Re: A little challenge for relativists.

shevek:
>
>> The entire point of the exercise
>> is that the coordinates I gave are the canonical coordinates for
>> three regions which cannot be connected by a lorentz transform.
>>
>
>I'm not sure I follow you Bilge. How does that differ from the case
>for a more standard SR approach? Can the three regions be connected by
>SR transformations?

No. That's the entire point.

>Because the transformations are identical it's hard for me to see the
>difference.. Will the two approaches disagree on measured intervals?

Here's a explicit picture. First, consider two observers with a
spacelike separation The diagonal line is the light cone.
t'
t ^ /
^ / You can perform a lorentz transform | /
| / + B such that B at (t,x) -> (0,x') => |/
|/ +-------+->x'
+-------> x A B
A

Spacelike intervals can always be made simultaneous, so all observers
who are separated by spacelike intervals differ only by a lorentz transform
and a space translation, which is okay if you assume invariance under
space translations. Lorentz had to assume this in order to define a frame
common to A and B. (It also gives conservation of momentum, but lorentz
lacked the means to prove that.)

What about two observers who are separated by a timelike interval?

t
^ +C/ If C is at (t,x), them obviously, there is
| / no lorentz transform (t,x) -> (0,x'). In this
| / case, you can perform a lorentz transform
|/ from (t,x) -> (t',0).
+------->
A

So, observers inside the light cone must differ by a time translation.
This is okay if spacetime is invariant under time translations. Now,
the quesion is, are there _any_ coordinates which could possibly define
a unique frame, common to A,B,C? The answer is no. (The obvious reason
is that your coordinates can't equate observers accross the light cone,
but in case that seems vague, I'll make it explicit).

Invariance under time translations implies conservation of energy.
Since energy is conserved in special relativity, conservation of
energy is required for equivalence. Since the total energy is defined
by the hamiltonian, H, Conservation of energy means dH/dt = 0.

This is straigtforward using coordinates (t,x) for observers A and B,
since those coordinates are familiar and used a lot. But, one also
expects conservation of energy to hold for observers A,C. After all,
observers A,C represent the world we actually observe, since those
coordinates are the ones which your past and future lie.

So, we need to find coordinates such that we can define a total
energy which is conserved, otherwise we would most certainly observe
that it isn't. Those coordinates are defined by, (T,\eta,\theta,\phi),
with,

t = Tcosh(\eta) = T\gamma

x = cT sin(\theta) cos(\phi) sinh(\eta)
= cT sin(\theta) cos(\phi) \gamma\beta

y = cT sin(\theta) sin(\phi) sinh(\eta)
= cT sin(\theta) sin(\phi)\gamma\beta

z = cT cos(\theta) sinh(\eta)
= cT cos(\theta) \gamma\beta

Which are just spherical coordinates that differ by a lorentz boost.
These are manifestly invariant. They also should look familiar.
Changing the coordinate \eta changes the velocity. The other
set of coordinates I gave would correspond to a fourth observer
who synchronizes watches with observer A. I'll skip it, since
the argument is the same and just defines a third frame to which
the rest below applies.

The last question is whether there exists _any_ frame which differs from
the two frames I defined by a lorentz transform? Obviously not. I chose
those coordinates on purpose because no lorentz transform connects them.
That, by the way is what I meant by not being able to connect coordinates
across the light cone. My first objection to lorentz' assertion of an
absolute frame (or unique if you prefer, but unique means only 1, and
therefore defines frame an absolute reference), is that he did exactly
that. He chose observers inside, outside and on the light cone to
derive length contraction, time dilation, etc., whichever fit the
derivation. He naturally found that the lorentz transforms were the
correct transform for each individual case, but then equivocated the
meaning of x,t across the light cone to conclude the existence of
a unique frame for which all observers differed by a lorentz transform.

Now a dilemna. Observer A is in both frames, so his measurements should
not depend upon which frame he chooses. This is the basis for my second
objection, that special relativity is not equivalent to LET. Since the two
frames do _not_ differ by a lorentz transform, the most obvious choice for
an ether theory is to decalare one of those frames to be ``preferred,'' in
which case, your choice of coordinates matters for _some_ measurement.
However, in this case, the measurements happen to be the energy and
momentum, so it should be straight forward to find the preferred frame.

If it isnt obvious, the reason is that the frames differ by a lorentz
transform _and_ a spacetime translation and the energy and momentum
correspond to invariance under spacetime translations. However, A is not
translated in spacetime with respect to himself. Using the ether, resolve
the dilemna.

By the way, if you've been reading eugene's posts regarding
his ``theory,'' his misconception is related to this.

>> Last time I checked, the group of transformations that defined
>> relativity was the poincare group, not the lorentz group.
>
>?? The Poincare group is just the Lorentz group plus translations,
>right? LET will also be "defined by" the Poincare group..

Why is that? Lorentz did not say that. A unique frame has an origin.
Translational invariance is not a given. In any case, the dilemna I gave
you above gives you the benefit of the doubt on that, since I allowed for
conservation of energy and momentum, which is a side effect of the
invariance. You might say that I relied on the fact that the poincare
invariance was assumed implicitly, but was never addressed so that the
interpretation is inconsistent with the assumptions.

>> >Yes, Lorentz used all of Maxwell's equations....
>>
>> But that isnt all. He also used the fact that div A = 0
>> in the coulomb gauge to insure the transversality of light.

>
>Incompressibility of the ether?

Nice try, but no cigar. The _ability_ to choose that gauge is certainly
essential (at least if light propagates at `c'). Unfortunately, the
ability to not choose it is also also essential. If you try to equate
gauge invariance to the compressibilty of the ether then the
compressibilty has to be anything you can imagine. The entire artifice
collapses if you try to apply that to the weak interaction, since that
isn't even an option unless the weak interaction has an infinite range.
Since it doesn't, you're stuck trying to figure out how to define infinite
to mean 10^-18 meters and a different constant `c' that applies to weak
interactions, so that the W,Z can be massless for that definition of
infinite, along with the definition infinite as infinite or else come up
with something else.

That of course, doesn't include the multitude of questions related
to how the ether can be incompressible yet so unnoticeable that no
one can measure the incompressibilty. Don't forget that if light
is a wave in the ether, then we could only observe it we interact
with the ether. I haven't noticed difficulty moving.

>It seems like Maxwell's equations already only allow transverse
>waves, regardless of choice of guage or reference frame.

Not true. In general you have to perform a gauge transformation
as well.

.

Relevant Pages

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