Re: A little challenge for relativists.

Bilge wrote:
Tom Roberts: >Bilge wrote:
>> 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.
#1 is false, as the coordinates I've given show.

Not true. Funky coordiantes have no power over a physical assumption like this. Just because you can display a set of coordinates does not mean that they correspond to any physical system, and yours don't.


They cannot
be reached by a lorentz transform.

But your coordinates do not represent any inertial frame (there is no time coordinate, and it is not possible to lay rulers down AT REST along one of the axes). See below for the futility in attempting to refute LET via strange coordinates.


#2 is meaningless as stated.
Lorentz relies on a great deal more than that.

Hmmm. This is enough to establish LET for correspondence with SR. Yes, Lorentz used all of Maxwell's equations....


The coordinates I gave define
an inertial frame 

No, they don't. See above and below.


I have another set of coordinates [...]

Attempting to find other coordinates has no power to refute a physical theory like LET.

When one says "inertial frame" and one says they are related by Lorentz transforms, that implicitly means Cartesian coordinates for space implemented with rulers at rest in the frame, and a time coordinate implemented by standard nutually-synchronized clocks at rest in the frame. THAT'S WHAT WE MEAN BY "INERTIAL FRAME". <shrug>


>> 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].

Nonsense -- both remain null. "spacelike" when applied to a coordainte means its basis vector has norm < 0, and "timelike" means its basis vector has norm > 0. But your x^0 and x^1 both have norm = 0. BOTH ARE NULL COORDINATES -- THAT'S WHAT THESE WORDS MEAN. <shrug>


>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.

Hmmm. It doesn't matter -- see above; none of your funky coordinates describe an INERTIAL FRAME. That does not mean they cannot be used, but it does mean they are useless in attempting to refute LET as I phrased it above.


t +/- x is not the light cone. t +/- r
is the light cone.

I'm not talking about light cones, I'm talking about the fact that g_00 = g_11 = 0 in those coordinates. That makes them null BY DEFINITION OF WHAT "NULL" MEANS. <shrug>


>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.

Quantum theory has nothing whatsoever to do with this. Neither SR nor LET deal with quantum phenomena.


>> 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.


But as you said, g+00 = g_11 = 0. THAT MAKES THEM NULL COORDINATES. <shrug>

For instance, a light pulse along the +x axis will have x^0=constant; one along the -x axis will have x^1=constant.


> 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. 

Yes.


>> 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.

But in neither of them can a timelike observer be at rest and be surrounded by a rigid framework of clocks and rulers as meant by the phrase "inertial frame". So they are NOT coordinates of any inertial FRAME. <shrug>


Tom Roberts	tjroberts@xxxxxxxxxx
.

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