a Cartesian coordinate
system. Nothing stops the space part of the coordinates being cylindrical
or spherical, whatever is most convenient.
The 4-coordinate system is not Cartesian. It looks Cartesian, but a metric
g_00 = -1, g_11 = g_22 = g_33 is not g_ij = d_ij.
Flat space, yes. Inertial in practice, but the Lorentz transforms
themselves say nothing about inertialness - one can use them to transform
between a set of non-inertial frames.
> > To consider more general
> > coordinate systems we have to explicitly include the metric in our
> > physical laws,
>
> So Nuv does not appear in SR?
Ah, but we already do! Check out the definitions of grad, div, curl when
the metric is not equal to d_ij. Stratton "Electromagnetic theory" has a
good coverage of coordinate systems and such things.
The standard notation of vector analysis is a really neat way of sweeping
ugly things like metric tensors under the carpet.
> > which maintains the appearance of coordinate equality,
> > but conceals the fact that some coordinate systems are more equal than
> > others.
Of course some coordinate systems are more equal than other, when it comes
down to the algebra or number-crunching. But that's problem dependent. And
often it's the (3D) coordinate systems with g not equal to d_ij that are
the most useful. I'm sure I could manage a good rant about why spherical
coordinates are better computationally than Cartesian coordinates.
--
Timo
Bill Hobba
"Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote in message
news:Pine.OSF.4.58.05...@dingo.cc.uq.edu.au...
> On Thu, 6 Jan 2005, Bill Hobba wrote:
>
> > "Edward Green" <spamsp...@netzero.com> wrote:
> > > Timo Nieminen wrote:
> > >
> > > > The aim is to provide a simple and general derivation of the
> > > > homogeneous Lorentz transformations, without assuming that axes are
> > > > parallel, or that motion is along the x-axis. (Perhaps a revision
> > > > to avoid the use of Cartesian coordinates might be useful?)
> > >
> > > The entire idea of "Lorentz transformation" in fact insists we will
> > > only consider coordinate systems which are, in their own internal way,
> > > Cartesian.
> >
> > No - it considers inertial coordinate systems. Inertial coordinates is
a
> > different concept than Cartesian coordinates.
>
> (mostly piggybacking, since I didn't see the original reply)
>
> Well, the space part of the coordinates can be a Cartesian system, but
> it's only a Cartesian coordinate system if it's a Cartesian coordinate
> system. Nothing stops the space part of the coordinates being cylindrical
> or spherical, whatever is most convenient.
You bet - which was part of what I meant by it is a different concept.
>
> The 4-coordinate system is not Cartesian. It looks Cartesian, but a metric
> g_00 = -1, g_11 = g_22 = g_33 is not g_ij = d_ij.
>
> Flat space, yes. Inertial in practice, but the Lorentz transforms
> themselves say nothing about inertialness - one can use them to transform
> between a set of non-inertial frames.
Hmmmmmm. I would like an example - can't see this right off. What comes to
mind is that in deriving the Lorentz transformations homogeneity and
isotropy is implicitly assumed - which is a defining property of an inertial
frame. But that is just off the top of my head - that does not imply the
converse - I am willing to be proven wrong.
>
> > > To consider more general
> > > coordinate systems we have to explicitly include the metric in our
> > > physical laws,
> >
> > So Nuv does not appear in SR?
>
> Ah, but we already do! Check out the definitions of grad, div, curl when
> the metric is not equal to d_ij. Stratton "Electromagnetic theory" has a
> good coverage of coordinate systems and such things.
>
> The standard notation of vector analysis is a really neat way of sweeping
> ugly things like metric tensors under the carpet.
Agreed.
>
> > > which maintains the appearance of coordinate equality,
> > > but conceals the fact that some coordinate systems are more equal than
> > > others.
>
> Of course some coordinate systems are more equal than other, when it comes
> down to the algebra or number-crunching.
You added context - and in that context I certainly agree. But general
statements like 'some coordinate systems are more equal than others' to me
are rather vacuous - which is why I object to them. But point taken - I
should object to them on that basis rather than assume they are trying to
say something I certainly do disagree with such as coordinate systems have
some kind of preference as far as the laws of nature are concerned.
>But that's problem dependent. And
> often it's the (3D) coordinate systems with g not equal to d_ij that are
> the most useful. I'm sure I could manage a good rant about why spherical
> coordinates are better computationally than Cartesian coordinates.
And I suspect your rant would be valid. Certainly some problems are better
done that way eg electrostatic problems involving round objects such as
metal spheres placed in electric fields.
Thanks
Bill
>
> --
> Timo
Timo Nieminen
> "Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote:
> > > >
> > Flat space, yes. Inertial in practice, but the Lorentz transforms
> > themselves say nothing about inertialness - one can use them to transform
> > between a set of non-inertial frames.
>
> Hmmmmmm. I would like an example - can't see this right off. What comes to
> mind is that in deriving the Lorentz transformations homogeneity and
> isotropy is implicitly assumed - which is a defining property of an inertial
> frame. But that is just off the top of my head - that does not imply the
> converse - I am willing to be proven wrong.
OK, see the last section in my original post. If one frame of reference
is inertial, every frame of reference obtained by a Lorentz transformation
is inertial. Conversely, if one frame of reference is not inertial, then
all frames of reference obtained by Lorentz transforms from it are also
not inertial. I can't think of any practical application of that.
The Lorentz transformations are just the transformations that preserve the
scalar product with a diagonal metric tensor (-1,+1,+1,+1).
This, in itself, does not require any laws of physics to be isotropic or
homogeneous, or even to hold. It does require the metric tensor to be
independent of the coordinates (so homogeneous in spacetime and
isotropic), but I would need to think somewhat more than I'm willing at
this time of night to decide whether an inertial frame requires isotropy
(given that Newton's laws - the validity of which defines an inertial
frame - are a statement of the conservation of momentum, and hence require
homogeneity in space, but what of homogeneity in time and isotropy?).
For example, the Galilean transformation also relate a set of inertial
frames, also with a metric tensor that is homogeneous and isotropic, or
given a non-inertial frame, a set of non-inertial frames.
Of course, the above probably require some qualification as to exactly
what is meant by Newton's laws, and the laws of mechanics being the same
in all inertial frames. Not for no reason is a distinction made between
coordinate time and proper time.
Basically, a metric tensor is not physics, any more than addition is
physics. The physics might well depend intimately on it, but that's
another matter.
> >But that's problem dependent. And
> > often it's the (3D) coordinate systems with g not equal to d_ij that are
> > the most useful. I'm sure I could manage a good rant about why spherical
> > coordinates are better computationally than Cartesian coordinates.
>
> And I suspect your rant would be valid. Certainly some problems are better
> done that way eg electrostatic problems involving round objects such as
> metal spheres placed in electric fields.
The core of my rant is that a Cartesian basis set for a general solution
of the Laplace or scalar or vector Helmholtz equation is continuous,
whereas the spherical coordinate basis set (solid spherical harmonics,
spherical wavefunctions j_n Y_{nm} or vector spherical wavefunctions) are
discrete. The Cartesian basis is a PITA to represent computationally,
never (for physically feasible cases) converges in the far field, while
the spherical basis has such really well-behaved convergence properties.
Beyond that, every plane wave of non-zero amplitude has infinite energy,
infinite momentum, and either undefined or zero or infinite angular
momemtum. That really sucks, when you're trying to find energy, momentum,
and angular momentum transfer during scattering.
If plane waves weren't the mathematically simplest solutions in the
mathematically simplest coordinate system, and a close enough
approximation to the local behaviour of a very large number of waves in
practice, why, they'd be confined to the dustbin of history!
--
Timo
Dirk Van de moortel
"Timo Nieminen" <ti...@physics.uq.edu.au> wrote in message news:Pine.LNX.4.50.0501061128280.13596-100000@localhost...
> transformations that seem to be making slow progress, I thought this
> might be a worthwhile contribution. Feel free to copy and inflict on
> correspondents in such threads!
Haven't checked very thoroughly but here's just a few
obvious math remarks...
[snip]
> 1.1.2 2D
>
> The transformation A has 4 matrix elements, but the orthogonality
> relations provide 3 equations relating these, so only one free
> parameter is required to describe a rotation. Therefore, we can give
> a single element of SO(2), and generate all other elements by raising
> it to a power. That is, given G, an element of SO(2), G^a is also an
> element. We can proceed by choosing an "infinitesimal generator" S such
>
> G = exp(-S)
>
> Thus, we have
>
> G^a = exp( - a S )
Perhaps it's better to replace all the G with A in the above, to
remain consistent with what follows:
>
> Noting that |A| = exp(Tr(S)), the requirement that |A| = 1 means that
> Tr(S) = 0. Since inv(A) = exp(S), and inv(A) = AT, we must have
> ST = -S, so S is antisymmetric. Since this requires all diagonal
> elements to be zero, we also have Tr(S) = 0
>
> The matrix
>
> S = [ 0 -1; 1 0 ]
>
> is a suitable infinitesimal generator, since any 2x2 antisymmetric
> matrix can be written as the product a S
>
> S has an interesting property:
>
> S^2 = [ -1 0; 0 -1], S^3 = [ 0 1; -1 0 ] = -S, S^4 = -S^2 = I
>
> Therefore, if we write the series expansion for exp(-aS), all of the
> higher powers of S can be reduced to S and S^2. Using this, we find
>
> exp(-aS) = - sin(aS) - cos(a S^2)
Typo: that should be
exp(-aS) = - sin(aS) + cos(a S^2)
or better
exp(-a S) = - sin(a S) + cos(a S^2)
[snip]
> 3. Lorentz transformations in physics
>
> To make use of the above mathemachinery, we note that we can specify
> an event - a combination of a position vector and a time - as a 4D
> vector (at,r) = (ar,x,y,z) where a is a scale factor so that ar and
Typo: "ar" should be "at" twice.
Don't forget that you use a space for scalar multiplication ;-)
Cheers,
Dirk Vdm
Bilge
>transformations that seem to be making slow progress, I thought this
>might be a worthwhile contribution. Feel free to copy and inflict on
>correspondents in such threads!
psychology than with physics.
[...]
>1. Rotations in 3D space
>
>Consider a 3D Euclidean space with a Cartesian coordinate system such
>that the distance between two points r1 and r2 is
>
>ds = sqrt( (r1 - r2).(r1 - r2) )
>
>Note that the scalar product is, in terms of coordinates,
>
>r1.r2 = g_11 x1 x2 + g_22 y1 y2 + g_33 z1 z2
Slight notational difficulty. You might want to write g_xx, g_yy, g_zz,
if you use numerical subscripts to denote the vector with which the
components are associated. Either g_xx x1 x2 + ... or something like,
A.B = g_11 A_1 B_1 + .... I also think you've misjudged your target
audience. Those readers who do not understand the lorentz transforms,
already think relativity is just mathematics. Writing down the properties
of exponentiated matrices and traces of matrices is not going to dispell
that notion at all. I also think you skip mentioning any relationship
between special relativity and the speed of light. The physics related
to special relativity and the lorentz transforms is contained entirely
in the first postulate. The second postulate logically belongs in
a theory of E&M, since it's easy to construct a theory of E&M which
is relativistically correct, but in which photons are not massless.
Einstein's motivation for the second postulate was to give maxwell's
equations a natural explanation in geometry. The second postulate is
basically an historical artifct of the era in which einstein published
his paper.
Dirk Van de moortel
"Bilge" <dub...@radioactivex.lebesque-al.net> wrote in message news:slrnctqo36...@radioactivex.lebesque-al.net...
> Timo Nieminen:
> >Given the rather long threads on derivations of the Lorentz
> >transformations that seem to be making slow progress, I thought this
> >might be a worthwhile contribution. Feel free to copy and inflict on
> >correspondents in such threads!
>
> Unfortunately, the length of the threads has more to do with
> psychology than with physics.
>
>
> [...]
> >1. Rotations in 3D space
> >
> >Consider a 3D Euclidean space with a Cartesian coordinate system such
> >that the distance between two points r1 and r2 is
> >
> >ds = sqrt( (r1 - r2).(r1 - r2) )
> >
> >Note that the scalar product is, in terms of coordinates,
> >
> >r1.r2 = g_11 x1 x2 + g_22 y1 y2 + g_33 z1 z2
>
> Slight notational difficulty. You might want to write g_xx, g_yy, g_zz,
> if you use numerical subscripts to denote the vector with which the
> components are associated. Either g_xx x1 x2 + ... or something like,
> A.B = g_11 A_1 B_1 + .... I also think you've misjudged your target
> audience. Those readers who do not understand the lorentz transforms,
> already think relativity is just mathematics. Writing down the properties
> of exponentiated matrices and traces of matrices is not going to dispell
> that notion at all.
I think (and surely hope) that Timo was talking ironically
if not sarcastically :-)
Dirk Vdm
Bill Hobba
"Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote in message
news:Pine.OSF.4.58.05...@dingo.cc.uq.edu.au...
>
> > "Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote:
> > > > >
> > > Flat space, yes. Inertial in practice, but the Lorentz transforms
> > > themselves say nothing about inertialness - one can use them to
transform
> > > between a set of non-inertial frames.
> >
> > Hmmmmmm. I would like an example - can't see this right off. What
comes to
> > mind is that in deriving the Lorentz transformations homogeneity and
> > isotropy is implicitly assumed - which is a defining property of an
inertial
> > frame. But that is just off the top of my head - that does not imply
the
> > converse - I am willing to be proven wrong.
>
> OK, see the last section in my original post. If one frame of reference
> is inertial, every frame of reference obtained by a Lorentz transformation
> is inertial. Conversely, if one frame of reference is not inertial, then
> all frames of reference obtained by Lorentz transforms from it are also
> not inertial. I can't think of any practical application of that.
>
> The Lorentz transformations are just the transformations that preserve the
> scalar product with a diagonal metric tensor (-1,+1,+1,+1).
>
> This, in itself, does not require any laws of physics to be isotropic or
> homogeneous, or even to hold.
Ahhhhh - I think I see where you are coming from now. But I am also
thinking of Wald - General Relativity where he says page 57 - 'The principle
of general covariance in this context states that the metric of space is the
only quantity pertaining to space that can appear in the laws of physics'.
This would seem to imply if the metric was Nub throughout the frame for all
time then any law of physics that did exist must not depend on position,
direction or when it was done. And as I argue later I think this is in fact
a defining property of an inertial frame.
> It does require the metric tensor to be
> independent of the coordinates (so homogeneous in spacetime and
> isotropic), but I would need to think somewhat more than I'm willing at
> this time of night
Saw your post as I was about to head off to bed. Feeling a bit tired so I
hope what I say is not too muddled.
>to decide whether an inertial frame requires isotropy
> (given that Newton's laws - the validity of which defines an inertial
> frame - are a statement of the conservation of momentum, and hence require
> homogeneity in space, but what of homogeneity in time and isotropy?).
This is examined in Rindler - Introduction to Special Relativity - page 6.
Rindler claims this follows immediately from the POR. Also see Landau -
Mechanics page 5 where Landau defines an inertial frame by its symmetry
properties - i.e. it must be homogenious in space and time and isotropic in
space. In fact I prefer that definition to ones based on free particles
because it avoids the problem of exactly what is meant by free.
>
> For example, the Galilean transformation also relate a set of inertial
> frames, also with a metric tensor that is homogeneous and isotropic, or
> given a non-inertial frame, a set of non-inertial frames.
>
Yes - in fact they result if that speed limit you used in your derivation
was infinity - thus in a sense it is a specific example of the Lorentz
transformations.
>
> Of course, the above probably require some qualification as to exactly
> what is meant by Newton's laws, and the laws of mechanics being the same
> in all inertial frames. Not for no reason is a distinction made between
> coordinate time and proper time.
Absolutely. This is another reason why I prefer Landaus definition. It
make it clear it is the symetry properties of an inertial frame than leads
to the conservation laws via the PLA and good old Noethers theorem.
>
> Basically, a metric tensor is not physics, any more than addition is
> physics. The physics might well depend intimately on it, but that's
> another matter.
>
> > >But that's problem dependent. And
> > > often it's the (3D) coordinate systems with g not equal to d_ij that
are
> > > the most useful. I'm sure I could manage a good rant about why
spherical
> > > coordinates are better computationally than Cartesian coordinates.
> >
> > And I suspect your rant would be valid. Certainly some problems are
better
> > done that way eg electrostatic problems involving round objects such as
> > metal spheres placed in electric fields.
>
> The core of my rant is that a Cartesian basis set for a general solution
> of the Laplace or scalar or vector Helmholtz equation is continuous,
> whereas the spherical coordinate basis set (solid spherical harmonics,
> spherical wavefunctions j_n Y_{nm} or vector spherical wavefunctions) are
> discrete. The Cartesian basis is a PITA to represent computationally,
> never (for physically feasible cases) converges in the far field, while
> the spherical basis has such really well-behaved convergence properties.
> Beyond that, every plane wave of non-zero amplitude has infinite energy,
> infinite momentum, and either undefined or zero or infinite angular
> momemtum. That really sucks, when you're trying to find energy, momentum,
> and angular momentum transfer during scattering.
Yep.
>
> If plane waves weren't the mathematically simplest solutions in the
> mathematically simplest coordinate system, and a close enough
> approximation to the local behaviour of a very large number of waves in
> practice, why, they'd be confined to the dustbin of history!
Thanks
Bill
>
> --
> Timo
Franz Heymann
"Timo Nieminen" <ti...@physics.uq.edu.au> wrote in message
news:Pine.LNX.4.50.0501061128280.13596-100000@localhost...
[snip]
Timo, thanks for a nice derivation. I am saving it. However, it is
unlikely that it
will be helpful in the case
of the present disputation between Randy and Androcles, as the latter
appears not even to be on close terms with teenage algebra, as evinced
by the fact that he does not understand the one-dimensional derivation
of the Lorentz transform, let alone the mathematical concepts you use
in your derivation.
In any case, Androcles is not actually interested in the truth. He is
simply grinding an axe, come what may.
Franz
John Schoenfeld
mme...@cars3.uchicago.edu wrote:
> In other words to convey the point
> that the proper statement is not "c is the speed of light" but "the
> speed of light is c".
Wrong. Equality is symmetric.
Dirk Van de moortel
"John Schoenfeld" <j.scho...@programmer.net> wrote in message news:1105036434.3...@c13g2000cwb.googlegroups.com...
And symmetry is equal?
Or simply, symmetric is equality?
Dirk Vdm
Randy Poe
Franz Heymann wrote:
> "Timo Nieminen" <ti...@physics.uq.edu.au> wrote in message
> news:Pine.LNX.4.50.0501061128280.13596-100000@localhost...
>
> [snip]
>
> Timo, thanks for a nice derivation. I am saving it. However, it is
> unlikely that it
> will be helpful in the case
> of the present disputation between Randy and Androcles
That "disputation" is prety much at an end. Androcles'
rock-hard self-contradictory position that he accepts
the PoR but not the invariance of c has erected a
wall of ignorance which leaves no further chinks
to explore.
However, I would find an alternate derivation of SR
very helpful. What is the "modern", non-electrodynamic
statement of the principal of relativity? I noted that
Timo ultimately ended up using the invariance of c in
deriving the Lorentz transformation. Is there another
approach that avoids that?
> , as the latter
> appears not even to be on close terms with teenage algebra, as
evinced
> by the fact that he does not understand the one-dimensional
derivation
> of the Lorentz transform,
Worse than that. I've been unable to get him to accept
a definition of "length of a moving object".
> let alone the mathematical concepts you use
> in your derivation.
>
> In any case, Androcles is not actually interested in the truth. He
is
> simply grinding an axe, come what may.
Androcles has a reflexive need to disbelieve anything
certain people say, no matter how elementary. It reminds
me of a story I read once about some old codger with
such an extreme level of disbelief that he eventually
disbelieved himself out of existence.
- Randy
Randy Poe
Franz Heymann wrote:
> "Timo Nieminen" <ti...@physics.uq.edu.au> wrote in message
> news:Pine.LNX.4.50.0501061128280.13596-100000@localhost...
>
> [snip]
>
> Timo, thanks for a nice derivation. I am saving it. However, it is
> unlikely that it
> will be helpful in the case
That "disputation" is prety much at an end. Androcles'
rock-hard self-contradictory position that he accepts
the PoR but not the invariance of c has erected a
wall of ignorance which leaves no further chinks
to explore.
However, I would find an alternate derivation of SR
very helpful. What is the "modern", non-electrodynamic
statement of the principal of relativity? I noted that
Timo ultimately ended up using the invariance of c in
deriving the Lorentz transformation. Is there another
approach that avoids that?
> , as the latter
> appears not even to be on close terms with teenage algebra, as
evinced
> by the fact that he does not understand the one-dimensional
derivation
> of the Lorentz transform,
Worse than that. I've been unable to get him to accept
a definition of "length of a moving object".
> let alone the mathematical concepts you use
> in your derivation.
>
> In any case, Androcles is not actually interested in the truth. He
is
> simply grinding an axe, come what may.
Androcles has a reflexive need to disbelieve anything
Dirk Van de moortel
"Randy Poe" <poespa...@yahoo.com> wrote in message news:1105038227....@f14g2000cwb.googlegroups.com...
>
> Franz Heymann wrote:
> > "Timo Nieminen" <ti...@physics.uq.edu.au> wrote in message
> > news:Pine.LNX.4.50.0501061128280.13596-100000@localhost...
> >
> > [snip]
> >
> > Timo, thanks for a nice derivation. I am saving it. However, it is
> > unlikely that it
> > will be helpful in the case
> > of the present disputation between Randy and Androcles
>
> That "disputation" is prety much at an end. Androcles'
> rock-hard self-contradictory position that he accepts
> the PoR but not the invariance of c has erected a
> wall of ignorance which leaves no further chinks
> to explore.
>
> However, I would find an alternate derivation of SR
> very helpful. What is the "modern", non-electrodynamic
> statement of the principal of relativity? I noted that
> Timo ultimately ended up using the invariance of c in
> deriving the Lorentz transformation. Is there another
> approach that avoids that?
Randy, I think you might like to have a look at section 10.8 of
http://www.courses.fas.harvard.edu/~phys16/Textbook/ch10.pdf
referring to section 14.9 of
http://www.courses.fas.harvard.edu/~phys16/Textbook/ch14_appendices.pdf
up to equation (14.83).
>
> > , as the latter
> > appears not even to be on close terms with teenage algebra, as
> > evinced
> > by the fact that he does not understand the one-dimensional
> > derivation of the Lorentz transform,
>
> Worse than that. I've been unable to get him to accept
> a definition of "length of a moving object".
>
> > let alone the mathematical concepts you use
> > in your derivation.
> >
> > In any case, Androcles is not actually interested in the truth. He
> > is simply grinding an axe, come what may.
>
> Androcles has a reflexive need to disbelieve anything
> certain people say, no matter how elementary. It reminds
> me of a story I read once about some old codger with
> such an extreme level of disbelief that he eventually
> disbelieved himself out of existence.
Good one :-))
Cheers,
Dirk Vdm
Uncle Al
1) Descartes enters a Parisian cafe. Some hours later a waiter
wanders over and inquires as to whether Descartes would like an
aparitif.
"Descartes, "I think not."
and poof! he was gone.
2) God was an atheist. Poof!
You've done a noble deed, Randy. You have demonstrated well beyond
any rational doubt that stupidity is religion not science. Stupidity
is faith-based. Any evidence broached in opposition to stupidity's
inerrant path is a test of faith and therefore supportive of the
stupid position. "The Wicker Man"'s denoument has a devout Christian
in a shouting match with a devout pagan. The audience soon starts
laughing. They are indistinguishable! The Christian is subsequently
burned alive as sacrifice, all the while in indomitable denial of
pagan worship singing a hymn about Christ's sacrifice. Too funny!
Uncle Al suspects archeologists should be searching for remains of the
really large button battery that powered Stonehenge.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz.pdf
Daryl McCullough
>Androcles has a reflexive need to disbelieve anything
>certain people say, no matter how elementary. It reminds
>me of a story I read once about some old codger with
>such an extreme level of disbelief that he eventually
>disbelieved himself out of existence.
I think the story you are thinking of is
"Obstinate Uncle Otis", found in the collection
Hitchcock Ghostly Gallery (Alfred Hitchcock's Story Collection for Young
Readers).
--
Daryl McCullough
Ithaca, NY
Androcles
"Uncle Al" <Uncl...@hate.spam.net> wrote in message
news:41DD92F5...@hate.spam.net...
> Randy Poe wrote:
>>
>> Franz Heymann wrote:
>> > "Timo Nieminen" <ti...@physics.uq.edu.au> wrote in message
>> > news:Pine.LNX.4.50.0501061128280.13596-100000@localhost...
>> >
>> > [snip]
>> >
>> > Timo, thanks for a nice derivation. I am saving it. However, it
>> > is
>> > unlikely that it
>> > will be helpful in the case
>> > of the present disputation between Randy and Androcles
>>
>> That "disputation" is prety much at an end. Androcles'
>> rock-hard self-contradictory position that he accepts
>> the PoR but not the invariance of c has erected a
>> wall of ignorance which leaves no further chinks
>> to explore.
Too right it does.
Relativity: The Special and General Theory. 1920.
VII. The Apparent Incompatibility of the Law of Propagation of Light
with the Principle of Relativity
Einstein.
>>
>> However, I would find an alternate derivation of SR
>> very helpful. What is the "modern", non-electrodynamic
>> statement of the principal of relativity? I noted that
>> Timo ultimately ended up using the invariance of c in
>> deriving the Lorentz transformation. Is there another
>> approach that avoids that?
>>
>> > , as the latter
>> > appears not even to be on close terms with teenage algebra, as
>> evinced
>> > by the fact that he does not understand the one-dimensional
>> derivation
>> > of the Lorentz transform,
>>
>> Worse than that. I've been unable to get him to accept
>> a definition of "length of a moving object".
>>
>> > let alone the mathematical concepts you use
>> > in your derivation.
>> >
>> > In any case, Androcles is not actually interested in the truth. He
>> is
>> > simply grinding an axe, come what may.
>>
>> Androcles has a reflexive need to disbelieve anything
>> certain people say, no matter how elementary.
Any relativist says, sure. A load of ignorami, the lot of them.
[snip crap]
Stooopid ignoramus Schwartz.
You boy Poe is losing, contradicting himself constantly.
Even Einstein tries to change the PoR, KNOWING
it is incompatible with his second postulate.
You and arseholes like you have had your head up your
wall of ignorance since 1905.
Androcles.
John Schoenfeld
Almost a paradox. But,
http://mathworld.wolfram.com/Equal.html
http://mathworld.wolfram.com/Defined.html
> Dirk Vdm
Bill Hobba
"Bill Hobba" <bho...@rubbish.net.au> wrote in message
news:75cDd.107056$K7.6...@news-server.bigpond.net.au...
As mentioned a bit later I posted this just before I headed to bed and was a
bit tired. Ignore the above. A simple counter example proves it wrong eg
LET. Wald assumed the principle of general covariance - theories exist
where the Lorentz transformations are true and it does not hold. I
understand where Timo was coming from now.
>
> > It does require the metric tensor to be
> > independent of the coordinates (so homogeneous in spacetime and
> > isotropic), but I would need to think somewhat more than I'm willing at
> > this time of night
>
> Saw your post as I was about to head off to bed. Feeling a bit tired so I
> hope what I say is not too muddled.
Yea - as I am now acutely aware.
>
> >to decide whether an inertial frame requires isotropy
> > (given that Newton's laws - the validity of which defines an inertial
> > frame - are a statement of the conservation of momentum, and hence
require
> > homogeneity in space, but what of homogeneity in time and isotropy?).
>
> This is examined in Rindler - Introduction to Special Relativity - page 6.
> Rindler claims this follows immediately from the POR. Also see Landau -
> Mechanics page 5 where Landau defines an inertial frame by its symmetry
> properties - i.e. it must be homogenious in space and time and isotropic
in
> space. In fact I prefer that definition to ones based on free particles
> because it avoids the problem of exactly what is meant by free.
Again I forgot about specific counter examples eg LET and aether drag type
theories.
Thanks
Bill
Bill Hobba
>
> Franz Heymann wrote:
> > "Timo Nieminen" <ti...@physics.uq.edu.au> wrote in message
> > news:Pine.LNX.4.50.0501061128280.13596-100000@localhost...
> >
> > [snip]
> >
> > Timo, thanks for a nice derivation. I am saving it. However, it is
> > unlikely that it
> > will be helpful in the case
> > of the present disputation between Randy and Androcles
>
> That "disputation" is prety much at an end. Androcles'
> rock-hard self-contradictory position that he accepts
> the PoR but not the invariance of c has erected a
> wall of ignorance which leaves no further chinks
> to explore.
>
> However, I would find an alternate derivation of SR
> very helpful. What is the "modern", non-electrodynamic
> statement of the principal of relativity? I noted that
> Timo ultimately ended up using the invariance of c in
> deriving the Lorentz transformation. Is there another
> approach that avoids that?
Take a peek at the following I have posted many times. Not as elegant
mathematically as what Timo wrote - but still ok IMHO. Indeed what Tom
Roberts wrote can be very easily extended using Timo's methods (which are
usually done in reverse - eg the Poincare group is derived from the Lorentz
transformations on page 171 of Classical Mechanics, Quantum Mechanics and
Filed Theory by Amnon Katz).
http://arxiv.org/abs/physics/0110076,
and ancient, but I still think excellent post by Tom Roberts
http://groups.google.com/groups?hl=en&lr=&c2coff=1&selm=54jfst%24glp%40ssbunews.ih.lucent.com
and chapter 10 of
http://www.courses.fas.harvard.edu/~phys16/Textbook/
under the heading of Relativity without c.
Timo Nieminen
At first thought I agree. However, this is bringing in some physics in
addition to the geometric statement made by the metric tensor.
Consider the following from M. Bunge, Foundations of Physics, pp 131-132:
Note the difference between:
(a) (i) E3 is a three-dimensional Euclidean space (ii) E3 maps physical
space
(b) Physical space has a Euclidean structure.
(a) separates the mathematical assumptions and their physical
interpretation.
Really, what you note above is the justification for restricting ourselves
to reference frames with uniform metric tensor when dealing with inertial
reference frames. Perhaps this a good thing to make explicit?
I think that our above statements only apply when the space coordinates
are Cartesian.
> And as I argue later I think this is in fact
> a defining property of an inertial frame.
At least for some definitions of inertial frame. More below.
> > (given that Newton's laws - the validity of which defines an inertial
> > frame - are a statement of the conservation of momentum, and hence require
> > homogeneity in space, but what of homogeneity in time and isotropy?).
>
> This is examined in Rindler - Introduction to Special Relativity - page 6.
> Rindler claims this follows immediately from the POR. Also see Landau -
> Mechanics page 5 where Landau defines an inertial frame by its symmetry
> properties - i.e. it must be homogenious in space and time and isotropic in
> space. In fact I prefer that definition to ones based on free particles
> because it avoids the problem of exactly what is meant by free.
If we state that an inertial frame is one in which all of the laws of
mechanics hold good, then we have conservation of energy, momentum, and
angular momentum, and thus space-time homogeneity and isotropy.
If we state that an inertial frame is one in which Newton 1-2 hold (which,
without Newton 3, doesn't even give us conservation of momentum), then we
don't yet have those requirements. Adding that the laws of physics must
hold in all inertial frames, and that conservation of energy, momentum,
and angular momentum are laws of physics, then the PoR requires
homogeneity and isotropy.
The first is neater, but requires a much stronger initial assumption.
Timo Nieminen
> Timo Nieminen:
>
> [...]
> >1. Rotations in 3D space
> >
> >Consider a 3D Euclidean space with a Cartesian coordinate system such
> >that the distance between two points r1 and r2 is
> >
> >ds = sqrt( (r1 - r2).(r1 - r2) )
> >
> >Note that the scalar product is, in terms of coordinates,
> >
> >r1.r2 = g_11 x1 x2 + g_22 y1 y2 + g_33 z1 z2
>
> Slight notational difficulty. You might want to write g_xx, g_yy, g_zz,
> if you use numerical subscripts to denote the vector with which the
> components are associated. Either g_xx x1 x2 + ... or something like,
> A.B = g_11 A_1 B_1 + ....
My typeset original used dashes. ASCII-all-on-one-line makes me a bit wary
of having both subscripts and superscripts on the one entity. Your
suggestion of g_xx is good except I'd have to think about g_ctct vs g_tt.
g_ww is tempting ...
> I also think you've misjudged your target
> audience. Those readers who do not understand the lorentz transforms,
> already think relativity is just mathematics. Writing down the properties
> of exponentiated matrices and traces of matrices is not going to dispell
> that notion at all.
Well, the real target audience is students in a graduate EM course, but I
thought that some people here might be interested in it. The thought of
the most-persistent Randy Poe dumping it verbatim into his threads amused
me, though.
> I also think you skip mentioning any relationship
> between special relativity and the speed of light. The physics related
> to special relativity and the lorentz transforms is contained entirely
> in the first postulate. The second postulate logically belongs in
> a theory of E&M, since it's easy to construct a theory of E&M which
> is relativistically correct, but in which photons are not massless.
> Einstein's motivation for the second postulate was to give maxwell's
> equations a natural explanation in geometry. The second postulate is
> basically an historical artifct of the era in which einstein published
> his paper.
Yes. Discussed this somwehat with Mati.
Eugene Shubert
school, no doubt, in the physics department, is a mathematically
inept perspective on physics.
An elegant and mathematically correct approach is given here:
http://www.everythingimportant.org/relativity/special.pdf
Timo Nieminen
> >
> > However, I would find an alternate derivation of SR
> > very helpful. What is the "modern", non-electrodynamic
> > statement of the principal of relativity? I noted that
> > Timo ultimately ended up using the invariance of c in
> > deriving the Lorentz transformation. Is there another
> > approach that avoids that?
First, note my discussion with Mati on alternative postulates, and Bilge's
comment.
Secondly, note that the derivation of the Lorentz transformation doesn't
make of the invariance of c. Now, I did use the invariance of c to show
that, given Maxwell's equations as a correct law of physics, then the
Lorentz transformations are the correct transformations between inertial
frames - ie tying the maths to the physics. But the Lorentz
transformations themselves don't require it.
I was going to feed that derivation past Androcles (in smaller portions),
but he wanted me to adopt the Lorentz transformations as a starting
assumption. Since that seemed a truly bizarre thing to do if one is
wanting to derive them, I refused, and he killfiled me.
Maybe you can have similar luck?
David McAnally
>"Uncle Al" <Uncl...@hate.spam.net> wrote in message
>news:41DD92F5...@hate.spam.net...
>> Randy Poe wrote:
<snip>
>>> That "disputation" is prety much at an end. Androcles'
>>> rock-hard self-contradictory position that he accepts
>>> the PoR but not the invariance of c has erected a
>>> wall of ignorance which leaves no further chinks
>>> to explore.
>Too right it does.
>Relativity: The Special and General Theory. 1920.
>VII. The Apparent Incompatibility of the Law of Propagation of Light
>with the Principle of Relativity
>Einstein.
It looks like Androcles does not know the meaning of the word "apparent".
He would appear to think that the above means
"The Incompatibility of the Law of Propagation of Light
with the Principle of Relativity",
completely ignoring the word "apparent".
<snip>
>Stooopid ignoramus Schwartz.
>You boy Poe is losing, contradicting himself constantly.
>Even Einstein tries to change the PoR, KNOWING
>it is incompatible with his second postulate.
Thus demonstrating that either Androcles is stupid enough to believe that
Einstein genuinely meant
"The Incompatibility of the Law of Propagation of Light
with the Principle of Relativity",
when he wrote
"The Apparent Incompatibility of the Law of Propagation of Light
with the Principle of Relativity",
in which Androcles has shown that he cannot comprehend simple English, or
he is deliberately lying in order to promote his own point of view, in
which case, he is stupid enough to think that his audience will not pick
up his deliberate misinterpretation of what Einstein wrote.
This is even more stupid than Andocles' interpretation of "The speed of
light plays the part of infinite speed" as "The speed of light is infinite
speed".
David
-----
Ken S. Tucker
The originator of the unification of space and time ,
Minkowski, enabled a *non-orthogonal* relation
of time and *orthogonal* space, per his 1908
lecture.
Unfortunately that genius died suddenly, and was
unable to evolve his work.
For what it's worth, I find spacetime and relativity
are unified by the need to rationalize Einstein's
postulate, ***no such thing as absolute motion***.
That's where I suggest you begin.
Regards
Ken S. Tucker
Timo Nieminen
> Just want to tell you that W. Pauli presents the general form of the
> transformation equations as a footnote (pages 10 and 11) in his book "Theory
> of Relativity", Pergamon Press, 1958.
Not quite, just the general pure boost transformation. (Jackson also gives
this, writing down the transformation matrix.)
Pauli's book looks quite good. I think it'll be my commuting reading for a
while. Thanks for the pointer.
Bill Hobba
Yep - as I realized once I slept on it.
>
> Consider the following from M. Bunge, Foundations of Physics, pp 131-132:
>
> Note the difference between:
> (a) (i) E3 is a three-dimensional Euclidean space (ii) E3 maps physical
> space
> (b) Physical space has a Euclidean structure.
>
> (a) separates the mathematical assumptions and their physical
> interpretation.
>
> Really, what you note above is the justification for restricting ourselves
> to reference frames with uniform metric tensor when dealing with inertial
> reference frames. Perhaps this a good thing to make explicit?
Yep.
>
> I think that our above statements only apply when the space coordinates
> are Cartesian.
I agree.
>
> > And as I argue later I think this is in fact
> > a defining property of an inertial frame.
>
> At least for some definitions of inertial frame. More below.
>
> > > (given that Newton's laws - the validity of which defines an inertial
> > > frame - are a statement of the conservation of momentum, and hence
require
> > > homogeneity in space, but what of homogeneity in time and isotropy?).
> >
> > This is examined in Rindler - Introduction to Special Relativity - page
6.
> > Rindler claims this follows immediately from the POR. Also see Landau -
> > Mechanics page 5 where Landau defines an inertial frame by its symmetry
> > properties - i.e. it must be homogenious in space and time and isotropic
in
> > space. In fact I prefer that definition to ones based on free particles
> > because it avoids the problem of exactly what is meant by free.
>
> If we state that an inertial frame is one in which all of the laws of
> mechanics hold good, then we have conservation of energy, momentum, and
> angular momentum, and thus space-time homogeneity and isotropy.
Hmmmmm. Interesting take - the converse of the way I usally look at it. At
first sight I am inclined to agree but will think about it some more.
>
> If we state that an inertial frame is one in which Newton 1-2 hold (which,
> without Newton 3, doesn't even give us conservation of momentum), then we
> don't yet have those requirements. Adding that the laws of physics must
> hold in all inertial frames, and that conservation of energy, momentum,
> and angular momentum are laws of physics, then the PoR requires
> homogeneity and isotropy.
Yes - I think I see your point (although I suspect a bit more thought is
required on my part to fully appreciate it). I think adding the PLA into
the mix (and hence Noethers powerful result) makes the idea of symmetry
implying the conservation laws more natural - at least to me. In fact now I
think about it that seems to be the real key - we really need to add some
physical principle into the mix eg the PLA, QM (from which the PLA follows
anyway via Feymans sum over histories), or whatever. I am not a big fan of
the usual treatment of classical mechanics via Newton's laws - but that
purely a personal thing - treatments avoiding all the usual problems are
readily available.
Thanks for the very enjoyable posts.
Bill
Franz Heymann
"David McAnally" <D.McAnally@i'm_a_gnu.uq.net.au> wrote in message
news:crkm9a$4ud$1...@bunyip.cc.uq.edu.au...
[snip]
> Androcles has shown that he cannot comprehend simple English, or
> he is deliberately lying in order to promote his own point of view,
Both
[snip]
Franz
Bjoern Feuerbacher
> "Uncle Al" <Uncl...@hate.spam.net> wrote in message
> news:41DD92F5...@hate.spam.net...
>
>>Randy Poe wrote:
>>
>>>Franz Heymann wrote:
>>>
>>>>"Timo Nieminen" <ti...@physics.uq.edu.au> wrote in message
>>>>news:Pine.LNX.4.50.0501061128280.13596-100000@localhost...
>>>>
>>>>[snip]
>>>>
>>>>Timo, thanks for a nice derivation. I am saving it. However, it
>>>>is
>>>>unlikely that it
>>>>will be helpful in the case
>>>>of the present disputation between Randy and Androcles
>>>
>>>That "disputation" is prety much at an end. Androcles'
>>>rock-hard self-contradictory position that he accepts
>>>the PoR but not the invariance of c has erected a
>>>wall of ignorance which leaves no further chinks
>>>to explore.
>
>
> Too right it does.
> Relativity: The Special and General Theory. 1920.
>
> VII. The Apparent Incompatibility of the Law of Propagation of Light
> with the Principle of Relativity
>
> Einstein.
Have you troubles understanding the meaning of the word
"apparent"?
[snip]
> Even Einstein tries to change the PoR, KNOWING
> it is incompatible with his second postulate.
Please present evidence that Einstein ever tried to change
the PoR.
[snip]
Bye,
Bjoern
Bilge
Randy Poe
Timo Nieminen wrote:
> I was going to feed that derivation past Androcles (in smaller
portions),
> but he wanted me to adopt the Lorentz transformations as a starting
> assumption. Since that seemed a truly bizarre thing to do if one is
> wanting to derive them, I refused, and he killfiled me.
>
> Maybe you can have similar luck?
Yes. Androcles persisted in misreading the most trivial
of definitions and derivations, and insists that I can
not use the speed of light as c in the rest frame. I dropped
the effort to derive the Lorentz transformation (all I
ever hoped from that was a chance to explain the symbols
in a single line of Einstein's paper, but I've abandoned
even that). For a little while, I hoped to walk Androcles
through some simpler non-relativistic calculations involving
sound waves or mosquitos, but I've abandoned that as well.
- Randy
Randy Poe
I think so, though that may not have been the collection.
I have long wished for a forum dedicated to helping
people locate dimly recalled books, stories, and films.
- Randy
RP
Timo Nieminen wrote:
> Given the rather long threads on derivations of the Lorentz
> transformations that seem to be making slow progress, I thought this
> might be a worthwhile contribution. Feel free to copy and inflict on
> correspondents in such threads!
It's typically the nature of the transform that is argued, not its
derivation. There are plenty of derivations to be found with the
stroke of a few keys. Google "derivation of the lorentz transform",
31,800 hits.
What most opponents of the theory have difficulty with is the
reciprocity of relativistic effects. These cannot be resolved without
the use of general relativity, though many adherents of special
relativity believe otherwise, i.e. that general relativity is
unnecessary to resolve such paradoxes. Given this false perception of
special relativity by its own adherents, the opponents have valid
counter-arguments in many cases. They are not however showing internal
inconsistency of the theory, but rather that the claims of the
adherents are inconsistent with the theory.
If boosts are omitted from the arguments intended to resolve the
paradoxes, then the lorentz transformation is consistent, but this
approach omits the processes occurring during
acceleration/deceleration, and in doing so introduces seemingly
magical, sometimes very drastic instantaneous changes in distances
and/or time readings that, without general relativistic effects
accounted for, seem to be the result of nothing more than wishful
thinking, i.e. these "jumps" have no logically consistent basis.
The argument that acceleration can be discretized into what are
referred to as MCIFs, and that this approach allows the lorentz
transform to apply equally to accelerated frames, is incorrect.
Acceleration requires a transverse length contraction and/or
variations in light speed to occur in order to resolve reciprocity
arguments, especially those involving light clocks with beams
propagating transversely.
This is however only one of the several issue raised in
counter-arguments. I won't go into the other types of arguments, I'll
only take time to note it is typically either the bogus or else
botched claims of adherents that generates much of the confusion.
There are also those opponents that are simply idiots by birth.
Nothing can be done to persuade the latter, but improved arguments
could do much to bring the former into a better understanding of the
principles and of the theory.
Richard Perry
Dirk Van de moortel
"RP" <no_mail...@yahoo.com> wrote in message news:34816vF...@individual.net...
>
>
> Timo Nieminen wrote:
> > Given the rather long threads on derivations of the Lorentz
> > transformations that seem to be making slow progress, I thought this
> > might be a worthwhile contribution. Feel free to copy and inflict on
> > correspondents in such threads!
>
> It's typically the nature of the transform that is argued, not its
> derivation. There are plenty of derivations to be found with the
> stroke of a few keys. Google "derivation of the lorentz transform",
> 31,800 hits.
>
> What most opponents of the theory have difficulty with is the
> reciprocity of relativistic effects. These cannot be resolved without
> the use of general relativity, though many adherents of special
> relativity believe otherwise, i.e. that general relativity is
> unnecessary to resolve such paradoxes.
Perry, how would *you* know? You haven't understood the
first thing about relativity to begin with:
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/HowdyDoo.html
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/LorentzPerry.html
http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/SimplyPut.html
You are just stupid.
Please take your medicine before you go delirious again?
Dirk Vdm
Androcles
Poe has capitulated.
Who's next?
Androcles.
Bilge
>> This is examined in Rindler - Introduction to Special Relativity - page 6.
>> Rindler claims this follows immediately from the POR. Also see Landau -
>> Mechanics page 5 where Landau defines an inertial frame by its symmetry
>> properties - i.e. it must be homogenious in space and time and isotropic in
>> space. In fact I prefer that definition to ones based on free particles
>> because it avoids the problem of exactly what is meant by free.
>
>If we state that an inertial frame is one in which all of the laws of
>mechanics hold good, then we have conservation of energy, momentum, and
>angular momentum, and thus space-time homogeneity and isotropy.
something that can be quantified independent of the coordinates you choose
to define energy and momentum. There is really no reason to say anything
about inertial frames, up front.
Einstein's wording of the first postulate is nothing more than a statement
about what we implicitly require in order to apply the scientific method.
In order to accept the outcome of experiments, we expect experiments to
be repeatable, so that we may verify them. If an experiment is repeatable,
then physics hasn't changed under a time displacement. In other words,
our definition of time is homogeneous by definition. (One could further
argue this from the standpoint of quantum mechanics on the basis that
equal probabilities for a process to occur in any time interval imply
a poisson process).
The same argument applied to location and orientation implies spatial
homogeneity and isotropy, which are alreay present in newtonian mechanics,
implicity.
Einstein may have made some assumptions about what the laws of physics
were, but it is unnecessary to do so. Einstein lacked the means to
construct conservation laws from symmetries in 1905, so conservation laws
remained a mystery until noether came along. Simply assume that whatever
the laws, the laws are the same, independent of time, location or
orientation. If you require this globally, you have special relativity. If
you require this to only hold locally, you get general relativity.
Another way to look at those requirements is that homogeneity and
isotropy _define_ an inertial frame. So, one can state that in terms
of an infiniesimal displacement,
x -> x' = x + \delta x
Or, in matrix notation:
x_i = I_ij x_j + a_ij x_j
where I_ij is the identity and a_ij is the infinitesimal transformation.
(I hate to repeat a simple derivation, but I think it reflects the logical
constryction that follows from what bill wrote above).
Since I haven't yet defined the metric other than to require homogeneity
and isotropy, this is a perfectly general coordinate transform in four-
dimensions. You can construct the scalar product (I resort to upper and
lower indices here):
g_ik x^i x^k = g_ik I^i_j I^k_l x^j x^l
+ g_ik I^i_j a^k_l x^j x^l + g_ik a^i_j I^k_l x^j x^l
+ (O(a^))
Since I is the identity,
g_ik I^i_j I^k_l = g_jl
g_ik I^i_j a^k_l = g_jk a^k_l
g_ik a^i_j I^k_l = g_il a^i_j
giving,
g_ik x^i x^k = g_jl x^j x^l + g_il a^i_j + g_jk a^k_l + O(a^2)
Since x'^2 = x^2,
g_ik x^i x^k - g_jl x^j x^l = 0
and therefore,
g_il a^i_j + g_jk a^k_l = 0
From that, you can obtain the symmetry of a_ij. In four dimensions, the
possible signatures are, ++++, -+++, --++, +--- and ----. Agreement with
observation restricts the choices to +--- or ---+, (e.g, ++++ would imply
being able to turn around in time), so the metric has to be lorentzian.
To construct global inertial frames, the metric has to be constant.
The antisymmetry of a_ij implies six independent components. For each,
the finite transform is obtained from the infinite product of the in-
finitesimal transforms, which give you the exponential form of the
transforms.
>If we state that an inertial frame is one in which Newton 1-2 hold (which,
>without Newton 3, doesn't even give us conservation of momentum), then we
>don't yet have those requirements. Adding that the laws of physics must
>hold in all inertial frames, and that conservation of energy, momentum,
>and angular momentum are laws of physics, then the PoR requires
>homogeneity and isotropy.
>The first is neater, but requires a much stronger initial assumption.
Noether's theorem only works one direction, i.e., contiuous symmetries
of the lagragian imply conservation laws, but conservation laws don't
necessarily imply symmetries. Since the assumption of isotropy and
homogeneity used to obtain the lorentz transforms _are_ symmetries
of the lagrangian, the conservation laws follow immediately by application
of noether's theorem to a lagrangian that is invariant under a spacetime
displacement,
L[x^u] -> L'[x^u] = L[x^u'] = L[x] + a^u d_u L
\delta L = a^u d_u L
where a^u is an infinitesimal vector. You can equivalently write the
infinitesimal change in the lagrangian in terms of some generalized
coordinate (or field) q(x^u),
\delta L = (dL/dq)\delta q + (dL/d(d_u q)) \delta (d_u q)
If you rewrite the second term via differentiating d_v [(dL/d(d_u q)) q]
by parts, use the euler lagrange equations to eliminate two of the terms
in the result and then set the expressions for \delta L equal to each
other, you obtain a conserved current, T^uv such that,
d_u T^uv = 0
The quantitiy T^00 is the hamiltonian density from which you obtain
the conserved four-momentum.
Ken S. Tucker
Bilge wrote:
[...]
> observation restricts the choices to +--- or ---+, (e.g, ++++ would
imply
> being able to turn around in time), so the metric has to be
lorentzian.
Ah, is Parity Time conserved? No, we expect CPT conservation.
Bilges statement presumes (wrongly) PT conservation.
I'm curious though why Bilge thinks (++++) enables a reversal
of time, the subject does relate to CPT conservation.
[...]
Snipped good stuff...
Ken S. Tucker
Ken S. Tucker
Ken
Bilge
>Bilge wrote:
>[...]
>> observation restricts the choices to +--- or ---+, (e.g, ++++ would
>imply
>> being able to turn around in time), so the metric has to be
>lorentzian.
>
>Ah, is Parity Time conserved? No, we expect CPT conservation.
>Bilges statement presumes (wrongly) PT conservation.
No, it doesn't assume any such thing. It assumes nothing about
c,p or t. The cpt theorem is a consequence of what I _did_ assume.
Ken S. Tucker
Would you enlighten us why you made this statement...
I'm curious though why "Bilge thinks (++++) enables a reversal
of time,"
Honestly, I don't quite catch why you would say that except
in relation to CPT conservation.
Ken
Daryl McCullough
>Ah, is Parity Time conserved? No, we expect CPT conservation.
>Bilges statement presumes (wrongly) PT conservation.
>I'm curious though why Bilge thinks (++++) enables a reversal
>of time, the subject does relate to CPT conservation.
It seems to me that if the signature were (++++) then there would
be no difference between the time axis and a space axis. So turning
around in time would be no harder than turning around in space.
--
Daryl McCullough
RP
Dirk Van de moortel wrote:
How would *I* know, you ask? The answer isn't complicated, it's
because I had a few good teachers on the subject, none of which were
you. OTOH, it was in part your arguments that steered me into an
understanding of the theory. You weren't entirely helpful, even though
mathematically consistent. You are one of those who suck at relating
the concepts to others. Nevertheless, I thank you for your minor
contributions to the free education.
With my last counter argument, and argument that was valid when
restricted to the confines of special relativity, I put down the
equations provided by gtr, accounting for gravitational clock ticking
rate offsets, and was surprised to find that there were no longer any
discrepancies. The argument in inequalities seemed to preclude this,
but it was just an illusion, a good illusion. I realized that I had no
more arguments left, save the one posted above, and this one:
It's a model Dirk, and Ritz proved this adequately by accounting for
all known electromagnetic phenomena from within a Galilean context
(And BTW, apparently this guy beat me to the idea that the
electrostatic field is the result of relative motions of charges.
Hell, I could have spent that time studying Weber and Ritz. Hindsight
is rarely joyous.) Empirically falsified? Maybe, maybe not; aren't we
free to modify and improve the theory to account for minor
discrepancies, as has been done with virtually every other theory
whose initial conclusions were slightly off? Through how many
revisions of existing theories have they been claimed to be complete?
How many are considered to be complete now? And, furthermore, have you
single-handedly solved the discrepancies between gtr and QM, or any of
the other as yet unsolved puzzles? What's your take on Aspect's
experiment, on the whole EPR issue?
Many of my arguments were, and still are valid, many were not. And I
don't hesitate to admit error when I understand that I've made one. If
idiot is the name that you apply to those who have come to understand
an issue, as opposed to those who still wallow in confusion, then it
seems to me that you've a bit of maturing left to do, but then I've
noted this several times in the past, so take it or leave it. Not
everyone was born with all the knowledge of the universe, nor were
you. Most people were necessarily idiots about a given subject until
they put their mind to studying it. The act, the mere decision, to
learn a subject, removes them from the idiot list. So what that I
don't just study a subject, but instead look into every corner of a
notion before accepting it as logically sound? Isn't that the same
sort of tendency that brought you all of those wonderful theories in
the first place?
I have better things to do than waste my time with educated idiots
such as yourself. And since you obviously don't understand the
concept, I'll outline it for you. An educated idiot is a person who
can recite the Encyclopedia Britannica, but who regularly demonstrates
a lack in common sense and good reason, and the lack of a desire to
learn the difference.
Richard Perry
Ken S. Tucker
Recall our metric is derived from a scalar product
of unitary base vectors i.e,
g_uv = e_u . e_v
The signature (+ - - - ) doesn't work, check out the
Kronecker delta, for the following scalar products,
e_0. e^0 = 1 == delta_0^0
e_1. e^1 = -1 == delta_1^1
In GR and tensor analysis the Kronecker has TWO
solutions, 0,1... NOT three, 1,0,-1. I'm not selling that
solution, I'll sell U_i =0, where,
g = |+1 -v|
|-v +1|
in accord with U_i=0.
Regards
Ken S. Tucker
Ken S. Tucker
Ken
Timo Nieminen
> That is more or less a tautology as it assumes ``inertial frame'' means
> something that can be quantified independent of the coordinates you choose
> to define energy and momentum. There is really no reason to say anything
> about inertial frames, up front.
[moved:]
> Another way to look at those requirements is that homogeneity and
> isotropy _define_ an inertial frame.
I am uncomfortable with the definition of an inertial frame as a frame of
zero intrinsic curvature with a homogeneous and isotropic metric, in that
not all such frames are inertial, in the sense of Newton's laws. For
example, I suspect (but haven't checked, so feel free to correct me if I'm
wrong) that an accelerating frame admits a coordinatisation such that the
metric is homogeneous and isotropic, and such a frame is clearly not
inertial in the mechanical sense. While such a coordinatisation might well
be pointless and bizarre, how can it be excluded without bringing in some
real physics?
What is the minimum physics needed to uniquely define inertialness in a
manner that must be consistent with mechanics?
Newton 1 only requires force as a concept (well, force-free, but that
appears to require force). (I think I was writing Newton 1 and 2 earlier,
but is not Newton 1 sufficient to define inertialness?)
Perhaps this isn't the most useful approach.
The PoR defines an inertial frame as one in which the laws of physics (in
a particular form - OK, here is the tautology, but can it be avoided ?)
hold. Since, empirically, this requires a homogeous and isotropic metric
(in the SR-regime) with signature (-+++) or (+---), then the validity and
necessity of the Lorentz transformations is obvious, and we're simply not
interested in frames where the laws of physics don't hold.
The PoR being sufficient definition in and of itself, no further
definition is needed. (As you noted, a futher definition is more or less a
tautology.)
I'd rather just stop at that point than adopt a broader definition of
inertialness.
--
Timo
Bilge
axis would be just like the space axes, so turning around in time
would be no different than turning around in space. The reason
time reversal is a discrete symmetry is because of the odd
sign in the signature.
Bill Hobba
"Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote in message
news:Pine.OSF.4.58.05...@dingo.cc.uq.edu.au...
> On Fri, 7 Jan 2005, Bilge wrote:
>
> > That is more or less a tautology as it assumes ``inertial frame''
means
> > something that can be quantified independent of the coordinates you
choose
> > to define energy and momentum. There is really no reason to say anything
> > about inertial frames, up front.
> [moved:]
> > Another way to look at those requirements is that homogeneity and
> > isotropy _define_ an inertial frame.
>
> I am uncomfortable with the definition of an inertial frame as a frame of
> zero intrinsic curvature with a homogeneous and isotropic metric, in that
> not all such frames are inertial, in the sense of Newton's laws.
But such is not the usual definition. My posts were concerted about if a
frame did have such then I think it may be inertial. The usual definition
is one in which Newton's first law holds ie free particle moving at constant
velocity. I do not like that definition because you have the difficultly in
defining what free is exactly. The other definition is the one by Landau -
a frame that is homogeneous in space and time and isotropic in space. This
IMHO avoids the problems of the usual definition so is to be preferred - but
it has problems of its own.
> For
> example, I suspect (but haven't checked, so feel free to correct me if I'm
> wrong) that an accelerating frame admits a coordinatisation such that the
> metric is homogeneous and isotropic, and such a frame is clearly not
> inertial in the mechanical sense.
Hmmmmmm. Interesting point. One thing for sure is that in such a frame
clocks tick at different rates depending on position so space is not
homogeneous - so is not inertial via the usual dentitions. I in fact may be
wrong in thinking a knowledge of the metric alone is enough. In fact I now
think Wald may not be 100% accurate in what he said - then again I am loath
to dismiss what a physicist of his stature says. I need to think some more
I suspect - and hope others more knowledgeable (than me) could post
something useful. Do others have any thoughts? No cranks please (although
saying such is doubtful to do any good).
>While such a coordinatisation might well
> be pointless and bizarre, how can it be excluded without bringing in some
> real physics?
I do not think it would be pointless or bizarre - accelerating frames are
very useful in understanding for example the Pound-Rebka experiment.
>
> What is the minimum physics needed to uniquely define inertialness in a
> manner that must be consistent with mechanics?
>
I still think the symmetry properties are what is required. The problem
comes with given a real frame in proving it has such properties. For
example we can not prove it is isotropic because, for example, we can not in
principle measure OWLS. I think all we can do is when given an actual frame
is construct some coordinates by reasonable means and carry out experiments
with say atomic clocks, accelerometers, gyroscopes, pendulums etc and see if
they are in accord with the symmetry assumptions. If to the accuracy of the
experiments they are then it can be consderiered inertial. I think the
repeatability necessary for the scientific enterprise to make sense more or
less demands frames with the required symmetry properties exist in principle
if not in fact. The other possible way to define an inertial frame is via
the EEP. Imagine a freely floating platform far from any masses in
intergalactic space screened off from EM Then that frame should be
inertial to a very high degree of accuracy. To me it is a lot like the idea
of a point having no size but position. Such is very useful as a concept
but finding one is not possible.
>
> Newton 1 only requires force as a concept (well, force-free, but that
> appears to require force). (I think I was writing Newton 1 and 2 earlier,
> but is not Newton 1 sufficient to define inertialness?)
>
> Perhaps this isn't the most useful approach.
Yes Newtons first law looks rather suspect. Of course what you need to do is
look a little deeper to see what it is 'trying' to tell us. This is usually
expressed by saying free particles move with constant velocity - but then
you face the embarrassing problem of defining what one means by 'free'. So
one says - not acted on by
external forces - but what is an external force eg do the fictitious forces
of acceleration count as an external force? Indeed to even see if a particle
moves with constant velocity you need a coordinate system and synced clocks
so you need to specify how you do that.
>
> The PoR defines an inertial frame as one in which the laws of physics (in
> a particular form - OK, here is the tautology, but can it be avoided ?)
> hold.
I am not so sure the POR is a definition of an inertial frame. It is a
hypothesized principle they are supposed to possess. It is refutable
experimentally eg a positive result on the Trouton-Noble experiment would
have cast strong doubt on it.
> Since, empirically, this requires a homogeous and isotropic metric
> (in the SR-regime) with signature (-+++) or (+---), then the validity and
> necessity of the Lorentz transformations is obvious, and we're simply not
> interested in frames where the laws of physics don't hold.
But what would have happened if say the Trouton-Noble experiment yielded a
positive result? Would you simply have declared it as of no import since we
are not interested in frame where such experiments yield positive results?
This is a lot like a dicussion that sometimes pops up on
sci.physics.relativity about the POR. If we ever find it violated then you
simply declare the violation is not a law of physics. Such a view is
obviously not correct because at certain points in the derivation of the
Lorentz transforms you need to invoke it - Timo certainly invoked in section
3 of his derivation.
>
> The PoR being sufficient definition in and of itself, no further
> definition is needed. (As you noted, a futher definition is more or less a
> tautology.)
>
> I'd rather just stop at that point than adopt a broader definition of
> inertialness.
I still think the definition based on symmetry properties is the best one.
Still a very interesting discussion. Thanks
Bill
>
> --
> Timo
Timo Nieminen
> "Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote:
> > On Fri, 7 Jan 2005, Bilge wrote:
> >
> > > Another way to look at those requirements is that homogeneity and
> > > isotropy _define_ an inertial frame.
> >
> > I am uncomfortable with the definition of an inertial frame as a frame of
> > zero intrinsic curvature with a homogeneous and isotropic metric, in that
> > not all such frames are inertial, in the sense of Newton's laws.
>
> But such is not the usual definition. My posts were concerted about if a
> frame did have such then I think it may be inertial. The usual definition
> is one in which Newton's first law holds ie free particle moving at constant
> velocity. I do not like that definition because you have the difficultly in
> defining what free is exactly.
Yes. "Let be be a force-free particle."
> The other definition is the one by Landau -
> a frame that is homogeneous in space and time and isotropic in space. This
> IMHO avoids the problems of the usual definition so is to be preferred - but
> it has problems of its own.
If the definition includes that the laws of physics are homogeneous and
isotropic, sure. Problems and advantages already discussed earlier.
> > For
> > example, I suspect (but haven't checked, so feel free to correct me if I'm
> > wrong) that an accelerating frame admits a coordinatisation such that the
> > metric is homogeneous and isotropic, and such a frame is clearly not
> > inertial in the mechanical sense.
>
> Hmmmmmm. Interesting point. One thing for sure is that in such a frame
> clocks tick at different rates depending on position so space is not
> homogeneous - so is not inertial via the usual dentitions. I in fact may be
> wrong in thinking a knowledge of the metric alone is enough.
Well, a physical clock will tick different rates, as measured by the
coordinate time, even if stationary in the coordinate system, at different
positions, and a physical ruler will be different lengths, as measured by
the space coordinates, at different times, even if stationary.
But is that any more than expected in a frame where our (inertial) laws of
physics don't hold?
> >While such a coordinatisation might well
> > be pointless and bizarre, how can it be excluded without bringing in some
> > real physics?
>
> I do not think it would be pointless or bizarre - accelerating frames are
> very useful in understanding for example the Pound-Rebka experiment.
It was not to say that accelerating frames are pointless or bizarre, just
that the choice of a coordinate system in an accelerating frame to give a
homegeneous and isotropic metric might be pointless and bizarre.
Well, not necessarily. The mathematical description of motion doesn't
require any possibility of actually physically measuring the coordinates
of events (a useful mathematical description of motion is another matter).
> > The PoR defines an inertial frame as one in which the laws of physics (in
> > a particular form - OK, here is the tautology, but can it be avoided ?)
> > hold.
>
> I am not so sure the POR is a definition of an inertial frame. It is a
> hypothesized principle they are supposed to possess. It is refutable
> experimentally eg a positive result on the Trouton-Noble experiment would
> have cast strong doubt on it.
PoR could still hold. For example, if Trouton-Noble is in agreement with a
stationary ether, then a more complete EM theory, accounting for velocity
relative to the ether, simply leads to Galilei-symmetry. The PoR in itself
doesn't specify the transformation law between inertial frames.
> > Since, empirically, this requires a homogeous and isotropic metric
> > (in the SR-regime) with signature (-+++) or (+---), then the validity and
> > necessity of the Lorentz transformations is obvious, and we're simply not
> > interested in frames where the laws of physics don't hold.
>
> But what would have happened if say the Trouton-Noble experiment yielded a
> positive result? Would you simply have declared it as of no import since we
> are not interested in frame where such experiments yield positive results?
Non-null Trouton-Noble would tell us that Maxwell's equations +
consitutive equations for free space as currently used are incomplete as
far as being a law of physics.
Lorentz-invariance being an empirical result, there's no fundamental
reason why it shouldn't be discarded when results show otherwise.
> > The PoR being sufficient definition in and of itself, no further
> > definition is needed. (As you noted, a futher definition is more or less a
> > tautology.)
> >
> > I'd rather just stop at that point than adopt a broader definition of
> > inertialness.
>
> I still think the definition based on symmetry properties is the best one.
In a specialised relativity course, or in a mechanics course, I think that
might be a better definition. I want to do relativity in 2 weeks, 1/2 of
that being specifically EM, so I want to avoid mechanics beyond that
required for motion of charged particles.
Of course, what is good in a course for a particular set of students is
not necessarily the "best" in whatever absolute sense "best" might be
valid in.
--
Timo
Bilge
>
>> That is more or less a tautology as it assumes ``inertial frame'' means
>> something that can be quantified independent of the coordinates you choose
>> to define energy and momentum. There is really no reason to say anything
>> about inertial frames, up front.
>[moved:]
>> Another way to look at those requirements is that homogeneity and
>> isotropy _define_ an inertial frame.
>
>I am uncomfortable with the definition of an inertial frame as a frame of
>zero intrinsic curvature with a homogeneous and isotropic metric, in that
>not all such frames are inertial, in the sense of Newton's laws.
newton's laws in a way which isn't circular.
>For example, I suspect (but haven't checked, so feel free to correct
>me if I'm wrong) that an accelerating frame admits a coordinatisation
>such that the metric is homogeneous and isotropic, and such a frame is
>clearly not inertial in the mechanical sense.
possible to define an inertial frame in ``the mechanical sense.''
>While such a coordinatisation might well be pointless and bizarre, how
>can it be excluded without bringing in some real physics?
It can't. All you can do is specify what properties those coordinates
have and find the physics that satisfies those properties. If you don't
exploit the isotropy and homogeneity, you can't define a conserved
energy or momentum.
>What is the minimum physics needed to uniquely define inertialness in a
>manner that must be consistent with mechanics?
Probably the principle of equivalence.
>Newton 1 only requires force as a concept (well, force-free, but that
>appears to require force). (I think I was writing Newton 1 and 2 earlier,
>but is not Newton 1 sufficient to define inertialness?)
No, it isn't. Newton's first law, is:
``Every object in a state of uniform motion, remains in uniform
motion unless acted upon by an external force.''
The law is completely circular as it does not say how to determine
if the motion is uniform except that it's motion which is force free.
We define a straight line as the path of an inertial object and we define
an inertial object as one which moves in a straight line.
>Perhaps this isn't the most useful approach.
>
>The PoR defines an inertial frame as one in which the laws of physics (in
>a particular form - OK, here is the tautology, but can it be avoided ?)
>hold.
Not very easily. Most every definition of ``inertial'' reies on
some intuitive notion of force. Ialso don't think trying to find
coordinates for a particular set of ``laws of physics'' is the right
way to go. Symmetry and conservation laws have always been part of
classical physics, but up until noether's theorem came along the
connection wasn't evident and conservation laws were pretty much
a mystery.
>Since, empirically, this requires a homogeous and isotropic metric
>(in the SR-regime) with signature (-+++) or (+---), then the validity and
>necessity of the Lorentz transformations is obvious, and we're simply not
>interested in frames where the laws of physics don't hold.
Not in special relativity, anyway. In general relativity, those laws
hold at a point, but for a general metric, energy and momentum aren't
well defined for that reason.
RP
You didn't get past the introductory arguments that (historically) led
to Lornetz's postulation of length contraction along the line of
motion, but....
I've gone back over Ritz, and so far I have this one serious issue
with his theory. He posits an emission of light particles at c wrt the
source only. He regards the speed of these particles to be c+v wrt
other frames. Now, the main problem connected to these ideas is that
you must explain what it is that holds the speed of the particles at c
wrt the source, that is, since there is no longer a physical
connection between the source and the particle. What I'm inferring
here is that you must assume that space is an objective entity,
existing independently of the matter that exists within it, and that
the particle is moving at a constant velocity wrt *it* after its
emission from the source. Let me qualify this conclusion before you
jump to your own conclusions, which more than likely weren't implied
in the argument:
Because the source may accelerate at any moment after emitting the
particle, and in so doing enter into a frame in which, according to
Ritz, the particle was moving at c+v, then either the particle
accelerates along with the source, or it is maintained in its motion
at c wrt a frame in which no observer any longer exists. Thus what
maintains the particle's speed of c wrt nothing? Now you'll probably
say something like, "it was moving at constant speed wrt every other
inertial observer too, and thus it is these observers that maintain
the uniform and consistent motion of the particle, i.e. it is
maintained wrt them". To this I'll say, Aha! Then why don't the
particles move at c wrt these other masses as well, that is, since the
particles are equally influenced and regulated in their propagation by
them.
Are you going to argue that the light particles are moving uniformly
wrt nothing, or is it that it is moving wrt a separate ether, or are
you rather going to state sanely that it is uniform only wrt the
collective sum of the matter in the universe, i.e. that motion is Machian.
Now before you go comparing these particles to balls or trucks or
mosquitos, and assuming that the same must apply to them, i.e. that
their speeds must also be invariant, let's first note that in this
case the argument does apply to them in at least one sense, that is,
they too must be assumed to either be moving wrt an independent space,
or else relative to other matter in the universe, in that there must
be some mechanism for them to continue along at a uniform velocity. In
the fist alternative (ether), the particles are separated causally
from any other neutral objects, and in the second they are causally
connected to all of them in every instant of their existence. IOW,
there are only two possibilities providing for propagation uniformity
(otherwise called the law of inertia), and unfortunately Ritz's
ballistic model of light doesn't correspond consistently with either
of these remaining alternatives.
Now to answer the question, why doesn't the mosquito propagate at the
same speed wrt every inertial frame, note that this simply isn't what
the lorentz transform predicts. If you can't understand the difference
between c and 5m/s, and between bosons and fermions, in the
qualitative sense of the theory , then you can't understand the
theory. c is the maximum speed, and 5 m/s isn't. As c is approached
the speed of a point comes closer and closer to being invariant, but
doesn't actually reach invariance until it achieves c.
There is another alternative that I mentioned, but didn't address
above, but it is so stupid as to barely be worthy of discussion, to
wit, it is the assumption that the light particle maintains a causal
connection to the source throughout its flight, i.e.. that it does
indeed accelerate in tandem with the source.
Such a behavior would preclude the particle from being absorbed or
otherwise disturbed by other masses, since their interaction with it
is contradictory to the premise that the particle is bound in its
behaviors to the motion of the source. That is, in order to have a
light particle that interacts with masses other than the source, both
the source and the detector must have equal rights to the properties
of the particle, that is, insofar as its propagation is concerned.
Thus either the particle's propagation is dependent upon the motions
of both source and detector, or IOW these masses cooperate to
establish the space-time medium<sic> of propagation, or else there is
a separate medium, i.e. an ether.
So regardless of Ritz's claims, we have logically arrived at the
special relativity/ether debate, and the ballistic model that you
propose has been eliminated as a viable possibility, by reductio.
Richard Perry
RP
Bilge wrote:
> Timo Nieminen:
> >On Fri, 7 Jan 2005, Bilge wrote:
> >
> >> That is more or less a tautology as it assumes ``inertial frame'' means
> >> something that can be quantified independent of the coordinates you choose
> >> to define energy and momentum. There is really no reason to say anything
> >> about inertial frames, up front.
> >[moved:]
> >> Another way to look at those requirements is that homogeneity and
> >> isotropy _define_ an inertial frame.
> >
> >I am uncomfortable with the definition of an inertial frame as a frame of
> >zero intrinsic curvature with a homogeneous and isotropic metric, in that
> >not all such frames are inertial, in the sense of Newton's laws.
>
> I'm willing to bet that you cannot define an inertial frame using
> newton's laws in a way which isn't circular.
Frames K and K' are inertial, iff K' emits em waves of constant
frequency wrt itself that are observed to be of constant frequency wrt
K as well, and if K' is changing in displacement wrt K over time.
An inertial frame has no intrinsic defining properties.
Richard Perry
Timo Nieminen
> Timo Nieminen:
> >
> >I am uncomfortable with the definition of an inertial frame as a frame of
> >zero intrinsic curvature with a homogeneous and isotropic metric, in that
> >not all such frames are inertial, in the sense of Newton's laws.
>
> I'm willing to bet that you cannot define an inertial frame using
> newton's laws in a way which isn't circular.
Using only Newton's laws, yes, that might well be a safe bet.
> >For example, I suspect (but haven't checked, so feel free to correct
> >me if I'm wrong) that an accelerating frame admits a coordinatisation
> >such that the metric is homogeneous and isotropic, and such a frame is
> >clearly not inertial in the mechanical sense.
>
> If that is true, which to some extent it is, then it's also not
> possible to define an inertial frame in ``the mechanical sense.''
... without circularity, I presume you mean.
> >Newton 1 only requires force as a concept (well, force-free, but that
> >appears to require force). (I think I was writing Newton 1 and 2 earlier,
> >but is not Newton 1 sufficient to define inertialness?)
>
> No, it isn't. Newton's first law, is:
>
> ``Every object in a state of uniform motion, remains in uniform
> motion unless acted upon by an external force.''
>
> The law is completely circular as it does not say how to determine
> if the motion is uniform except that it's motion which is force free.
> We define a straight line as the path of an inertial object and we define
> an inertial object as one which moves in a straight line.
Is not dr/dt = constant a sufficient definition of uniform motion (at
least in Cartesian coordinates)? True, this is not (as far as I've ever
seen) included as part of Newton's 1st law.
Of course, this doesn't avoid the problem of defining "force-free".
For which I think there is no easy solution within Newton's laws. Which
makes me think I should have written: but is not Newton 1, uniform motion
has constant dr/dt, and we'll worry about what force is later, but assume
an intuitive idea of "force-free" for the moment :)
--
Timo
Bilge
>Bilge wrote:
>> I'm willing to bet that you cannot define an inertial frame using
>> newton's laws in a way which isn't circular.
>
>Frames K and K' are inertial, iff K' emits em waves of constant
>frequency wrt itself that are observed to be of constant frequency wrt
>K as well, and if K' is changing in displacement wrt K over time.
That is circular.
>An inertial frame has no intrinsic defining properties.
If you believe that, the why are responding and why did you try to
specify just such a property?
Bill Hobba
"Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote in message
news:Pine.OSF.4.58.05...@dingo.cc.uq.edu.au...
>
> > "Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote:
> > > On Fri, 7 Jan 2005, Bilge wrote:
> > >
> > > > Another way to look at those requirements is that homogeneity and
> > > > isotropy _define_ an inertial frame.
> > >
> > > I am uncomfortable with the definition of an inertial frame as a frame
of
> > > zero intrinsic curvature with a homogeneous and isotropic metric, in
that
> > > not all such frames are inertial, in the sense of Newton's laws.
> >
> > But such is not the usual definition. My posts were concerted about if
a
> > frame did have such then I think it may be inertial. The usual
definition
> > is one in which Newton's first law holds ie free particle moving at
constant
> > velocity. I do not like that definition because you have the
difficultly in
> > defining what free is exactly.
>
> Yes. "Let be be a force-free particle."
Yes.
>
> > The other definition is the one by Landau -
> > a frame that is homogeneous in space and time and isotropic in space.
This
> > IMHO avoids the problems of the usual definition so is to be preferred -
but
> > it has problems of its own.
>
> If the definition includes that the laws of physics are homogeneous and
> isotropic, sure. Problems and advantages already discussed earlier.
Yes.
>
> > > For
> > > example, I suspect (but haven't checked, so feel free to correct me if
I'm
> > > wrong) that an accelerating frame admits a coordinatisation such that
the
> > > metric is homogeneous and isotropic, and such a frame is clearly not
> > > inertial in the mechanical sense.
> >
> > Hmmmmmm. Interesting point. One thing for sure is that in such a frame
> > clocks tick at different rates depending on position so space is not
> > homogeneous - so is not inertial via the usual dentitions. I in fact
may be
> > wrong in thinking a knowledge of the metric alone is enough.
>
> Well, a physical clock will tick different rates, as measured by the
> coordinate time, even if stationary in the coordinate system, at different
> positions, and a physical ruler will be different lengths, as measured by
> the space coordinates, at different times, even if stationary.
>
> But is that any more than expected in a frame where our (inertial) laws of
> physics don't hold?
Sure - if one constructed a coordinate system in such a frame in the usual
fashion (via rods that for all practical purposes are rigid) and synced
clocks the usual way (via light) we will see immediately free particles do
not move with constant velocity. Assuming of course we agree that 'free'
means we can not find something else in the frame that would be responsible
for the force. That is the problem with using 'free' - we must look around
and ensure other causes are not about and the reason I am not 100% happy
with it - you are making a physical assumption - forces are always caused by
something else in the frame - if not then the frame is not inertial. I
prefer the one based on symmetry because no other physical assumption is
made - construct the coordinate system in the usual way, sync clocks the
usual way and check if for example clocks tick at the same rate. Actually
even arguing like that is rather silly of me because if you were in such as
frame you would feel it quite readily in the seat of your pants - no further
experimentation necessary. But I am trying to think in terms of principles
rather than what is obvious.
>
> > >While such a coordinatisation might well
> > > be pointless and bizarre, how can it be excluded without bringing in
some
> > > real physics?
> >
> > I do not think it would be pointless or bizarre - accelerating frames
are
> > very useful in understanding for example the Pound-Rebka experiment.
>
> It was not to say that accelerating frames are pointless or bizarre, just
> that the choice of a coordinate system in an accelerating frame to give a
> homegeneous and isotropic metric might be pointless and bizarre.
Got it.
See what I said above. And yes I agree.
>
> > > The PoR defines an inertial frame as one in which the laws of physics
(in
> > > a particular form - OK, here is the tautology, but can it be avoided
?)
> > > hold.
> >
> > I am not so sure the POR is a definition of an inertial frame. It is a
> > hypothesized principle they are supposed to possess. It is refutable
> > experimentally eg a positive result on the Trouton-Noble experiment
would
> > have cast strong doubt on it.
>
> PoR could still hold. For example, if Trouton-Noble is in agreement with a
> stationary ether, then a more complete EM theory, accounting for velocity
> relative to the ether, simply leads to Galilei-symmetry. The PoR in itself
> doesn't specify the transformation law between inertial frames.
Ahhhhhhhhh. First the POR all by itself does specify the transformation law
up to an undetermined value of c - a value of infinity would give the
Galilean transformations - a finite value contains essentially the same
physics for any value because it can be set to one by a suitable choice of
units. This is discussed at length on page 19 of Rindler - Introduction to
Special Relativity under the heading of the role of the second axiom. Also
consider the following. If an aether did exist, was the medium that light
undulated in, and an inertial frame was stationary in, then a frame moving
at constant velocity to such a frame would have an aether wind breaking its
isotropy so it would not be inertial. Thus the POR would be invalidated.
If one conceptually removed such an aether then light itself would not exist
so the laws of physics as we know them would be down the tube. To me that is
something implicit in the idea of the POR - our usual laws of physics must
still hold. Indeed from Einstein's 1905 paper he said:
'They suggest rather that, as has already been shown to the first order of
small quantities, the same laws of electrodynamics and optics will be valid
for all frames of reference for which the equations of mechanics hold good.1
We will raise this conjecture (the purport of which will hereafter be called
the ``Principle of Relativity'') to the status of a postulate'
Thus if light could not propagate then the POR would be violated from the
outset because the laws of optics would not exist.
But I hasten to add Griffiths disagrees with me on this - see Griffith -
Introduction to Electrodynamics page 449:
'Ether is no more an absolute rest system than the water in a goldfish
bowl - which is a special system if you happen ot a goldfish but scacely
'absolute'
I remember when I first read this quite a few years ago now I did not agree
with it - and still do not. The aether was conceived of as the carrier of
light. If it did exist it would be fundamental to the existence of the laws
of nature as we understand them - specifically Maxwell's equations would not
exist without an aether - our understanding of matter as composed on
electrically charged particles would have problems - the world as we know it
would be fundamentally dependant on this aether - the laws of physics could
not exist without it. It would be an inseparable part of an inertial
frame - not like the water for goldfish.
>
> > > Since, empirically, this requires a homogeous and isotropic metric
> > > (in the SR-regime) with signature (-+++) or (+---), then the validity
and
> > > necessity of the Lorentz transformations is obvious, and we're simply
not
> > > interested in frames where the laws of physics don't hold.
> >
> > But what would have happened if say the Trouton-Noble experiment yielded
a
> > positive result? Would you simply have declared it as of no import
since we
> > are not interested in frame where such experiments yield positive
results?
>
> Non-null Trouton-Noble would tell us that Maxwell's equations +
> consitutive equations for free space as currently used are incomplete as
> far as being a law of physics.
Yes. But it would be strong evidence one frame is special with respect to
the laws of electrodynamics - removal of the special property of such a
frame would invalidate the laws of physics. And frames moving with respect
to such a frame would experience an 'aether' wind - a wind that could not
conceptually be removed without affecting the laws of physics. Indeed that
is exactly what Trouton-Noble was looking for - the effects of an aether
wind
>
> Lorentz-invariance being an empirical result, there's no fundamental
> reason why it shouldn't be discarded when results show otherwise.
>
> > > The PoR being sufficient definition in and of itself, no further
> > > definition is needed. (As you noted, a futher definition is more or
less a
> > > tautology.)
> > >
> > > I'd rather just stop at that point than adopt a broader definition of
> > > inertialness.
> >
> > I still think the definition based on symmetry properties is the best
one.
>
> In a specialised relativity course, or in a mechanics course, I think that
> might be a better definition. I want to do relativity in 2 weeks, 1/2 of
> that being specifically EM, so I want to avoid mechanics beyond that
> required for motion of charged particles.
Timo - IMHO your derivation was excellent. If I was in your class and
exposed to it I would be very happy indeed. The only reason I raised such
issues is that long postings on sci.physics.relativity have caused me to
reflect on the foundations of relativity so that I have reached a slightly
different view. If any student raises the issues I did in this post then
the proper place to handle it is outside class and you can refer the student
to other source material. Indeed for such an interested student it might
form the basis of an assignment or perhaps a class presentation if your
teaching style accommodates such. That usually shuts up wise ass students -
at least it did me when lecturers figured out it was the way to shut me up.
>
> Of course, what is good in a course for a particular set of students is
> not necessarily the "best" in whatever absolute sense "best" might be
> valid in.
IMHO your presentation is excellent. No need to change it on account of
what I say.
Thanks
Bill
>
> --
> Timo
Dirk Van de moortel
"RP" <no_mail...@yahoo.com> wrote in message news:348j0pF...@individual.net...
It was a rhetorical question.
But, yes, in case your question was not rhetorical, you amply proved
before that you never had a clue about relativity, and with that
sentence you wrote there, you prove it al over again.
After everything that has been told to you, this should even prove
to *yourself* that you are stupid. Not even *that* you understand.
See?
Dirk Vdm
Ken S. Tucker
See AE's GR1916 Eq.(43)...in "Dover's PoR",
the Riemann-Christoffel Tensor B^a_bcd=0
for SR to be valid, meaning there is no
accelerations permissible, the metrics may
be constant in all FoR's. ((Please do not
confuse this with the idea you can't have
acceleration in SR, for the purpose of
ad hoc mechanical problem solutions, we're
in the realm of describing general physical
laws and their boundary's)).
> What is the minimum physics needed to uniquely define inertialness in
a
> manner that must be consistent with mechanics?
>
> Newton 1 only requires force as a concept (well, force-free, but that
> appears to require force). (I think I was writing Newton 1 and 2
earlier,
> but is not Newton 1 sufficient to define inertialness?)
>
> Perhaps this isn't the most useful approach.
>
> The PoR defines an inertial frame as one in which the laws of physics
(in
> a particular form - OK, here is the tautology, but can it be avoided
?)
> hold. Since, empirically, this requires a homogeous and isotropic
metric
> (in the SR-regime) with signature (-+++) or (+---), then the validity
and
> necessity of the Lorentz transformations is obvious, and we're simply
not
> interested in frames where the laws of physics don't hold.
Signatures with (+---) don't work well in GR
except as a approximation.
A nonorthogonal (++++) works using U_i=0, by
vanishing absolute spatial motion, dx_i dx^i,
in all FoR's.
> Timo
Regards
Ken S. Tucker
Androcles
That's quit normal for a bullet from a gun, too.
> What I'm inferring here is that you must assume that space is an
> objective entity, existing independently of the matter that exists
> within it, and that the particle is moving at a constant velocity wrt
> *it* after its emission from the source.
What property of space maintains the speed of the planets in
their orbits?
> Let me qualify this conclusion before you jump to your own
> conclusions, which more than likely weren't implied in the argument:
Please do.
>
> Because the source may accelerate at any moment after emitting the
> particle, and in so doing enter into a frame in which, according to
> Ritz, the particle was moving at c+v, then either the particle
> accelerates along with the source, or it is maintained in its motion
> at c wrt a frame in which no observer any longer exists. Thus what
> maintains the particle's speed of c wrt nothing?
Since when did the speed of a bullet require something to maintain it?
> Now you'll probably say something like, "it was moving at constant
> speed wrt every other inertial observer too, and thus it is these
> observers that maintain the uniform and consistent motion of the
> particle, i.e. it is maintained wrt them".
Why would I say that? As an observer, I too can accelerate and that
changes the relative velocity of the bullet for me. It's called "running
away".
Once I've reduced the relative speed of the bullet to zero, it cannot
harm
me.
This is something that happens every day on the highway. The car
following
me at 60 mph relative to the road has a 1 mph relative velocity to me
travelling at 59 mph by my speedometer, and will eventually pull out
and pass me. The surrounding space doesn't have a darned thing to
do with maintaining a 1 mph speed.
> To this I'll say, Aha! Then why don't the particles move at c wrt
> these other masses as well, that is, since the particles are equally
> influenced and regulated in their propagation by them.
The car coming the other way with a speed of 61 mph relative to the
road has a speed of 120 mph relative to me, and a speed of 121 mph
with respect to the car following me. I do not influence the speed of
the other two cars relative to each other, I can only accelerate myself.
> Are you going to argue that the light particles are moving uniformly
> wrt nothing,
Good grief no, that's Einstein's second postulate.
"light is always propagated in empty space with a definite velocity c"
Reference :
http://www.fourmilab.ch/etexts/einstein/specrel/www/
> or is it that it is moving wrt a separate ether,
Good grief no, there is no aether.
> or are you rather going to state sanely that it is uniform only wrt
> the collective sum of the matter in the universe, i.e. that motion is
> Machian.
I've already stated it to be c relative to the source, as Ritz did.
Perhaps what you are missing here needs clarifying.
The velocity of a bullet is constant relative to the machine gun that
fired it
at the *instant* of pulling the trigger. What happens to the gun after
that isn't going to change the motion of the bullet relative to the
collective sum of the rest of the universe. Sanely.
> Now before you go comparing these particles to balls or trucks or
> mosquitos, and assuming that the same must apply to them, i.e. that
> their speeds must also be invariant, let's first note that in this
> case the argument does apply to them in at least one sense, that is,
> they too must be assumed to either be moving wrt an independent space,
> As I said, that is Einstein's insane notion.
> or else relative to other matter in the universe,
There isn't just one speed for all other matter in the universe.
For every pair of cars on the highway there is a unique relative speed.
> in that there must be some mechanism for them to continue along at a
> uniform velocity.
If the PoR is called a "mechanism", then ok, there is a mechanism.
> In the fist alternative (ether), the particles are separated causally
> from any other neutral objects, and in the second they are causally
> connected to all of them in every instant of their existence. IOW,
> there are only two possibilities providing for propagation uniformity
> (otherwise called the law of inertia), and unfortunately Ritz's
> ballistic model of light doesn't correspond consistently with either
> of these remaining alternatives.
I fail to follow your argument. Inertia is exactly the basis of Ritz
theory. The gun imparts the initial velocity, inertia does the rest.
>
> Now to answer the question, why doesn't the mosquito propagate at the
> same speed wrt every inertial frame,
Because it cannot. If I drive by the mosquito at 55 mph, then in my
frame of reference the mosquito is going away from me at 50 mph
yet still travels toward Joe.
> note that this simply isn't what the lorentz transform predicts.
The LT is the result of an error in Einstein's computation.
> If you can't understand the difference between c and 5m/s, and between
> bosons and fermions, in the qualitative sense of the theory , then you
> can't understand the theory.
If you can't understand the principle of relativity and think that
the light particles are moving uniformly wrt nothing,
then you cannot understand physics.
> c is the maximum speed, and 5 m/s isn't.
c is a symbol representing velocity. If you can't understand that,
you can't understand algebra.
> As c is approached the speed of a point comes closer and closer to
> being invariant, but doesn't actually reach invariance until it
> achieves c.
As I said, there is an error in Einstein's equation used to derive the
LTs.
>
> There is another alternative that I mentioned, but didn't address
> above, but it is so stupid as to barely be worthy of discussion, to
> wit, it is the assumption that the light particle maintains a causal
> connection to the source throughout its flight, i.e.. that it does
> indeed accelerate in tandem with the source.
Ritz doesn't claim that, and neither do I.
There is another alternative but it is so stupid as to barely be
worthy of discussion, to wit, it is the assumption that the light
particle maintains a causal connection to the OBSERVER
throughout its flight, i.e.. that it does indeed accelerate in tandem
with the OBSERVER.
Relativists claim that.
I agree it is not worthy of discussion, but we discuss it anyway.
>
> Such a behavior would preclude the particle from being absorbed or
> otherwise disturbed by other masses, since their interaction with it
> is contradictory to the premise that the particle is bound in its
> behaviors to the motion of the source. That is, in order to have a
> light particle that interacts with masses other than the source, both
> the source and the detector must have equal rights to the properties
> of the particle, that is, insofar as its propagation is concerned.
I agree, to claim that the speed of light is invariant and c with
respect
to either source or observer is absurd. Why do you do it, and why
do you have a preference for the observer rather than the source?
>
> Thus either the particle's propagation is dependent upon the motions
> of both source and detector,
Correct. That is the PoR in action.
> or IOW these masses cooperate to establish the space-time medium<sic>
> of propagation,
There is no cooperation in any fanciful spacetime between
gun and target. Once the bullet is on its way, either can move.
> or else there is a separate medium, i.e. an ether.
No, it needs no aether to keep a bullet moving.
>
> So regardless of Ritz's claims, we have logically arrived at the
> special relativity/ether debate, and the ballistic model that you
> propose has been eliminated as a viable possibility, by reductio.
>
> Richard Perry
WE? WE have arrived? Is that the royal "we" that Queen Victoria
used and Einstein adopted, meaning "I" ?
YOU have arrived at
"the light particles are moving uniformly wrt nothing"
and
"the assumption that the light particle maintains a causal
connection to the OBSERVER throughout its flight, i.e..
that it does indeed accelerate in tandem", which
"is so stupid as to barely be worthy of discussion".
Androcles.
Dirk Van de moortel
"Androcles" <du...@dummy.net> wrote in message news:UiODd.72508$Z7.4...@fe2.news.blueyonder.co.uk...
[snip]
> > So regardless of Ritz's claims, we have logically arrived at the
> > special relativity/ether debate, and the ballistic model that you
> > propose has been eliminated as a viable possibility, by reductio.
> >
> > Richard Perry
>
> WE? WE have arrived? Is that the royal "we" that Queen Victoria
> used and Einstein adopted, meaning "I" ?
>
> YOU have arrived at
> "the light particles are moving uniformly wrt nothing"
> and
> "the assumption that the light particle maintains a causal
> connection to the OBSERVER throughout its flight, i.e..
> that it does indeed accelerate in tandem", which
> "is so stupid as to barely be worthy of discussion".
>
> Androcles.
Androcles, be careful there, you are arguing with a fellow
crackpot!
Dirk Vdm
Dirk Van de moortel
"Ken S. Tucker" <dyna...@vianet.on.ca> wrote in message news:1105140030.6...@c13g2000cwb.googlegroups.com...
> I agree with Mr. Perry
You even talk to Sarfatti.
Dirk Vdm
Timo Nieminen
> "Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote:
> > On Sat, 8 Jan 2005, Bill Hobba wrote:
> > > "Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote:
> Assuming of course we agree that 'free'
> means we can not find something else in the frame that would be responsible
> for the force. That is the problem with using 'free' - we must look around
> and ensure other causes are not about and the reason I am not 100% happy
> with it - you are making a physical assumption - forces are always caused by
> something else in the frame - if not then the frame is not inertial. I
> prefer the one based on symmetry because no other physical assumption is
> made - construct the coordinate system in the usual way, sync clocks the
> usual way and check if for example clocks tick at the same rate.
Which is, basically, bringing a lot of physics into the definition - the
physics of rods and clocks that can be used to construct a coordinate
system is nontrivial.
> > > > The PoR defines an inertial frame as one in which the laws of physics
> (in
> > > > a particular form - OK, here is the tautology, but can it be avoided
> ?)
> > > > hold.
> > >
> > > I am not so sure the POR is a definition of an inertial frame. It is a
> > > hypothesized principle they are supposed to possess. It is refutable
> > > experimentally eg a positive result on the Trouton-Noble experiment
> would
> > > have cast strong doubt on it.
> >
> > PoR could still hold. For example, if Trouton-Noble is in agreement with a
> > stationary ether, then a more complete EM theory, accounting for velocity
> > relative to the ether, simply leads to Galilei-symmetry. The PoR in itself
> > doesn't specify the transformation law between inertial frames.
>
> Ahhhhhhhhh. First the POR all by itself does specify the transformation law
> up to an undetermined value of c - a value of infinity would give the
> Galilean transformations - a finite value contains essentially the same
> physics for any value because it can be set to one by a suitable choice of
> units.
At least the PoR + a reasonable guess at our laws of physics specifies it.
At first thought, I wouldn't rule out the posssibility of constructing a
hypothetical pathological set of laws of physics that gives another
transformation law.
But, more concretely, that a 3D Euclidean frame and a 1D Euclidean frame
can be used is a consequence of our physics, and not the PoR itself. The
PoR, sans consideration of what the laws of physics are, doesn't even tell
us that we have 3+1 dimensions.
But perhaps this is largely a matter of precisely what we mean by PoR.
I guess I might go and read next week:
> This is discussed at length on page 19 of Rindler - Introduction to
> Special Relativity under the heading of the role of the second axiom.
> Also
> consider the following. If an aether did exist, was the medium that light
> undulated in, and an inertial frame was stationary in, then a frame moving
> at constant velocity to such a frame would have an aether wind breaking its
> isotropy so it would not be inertial. Thus the POR would be invalidated.
[cut for brevity]
Well, I have to say that I agree with Griffiths. Given a measurable ether,
then the laws of electrodynamics must include velocity wrt the ether.
Given that eg Maxwell's original theory was presumably meant to hold in
the (local) ether rest frame, it is not necessarily even difficult to
write down laws of electrodynamics taking it into account.
Surely this is nothing more (in a real-ether-universe) than writing down
the laws of fluid dynamics for a moving fluid?
Also note that one can write down laws for fluid dynamics for moving fluid
assuming Galilei symmetry, or assuming Lorentz symmetry - ether would not
even automatically mean Galilei symmetry holds (although it's a safe guess
that it would have been assumed to hold).
The goldfish bowl analogy is appropriate, especially considering ethers
with no global state of rest. If an ether were to be a swirling fluid with
global rest frame, would it make any frame special?
> If one conceptually removed such an aether then light itself would not exist
> so the laws of physics as we know them would be down the tube. To me that is
> something implicit in the idea of the POR - our usual laws of physics must
> still hold.
The laws of physics must still hold - not necessarily what we think (or
thought) the laws of physics are (were).
> > > > The PoR being sufficient definition in and of itself, no further
> > > > definition is needed. (As you noted, a futher definition is more or
> less a
> > > > tautology.)
> > > >
> > > > I'd rather just stop at that point than adopt a broader definition of
> > > > inertialness.
> > >
> > > I still think the definition based on symmetry properties is the best
> one.
> >
> > In a specialised relativity course, or in a mechanics course, I think that
> > might be a better definition. I want to do relativity in 2 weeks, 1/2 of
> > that being specifically EM, so I want to avoid mechanics beyond that
> > required for motion of charged particles.
>
> Timo - IMHO your derivation was excellent. If I was in your class and
> exposed to it I would be very happy indeed. The only reason I raised such
> issues is that long postings on sci.physics.relativity have caused me to
> reflect on the foundations of relativity so that I have reached a slightly
> different view. If any student raises the issues I did in this post then
> the proper place to handle it is outside class and you can refer the student
> to other source material. Indeed for such an interested student it might
> form the basis of an assignment or perhaps a class presentation if your
> teaching style accommodates such. That usually shuts up wise ass students -
> at least it did me when lecturers figured out it was the way to shut me up.
My students are perfectly welcome to choose such assignment topics (and,
indeed, one did something vaguely along those lines last year). Especially
since I only cover it very briefly. But that's the point of assignments
with free (within limits) choice of topics - explore things not covered in
the basic material. (And all assignments include class presentations, so
that covers all of your suggestions :) )
Last year's crop of assignments was diverse, and I did learn from them
too. Who could ask for more?
--
Timo
Bill Hobba
"Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote in message
news:Pine.OSF.4.58.05...@dingo.cc.uq.edu.au...
>
> > "Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote:
> > > On Sat, 8 Jan 2005, Bill Hobba wrote:
> > > > "Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote:
>
> > Assuming of course we agree that 'free'
> > means we can not find something else in the frame that would be
responsible
> > for the force. That is the problem with using 'free' - we must look
around
> > and ensure other causes are not about and the reason I am not 100% happy
> > with it - you are making a physical assumption - forces are always
caused by
> > something else in the frame - if not then the frame is not inertial. I
> > prefer the one based on symmetry because no other physical assumption is
> > made - construct the coordinate system in the usual way, sync clocks the
> > usual way and check if for example clocks tick at the same rate.
>
> Which is, basically, bringing a lot of physics into the definition - the
> physics of rods and clocks that can be used to construct a coordinate
> system is nontrivial.
Yep.
Good point. That would seem to be part of the whole thing - we must make
reasonable assumptions about what is meant by the POR. When faced with that
type of thing I am usually in the camp if someone says something I do not
necessarily agree with, but it is reasonable, then that is fine by me - they
may in fact be correct.
> At first thought, I wouldn't rule out the posssibility of constructing a
> hypothetical pathological set of laws of physics that gives another
> transformation law.
Yes - Tom Roberts has posted they in fact do exist and are experimentally
indistinguishable from SR.
>
> But, more concretely, that a 3D Euclidean frame and a 1D Euclidean frame
> can be used is a consequence of our physics, and not the PoR itself. The
> PoR, sans consideration of what the laws of physics are, doesn't even tell
> us that we have 3+1 dimensions.
Yep.
>
> But perhaps this is largely a matter of precisely what we mean by PoR.
Yep.
Yes that would seem a crucial point in the whole issue. As I said above I
am happy with any reasonable position. Now if you are willing to accept
that the POR may include situations where what we currently think the laws
of physics are (eg the laws of optics as we currently understand them could
conceivably not even exist) then I would say you and Griffths have a
definite point. But as you probably have guessed that is not part of what I
would consider a reasonable interpretation of what the POR is saying.
That is basically what real learning is all about. Of course I was being a
bit tongue in cheek. I know from previous posts you have a teaching style
that includes such things. Indeed I was being a bit cheeky about when I was
studying as well - while I was a 'wise ass' in class it was because I was
actually interested in the material. The lecturers knew that and encouraged
me to investigate stuff outside the class. They were also very liberal
about assignments and presentations were part of what was required -
excellent preparation for actual work.
Thanks
Bill
--
> Timo
Ken S. Tucker
Dirk Van de moortel wrote:
Oh Yeah, you're worse, you even talk to Tucker!
jem
Remember this proposed definition? An inertial reference frame is a
reference frame in which all stationary standard clocks tick at the same
rate.
RP
Bilge wrote:
> RP:
> >Bilge wrote:
>
> >> I'm willing to bet that you cannot define an inertial frame using
> >> newton's laws in a way which isn't circular.
> >
> >Frames K and K' are inertial, iff K' emits em waves of constant
> >frequency wrt itself that are observed to be of constant frequency wrt
> >K as well, and if K' is changing in displacement wrt K over time.
>
> That is circular.
There is nothing circular about it, it is measurable. If you have a
clock then you can determine whether you are in an inertial frame, but
only if you are clocking the radiation from an inertial source that is
in motion wrt you.
>
> >An inertial frame has no intrinsic defining properties.
>
> If you believe that, the why are responding and why did you try to
> specify just such a property?
I was only implying that inertia is a Machian effect, i.e. that frames
cannot be defined except wrt other frames. As you can see above, I did
define an inertial frame in terms of another inertial frame, thus it
was implicit within that definition that a single frame has no
intrinsic defining properties. IOW, you cannot define an inertial
frame without referencing other frames. This is why Newton's
definition falls short. "Force-free motion", is equivalent to
"inertial motion", just different terminology applied there, and this
defining property of inertial frame doesn't reference other frames,
i.e. there is nothing Machian about Newton's postulates. They've been
supplied with consistency, after the fact, by Noether's Theorem, which
in turn inserts the necessary symmetry into the mix. This was absent
in Newton's speculations, thus leaving him a bit empty handed.
Symmetry in turn, requires something external with which to be
symmetric wrt to, thus the requirement of at least one more frame in
order to define the first.
Richard Perry
>
RP
jem wrote:
There is no such frame in this universe. You could qualify it with
"in a small region of space". Now suppose, however, that we have an
inertial source and an inertial detector, but these are located such
that a galaxy is located between them. This isn't a small region of
space, and thus even the amended definition fails to correctly define
inertial frames. If OTOH, you allow extended regions of space, then
you've taken in the gravitational field of the galaxy, and all
stationary clocks within that region don't tick at the same rate. Hmmm.
I suggest the alternate, "An inertial reference frame is a reference
frame in which all stationary standard clocks tick at fixed rates wrt
each other, and at fixed rates wrt the clocks that are stationary wrt
a second inertial frame". This is however perfectly equivalent to my
previous definition.
Richard Perry
Edward Green
> On Thu, 6 Jan 2005, Bill Hobba wrote:
>
> > "Edward Green" <spamsp...@netzero.com> wrote:
> > > Timo Nieminen wrote:
> > >
> > > > The aim is to provide a simple and general derivation of the
> > > > homogeneous Lorentz transformations, without assuming that axes
are
> > > > parallel, or that motion is along the x-axis. (Perhaps a
revision
> > > > to avoid the use of Cartesian coordinates might be useful?)
> > >
> > > The entire idea of "Lorentz transformation" in fact insists we
will
> > > only consider coordinate systems which are, in their own internal
way,
> > > Cartesian.
> >
> > No - it considers inertial coordinate systems. Inertial
coordinates is a
> > different concept than Cartesian coordinates.
>
> (mostly piggybacking, since I didn't see the original reply)
>
> Well, the space part of the coordinates can be a Cartesian system,
but
> it's only a Cartesian coordinate system if it's a Cartesian
coordinate
> system. Nothing stops the space part of the coordinates being
cylindrical
> or spherical, whatever is most convenient.
>
> The 4-coordinate system is not Cartesian. It looks Cartesian, but a
metric
> g_00 = -1, g_11 = g_22 = g_33 is not g_ij = d_ij.
>
> Flat space, yes. Inertial in practice, but the Lorentz transforms
> themselves say nothing about inertialness - one can use them to
transform
> between a set of non-inertial frames.
Yes, and one can use an audio amplifier to process graphics, but the
result is unlikely to mean very much!
The world view of the Lorentz transformation is that we implicitly have
at least one good coordinate system where physics looks simple and we
would like to identify some others. Under the assumption that the
velocity of light will be isotropically "c" in these other coordinate
systems, out falls the Lorentz transmormation. "Physics looks simple"
is an undefined phrase so far, but at least it is an undefined phrase
which advertises itself as such! "Inertial" may suggest we know exactly
what we mean, but we may not.
We all know what a "non-inertial coorindate system" looks like for
example: a frame of spatial coordinates which may be rotating or
accelerating relative to our assumed at-least-one-good-frame. But are
cylindrical coordinates "inertial", even if fixed in an inertial
spatial frame? One might think so, but under the approach that
Newton's First law holds with uniform motion represented in simple
coordinate form "A + lambda(B-A)" [A,B triples of spatial coordinates]
they are not! So, one might say, that is simply not the appropriate
form to represent "uniform inertial motion" in these coordinates. Ok.
But the same might be said of any "non-inertial" frame: Newton's first
law holds provided we express uniform inertial motion in terms
appropriate to that frame! So I guess then all coordinate systems are
inertial? ;-)
We may make the distinction between coordinate systems where a
non-cartesian spatial part is piggy-backed on the cartesian part of a
proper Lorentz coordinate system, and all other "non-inertial"
coordinate systems, but we might also adopt the POV of KISS, and
consider the Lorentz transformations to interrelate coordinate systems
in which physics looks particularly simple for uniform inertial motion,
and leave all other developments to the machinery of GR, in principle,
which interrelates any damn coordinate system.
Anyway, Bill Hobba's critical "no" is unjustified, in retrospect,
because I didn't say that all coordinate systems with Cartesian spatial
parts are treated by Lorentz transformations, but that all coordinate
systems properly interrelated by Lorentz transformations are Cartesian.
That's true for a reasonable meaning of the words (spatial part is
Cartesian, which I think is what you meant). If you go beyond that to
relationships between more general coordinate systems, you are starting
down the slippery slope to the machinery of GR, and not considering
Lorentz transformations anymore, tautologically.
Franz Heymann
Androcles wrote:
> Poe has capitulated.
Randy capitulated because he realised the impossibility of arguing
with a baboon.
Have you coped with the mosquito problem yet?
Franz
Franz Heymann
"Timo Nieminen" <uqtn...@mailbox.uq.edu.au> wrote in message
news:Pine.OSF.4.58.05...@dingo.cc.uq.edu.au...
OT: I wonder why Androcles has not had something to offer in this
thread. After all, his specialist knowledge of the flaws in the
deduction of the Lorentz transform should stand him in good stead.
Franz
Franz Heymann
[snip]
> It's a model Dirk, and Ritz proved this adequately by accounting for
> all known electromagnetic phenomena from within a Galilean context
Prolate spheroids..
There are no known classical EM phenomena which cannot be handled
quantitatively and correctly by Maxwell's equations.
Maxwell's equations obey Lorentz invariance
Maxwell's equations do not obey Galilean invariance.
Do try proving this for yourself.
[snip]
Franz
RP
You're begging the question here Franz.
That Maxwell and Galilean invariance are mutually exclusive isn't an
argument, its a statement. The only conclusion that can be
validly drawn from your *argument* is....well.... that they are
mutually exclusive.
In the
Galilean domain, invariance, that is, wrt the laws of
electromagnetism, is a matter of the invariance of the velocities of
particles wrt each
other. The laws of physics are thus Galilean invariant if they exclude
quantities that are variable through transformations, or else contain
expressions that reduce to invariant quantities. That you haven't
managed to provide these Galilean solutions does nothing to invalidate
the possibility. Can you derive a proof that this isn't possible?
If OTOH, you introduce another premise such as "Galilean invariance is
empirically falsified", then you will have a complete argument, if
only you could provide a comprehensive empirical disproof of
Galilean invariance. In order to provide this you must show that a
form of the laws of
electromagnetism cannot possibly be derived that is consistent with
Galilean invariance. And in order to do this, you must first cough up the
Galilean theory that you wish to disprove.
In both forms of relativity, the transformation is from one Euclidean
frame to another, and simply provides the relative velocities and
positions of points that an observer in the new frame will perceive.
If you hold that each will perceive the speed of light to be equal to
c, then how is this contradictory to the Galilean transform? Even
given the existing qualitative interpretation of light transmission,
we can provide Galilean invariance of light speed by simply allowing
for physical alterations to the measuring devices used. This is what
Lorentz did, it's called LET.
Now as for the static laws of electromagnetism, the speed of
propagation doesn't even enter in, and it's a simple matter to provide
for the forces acting in these systems with Weber/Ritz type
approaches. I know this because I've derived just such a system.
Now, even though I've come to prefer the special relativistic version,
because of its logical basis, this doesn't alter my perspective that
its just a model. Try argue along the same vein with Uncle Al about
metric vs. affine theories of gravity and see whether or no he agrees
that there is anything absolute about a model.
Richard Perry
Ken S. Tucker
Open minded but precise.
Ken
Bilge
>Remember this proposed definition? An inertial reference frame is a
>reference frame in which all stationary standard clocks tick at the same
>rate.
Yes, I remember it. It's still circular and doesn't explain what
``ticks at the same rate'' means.
Bilge
>
>Bilge wrote:
>
>> RP:
>> >Bilge wrote:
>>
>> >> I'm willing to bet that you cannot define an inertial frame using
>> >> newton's laws in a way which isn't circular.
>> >
>> >Frames K and K' are inertial, iff K' emits em waves of constant
>> >frequency wrt itself that are observed to be of constant frequency wrt
>> >K as well, and if K' is changing in displacement wrt K over time.
>>
>> That is circular.
>
>There is nothing circular about it, it is measurable. If you have a
>clock then you can determine whether you are in an inertial frame, but
>only if you are clocking the radiation from an inertial source that is
>in motion wrt you.
Bill Hobba
"jem" <x...@xxx.xxx> wrote in message
news:sdSDd.25692$Q%4.10400@fed1read06...
How do you sync those clocks? I am pretty sure you will need the assumption
of space being homogeneous and isotropic to ensure your chosen method is
valid. When I first read his views on the matter quite some time ago now I
thought and thought about it and was forced to concede he had a point. For
example construct a coordinate system from practically rigid rods, sync
clocks via light (all of which involve further physical assumptions such as
homogeneity and isotropy of space to ensure the method of syncing works -
otherwise it is simply a convention that can be challenged - you see posters
doing it all the time), check if a particle is free then measure its motion.
If all the previously mentioned problems were not enough the problem now is
how do you ensure a particle is free? Exactly what does free mean? Does it
mean if we can not find something in the frame that will not affect its
motion then it is free? Exactly what is meant by something not affecting
its motion? - does that apply to forces of acceleration? One might like to
define an inertial frame by for example the absence of inertial or
gravitational forces as indicated by accelerometers. But again such a
definition relies on further physical assumptions based on our intuition
about force. That seems to be the problem with Newton's first law - one
really needs further physical assumptions to make it meaningful and those
assumptions are all based on physical intuition and/or some convention such
as the Einstein sync procedure. If you do not make these further
assumptions then you run into the problem that Bilge alludes to 'We define a
straight line as the path of an inertial object and we define an inertial
object as one which moves in a straight line' The best way to operationally
define an inertial frame I think is via the EEP.
Thanks
Bill
Bill Hobba
That is irrelevant from the point of view of defining a concept. Remember a
point is defined as having position but no size. Such does not exist but
that does not invalidate geometry. I think Jems proposed definition has
problems but they are not what you suggest.
Bill
Bilge
>> The law is completely circular as it does not say how to determine
>> if the motion is uniform except that it's motion which is force free.
>> We define a straight line as the path of an inertial object and we define
>> an inertial object as one which moves in a straight line.
>
>Is not dr/dt = constant a sufficient definition of uniform motion (at
>least in Cartesian coordinates)?
It's sufficient as a _mathematical_ definition, but so were the
others. It becomes circular when you assume that determining r
physically is ``obvious''. Just consider defining a _new_ vector
r' = f(r) where f(r) might be k ln(r/r_0). dr'/dt and dr/dt are
not both a description of uniform motion, but physically, you
have no obvious way to determime whether dr'/dt is the uniform motion
and r = r_0 exp(r'/k) or vice-versa.
>True, this is not (as far as I've ever seen) included as part of
>Newton's 1st law.
>
>Of course, this doesn't avoid the problem of defining "force-free".
>For which I think there is no easy solution within Newton's laws. Which
>makes me think I should have written: but is not Newton 1, uniform motion
>has constant dr/dt, and we'll worry about what force is later, but assume
>an intuitive idea of "force-free" for the moment :)
OK, but if I assume that, then I eliminate the problem, since it's
equivalent to assuming the intuitive notion of a straight line.
RP
Bilge wrote:
> RP:
> >
> >
> >Bilge wrote:
> >
> >> RP:
> >> >Bilge wrote:
> >>
> >> >> I'm willing to bet that you cannot define an inertial frame using
> >> >> newton's laws in a way which isn't circular.
> >> >
> >> >Frames K and K' are inertial, iff K' emits em waves of constant
> >> >frequency wrt itself that are observed to be of constant frequency wrt
> >> >K as well, and if K' is changing in displacement wrt K over time.
> >>
> >> That is circular.
> >
> >There is nothing circular about it, it is measurable. If you have a
> >clock then you can determine whether you are in an inertial frame, but
> >only if you are clocking the radiation from an inertial source that is
> >in motion wrt you.
>
> No, you can't.
I missed what it was that you were calling circular, and now that I've
caught it, I'll agree that it's circular in that time becomes
definable only in terms of inertial frames. OTOH, I won't agree that
an observer cannot determine that he is in an inertial frame via
measurements, any more than he can't measure any number of other
states using arguments that are equally as circular. Almost every law
of physics relies upon an intuitive understanding of time, so were not
alone in this circularity, i.e. why limit the complaint to the topic
of inertia?
We must assume constancy of time rate, but unless time varies in rate
randomly and differently in various frames, then for all practical
purposes, for us it is constant, that is, it would be impossible for
us to determine otherwise if it varied randomly, or even if it were
changing uniformly in rate. For this reason I believe that the
definition of inertial frame that I provided leaves no room for
ambiguity. A failure to detect ones inertial motion by the means that
I prescribed would require either that time isn't isotropic, or that
the observer isn't actually in an inertial frame. OTOH, if time isn't
isotropic then there are no inertial frames. It would follow that the
conclusion that he isn't in an inertial frame would be correct either
way. If OTOH there are inertial frames that correspond to my
definition, then they are detectable.
As for the circularity, I'm not convinced that this is such a bad
word. I recall stating recently something like "all definitions are
circular, they must be if they are to be unassailable". Not that this
singular property renders them necessarily correct or consistent, of
course.
Richard Perry
RP
Bill Hobba wrote:
I considered this very argument before drawing any conclusions, and I
decided to swing in favor of falsifiability. IOW, I'm not in favor of
idealism. I don't define a point as having position but no size, I
definite a point as simply "the position itself", thus avoiding
altogether the paradox that the standard definition implies. Picking
out ill-conceived definitions like this is one of the tasks that I
enjoy most. I'm a philosopher first I suppose, and philosophy is
nothing more than the process of honing down the language and
illuminating its misapplications, contradictions, redundancies and
oversights.
I specifically tried to find a limiting case of his definition that
would work, and I could not. And if it isn't even a limiting case,
then where am I to apply it? We could settle for something like
" An inertial reference frame is a reference frame in which all
stationary standard clocks tick at *fixed rates* wrt each other.",
but this doesn't do the trick, since all of these clocks could be
accelerating in tandem, and the definition would still apply, thus it
is still lacking.
We should also specify whether a clock that is at rest wrt an inertial
frame is itself in an inertial frame given any other circumstances. I
bring this up because the clock may be located within a different
gravitational potential than the other and still satisfy the
definition of inertial frame that I provided. My contention is that it
is in an inertial frame because it is acceleration rather than
potential that is the determining factor in this argument. I also
considered this argument before deriving my definition. If by doing so
I've derived a definition of inertial frame that differs somewhat from
Newton's interpretation, then so be it, it is the only definition that
I could derive that corresponded to "force free motion", and that
simultaneously provided empirical utility.
Richard Perry
no_mail...@yahoo.com
Thank you for your kind words, here, and in another post.
Unfortunately none of your posts to this thread have shown up on my
server. Had it not been for Dirk's reply to you I would have missed
them entirely, though you don't seem to be missing from any other
threads. Sorry for the delay, but I had to go to Google Beta to find
you, although it was worth the effort. As always, you've provided
intelligent arguments.
Thanks again.
Richard Perry
Tom Roberts
> The standard notation of vector analysis is a really neat way of sweeping
> ugly things like metric tensors under the carpet.
Only in flat manifolds.
But that vector analysis notation generalizes to tensor notation, which
works in any manifold and succeeds in sweeping ugly things like
COORDINATES under the carpet. The metric tensor is nowhere near as ugly
as coordinates -- in fact the metric is downright necessary if you ever
want to measure distances (which applies to virtually every experiment).
>>>which maintains the appearance of coordinate equality,
>>>but conceals the fact that some coordinate systems are more equal than
>>>others.
Only when you write equations that are valid only in flat manifolds,
ignoring various enormously-complicated terms involving curvature
tensors....
> Of course some coordinate systems are more equal than other, when it comes
> down to the algebra or number-crunching.
Yes. Generally because the coordinates themselves reflect some symmetry
(Killing vector) of the physical situation.
> I'm sure I could manage a good rant about why spherical
> coordinates are better computationally than Cartesian coordinates.
Only when the physical situation has spherical symmetry.
Tom Roberts tjro...@lucent.com
Tom Roberts
> mme...@cars3.uchicago.edu wrote:
>>In other words to convey the point
>>that the proper statement is not "c is the speed of light" but "the
>>speed of light is c".
>
> Wrong. Equality is symmetric.
You misunderstood, and misspoke. What you are thinking of is: "'=' is
symmetric", which displays your error -- mathematical equality is indeed
symmetric, but ENGLISH is not. "c is the speed of light" is defining c
but "the speed of light is c" is not. That was Mati's point: the value
of c is not really determined by light, but by the local symmetry
properties of spacetime; light "just happens" to have the same local speed.
Tom Roberts tjro...@lucent.com
Timo Nieminen
> Timo Nieminen wrote:
> > The standard notation of vector analysis is a really neat way of sweeping
> > ugly things like metric tensors under the carpet.
>
> Only in flat manifolds.
Sure.
> But that vector analysis notation generalizes to tensor notation, which
> works in any manifold and succeeds in sweeping ugly things like
> COORDINATES under the carpet. The metric tensor is nowhere near as ugly
> as coordinates -- in fact the metric is downright necessary if you ever
> want to measure distances (which applies to virtually every experiment).
Hmm. Does not vector analysis already hide coordinates (in flat
manifolds)?
Does tensor notation hide coordinates? I suppose it depends on what one
means by "hides" ...
> > I'm sure I could manage a good rant about why spherical
> > coordinates are better computationally than Cartesian coordinates.
>
> Only when the physical situation has spherical symmetry.
Even without spherical symmetry. Real, finite objects resemble spheres far
more than they resemble infinite slabs, or infinite half-spaces.
A discrete plane wave approximation to a spherical wave never converges in
the far field. That sucks.
Plane waves of non-zero amplitude have infinite energy flux, infinite
momentum flux, and (if applicable) zero or undefined angular momentum
flux. That sucks.
--
Timo
Tom Roberts
> On Fri, 7 Jan 2005, Bilge wrote:
>> Another way to look at those requirements is that homogeneity and
>>isotropy _define_ an inertial frame.
>
> I am uncomfortable with the definition of an inertial frame as a frame of
> zero intrinsic curvature with a homogeneous and isotropic metric,
Any 3-d Riemannian manifold having 6 Killing vectors is globally flat.
Homogeneity is a set of 3 Killng vectors, and isotropy is another set of
3 (in this 3-d space). In such a flat space there exists a global set of
coordinates in which the metric is diag(1,1,1). So a homogeneous and
isotropic 3-d space is either E^3 or a topological variation of it.
Because all pre-relativistic physics implicitly assumed Euclidean space,
I think it's quite reasonable to postulate Euclidean space in a
derivation of the Lorentz transform. IMHO doing so avoids several
problems, such as those being discussed in this thread.
> in that
> not all such frames are inertial, in the sense of Newton's laws.
In spaceTIME they are (see below).
> For
> example, I suspect (but haven't checked, so feel free to correct me if I'm
> wrong) that an accelerating frame admits a coordinatisation such that the
> metric is homogeneous and isotropic,
In space, sure. But not in spacetime. Applying accelerated coordinates
to Minkowski spacetime, the metric components are not homogeneous in space.
Someone thought it might be possible to E-synch clocks in
an accelerated system and mimic an inertial frame. This is
not possible -- basically clocks at rest in an accelerated
system "tick at different rates" depending on position, and
E-synching them is not possible. This is just a physical
consequence of the fact that in accelerated coordinates g_tt
is dependent on spatial position.
> and such a frame is clearly not
> inertial in the mechanical sense. While such a coordinatisation might well
> be pointless and bizarre, how can it be excluded without bringing in some
> real physics?
It can be excluded geometrically, in spaceTIME. And that's enough.
> What is the minimum physics needed to uniquely define inertialness in a
> manner that must be consistent with mechanics?
>
> Newton 1 only requires force as a concept (well, force-free, but that
> appears to require force). (I think I was writing Newton 1 and 2 earlier,
> but is not Newton 1 sufficient to define inertialness?)
>
> Perhaps this isn't the most useful approach.
A modern statement of Netwon's first law is probably more useful: a free
object moves in a uniform straight line relative to an inertial frame.
But I think you'll ultimately have to just define an inertial frame to
have Euclidean 3-space.
> I'd rather just stop at that point than adopt a broader definition of
> inertialness.
To me, the ultimate weakness of SR is its dependence on inertial frames,
and the difficulty in defining them. This is at base a geometrical
difficulty, and I don't think there is a non-geometrical solution.
That's why I think one must ultimately include Euclidean 3-space in the
definition of inertial frame.
Or try a whole different approach: assume GR and only use
SR as its local limit. This, of course, is non-trivial....
But the difficulties of defining inertial frames vanish (:-)).
Tom Roberts tjro...@lucent.com
Tom Roberts
> On Sun, 9 Jan 2005, Tom Roberts wrote:
>>But that vector analysis notation generalizes to tensor notation, which
>>works in any manifold and succeeds in sweeping ugly things like
>>COORDINATES under the carpet. The metric tensor is nowhere near as ugly
>>as coordinates -- in fact the metric is downright necessary if you ever
>>want to measure distances (which applies to virtually every experiment).
>
> Hmm. Does not vector analysis already hide coordinates (in flat
> manifolds)?
Yes.
> Does tensor notation hide coordinates?
Sure it does. For instance, here are Maxwell's equations in the notation
of differential forms (a specific type of tensor):
dF = 0
*d*F = J
Here F is the Maxwell 2-form, J is the current 1-form, d is the exterior
derivative, and * is the Hodge dual. Not a coordinate in sight. The
metric does not appear (though it is lurking inside the Hodge dual). As
a bonus, these equations are valid in curved manifolds.
You may be thinking of component notation, like
G^ab + k g^ab = T^ab
This is not tensor notation; the EFE in tensor notation is:
G + k g = T
Again, not a coordinate in sight. And the metric appears only
in its guise as a dynamical participant in the cosmological
term.
Tom Roberts tjro...@lucent.com
jem
> jem wrote:
>>
>> Remember this proposed definition? An inertial reference frame is a
>> reference frame in which all stationary standard clocks tick at the
>> same rate.
>
>
> There is no such frame in this universe. You could qualify it with
> "in a small region of space".
"Inertial frame" is defined within theory - there's bo need for such a
reference frame to exist in Nature.
> Now suppose, however, that we have an
> inertial source and an inertial detector, but these are located such
> that a galaxy is located between them. This isn't a small region of
> space, and thus even the amended definition fails to correctly define
> inertial frames. If OTOH, you allow extended regions of space, then
> you've taken in the gravitational field of the galaxy, and all
> stationary clocks within that region don't tick at the same rate. Hmmm.
>
> I suggest the alternate, "An inertial reference frame is a reference
> frame in which all stationary standard clocks tick at fixed rates wrt
> each other,
There are non-inertial reference frames where stationary clocks tick at
fixed rates wrt each other.
> and at fixed rates wrt the clocks that are stationary wrt
> a second inertial frame".
It's generally not a good idea to use the term being defined within its
definition.
jem
"Ticks at the same rate" means the number of ticks recorded on each
clock is equal over every time interval. The comparison between any 2
clocks is accomplished by sending signals from one to the other and
comparing the rate of the received signal with the tick rate of the
clock at the reception point. This comparison can be done without
synchronizing the clocks (although synchronizing them at the onset would
make an actual comparison easier).
Where do you see circularity?