Why Must The Lorentz Transformation Equations Be Linear?
Shubee
Lorentz transformation. Here it is:
http://www.everythingimportant.org/relativity/special.pdf
Igor
> I promised many years ago to publish my derivation of the nonlinear
> Lorentz transformation. Here it is:http://www.everythingimportant.org/relativity/special.pdf
Lorentz transformations are linear transformations by definition.
Nonlinear spacetime transformations are covered by General
Relativity. If you can find nonlinear transformations that preserve
the Minkowksi metric, fine. You just can't refer to them as Lorentz.
Guy Neapig
> I promised many years ago to publish my derivation of the nonlinear
> Lorentz transformation. Here it is:http://www.everythingimportant.org/relativity/special.pdf
I found it a nice idea to use the position graduation of the moving
reference frame as a clock
in the rest frame. However, when you generalize equs (4-5) to (6), you
introduce a dependency
of the g_i(x_i) upon the mu_ij (mutual speed of the frames) that
doesn't seem correct to me,
because the g_i functions are redefinitions of the zero of the clock
at each position, they
are a choice made by the observers using the clock, hence they don't
depend on theses speeds.
Guy
Shubee
Hi Guy,
Thanks for liking my new approach to thinking about a spacetime clock
at each point of an inertial frame of reference. I do believe that the
idea of replacing the clunky Einstein clocks with the abstract
mathematical clocks (and exchanging those bulky material rods for
those light ethereal rulers) was a big help in my elementary
integration of space and time.
I do, however, have a hard time understanding your objection to
equation (6). If you would solve the familiar equation x' = Y(v)(x-vt)
for t, thinking of Y(v) as gamma, t as t_i and x' as x_j, while
remembering that v stands for v_ij, then
t_i = -x_j/vY(v) + x_i/v
So, ultimately, the g_i(x_i) do indeed depend upon the mu_ij, which
are just constants.
BTW, I don't solve equation 6 in the paper. I just thought that I
should state my expectation of what the most general time equation in
1 spatial dimension should look like.
Shubee
Shubee
In the abstract on page 1, I called them “nonlinear Lorentz-equivalent
transformation equations.” If you can think of a more appropriate
name, I would love to consider it.
Shubee
Tom Roberts
First, let me answer the question in the subject (which is not addressed in the
linked paper):
Inertial frames are defined as coordinates relative to which free objects move
in uniform straight lines (i.e. Newton's first law holds). It is straightforward
to prove that the map between any pair of inertial frames must be linear --
linearity is required in order to map uniform straight lines to uniform straight
lines, as is required by the definition of inertial frame. Lorentz transforms
are defined as maps between inertial frames, and thus must be linear.
At least one person has claimed that the presence of the factor
1/sqrt(1-v^2/c^2) make them "nonlinear". This is irrelevant, as
"linear" applies to the map, not to coefficients of the map.
In the linked paper above, the "nonlinearity" is really just an arbitrary
function S(.) describing non-standard clock synchronizations. Most functions do
not yield an inertial frame (in the sense above). The trivial S(.)=0 does yield
an inertial frame; there are other forms of S(.) that yield an inertial frame,
but the resulting transform is linear, and is equivalent to either a) some other
inertial frame, b) to a change of units, or c) to a change in clock synchronization.
Note also that the approach of that paper has rather serious drawbacks:
A) such a "clock" is not implementable in the real world
B) it does not generalize to more than 1 spatial dimension
C) it does not give any way to relate the quite artificial constructs
to the real world -- i.e. it is not useful as a model
D) the author has a writing style that is both juvenile and excessively
naive (redefining common words in artificial ways), making the paper
difficult to read with a straight face
Shubee clearly does not understand the basics of modern physics. In particular
he does not realize the importance of providing a model of the world (rather
than just a cute construct). Note that he actually applied some group theory in
the paper, but did not realize it; its presence is the reason he could achieve a
recognizable result. Had he pursued it, he would have found that the freedom
represented by S(.) would yield the transforms of the equivalence class of
theories that are experimentally indistinguishable from SR (indeed he came
rather close to presenting the Tangherlini transforms, and at one point I
thought they were where he was headed).
In short, I see nothing useful here.
Tom Roberts
Dono.
> I promised many years ago to publish my derivation of the nonlinear
> Lorentz transformation. Here it is:http://www.everythingimportant.org/relativity/special.pdf
Your theory is falsified by the Ives-Stilwell experiment.
Rock Brentwood
> Inertial frames are defined as coordinates relative to which free objects move
> in uniform straight lines (i.e. Newton's first law holds). It is straightforward
> to prove that the map between any pair of inertial frames must be linear --
> linearity is required in order to map uniform straight lines to uniform straight
> lines, as is required by the definition of inertial frame. Lorentz transforms
> are defined as maps between inertial frames, and thus must be linear.
No. Actually, the condition you refer to defines an AFFINE map, not a
linear map. The Lorentz transformations are not the most general
transformations -- they're linear. The affine maps are the Poincare'
transformations. These are the most general case.
This too -- even -- requires qualification. If one only requires lines
to map to lines, one can go further into projective geometry. The
affine maps are only those which preserve the "points at infinity".
Other transformations that move this subspace are non-linear.
(Conformal transformations).
The characterization you provided is also problematic on another (more
fundamental) ground. That is: the preservation of inertial motion is
NOT the necessary condition for the Lorentz transformations or
Poincare' transformations.
Rather, one only needs for the "before-after" relation of events to be
preserved -- i.e. a completely non-geometrical relation in temporal
logic suffices to recover Minkowski geometry and the Lorentz
transformations.
(Event A is "before" event B iff in every frame of reference, the time
associated with A precedes that associated with B).
One can go further still: transformations that only preserve the null
intervals are linear/affine/conformal.
One only needs dt^2 = (1/c)^2 dr^2 to be preserved. In fact, based on
the null relation (A ~~ B iff A and B are separated by a null
interval), one can construct the definition of line, congruence,
angle, distance (up to a choice of length unit), duration (up to the
choice of a time unit), before-after (up to a global orientation of
the time-like direction) -- all of it solely from the ~~ relation and
nothing more.
When Einstein wrote his 1905 paper, he assumed differentiability of
the transformation function and proved linearity (by foxing a scale
and choice of origin). With the above result (first surmised around
1915 by A.A.Robb and later extended in the 1950's) you don't even need
differentiability. You get the affine and conformal nature of the
transformations for free.
Now ... having said all that: what the originator of the thread is
really getting at is the idea of non-linear representations of space-
time symmetry. The best way to handle this is also the proper way to
handle representations for transformation groups in general: via non-
linear representation theory.
In contrast to linear representation theory, which is fixed on the
idea of Hilbert spaces and is biased toward applications in quantum
theory, non-linear representations are fixed on the notion of Poisson
manifolds and apply universally, independent of paradigm, at both the
classical and quantum level -- both cases covered in the same
"classical" and intuitive language.
A transformation group is associated with a Poisson manifold in a
natural way. If the Lie group is G, its Lie algebra L and a basis for
L is given by (Y_a) with a vector in L written as e^a L_a (summation
convention used), then the dual algebra L* has (e^a) as a basis, Y_a
as the vector components.
The function space of smooth functions over L* is then a space that
includes all the linear functions (i.e. vectors) in L** = L. So it's
the non-linear extension of the original Lie algebra L.
A Lie bracket [Y_a, Y_b] = f^c_{ab} Y_c, represented in terms of the
structure coefficients (f^c_{ab}) then becomes the Poisson bracket
{Y_a, Y_b} = f^c_{ab} Y_c, which generalized to functions as:
{f(Y), g(Y)} = f^c_{ab} Y_c (@f/@Y_a) (@g/@Y_b)
(@ denotes partial derivative operator).
The vectors
Delta = {_, e^a Y_a}
then provide the infinitesimal forms of the transformations associated
with the basis element Y_a, and one has the transformation rule for
L*:
Delta (Y_b) = -f^c_{ab} Y_c e^b
which contracts the Lie vector components (e^b) into the co-vector
components (Y_c). This is referred to as the co-adjoint
representation.
The span of the flow lines generated by the vector fields then carve
out a subspace in the Poisson manifold. Each subspace is a layer
called a symplectic leaf. Each leaf corresponds to what in linear
representation theory is called an irreducible representation. Each
leaf defines a distinct type of elementary system.
The partition of the Poisson manifold gives a complete classification
of all the elementary systems -- all the "irreducible representations"
of the transformation group.
All of this is couched in the non-linear math of Poisson manifolds,
and so falls under the header of what I referred to as non-linear
representation theory.
The classification for the Poincare' group are:
0-dimensional: Vacuum (homogeneous, isotropic, boost-invariant,
stationary)
4-dimensional: (Unofficially named) "vacuon" (homogeneous, stationary)
6-dimensional: "Spin 0" systems (all have a 0 Pauli-Lubanski vector W)
-- subclasses: Tardions, Luxons, Tachyons (= slower, at, faster
than light)
6-dimensional: Helical luxons (W lies on a line and is parallel to the
momentum P)
-- in which the photon and Weyl neutrino fall
8-dimensional: Spin non-zero tardions (W lies on a sphere)
8-dimensional: "Cylindrical" luxons (W lies on a cylinder)
8-dimensional: Tachyon A (W is on a 2-sheeted hyperboloid)
8-dimensional: Tachyon B (W is on a double-cone)
8-dimensional: Tachyon C (W is on a 1-sheeted hyperboloid)
The number of dimensions is twice the number of conjugate coordinate
pairs associated with the system (by Darboux' Theorem). In particular,
photons have only THREE Heisenberg pairs, as do spin-0 systems, not 4,
as does the cylindrical luxon and spin non-zero system.
The 4th conjugate pairs -- at least for spin non-zero tardios -- is
the spin pair; i.e. the one whose angular coordinate is the azimuth,
and momentum component is S_z, and is quantized generally as "m".
For the Galilei group (with central charge) the classification can
also be done and is somewhat simpler:
0-dimensional: Vacuum (as above)
2-dimensional: Boost-invariant vacuon
4-dimensional: General vacuon
6-dimensional: Spin 0 synchron (zero mass systems, W is timelike)
6-dimensional: Spin 0 tardions
8-dimensional: Helical synchrons (zero mass, W is spatially non-zero)
8-dimensional: Spin non-zero tardions.
The spin 0 synchron is somewhat analogous to the helical luxon, as
well as the spin 0 luxon and tachyon; while the spin non-zero synchron
shares features of both the tachyons of types A, B and C, and
cylindrical luxon.
This is how you generalize the notion of linearity to non-linearity
for Lorentz (and Poincare') transformations.
Tom Roberts
quoted-printable stuff. -P.H.]
Rock Brentwood wrote:
> On Oct 10, 9:57 pm, Tom Roberts <tjroberts...@sbcglobal.net> wrote:
>> Inertial frames are defined as coordinates relative to which free objects move
>> in uniform straight lines (i.e. Newton's first law holds). It is straightforward
>> to prove that the map between any pair of inertial frames must be linear --
>> linearity is required in order to map uniform straight lines to uniform straight
>> lines, as is required by the definition of inertial frame. Lorentz transforms
>> are defined as maps between inertial frames, and thus must be linear.
>
> No. Actually, the condition you refer to defines an AFFINE map, not a
> linear map.
In this context, "affine" is merely another way of saying "linear" [#].
Remember the context is special relativity and transforms among its
inertial frames (coordinates) on Minkowski spacetime. That provides a
lot of additional context and conditions; the context either precludes
most of what you said, or makes it irrelevant (but not necessarily
uninteresting).
[#] Yes, there is ambiguity in the meaning of "linear". I meant
it in the sense of a linear function, not in the sense of a
linear map between vector spaces (which is apparently how you
interpreted it). Coordinate transforms are expressed in terms
of functions of the coordinates, so "linear" in the sense of
functions is directly applicable; interpreting the COORDINATES
as a vector space is not nearly so direct. (Yes, there is a lot
more here that just transforms among coordinates, but that is
irrelevant as we are just discussing transforms among
coordinates.)
[Thank you for reminding me that an alternate derivation of the
equations of SR requires basically just the assumption that the
time-ordering of events is independent of frame, plus the usual other
requirements. I have to think a bit about how that relates to another
alternate derivation based on the assumption of a maximum signal speed.
And to a third alternate derivation based on linearity plus experimental
evidence....]
> The Lorentz transformations are not the most general
> transformations -- they're linear. The affine maps are the Poincare'
> transformations. These are the most general case.
Sure. But the original poster used the term "Lorentz transform". The
fact that there is a larger transform group is irrelevant -- I was
discussing HIS claims. In the sense I used the term, the Poincar�
transforms are linear.
> This too -- even -- requires qualification. If one only requires lines
> to map to lines, one can go further into projective geometry.
> [... wide digression]
But the context is SR and all that entails, which precludes things like
projective geometry and conformal maps (we physicists always use the
same units in all inertial frames). It is not particularly useful that
other derivations are possible that do not assume linearity, as they
necessarily end up with a linear set of transforms -- whether linearity
is a postulate or a theorem is irrelevant here. And we are concerned
with Minkowski spacetime (a Lorentz manifold), not a Poisson manifold --
yes you can ignore the context and go off on that long tangent, but it
is not relevant here.
Tom Roberts