Re: Lorentz transformations - a derivation

From: Bill Hobba (bhobba_at_rubbish.net.au)
Date: 01/07/05 

Date: Fri, 07 Jan 2005 05:23:23 GMT

"Timo Nieminen" <timo@physics.uq.edu.au> wrote in message
news:Pine.LNX.4.50.0501070847300.26375-100000@localhost...
> On Thu, 6 Jan 2005, Bill Hobba wrote:
>
> > "Timo Nieminen" <uqtniemi@mailbox.uq.edu.au> wrote:
> > > On Thu, 6 Jan 2005, Bill Hobba wrote:
> > >
> > > > "Timo Nieminen" <uqtniemi@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.
>
> At first thought I agree. However, this is bringing in some physics in
> addition to the geometric statement made by the metric tensor.

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

>
> The first is neater, but requires a much stronger initial assumption.
>
> --
> Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
> Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html

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