Not Yet Optimally Understood

 ...there's one expression which I readily recognize: functions 
 f: X->Y preserving realizations: f�(f_a)=(h_a)�f, f_a: X->X, 
 h_a: Y->Y.  Not long ago, I derived (independently) a similar 
 expression...   (Frank Wappler)

It almost looks as though the "expression" you're referring 
to is the simple conjugacy relation, which appears in all 
branches of mathematics in one form or another.  Two functions 
(or operators or whatever) u and v of the set S are said to 
be conjugates if there exists an element f of S such that 
fu = vf (where juxtaposition signifies composition).  This 
is typically an equivalence relation, splitting up the set S 
into equivalence classes of mutually conjugate elements.

This has applications for things like Mobius transformations
of the complex plane, where S is the set of meromorphic 
one-to-one mappings of the extended complex plane to itself.
The conjugacy classes consist of mappings that are essentially
just translated, rotated, and/or re-scaled versions of each
other, i.e., they all induce the same transformation of the
complex plane, up to translation, rotation, and scale.  The
function f is, in a sense, the mapping between the conjugate
mappings u and v.  A discussion of this subject can be found
on the web at http://www.mathpages.com/mobius/mobtoc.htm .

 Is it correct to say that in general f must map points of equal
 periodicity between a and b (i.e. fixpoints to each other;
 points of periodicity 2 to each other, etc.)? (Frank Wappler)

There are two different kinds of periodicity here.  The web page
noted above focuses on the periodicity of the transformations a() 
and b() themselves, i.e., the period of a function a(z) is the 
smallest integer n such that a_n(z) = z, where a_n() denotes the 
nth iteration of a().  Thus, every point of the extended complex 
plane is returned to its original position by n applications of 
a().  In a sense, these functions constitute all the meromorphic
"nth roots" of the identity mapping.

However, it sounds like you're focusing on the periodicity of 
individual points in the domain under the action of some specific 
function.  If we consider just linear fractional transformations 
(LFTs), it's clear that each LFT has two fixed points (excluding 
degenerate cases where there is just a single fixed point).  Of 
course, the composition of any two LFTs is also an LFT, so it has 
exactly two fixed points.  Thus, in general (discounting degenerate 
cases), there are two unique points that are returned to their 
original positions by k applications of any given LFT for any 
given k.  If a() and b() are conjugate LFTs related by f(), then 
the mappings they induce (and the iterations of those mappings) 
are identical up to translation, rotation, and scale, so all their 
fixed points necessarily map according to f().  Whether this would 
apply to conjugacy or "similarity" relations involving more general 
functions, I don't know.

 Is there a general way to compare functions R->R (or perhaps C->C)
 for the number of points with same periodicity? (Frank Wappler)

Clearly the comparison is trivial for LFTs, but I take it you're 
wondering about more general functions.  I doubt there is a general 
way to compare the numbers of points of a given periodicity, because 
it's often extremely difficult to even figure out how many periodic 
points are possessed by a given transcendental function.  There may 
be infinitely many.  As an example, see the web page at

http://www.mathpages.com/home/kmath002.htm

which barely scratches the surface of the periodicity and stability 
properties of a very simple function.

 Frank Wappler asked: And if they are "similar in this sense", how 
 can f be derived?

Without stipulating the form of the functions, there's no finite
way of determining that they are in fact conjugate, so even the
existence of a linking function f would be indeterminate.

 Albro Swift wrote: [That relation] appears in all branches of 
 mathematics in one form or another.

 Frank Wappler replied: In my humble opinion, where it doesn't 
 yet appear in physics that's where physics is not yet (optimally) 
 understood.

I see.  It actually figures prominently in many areas of physics 
already.  For example, you might recall that when we use density 
operators to represent states, Schrodinger's equation takes the 
form  Dt Ut = Ut D0  where Ut is the unitary operator exp(-iHt).
Also, one could regard Einstein's general covariance requirement 
as perhaps the purest expression of this schema in physics
(although the complications of applying it to 2nd order tensors 
in 4 dimensions tends to obscure the underlying "DA=BD" structure 
of the relation between two arbitrary sets of coordinates).  Of
course, we now understand that ANY set of physical laws can be
expressed in covariant form, and Einstein's requirement really 
only has teeth when combined with the requirement of simplicity.

Incidentally, another application of the "conjugacy relation" 
(known as the "similarity relation" in the world of matricies) 
to a more general class of functions (those with an "addition 
law") and iterations of those functions is discussed on the web
at  http://www.mathpages.com/home/kmath067.htm.

 It is a very interesting suggestion to relate that to "general 
 covariance", but your description is a little too cryptic for me 
 to decipher.  How is a distance between a (any) pair of observers 
 to be measured in GR? (Frank Wappler)

The general theory of relativity has no simple concept of 
the "distance between two separate objects", for two reasons.  
First, there's no absolute simultaneity, so we can't say 
unambiguously which points on the worldlines of those two 
objects should be regarded as representing their locations 
"now".  (In fact, it's typically the case - even in special 
relativity - that the rest frames of two objects disagree as 
to the spatial distance between them, simply because they 
can't agree on a surface of simultaneity.)  Thus, it really 
only makes sense to talk about the distance between EVENTS 
rather than objects.

Second, although the special theory assigns an unambiguous 
absolute separation to every pair of EVENTS, the general 
theory does not.  In view of the Equivalence Principle and 
the presence of gravitating matter in the universe, it turns 
out that no system of absolute intervals (such as in the 
special theory) can coherently represent the set of all 
possible mutual observations.  It is possible, however, to 
maintain the principles of special relativity for sufficiently 
small regions around any given point, so there exists a 
definite absolute "length" along any specified PATH in space-
time (from one event to another), defined as the integral of 
the incremental absolute intervals (from SR) along that path.

For any specific observer (or particle), the only definite 
events are direct interactions with other entities, such as 
photons, electrons, etc, (i.e., no action at a distance) and 
the only directly measureable intervals for a given observer 
are the lapses of "proper time" between its interactions.  
Notice that the theory ASSUMES the existence (at least in 
principle) of ideal clocks that measure the lapse of proper 
time along their own respective worldlines.  In practice, 
every corporeal entity - whether it's actually a clock or 
not - is assumed to undergo the same lapse of proper time 
that would be shown by an ideal clock travelling that path.  
What it physically means for an entity to "undergo" a lapse 
of proper time depends on the entity itself.  For example, 
certain identifiable particles are known to "decay" in a 
characteristic amount of proper time, regardless of the paths 
they travel.  Phenomena like this are what give "proper time" 
its physical significance.

The only direct way of "measuring" the length of a certain 
path between two events is by means of an entity that traverses 
the path.  Essentially, we send a "clock" from here to there 
along the path, and the lapse of proper time for that clock 
provides a measure of the length of the path.  However, it's 
important to bear in mind that there are multiple paths between 
events A and B.  In fact, there are generally multiple FREE FALL 
paths from A to B, and there can even be more than one NULL path 
(i.e. light ray) from A to B (as shown by the double images of 
some distant galaxies seen through a "gravitational lense").  
This is why we have to be careful not to think there is some 
specific "distance" between events A and B (let alone between 
two objects).

Needless to say, the processes by which high-level conscious 
observers infer the relative locations of other objects, with 
which they are not in direct contact, based on their direct 
interactions, is far more complex, consisting of a vast number 
of primitive interactions between the particles comprising the 
sense organs and brains, and ultimately relying (as with all 
knowledge) on an incomplete induction.  The general theory of 
relativity provides a comprehensive and realistic (in the 
philosophical sense) description of the primitive physical 
processes of dynamics and gravity, but higher-order consider-
ations of how (provisional) knowledge is acquired by conscious 
observers belong to the realms of epistemology and psychology. 
(Much is known in these areas, but it's probably off-topic for 
this newsgroup, and the issues are certainly not unique to
the relativistic framework.)

 Frank Wappler asked: How is a distance between a pair of observers 
 to be measured in GR?

Albro Swift replied:  The general theory of relativity has no 
simple concept of the "distance between two separate objects", for 
two reasons.  First, there's no absolute simultaneity... (In fact, 
it's typically the case - even in special relativity - that the rest 
frames of two objects disagree as to the spatial distance between 
them, simply because they can't agree on a surface of simultaneity...

 Frank Wappler responded: That's why I was only asking for a 
 definition of simulateity for any _one pair_ of observers.

The case of a single pair of observers was addressed in my previous 
answer.  In the theory of relativity (both special and general), two 
observers are not necessarily stationary with respect to each other.  
In fact, the purpose of the theory of relativity is to accommodate 
cases when two observers are not co-stationary.  In those 
circumstances, a pair of "observers" will not agree on simultaneity 
relations between events, and consequently they will not agree on 
the distances between objects (such as themselves).

 Frank Wappler wrote: That agreement or disagreement between any 
 pair of observers is in SR based on conducting Einstein's calibration 
 procedure... successful or not.

The motivation for Einstein's theory of relativity was the realization 
that the procedure you described does not give mutually consistent 
results for a pair of observers, because observers are not, in general,
mutually stationary.  The "procedure" you described is really just a 
pedegogical device that Einstein used in some of his expositions to 
illustrate how a simple operation involving space and time implies 
the selection of a *basis* for space-time evaluations, and he went 
on to explain how this selection differs (under fixed assumptions 
of dynamical isotropy) when it is regarded from different frames 
reference (i.e. by different observers).

You may think that the results of applying such a "procedure"
to various co-stationary objects could be used to somehow build
up the spatial relations between entities in motion, but that's 
precisely what special relativity says cannot be done.  The
peculiar nature of the Minkowski metric (non-positive definite)
is only manifest when evaluating spatial and temporal intervals
in relatively moving frames of reference.  Thus, the absolute 
measureable intervals in special relativity are necessarily
between events, not objects.

What you call "Einstein's calibration procedure" does not embody
the essential content of relativity.  Rather, it illustrates the
NEED for relativity in order to place the observations of co-
moving observers within a consistent context.  Of course, the
special theory succeeds in this only for UNIFORMLY co-moving 
observers, whereas the general theory is applicable to the
reference frames of observers in any kind of motion whatsoever.

 Frank Wappler wrote: Note that the procedure succeeds for (momentary) 
 zero roundtrip time.

How do you know it "succeeds", i.e., how do you know the indication
it give you is correct?  Also, what constitutes a trip (round or 
otherwise) of zero extent?  One would think that a "trip", by 
definition, would have some non-zero extent.

 Albro Swift wrote: Thus, it really only makes sense to talk about 
 the distance between EVENTS rather than objects.

 Frank Wappler wrote: Events occur to individual observers (perhaps 
 individual systems of observers), and can be expressed as their 
 states, no?

No, events are interactions, not "states". The concept of a "state"
(in this context) is of a higher order.

 Albro Swift wrote: It is possible, however, to maintain the 
 principles of special relativity for sufficiently small regions 
 around any given point...

 Frank Wappler wrote: What does "small" mean?

The expression "sufficiently small" is a mathematical term,
signifying not a specific magnitude, but rather the concept
that there is SOME magnitude m greater than zero such that the 
statement applies for all magnitudes from 0 to m.  It is neither 
assseted nor assumed that we know or can determine the value of
m.  Needless to say (or so one would have thought), we typically
DO have provisional knowledge of m, based (as with all knowledge)
on an incomplete induction, but that knowledge is not required
to assert that the length of a path is the integral of the 
infintessimal absolute intervals along the path.  (See calculus.)

 Frank Wappler wrote: If you could give a procedure to determine 
 _which_ intervals are "small enough" and which are not, wouldn't that 
 already go a long way towards a procedure to determine distances/
 intervals, about which I was asking initially?

No, an ability to assess the flattness of the manifold in various
regions would not enable us to unambiguously assign a "distance" to
any pair of events, for the reasons explained previously.  It isn't 
a matter of us being unable to figure out what "the distance" is, 
but that there simply is no single "distance" between points on a 
curved manifold.

Of course, if you postulate some embedding of that manifold within a 
higher-dimensional flat manifold, then you could assign a "distance" 
to every pair of points based on the high-dimensional space, but 
that "distance" would have no definite physical significance, and, 
moreover, the possible embeddings are not unqiuely determined by 
the manifold, so the "distance" remains indeterminate.

Furthermore, we CAN give procedures that will determine flatness
provisionally, but of course they will only be effective if we
happen to reside in a manifold with sufficiently well-behaved
metrical properties.  It's always possible to imagine a manifold
sufficiently bizzare that it will defeat any finite inductive
attempts to infer its structure from within it, and it's even
possible that we live in such a manifold, and have only been
fooled by a vast body of experience showing apparent conformity
with a simple inductive model.  ("The most incomprehensible thing
about the world is that it is comprehensible.")  Nevertheless,
we are unlikley to improve the quality of our inductive model by
including within our data base the most bizzare hypothetical
observations we can imagine.

 Albro Swift wrote: Notice that the theory ASSUMES the existence 
 (at least in principle) of ideal clocks that measure the lapse of 
 proper time along their own respective worldlines.

 Frank Wappler wrote:  At least Einstein's procedure doesn't 
 seem to require such an assumption.

That's not true.  This is called the "clock hypothesis" in the
relativistic literature, and it is most definitely required in
order to relate the theoretical model to experience.

 Albro Swift wrote: However, it's important to bear in mind that 
 there are multiple paths between events A and B. In fact, there are 
 generally multiple FREE FALL paths from A to B, and there can even 
 be more than one NULL path (i.e. light ray) from A to B (as shown 
 by the double images of some distant galaxies seen through a 
 "gravitational lense").

 Frank Wappler wrote: Is it not necessary to express distances 
 between that galaxy, that lense and "us" to attribute whichever 
 effect to that lense?

Not distances, but path lengths.  There is no concept of a unique 
point-to-point distance between events in general relativity, for
the reasons noted in the above-quoted text (and previously).

 Albro Swift wrote: This is why we have to be careful not to think 
 there is some specific "distance" between events A and B (let alone 
 between two objects).

 Frank Wappler wrote: But one might expect that there is one 
 shortest distance (value), between two observers at some "moment"/
 pair of states which they have calibrated to each other.

One might expect this if one didn't know about special relativity, 
according to which the minimum path length between each pair of
events is zero (because of the non-positive definite nature of
the Minkowski metric).  Of course the maximum path length is
infinite, so this doesn't narrow it down very much.  Also,
your comment is based on the premise of mutual simultaneity for
separated observers.  This, as explained above, is an erroneous
premise.