Re: Lorentz transformations - a derivation

From: RP (no_mail_no_spam_at_yahoo.com)
Date: 01/09/05 

Date: Sun, 09 Jan 2005 13:55:02 -0600

jem wrote:
> Bilge wrote:
>
>> jem: >Bill Hobba wrote:
>> >> "jem" <xxx@xxx.xxx> wrote in message
>> >> news:sdSDd.25692$Q%4.10400@fed1read06...
>> >
>> >>>Remember this proposed definition? An inertial reference frame is a
>> >>>reference frame in which all stationary standard clocks tick at
>> the same
>> >>>rate.
>> >> >> >> How do you sync those clocks? >
>> >It's not necessary to synchronize clocks in order to compare their
>> tick >rates.
>>
>> Synchronizing clocks _is_ comparing their tick rates.
>>
>
> Not hardly. Synchronizing 2 clocks means setting them so that exactly
> one tick on each occurs simultaneously. Whether the clocks tick at the
> same rate can only be determined by testing whether subsequent clock
> ticks also occur simultaneously.

That's a good one. First let's note, just to be pedantic, that ideal
clocks don't actually tick, they read successive infinitesimal
instants, but I digress. You must amend your definition to read
"[...]exactly *the same* tick on each occurs simultaneously (wrt the
inertial frame in which they are both at rest)[...]

OTOH, Bilge's definition falls somewhat short as well. Synchronizing
two clocks, at least according to Einstein's definition, means to set
them so that their readings agree in every instant, which of course he
applied to the exclusive case of identical clocks at rest wrt each
other in an inertial frame. This of course omits the possibility of
gravitational offsets, though he covered this with the proviso "under
otherwise identical conditions".

If you'll read Tom Roberts' responses to this thread, then you'll find
that he defined "force free motion", as motion that proceeds along a
straight line wrt an inertial frame. Thus not only does this concur
with my premise that another frame is required in in order to define
the first, but it also concurs with my convention that gravitational
potential is immaterial to the definition inertial frame. Your
definition of inertial frame, OTOH, cannot subsume force free motion
in the presense of a gravitational field, since though it is true that
all of the clocks that are at rest would
not be observed to tick at the same rate in an accelerated frame,
neither would they necessarily be observed to tick at the same rate in
an inertial frame.

Keep in mind that a frame is infinite in extent, and thus it contains
the entirety of the universe. It follows that all of the variations in
gravitational potential that occur therein are contained within that
coordinate frame. Once again, you are thus limited to very small
regions of space with your definition, and will obtain accuracy only
when that volume approaches zero, or perhaps along an equipotential
surface if one intersects the location of the clock. You thus have no
method, even in theory, of detecting inertial motion, and because of
this the definition is useless from an engineering as well as
theoretical standpoint.

If you want to assume a force free *universe*, then your definition is
fine when applied to such an idealistic entity. OTOH all frames will
be constrained by very definition to be inertial in that universe, so
the definition of inertial frame becomes a bit redundant in this
idealistic context.

Richard Perry

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