That Largeness of Meaning

Edward Green wrote:
 I don't think the 3rd law follows strongly from the principle
 of relativity, unless we add some other assumptions...
 For example, let identical bodies in relative motion A and B
 interact, and suppose that labeled appropriately to A's rest
 frame we had some item called 'force' whose magnitude on A
 (stationary) were 5, but on B (moving) were 7.  Now, what the
 principle of relativity (possibly throwing in spatial isotropy)
 requires is that from _B's_ rest frame, we would then be able
 to correctly label the force on _B_ as 5, but the magnitude of
 force on A as 7.  This satisfies the principle of relativity...

The symmetry used in the derivation of mass-energy equivalence
isn't between the force exerted by A as seen from B's reference
frame and vice versa, it's between the force exerted by A as seen
from A's own reference frame and the same for B.  Your example
agrees with this, because both A and B (from their respective
frames of reference) sense that they are exerting a force of 5
newtons on the "other guy".  This is the sense of the 3rd law 
that is exploited in the mass-energy derivation, where we argued 
that if A senses he is subject to 7 newtons of force at the point 
of contact with B, then by symmetry B must sense that he is 
subject to 7 newtons of force from contact with A at the same 
point of contact, just as in your example.


Edward Green wrote:
 The large degree of symmetry implied in the third law is already 
 interesting from Newtonian physics:  that for any two bodies at 
 all, of whatever composition, mass, and whatever properties you 
 like, it is possible to assign a single magnitude 'f' to their 
 interaction, entering identically in their several equations of 
 motion...

Yes, the concept of 'force' is one of the most peculiar in all
of physics, and has a fascinating history.  It is, in one sense,
the most viscerally immediate concept in classical mechanics,
and seems to serve as the essential 'agent of causality' in all
interactions, and yet the ontological status of 'force' has
always been highly suspect.  For example, we typically regard 
force as the "cause" of changes in motion, and assert that those
changes would not occur in the absense of the forces, but this
"causitive" aspect of force is not really part of its technical
definition, and we can equally well regard the force as a product 
of changes in motion, or even as merely a descriptive parameter 
with no independent ontological standing at all.  

In fact, the concept of 'force' could *almost* be eliminated 
entirely from classical mechanics, but for the problems of 
absolute rotation and the infamous "static solution" in linear 
motion (the confronting of which - in different ways - seems 
to have accompanied each new fundamental theory of physics).  
Newton certainly wrestled with such questions as whether force 
should be regarded as an observable or simply a relation between
observables.  It's interesting that Mach regarded the 3rd Law 
as Newton's most important contribution to mechanics, even 
though other's have criticized it as being more a definition 
than a law.

On the other hand, in the modern formulation of relativistic 
mechanics the concept of Force is an anachronism, and its various
generalized definitions are introduced only for the purpose of
relating relativistic descriptions to their classical counterparts.  

Needless to say, the word 'Force' has many non-technical meanings 
and connotations (often related to notions of "causation") in 
addition to its strict scientific definition(s), and those non-
technical meanings sometimes color people's thinking on the 
subject.  This reminds of Maxwell's commentary on Herbert 
Spenser's talk before the Belfast Section of the British 
Society in 1874:

  "Mr Spenser in the course of his remarks regretted that so
   many members of the Section were in the habit of employing
   the word Force in a sense too limited and definite to be of 
   any use in a complete theory.  He had himself always been 
   careful to preserve that largeness of meaning which was too 
   often lost sight of in elementary works.  This was best done 
   by using the word sometimes in one sense and sometimes in 
   another, and in this way he trusted that he had made the 
   word occupy a sufficiently large field of thought."


Albro Swift wrote:
 there's another assumption lurking here as well, namely, the
 assumption of physical equivalence between instantaneously
 co-moving frames, regardless of acceleration.

Edward Green wrote:
 You could say that it's not so much that we assume instantaneously
 co-moving clocks keep time at the same rate, as that we assume that 
 to the extent they don't we can calculate this effect via 'the 
 ordinary physical effects of acceleration'...

Yes, that's pretty much the standard view, i.e., the "clock
hypothosis" states that an *ideal* clock is unaffected by
acceleration, and this can, in a sense, be regarded as simply 
the definition of an "ideal clock", which is one that compensates 
for any effects of 2nd (or higher) derivatives.  But of course
the physical significance of this definition arises from the
hypothesized fact that acceleration is absolute, and therefore
perfectly detectable (in principle).  In contrast, we 
hypothesize that velocity is perfectly UNdetectable, which
explains why we cannot define our "ideal clock" to compensate
for velocity (or, for that matter, position).  The point is 
that these are both assumptions invoked by relativity: (1) the 
zeroth and first derivatives of position are perfectly relative 
and UNdetectable, and (2) the second and higher derivatives of
position are perfectly absolute and detectable.  We're all 
mindful of the first assumption, but we sometimes overlook 
the second.

The notion of an ideal clock takes on even more physical
significance from the fact that there exist physical entities
(such a vibrating atoms, etc) in which the intrinsic forces
far exceed any accelerating forces we can apply, so that we
have in fact (not just in principle) the ability to observe
virtually ideal clocks.  For example, in the Rebka and Pound
experiments it was found that nuclear clocks were slowed by
precisely the factor gamma(v), even though subject to
accelerations up to 10^16 g (which is huge in normal terms,
but of course still small relative to nuclear forces).


Albro Swift wrote:
 ...mass-energy equivalence not required by special relativity, 
 it is actually inconsistent with it when combined with the 
 equivalence principle...

Edward Green wrote:
 This I don't follow.  Can one indicate briefly how mass-energy
 equivalence and the (other) equivalence principle conflict?

It's closely related to the problem that underlies the questions
appearing in these newgroups from time to time about why a particle
moving at sufficiently high speeds doesn't become a black hole -
and since every particle is moving at near the speed of light
relative to SOME frame, they should ALL be black holes - at
least with respect to some frames of reference (whatever that
might mean).

In 1912 (after special but before general relativity) Einstein 
gave a brief explanation of the problem in a letter to Max 
Abraham

  "One of the most important results of relativity theory
   is the knowledge that every form of energy E possesses
   inertia E/c^2 proportional to E.  Since, as far as our
   experience goes, inertial mass is at the same time
   gravitational mass, we cannot but attribute to every
   form of energy E a *gravitational mass* equal to E/c^2.
   From this it immediately follows that the gravitational
   force acting on a body is greater when the body is in
   motion than when it is at rest.

   If the gravitational field were to be accounted for in
   present day relativity theory, it would have to be
   regarded as either a 4-vector or an antisymmetric tensor 
   of the 2nd order...  but one thereby obtains results 
   which contradict the above-mentioned consequences 
   concerning the gravitational mass of energy...  It 
   therefore looks as if the gravitational vector cannot 
   be consistently fitted into the relativistic scheme 
   as it stands at the present moment."

As Zahar said, "Einstein had two reasons for giving up special
relativity as a suitable framework for physics.  First, 
philosophical dissatisfaction with having given a privliged 
status to the set of inertial frames; second, the technical
difficulty, arising from E=mc^2, of accommodating gravitational 
theory within special relativity.  The second reason appears to 
have been the more decisive one".


Albro Swift wrote:
 how the excess binding energy was originally a "rest property" 
 representing "real mass" with isotropic inertia, and then becomes 
 a kinetic property representing archaic old relativistic mass 
 with anisotropic inertia, is not well understood (at least not 
 by me).

Edward Green wrote:
 As to how we are to picture this hypothetical mathematical model, 
 this conserved integral seems likely to correspond to some local
 'curvature' or 'bunchiness' of the field, rest mass representing a
 stable and localized configuration of this bunchiness (perhaps held 
 in bound by some topological defect), and energy representing this 
 effect set free to run wild at c.  

Yes, the 'bunchiness' you refer to, if applied to things that 
could in some circumstances "run wild at c" would seem to require 
some kind of non-linear interaction between photons or EM fields.
There has certainly been no shortage of suggestions that perhaps
matter is ultimately electromagnetic in origin, or that it is a
manifestation of some "curvature" or other anomaly in an underlying
field.  

Along these lines it's interesting to review Einstein's 2nd 
derivation of mass-energy equivalence, in which he considered a 
bound "swarm" of particles buzzing around with some average velocity,
and if the swarm is heated (energy E is added) the particles move
faster and thereby gain both longitudal and transverse relativistic
mass, which is anisotropic, but since they are all buzzing around 
in random directions, the net effect on the stationary swarm (bound
together by some unspecified means) is that its resistance to
acceleration is isotropic, and its "rest mass" has effectively 
been increased by E/c^2.  Of course, such a composite object still
consists of elementary particles with some irreducible rest mass,
which could never run wild at c.  To get complete equivalence
you almost need to imagine *photons* bound together is a swarm, 
but no one (except possibly God) has ever been able to figure out 
how to bind pure photons together in a "stationary" configuration
(because EM waves are linear).


Edward Green wrote:
 Any sufficiently complex logical system, like a cellular automaton,
 can be explored into arbitrary and limitless realms of specific
 taxonomy of structure, without having therefore somehow gained any 
 new understanding of how the rules may arise from simpler rules...

True.  This is what makes so impressive the accomplishments of men
like Newton and Einstein who, in the full glare of these limitless
possibilities, manage somehow to muster the spiritual and intellectual
resources necessary to *make decisions* on the basis of insufficient
information, and synthesize some non-trivial understanding that has
(or at least seems to have) meaning.