Initial definition of an inertial coordinate system...
Sven or Liz
such a basic question to such a high-powered forum - this seems so
fundamental that I thought it must surely be in the FAQ - or somewhere
else on the 'net. But lo - it seems I'm either seeing problems where
there are none or I'm uncovering problems nobody else has noticed
yet. (And boy, do I hope it is the former... :-)
At the bottom, I just wanted to write down a somewhat more rigorous
formulation of special relativity than the somewhat wishy-washy things
I had seen in the pretty books from which I learned it and which
seemed to take particular care to talk around the actual points of the
matter. But then I got seriously stuck in the very first step: Define
an inertial coordinate system.
The "common notion" I'd have told you before I thought about it
would've been something like this: "an inertial coordinate system is
one in which force-free bodies exhibit uniform motion". This, however,
only shifts the question to "how do I know if a body is force-free"?
In the context of classical mechanics I'd say that a body is
force-free when it exhibits uniform motion in an inertial frame of
reference, but that would leave me with a circular definition...
I grabbed Taylor&Wheeler's "Spacetime Physics" to find them defining
inertial systems as "freely falling coordinate systems". But while
they correctly note that an inertial system is a *local* conception,
they make no mention how one should test whether or not a coordinate
system is, in fact, freely falling. Their poor Einsteinian observer
would have to have some test available to him to assess whether or not
his elevator cabin was in fact freely falling. But if I rely (again)
on the notion of a force-free body that would have to move uniformly
relative to an inertial frame of reference, then I'd have to be able
to decide if this here non-uniformly moving body was due to some force
upon it or due to some acceleration of my elevator cabin. In fact, to
tell whether there was a magnet out there pulling it I'd have to leave
the locality of my elevator cabin.
My old physics lecture notes try the observational approach: we
realize that "forces" are notions we invent to describe the behavior
of particles that are in interaction with other things, and you test
for interaction by doing something to one body and checking whether or
not something happens at some other body. Then you realize
(experimentally again) that all interactions diminish with distance
which allows you to define an "isolated body" which is sufficiently
far away from all others to have sufficiently small interactions. For
any number of such free particles, you then observe, that the location
of each of them can be used to define a "time" and that all the others
then have locations that are linear in this "time". You call this
behavior "uniform motion" and define "inertial frames" as those in
which isolated particles exhibit uniform motion.
But you noted already (I presume) that this presupposes the
measurement of a "sufficient distance" before there is a
metric. Before there is a coordinate system, to be precise. I could
just pick some random coordinate system "for the time being", but
relativity actually shows me how tricky it can be to measure distances
(and lengths and such) in some random oddball coordinate system, and I
fail to see how I could establish causal relationships to get from the
post-hoc of something happening after something else to the propter
hoc of an "interaction" in a coordinate system in which I have no
control of time and space (i.e. Of which I don't know whether it is
inertial). And a group of free particles would have to be widely
spaced so as to be 'sufficiently far apart' from each other, which
negates locality in the very definition of the inertial
system. Testing for *all* other particles to be far away requires
nothing less than a global measurement.
By this time I had managed to confuse myself entirely, so I decided to
take it out on people who are pros in confusing themselves. Hence this
post. ;-) So: does anybody have a reasonably self-consistent bootstrap
for an inertial coordinate system?
-- S
Oh, and while I'm here: is there a non-axiomatic way to get to the
relativity postulate, i.e. can symmetry under coordinate
transformations be formulated other than just something we'd like to
have? The Galilean notion was borne from observation, but then that
doesn't lead to things like time-dilation. On small-velocity scales
it is indeed observed that the laws of physics are the same to all
(inertial, however defined) observers. For high velocities, my sources
invariably start with Michaelson-Moreley, but showing that there is no
ether-frame (or that Maxwell's "c" is indeed the same in all
coordinate systems) does by no means preclude the existence of *some*
absolute coordinate system. As a matter of fact, the cosmic microwave
background looks to me as if it established quite a formidable
absolute coordinate system: namely that in which the CMB is isotropic
(or most nearly isotropic in terms of received power per unit solid
angle). As far as I can tell, the proper order of steps would have to
be 1) establishing relativity of the physical laws (somehow), 2) then
deriving the attendant formulae for the case of a)"no speed limit" and
b) a "speed limit 'c'", which has to be the same in all coordinate
systems for otherwise one could make it higher by switching frames --
thereby invoking the previous, and 3) *then* invoking
Michaelson-Moreley to decide experimentally between the two
possibilities, noting that there is at least one speed that is in fact
the same in all frames and that the equations make it clear that this
has to be the speed limit if I want to avoid for things to go
backwards in time just by changing reference frames. I find the math
in all this rather straightforward, (it can be done easily without
vectors, using just a little algebra and no pre-formed concept of a
"clock") but have been stumped by these presumably basic conceptions.
Toby Bartels
>At the bottom, I just wanted to write down a somewhat more rigorous
>formulation of special relativity than the somewhat wishy-washy things
>I had seen in the pretty books from which I learned it and which
>seemed to take particular care to talk around the actual points of the
>matter. But then I got seriously stuck in the very first step: Define
>an inertial coordinate system.
In a rigorous axiomatic notion of special relativity,
I've always seen the existence of inertial coordinate systems
introduced simply as a postulate, not constructed in any way.
Indeed, you must stick in some axioms somewhere --
SR is not deducible from pure logic alone
(which is a good thing since it's not quite true!) --
and this seems to be a popular one.
(I don't have access to a rigorous development of SR now,
or else I'd write down some axioms at this point.)
>The "common notion" I'd have told you before I thought about it
>would've been something like this: "an inertial coordinate system is
>one in which force-free bodies exhibit uniform motion". This, however,
>only shifts the question to "how do I know if a body is force-free"?
>In the context of classical mechanics I'd say that a body is
>force-free when it exhibits uniform motion in an inertial frame of
>reference, but that would leave me with a circular definition...
Right -- one or the other of these (or something!) must be axiomatic.
>I grabbed Taylor&Wheeler's "Spacetime Physics" to find them defining
>inertial systems as "freely falling coordinate systems".
This is a good expression for getting the correct intuitive picture
(or at least Taylor's & Wheeler's intuitive picture, which IMO is correct),
but it's not suitable for a rigorous axiomatic foundation.
Then again, that's not Taylor & Wheeler's goal.
>My old physics lecture notes try the observational approach: we
>realize that "forces" are notions we invent to describe the behavior
>of particles that are in interaction with other things, and you test
>for interaction by doing something to one body and checking whether or
>not something happens at some other body. [...]
This is how we decide *in*practice* if a coordinate system is inertial,
but as you note, such practical matters are insufficient for
a rigorous axiomatic development. We need both theory and practice;
the rigorous axiomatic development defines the theory,
and then these observations show that the theory closely matches reality.
>By this time I had managed to confuse myself entirely, so I decided to
>take it out on people who are pros in confusing themselves. Hence this
>post. ;-) So: does anybody have a reasonably self-consistent bootstrap
>for an inertial coordinate system?
You can bootstrap it if you accept some other axioms;
for instance, if you assume that spacetime has the structure of E^(3,1),
then you can define the inertial coordinate systems as
certain well defined coordinate systems on E^(3,1).
But this is probably backwards from the order that you want to do things.
(E^(3,1) is a 4D affine space with a linear metric with signature (3,1),
that is Minkowski space.)
>Oh, and while I'm here: is there a non-axiomatic way to get to the
>relativity postulate, i.e. can symmetry under coordinate
>transformations be formulated other than just something we'd like to
>have?
Again, you can't do it *ultimately* nonaxiomatically,
since this fact about the universe is contingent
and might have been false had the world been different.
You can't deduce it from pure air ^_^.
>As a matter of fact, the cosmic microwave
>background looks to me as if it established quite a formidable
>absolute coordinate system: namely that in which the CMB is isotropic
>(or most nearly isotropic in terms of received power per unit solid
>angle).
Well, this one *is* in the FAQ -- the Cosmic Microwave Background FAQ,
<http://www.astro.ubc.ca/people/scott/faq_basic.html>.
Basically, the CMB is a physical object in the universe,
so naturally it defines a coordinate system (up to origin and rotation),
but this is no more fundamental to physics than
an inertial coordinate system defined by my personal velocity right now
(as I am also just another physical object in the universe).
BTW, a fun fact: the coordinate system defined by the CMB is not inertial!
>As far as I can tell, the proper order of steps would have to
>be 1) establishing relativity of the physical laws (somehow), 2) then
>deriving the attendant formulae for the case of a)"no speed limit" and
>b) a "speed limit 'c'", which has to be the same in all coordinate
>systems for otherwise one could make it higher by switching frames --
>thereby invoking the previous, and 3) *then* invoking
>Michaelson-Moreley to decide experimentally between the two
>possibilities, [...]
Yes, this is the correct way to establish the (approximate) truth of SR.
But notice that this is a very different prospect from
a rigorous development of the theory of SR in the first place.
In any valid physical theory, there are 2 parts:
the mathematical development of the theory
(preferably rigorous, although this isn't absolutely necessary)
*and* establishing by experiment that this theory matches reality.
Your rigorous formulation of the theory is the first part,
but this last paragraph is the last part.
BTW, if you want to establish the *truth* of SR rigorously,
then don't bother -- it's not exactly true in the first place.
Really, no theory can be rigorously proved exactly true
on the basis of a finite number of observations,
other than a theory that merely lists the results of those observations.
If you want something that allows you to predict further observations,
then you must live with some doubt about its correctness.
-- Toby
to...@math.ucr.edu
Frank Wappler
Sven or Liz Lag...@mailandnews.com wrote:
> [...] Define an inertial coordinate system.
> The "common notion" I'd have told you before I thought about it
> would've been something like this: "an inertial coordinate system is
> one in which force-free bodies exhibit uniform motion". This, however,
> only shifts the question to "how do I know if a body is force-free"?
> In the context of classical mechanics I'd say that a body is
> force-free when it exhibits uniform motion in an inertial frame of
> reference, but that would leave me with a circular definition...
Not at all.
The notion of (pairwise membership of observers together in one)
"reference frame",
and the notion of "motion" of somebody particular
wrt. members of one particular reference frame
(principally wrt. those members who met this somebody
at least momentarily),
and the notion of "uniformity" of such motion,
can be (and conventionally are) all defined and determinable
in a _coordinate free_ (or coordinate independent) manner,
i.e. through reproducible measurement procedures.
The notion of this somebody "having been force-free", or not,
follows then per definition directly from whether or not
this somebody had been measured moving uniformly
by all reference frames containing members who met this somebody
at least momentarily.
Further, if indeed somebody particular was found
having moved uniformly wrt. various reference frames,
then one can generally prove
(based on the detailed procedure by which pairwise membership
together in one frame is defined and to be determined)
that this somebody him/her/itself, too, can measure
having belonged to some particular frame
together with particular other observers.
Finally (and only subsequently)
coordinate labels may be assigned; for instance:
Given the worldline { B_k } of a body,
where { k } is an ordered set,
and having determined through a coordinate _independent_
measurement procedure (i.e. conventionally those of SR)
that this body/worldline had been moving uniformly
wrt. all reference frames containing members who met { B_k }
at least momentarily, at particular states/events B_k,
then a corresponding assignment of real number quadruples
(t( k ), x( k ), y( k ), z( k )) as coordinate labels
together with the assignment of real number 9-tuples
(g_xx( k ), g_xy( k ), ... g_zz( k ))
may by called an "inertial coordinate system" and
"exhibiting the uniform motion" of { B_k }
for instance if for all pairs of distinct states/events
B_k_initial and B_k_final within { B_k }:
Int_{ j == k_initial ... k_final }_( dj
sqrt( Sum_{ a == x, y, z }_(
Sum_{ b == x, y, z }_(
g_ab( k ) d/dk( a( k ) ) |_(k == j) d/dk( b( k ) ) |_(k == j)
) ) )
/ (d/dk( t( k ) ) |_(k == j)) )
= constant.
In particular, the numbers t( k ) might be assigned such that
for every three members N, P, and Q who each met
{ B_k } at least momentarily, at particular states/events B_k
(say at B_n, B_p, and B_q; in that order, as asserted by { B_k }),
and who measured having belonged together to one particular
reference frame throughout the calibrated interval { B_q, B_n }
if the calibrated interval { Q_met_B_q, P_met_B_p }
is measured equal to the calibrated interval { P_met_B_p, N_met_B_n }
(through a coordinate independent measurement procedure)
then for the corresponding assigned coordinate numbers holds
t( q ) - t( p ) = t( p ) - t( n )
(i.e. such that the parametrization of { B_k } through
the numbers t( k ) were "affine");
and the numbers x( k ), y( k ), z( k ) and
(g_xx( k ), g_xy( k ), ... g_zz( k )) might be assigned such that
(for observers N, P, and Q as defined above)
if the distance( Q, P ) is measured equal to the distance( P, N )
(through a coordinate independent measurement procedure)
then for the corresponding assigned coordinate numbers holds
Int_{ j == p ... q }_( dj
sqrt( Sum_{ a == x, y, z }_(
Sum_{ b == x, y, z }_(
g_ab( k ) d/dk( a( k ) ) |_(k == j) d/dk( b( k ) ) |_(k == j)
) ) ) )
=
Int_{ j == n ... p }_( dj
sqrt( Sum_{ a == x, y, z }_(
Sum_{ b == x, y, z }_(
g_ab( k ) d/dk( a( k ) ) |_(k == j) d/dk( b( k ) ) |_(k == j)
) ) ) )
(i.e. such that the assignments were such that
g_ab( k ) d/dk( a( k ) ) |_(k == j) d/dk( b( k ) ) |_(k == j)
were proportional to components of the "metric tensor" which was
determined through a coordinate independent measurement procedure).
As a further specification,
with the assignments x( k ) = y( k ) = z( k ) = 0
equality of the indicated integrals is manifest and
independent of any particular values assigned to g_ab( k ).
But of course anyone is free to assign coordinate labels
whichever way they prefer, as long as distinct states/events B_k
remain distinctly labelled through t( k );
physics/measurements must remain independent of any particular
such assignment.
> Taylor&Wheeler's "Spacetime Physics" [defines]
> inertial systems as "freely falling coordinate systems".
This requires of course a definition and corresponding reproducible
measurement procedure for the determination of whether (and wrt. whom)
anybody particular had been "freely falling".
Perhaps that's (meant to be) equivalent to the above defined notion
of "moving uniformly", i.e. (meant to be) defined and to be determined
through the same/reproducible measurement procedure, namely
the determination of whether or not any two particular somebodies
belonged together to one "reference frame", in any particular trial.
For this in turn, Taylor&Wheeler seem to indicate in sect. 3.4
the conventional measurement procedure of SR,
i.e. Einstein's calibration procedure,
namely that two separate cars/observers belonged together
to one train/frame, in any particular (individually observed) trial,
if they can determine and agree upon that there existed
an auxiliary observer who constituted the "middle between"
those two cars, in that trial,
and who can therefore assert that a signal stated by one
of the two cars, in a (individually observed) trial, and
a signal stated by the other car, in a (individually observed) trial,
actually correspond to each other, as simultaneous and concerning
the same one trial.
> they correctly note that an inertial system is a *local* conception
I note that pairwise membership together in one frame is
correctly and generally a relation between _at least two distinct_
members, e.g. (at least) _two_ cars of a train,
or similarly (at least) two ends of one car, etc.
Apparently Taylor&Wheeler were considering
somebody particular being determined to have moved
_only approximately_ uniformly,
and they were characterizing the degree of approximation
as a function of the distance between members of a frame
who met that particular somebody momentarily.
> Their poor Einsteinian observer would have to have
> some test available to him to assess whether or not
> his elevator cabin was in fact freely falling.
That poor Einsteinian observer and the various constituents
of the elevator cabin would simply have to determine
whether or not they (pairwise) belonged together to one frame
throughout their experiment;
i.e. by the conventional procedure of SR, they'd have to determine
trial by trial, whether or not they (pairwise) could identify
some auxiliary observer as the "middle between" each other.
If so, they'd find having all been members together of one
reference frame, in every trial;
and, given sufficiently many consecutive trials,
they'd find having moved uniformly wrt. each other,
with velocity zero, and hence "force-free" and/or "freely falling".
AFAIK, (some approximation of) such a measurement is to be
carried out with the "Gravity Probe B" experiment.
> My old physics lecture notes try the observational approach:
> we realize that "forces" are notions we invent to describe the
> behavior
... or rather more precisely: measured mutual geometric relations ...
> of particles
... (or of sets of particles) wrt. each other ...
> that are in interaction with other things
... which are presumably considered to be particles (or sets thereof)
as well ...
> and you test for interaction by doing something to one body
> and checking whether or not something happens at some other body.
You can characterize the distribution of the (most probable) potential
in the experimental region, derived from the measured geometric
constraints, using variational calculus.
> Then you realize (experimentally again) that all interactions
> diminish with distance
... possibly requiring "conservation of charge",
for normalization and comparison ...
> which allows you to define an "isolated body" which is sufficiently
> far away from all others to have sufficiently small interactions.
... and surely having determined or having to assume
correspondingly small (sums of) charges.
> For any number of such free particles, you then observe,
> that the location of each of them
As far as "location" denotes a geometric relation of
(at least) _two_ wrt. each other, it is the result of a
_measurement_, which must have been obtained and agreed upon
by _both_ involved in that relation, at least in principle,
as derived from their individual observations mutually of each other
through a reproducible measurement procedure.
> [...]
> I could just pick some random coordinate system "for the time being",
You and/or anyone else can pick any individual coordinate system
as you and they wish; but obviously any measured/agreed_upon
geometric relations (of two or more wrt. each other)
_cannot_ depend on and change with those individual choices, and
neither can the subsequently derived physics relations and quantities
(for instance: the measurement of whether two charges are of equal
or of opposite "sign").
> [...]
> is there a non-axiomatic way to get to the relativity postulate,
> i.e. can symmetry under coordinate transformations be formulated
> other than just something we'd like to have?
Well - the distinction between what's
unambiguously derived from observations (i.e. what's measured)
and what's just arbitrarily labelled
appears quite self-evident, and beneficial to be made.
> As far as I can tell, the proper order of steps would have to be
> 1) establishing relativity of the physical laws (somehow)
That's implicit in the requirement that measured values
ought to be unambiguous/reproducible
and that any laws/algorithms/theories summarizing those values
ought to be definite and falsifiable.
> 2) then deriving the attendant formulae for the case of
> a) "no speed limit" and b) a "speed limit 'c'",
The "speed limit symbol", "c", is introduced through the
SR definition of a pairwise measured distance value as
"c/2 calibrated_lightsignal_roundtrip_interval",
if the two whose distance wrt. each other is thus evaluated
indeed succeeded conducting calibration procedures
with each other throughout their individually observed
lightsignal roundtrip intervals,
specificly having used Einstein's calibration procedure.
How do you suggest to define, measure and express
values of pairwise distance otherwise ?
> 3) *then* invoking [Michelson]-Moreley to decide experimentally
> between the two possibilities
Should Michelson and Moreley have measured the distances
(and/or other geometric relations) of the constituents of their
experimental apparatus wrt. each other ?
If so, how should they have done so
(i.e. using which measurement procedure for defining and
determining values of pairwise distance) ?
AFAIU their paper in Am. Jour. Sci. (34), no. 203., p. 333 (1887)
they approximated the conventional measurement procedure of SR,
and found their experimental constituents to be nearly
belonging together to the same one frame,
throughout the experimental trials they chose to report in detail
(i.e. not those trials which they considered for "adjustments").
Given Michelson and Moreley's results _that_ certain
constituents of their experimental apparatus were found having
belonged together to one frame, throughout some particular trials,
and also that those same constituents may well have been found
_not_ having belonged together to one frame throughout other trials,
can the decision you indicated be resolved through those results ?,
or is it instead to be made a priori, through the definitions and
choices of measurement procedures before having derived
particular experimental results ?
Best regards, Frank W ~@) R
Miguel Carrion
Toby Bartels <to...@math.ucr.edu> wrote:
>Sven or Liz wrote:
>In a rigorous axiomatic notion of special relativity,
>I've always seen the existence of inertial coordinate systems
>introduced simply as a postulate, not constructed in any way.
>Indeed, you must stick in some axioms somewhere --
>SR is not deducible from pure logic alone
>(which is a good thing since it's not quite true!) --
>and this seems to be a popular one.
>(I don't have access to a rigorous development of SR now,
>or else I'd write down some axioms at this point.)
>>The "common notion" I'd have told you before I thought about it
>>would've been something like this: "an inertial coordinate system is
>>one in which force-free bodies exhibit uniform motion". This, however,
>>only shifts the question to "how do I know if a body is force-free"?
>>In the context of classical mechanics I'd say that a body is
>>force-free when it exhibits uniform motion in an inertial frame of
>>reference, but that would leave me with a circular definition...
>Right -- one or the other of these (or something!) must be axiomatic.
Where you say "axiomatic" read "determined experimentally". Also, as
Toby stresses repeatedly, even if your experiments today support the
applicability of the axiom to the real world, more accurate future
experiments may show that it is wrong (and it is, General and not
Special relativity is correct at large scales, and some day Quantum
gravity may take over as the preferred theory).
A fairly rigorous development of SR is given in the last few chapters
of Keith R. Symon's "Mechanics". Basically what he does is say:
1) the speed of light in every direction is equal to a universal
constant c for any inertial observer,
2) inertial observers see each other moving at constant velocity
and go from there. Surprisingly, that's all that's needed to derive
special relativity.
One can obtaina development that is both rigorous and operational in
the following way:
a) It is an experimental question to decide whether there actually
exists a pair of observers moving with constant velocity relative to
each other, and such that both of them see the same pulse of light
expand as a sphere with speed c. In fact, in our universe it is false
in general, but it holds to a very good approximation if the two
observers are relatively close to each other and to the center of the
pulse of light, and we only care about what happens for very short
periods of time after the pulse of light is emitted. Also, the whole
experiment must happen "in vacuum" - this also gives an operational
definition of vacuum, since you might find yourself in some kind of
birrefringent fluid which made the speed of light different in
different directions.
b) Because the region of spacetime where the experiment a) is being
conducted is very small, each observer will see a locally euclidean
space and uniform time. If is also an experimental question to decide
whether a region of space is euclidean to some specified accuracy.
c) Each of the observers will have a Euclidean reference frame and a
clock. Write the conditions 1) and 2) (verified by a) in terms of both
coordinate systems, and you will find a Lorentz transformation
relating the space coordinates and time measurements of both
observers. The details of the algebra are in Symon's book.
>>Oh, and while I'm here: is there a non-axiomatic way to get to the
>>relativity postulate, i.e. can symmetry under coordinate
>>transformations be formulated other than just something we'd like to
>>have?
One way is to use the constancy of the speed of light, but you need a
way to measure velocities, and that requires coordinates, so I don't
know to what extent your question even has an answer.
Regards,
Miguel
--
}---------------------------------------------------------------------
| homepage: <http://www.math.ucr.edu/~miguel/>
| International Association of Physics Students, IAPS
| url: <http://www.iaphys.org/> e-mail: in...@iaphys.org
Charles Francis
<Lag...@mailandnews.com> writes
>I am feeling the strong urge to apologize for posting what seems like
>such a basic question to such a high-powered forum - this seems so
>fundamental that I thought it must surely be in the FAQ - or somewhere
>else on the 'net. But lo - it seems I'm either seeing problems where
>there are none or I'm uncovering problems nobody else has noticed
>yet.
Actually this is a rather tricky and intricate area, and many physicists
avoid the issue by saying that a physical theory is correct if it gives
correct predictions, and avoiding such issues as describing the first
most fundamental steps.
<snip>
>I got seriously stuck in the very first step: Define
>an inertial coordinate system.
>
>The "common notion" I'd have told you before I thought about it
>would've been something like this: "an inertial coordinate system is
>one in which force-free bodies exhibit uniform motion". This, however,
>only shifts the question to "how do I know if a body is force-free"?
>In the context of classical mechanics I'd say that a body is
>force-free when it exhibits uniform motion in an inertial frame of
>reference, but that would leave me with a circular definition...
Precisely. One should avoid the notion that an inertial frame is defined
by uniform motion. It is necessary to establish the absence of (a
significant amount of) electromagnetic or mechanical action on the
matter used to define the frame, then we can define:
An inertial object is one whose motion is not affected by the direct
action of other objects (to the limits of experimental accuracy). An
inertial reference frame is one defined with reference to inertial
matter.
For many people this definition immediately raises the issue you mention
"how do I know a body is force free", but really it should not be so
difficult. We can observe mechanical and electromagnetic forces fairly
directly, which merely raises the issue as to whether gravitational
force is an impressed force, or an inertial force (one due to choice of
coordinates, not action of another body). What we find is that if we
exclude mechanical and electromagnetic forces, and do not assume any
other impressed force, then the motions of inertial bodies are actually
the motions which we observe for a gravitational field according to the
assumptions of gtr. So we may conclude that gravity is in fact an
inertial force.
>I grabbed Taylor&Wheeler's "Spacetime Physics" to find them defining
>inertial systems as "freely falling coordinate systems". But while
>they correctly note that an inertial system is a *local* conception,
>they make no mention how one should test whether or not a coordinate
>system is, in fact, freely falling. Their poor Einsteinian observer
>would have to have some test available to him to assess whether or not
>his elevator cabin was in fact freely falling.
He would have to look outside to see that the cables holding the lift
were in fact detached.
>Oh, and while I'm here: is there a non-axiomatic way to get to the
>relativity postulate, i.e. can symmetry under coordinate
>transformations be formulated other than just something we'd like to
>have?
You have to take it back to the principle homogeneity of physical law -
not to be confused with the (ultimately artificial) homogeneous matter
distribution of Friedman cosmologies to which the CMB refers (see
below). It is perhaps preferential to use a general relativistic way of
looking at this, leading ultimately to the principle of general
covariance
The principle of general relativity states the laws of physics should be
the same irrespective of the coordinate system which a particular
observer uses to quantify them. This is a form of the principle of
homogeneity, that the behaviour of matter is everywhere the same. Laws
which are the same in all coordinate systems are most easily expressed
in terms of invariants, quantities which are the same in all coordinate
systems. The simplest invariant is an ordinary number or scalar. Another
invariant, familiar from classical mechanics, is the vector. A change of
coordinates has no effect on a vector, but it changes the description of
a vector in a coordinate system. Vector transformation laws are found by
defining displacement vectors in Minkowski space, and used to define
more general vector and tensor quantities which are the same in all
coordinate systems. Then the form of the principle of homogeneity most
directly applicable in relativity is the principle of general
covariance, The equations of physics have tensorial form.
> As a matter of fact, the cosmic microwave
>background looks to me as if it established quite a formidable
>absolute coordinate system: namely that in which the CMB is isotropic
And it is used as such in Friedman models. Or more strictly the proper
time of particles on geodesics from the big bang is used to define a
"cosmic time" or "world time". But this is an approximate principle.
Strictly we do not know that there are any particles on geodesics, and
galaxies move (relatively slowly) off this special class of geodesics.
Regards
- --
Charles Francis
Frank Wappler
> Given the worldline { B_k } of a body,
> where { k } is an ordered set,
> and having determined through a coordinate _independent_
> measurement procedure (i.e. conventionally those of SR)
> that this body/worldline had been moving uniformly
> wrt. all reference frames containing members who met { B_k }
> at least momentarily, at particular states/events B_k,
> then a corresponding assignment of real number quadruples
> (t( k ), x( k ), y( k ), z( k )) as coordinate labels
> together with the assignment of real number 9-tuples
> (g_xx( k ), g_xy( k ), ... g_zz( k ))
> may by called an "inertial coordinate system" and
> "exhibiting the uniform motion" of { B_k } for instance if [...]
While my earlier suggestion for identifying/defining some assignment
of real numbers as "exhibiting the uniform motion" of { B_k }
has appeared acceptable already,
one can of course also, or indeed rather, consider that
> [...] for all pairs of distinct states/events
> B_k_initial and B_k_final within { B_k }:
Integral_{ j == k_initial ... k_final }_( dj
sqrt( Sum_{ a == x, y, z }_(
Sum_{ b == x, y, z }_(
g_ab( k ) d/dk( a( k ) ) |_(k == j) d/dk( b( k ) ) |_(k == j)
) ) ) ) /
Integral_{ j == k_initial ... k_final }_( dj
(d/dk( t( k ) ) |_(k == j)) )
= constant.
Frank W ~@) R, Albany, Oct. 25th, 2001
[Moderator's note: Quoted text trimmed. -MM]
Frank Wappler
Miguel Carrion wrote:
> [...] One can obtain a development that is both rigorous
> and operational
Presumably. One can certainly ask _whether_ some particular
prescription to develop/define/measure geometric relations
is indeed rigorous, definite, unambiguous, operationally
reproducible; or in which aspects it might not be.
> in the following way:
> a) It is an experimental question to decide whether
> there actually exists a pair of observers
> moving with constant velocity relative to each other
Yes, velocity values can surely be experimentally determined;
it should be possible to compare a value found in one trial
to a value found in another trial,
and they should not necessarily be equal to each other.
However, a rigorous/reproducible development would require
a prescription for how to define and determine
any one value of "velocity" in the first place.
We might consider for instance the (formal) definition of
"average speed of P, as determined by A (auxiliary B) ==
distance from A to B /
interval between B_met_P and A_met_P",
"vA( P ) == distanceA( B ) / { B_met_P, A_met_P }".
If so, we'll have to ask for rigorous/reproducible procedures
to define, determine, and compare "distance" values as well as
to define, determine, and compare "intervals",
at least such that the suggested operation " / " can be
unambiguously evaluated.
For this, we might turn to the measurement procedures of SR.
Anticipating that distance values will be expressed
in the form of "c * interval", we can begin by asking
how intervals are to be defined, determined, and compared
rigorously/reproducibly. (To be continued.)
Toby Bartels
>Toby Bartels wrote:
>>Right -- one or the other of these (or something!) must be axiomatic.
>Where you say "axiomatic" read "determined experimentally".
Well, when one speaks of a "rigorous" development of a theory,
then one is not merely speaking of what is determined experimentally.
The class of conclusions that can be drawn *rigorously*
(without logical possibility of doubt) from experiments is quite small.
So the rigorous part of the development of anything starts with axioms.
But then the question comes of why we believe in these axioms at all.
This of course is where experiment comes in,
by showing that the axioms (or their logical consequences)
closely match observed reality.
But the experiments don't *determine* the axioms,
not completely; they provide *evidence* for them.
Since we are doing physics and not pure mathematics,
that is sufficient (although it's always good to get even more evidence).
-- Toby
to...@math.ucr.edu
Charles Cagle
<to...@math.ucr.edu> wrote:
> Miguel Carrion wrote:
>
> >Toby Bartels wrote:
>
> >>Right -- one or the other of these (or something!) must be axiomatic.
>
> >Where you say "axiomatic" read "determined experimentally".
>
> Well, when one speaks of a "rigorous" development of a theory,
> then one is not merely speaking of what is determined experimentally.
> The class of conclusions that can be drawn *rigorously*
> (without logical possibility of doubt) from experiments is quite small.
> So the rigorous part of the development of anything starts with axioms.
I, for one, agree with that last sentence but I don't see any kind of
an intense or coordinated effort by the scientific community to simply
lay out for everyone to see just what all the axioms are.
We have lots of things that closely match observed reality (Newton's
third law, for example). I am inclined to believe that we don't milk
these findings nearly enough and that we've merely settled for the
secondary cheese or curds that we've unconsciously without hardly any
effort managed to skim off of the top. I say we need to dig in and
ask the universe what it is trying tell us by the fact that we see
articulated or physically expressed the manifestation of Newton's third
law.
C.C.
Charles Francis
>Miguel Carrion wrote:
>Well, when one speaks of a "rigorous" development of a theory,
>then one is not merely speaking of what is determined experimentally.
>The class of conclusions that can be drawn *rigorously*
>(without logical possibility of doubt) from experiments is quite small.
>So the rigorous part of the development of anything starts with axioms.
Then can you explain why the fashion in physics is to believe that
axiomatic treatments are no longer appropriate, and yet physicists still
have pretentions to rigour?
>But then the question comes of why we believe in these axioms at all.
>This of course is where experiment comes in,
>by showing that the axioms (or their logical consequences)
>closely match observed reality.
>But the experiments don't *determine* the axioms,
>not completely; they provide *evidence* for them.
It depends on what experiments, and on what axioms. Through most of
normal physics we can reasonably make certain assumptions, such as the
existence of background space. The measurements being performed do not
challenge or affect that assumption or axiom. But suppose we are
studying the fundamentals of measurement. Then we should start with
axioms which describe what we actually do in measurement. Every
measurement of distance in a comparison with a reference distance. The
comparison of one distance with another necessitates communication by
light. From ideas like this we can abstract axioms for relativity, and
even for quantum mechanics without requiring one bit of data from the
experiment. Then the data is not evidence for the axioms, but the axioms
must be understood to correctly interpret the data.
Regards
--
Charles Francis