Re: A little challenge for relativists.

Tom Roberts:
>Bilge wrote:
>> shevek:
>> >Which are the seriously flawed theoretical foundations?

>> List the theoretical foundations.
>
>1. There is an ether which permeates all of space and everywhere
> defines a unique intertial frame.
>
>2. Clocks and rulers in motion wrt the ether frame behave as
> described by a Lorentz transform from the ether frame.
>
>The problem with these assumptions is that because of the structure of

#1 is false, as the coordinates I've given show. They cannot
be reached by a lorentz transform. Neither can the coordinates
I give below as a second example. #2 is meaningless as stated.
Lorentz relies on a great deal more than that.

>the Lorentz transforms, _ANY_ inertial frame can be selected as "the
>ether frame" without changing the theoretical predictions for any
>measurement.

I just demonstrated that isn't true. The coordinates I gave define
an inertial frame with a well defined hamiltonian in which the
time coordinates are as stated and translations which leave p_0
invariant, conserve energy.

I have another set of coordinates from which neither the previous
coordinates nor the standard coordinates can be reached by a lorentz
transform:

t = T cosh(\eta)
x = T sinh(\eta)sin(\theta)cos(\phi)
y = T sinh(\eta)sin(\theta)cos(\phi)
z = T sinh(\eta)cos(\theta)

then x^u = (T, \eta, \theta, \phi), with

g_00 = 1,
g_11 = -T^2
g_22 = -(T\sinh\eta)^2
g_33 = -(T\sinh\eta\sin\theta)^2

Those coordinates also define an inertial frame (and in fact, this
set of coordinates is manifestly lorentz invariant). Find a lorentz
transform from these coordintes to the ether frame, (t,x).


> So "the ether frame" becomes a mere label, of no physical
>significance whatsoever.

I think you're trying to find some middle ground that doesn't exist,
here.

[...]
>> I also disagree that the two are experimentally indistinguishable.
>
>The structure of the Lorentz group ensures it.

I've now given 2 coordinate transformations which are unreachable
from each other and from (t,x,y,z) via _any_ lorentz transform.

>> Consider the following coordinate transformation:
>> x^0 = ct - x With the metric tensor given by,
>> x^1 = ct + x g_01 = g_10 = 1/2, g_22 = g_33 = -1
>> x^2 = y
>> x^3 = z
>

[correction: the timelike coordinate should be ct + x and the
spacelike coordinate should be ct - x].

>I think most people do not consider such null coordinates to be on the
>SR side of the boundary between SR and GR.

I don't think I've ever seen those coordinates anywhere _but_
in special relativity. t +/- x is not the light cone. t +/- r
is the light cone.

>That is, the semi-Riemannian geometry needed to justify them seems
>more in the spirit of GR than SR.

Only because you probably look at general relativity more than
at quantum theory.

>> x^0 is the time coordinate and x^1 is the space coordinate.
>
>No. Neither is timelike and neither is spacelike, they are both NULL.
>That is, the norm of d/dx^0 and the norm of d/dx^1 are both zero (NULL),
>because as you said, g_00 = g_11 = 0. That's why these are called "null
>coordinates".

It's not a 2-d transformation. t +/- x do not define the light
cone. If you don't believe me, look up relativistic hamiltonian
mechanics & dirac. He defined these three sets of coordinates
which exhausted the number of ways to construct a poincare invariant
hamiltonian. Since the lorentz transforms are already implicit in
any coordinates, the issue is reduced to coordinates which are
_unreachable_ by a lorentz transform.

> In GR such null basis vectors always come in pairs, as
> the signature of the metric is -2 and cannot be modified
> by a mere change of coordinates.

I haven't changed the signature. The signature is the sum of the
_eigenvalues_. The eigenvalues of that metric are, +/-1/sqrt(2), -1,-1.
That gives 1/sqrt(2) + (-1/sqrt(2)) + (-1) + (-1) = -2.

>> One of the assumptions of LET is the existence of absolute
>> simultaneaty, i.e., an absolute time which can be defined
>> for every observer,
>
>Yes (it's not really "defined for every observer", it's just defined
>once and everybody uses it; or better, it just _is_).

Right. The point being that everybody can't use it. I've just
given examples of 2 coordinate systems which cannot be reached via
lorentz transforms. Since each of those

>> and that time really slows down, lengths really contract, etc.,
>
>I don't think Lorentz would agree -- I have seen no such statements in
>any of his writing. He considers the moving coordinates {x',y',z',t'} as
>merely mathematical convieniences.

>But since the Maxwell's equations are
>identical in the ether and moving coordinates, it's clear that in a
>moving frame electromagnetic phenomena behave _exactly_ the same as in
>the ether frame when using those moving coordinates (that was the whole
>point of his 1904 paper).

I've just given you coordinates which are not reachable by a translation
along the velocity. If I can find my copy of his paper, (or you give
me something more specific to pin down what absolute frame means), I'll
point out exactly why the mathematical convenience fails.



.

Quantcast