| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 629.1 | Relatively minor disagreements. | PBSVAX::COOPER | Topher Cooper | Mon Jan 18 1988 19:11 | 217 |
| RE: 348.43
> More or less, but photons are considered *massless*!!
That refers to the mass property of the particle which is intrinsic to
the particle and independent of the inertial frame of reference: to wit,
the rest mass, which is as I said 0. For all observers traveling in
inertial frames of reference the photon has a measurable inertial mass.
This can be seen immediately from two of the important properties of
a photon: 1) It has energy which is equivalent to mass by M = E/c�;
2) It has momentum (transfer of which to baryonic matter produces the
effect of light pressure); inertial mass is defined in terms of the velocity
and the momentum.
> It is *impossible* for us to observe these theoretical entities as we
> are "on the down side" of the "impenetrable wall" (v=c) and they
> are "on the up side".
That *assumption* has been proposed to explain why tachyons have not been
observed, but, as far as I know, there is no a priori reason to expect
it to be true. It seems logical (by symmetry) that interactions involving
v=c (rest-massless) particles: photons, gravitons and neutrinos, should
be possible from both sides of the wall even if others *are* forbidden
somehow. Since most interactions we experience are due to photons
and gravitons, this is not overly restrictive -- it just means that
we would not find atomic nuclei containing both protons and tachy-protons.
One person I spoke to said that his (physics) MS thesis consisted of showing
what a tachyon would look like as seen by a non-tachyonic observer. The
answer was, according to him, that it would look like non-tachyonic matter.
In other words, tachyons are just another way of describing ordinary
matter. Presumably this result passed his thesis review committee but,
as far as I know has never been reviewed and published and so must be
taken with a grain of salt.
> If you have some info. on an actual experiment to detect them, I would
> be interested in hearing it.
There have been a bunch. Most of them made further assumptions about
the observed mass, the charge, etc. One I remember looked for Cherenkov
radiation in a vacuum.
The one I remember best was rather general and was apparently successful,
but has been unable to be replicated, and so it probably was a glitch
of some kind.
A secondary cosmic ray detector was set up with a device to temporarily
record a second or so of results. Whenever a strong event occurred the
temporary record was saved and made permanent. Now imagine that a high
energy cosmic ray hits the top of the atmosphere and creates a shower of
particles including some gamma rays and a tachyon. The tachyon continues
on and hits an air molecule lower in the atmosphere, creating some more
conventional particles. These particles could, however, reach the detector
sooner than the gamma ray: something which could not happen with any
sub-luminal particle. The signature would be a little precursor hump
in activity from the tachyon followed by the main spike from the gamma
rays, with the usual trailing tail of sub-luminal particles. In a year
of logging one such event with a statistically significant precursor was
found. No one has been able to duplicate the result, however.
> This is not where E = MC^2 comes from.
OK, if you say so. It is a simple derivation which comes directly out
of the mass transformation and the addition of velocities formula and
the assumption of the conservation of "energy" (maintenance of the last
being shown to require a broader interpretation of the term than
classically used, QED). I am sure there are other derivations (the whole
thing being a completely indivisible mathematical whole, there are many
paths to any point) and this may not have been the one Einstein used
(I know of no other general ones as simple though, and it is frequently
used in elementary expositions both popular and technical; including,
if I remember correctly, Einstein's own popular book on the subject).
> Naw. You have this a bit confused, but it is essentially the argument
> to support the first postulate of SR: the speed of light c is the same
> in all frames of reference.
I don't think I have it confused and it is not even close to "the argument
to support the first postulate of SR". The latter does not exist -- one
does not support postulates one postulates them. (This refers only to
the standard exposition of SR which uses this as one of the postulates,
with the Michaelson/Morley experiments as the justification for this
seemingly rather odd idea to simply "assume". Einstein claimed years
later to not have been consciously aware of the Michaelson/Morley experiment
and to have derived the constancy of the speed of light directly from
Maxwell's equations for electromagnetic propagation and from the "second"
postulate. This is certainly possible to do, but given the amount of
attention M/M was getting in all the physics journals at the time, it
is not particularly plausible that Einstein hadn't seen it mentioned.
This can lead to a lot of interesting -- but ultimately unconfirmable --
speculation about subconscious sources of inspiration.
The derivations from the postulates of SR unambiguously lead to the
conclusion that a signal of any kind propagating at faster than the speed
of light within any frame of reference (more specifically, an event
having an effect at a remote location before a photon could reach it)
can be observed within another, equally valid, framework as having
arrived before it was sent. If (local) time is taken to be one-dimensional
and linear -- that is if all events can be unambiguously identified within
any frame of reference by a 4-vector -- then this leads to situations
where an event can cause its own contradiction. This assumption is one
of those which are lumped, in your phrase, with "the other common ones."
One of the "laws" taken to be invariant to frame of reference is the absolute
distinction between "space-like" and "time-like" intervals. Superluminal
velocities -- however achieved -- lead to a breakdown of this law, which
is the foundation of the relativistic notion of causality.
> It's not a case of breaking down, it's a case of denying the postulates
> upon which it is based. Clearly, if you change the axioms you're going
> to get a different system.
Which postulate did I deny? I said that if you take the standard
assumptions of relativity (the two you described and the "other common
ones" you also referred to) and further postulate a real superluminal
velocity it leads to paradox. No interpretation of "complex mass" will
change that unless it results in no effect propagating at faster than
the speed of light, in which case it is a little hard to see in what
sense we are talking about faster than light travel at all.
The traditional resolution of the contradiction is to deny the postulate
of FTL signal propagation. If we have reason to accept the existence
of FTL signal propagation, or equivalently of retro-causality, then
we have to deny some other postulate. The cleanest way to deal with this
is to generalize the description of time to allow a branching structure
or some other similar mechanism.
> Unless you're a genius you can't understand [SR] without understanding
> the mathematical description.
Perhaps, but that's really not a very big deal. Einstein's formulation
is completely in terms of high-school algebra. Physicists generally use
a more "efficient" for calculation form based on "tensor algebra" which
is less familiar, but not actually too hard to learn. In any case it
can all (I'm only talking SR here, not GR) be done with simple algebra.
The hard part to learn is the different, less absolute, view of such
seemingly concrete concepts as space, time, mass and velocity -- especially
time and the elimination of the idea that two different things can be
said to happen at the same time. The usual way of learning this is to
work lots of problems, and working the problems require using the formulas
to get "real" answers -- and hence the need for the math.
I have sometimes wondered whether a well-designed graphical presentation
might not allow some visually oriented people of average intelligence to
skip the equations and go directly to the "feel" of it. Einstein claimed
to have mentally seen the results and only then to have written the
equations to fit the images.
> I've recently moved and all my journals (and everything else *sigh*)
> are in boxes.
Wait a minute! Which of us wrote that! :-) (In other words, we're in the
same boat).
> In a nutshell, division = f(x,y)... This is *the* best way of
> understanding this...
It is certainly is a way, and for some purposes it may even be "*the* best"
way, but it completely obscures the point I was trying to make, which is
a real one (so to speak :-).
If we use a more classical development and define division as the inverse
*operation* of the axiomatically foundational operation of multiplication
then we can see that there is a fundamental difference between division
of *zero* by zero and division of other real values by zero.
Let's look at the mass transformation equation from this viewpoint and
see the insight it gives us. You look at the equation as a *formula*:
a rule for computing one quantity in terms of another, and which can
be algebraically manipulated to produce other consistent formulae which
can be used to compute other of those quantities. I prefer to look at
it as an *equation* which embodies a physical law which places constraints
on what values can be consistently assigned to the variables in a physical
model. From this equation I can derive various formulae which I can
use to calculate one quantity in terms of the others. I can also derive
other equivalent constraint equations or inequalities.
I will choose, for clarity, a form for the equation which eliminates the
inversely defined operation of division (and, for good measure the inversely
defined operation of square-root; I will retain, subtraction, however):
m�(c� - v�) = m0�c�
Let's look at various further constraints on m0 and v (c is, of course, a
constant).
If we constrain m0 to be a positive real and v to be between zero (inclusive)
and c (exclusive) then m is also constrained to be a positive real (a
measurable mass).
If we constrain m0 to be zero and v to be as before, then m is also constrained
to be zero. We see nothing -- no energy, no mass nothing. (Rest) massless
particles traveling at sub-light speeds have no effect -- are unobservable
and therefore do not exist.
If we constrain m0 to be a positive real value and v to be c then we find
that *no* value for m, real or complex, is consistent. No positively
massed object can travel at the speed of light and meet this constraint.
If we constrain m0 to be a positive real value and v to be greater than
c than we find that no *real* value for m is consistent. Complex values
for m are consistent under these conditions. However, another constraint
generally used is that observed quantities must be real valued. As you
pointed out, we can relax this with interesting results, but other constraints
in SR are fatal to this approach.
If we constrain m0 to be zero, however, and v to equal c something different
happens. Positive real values for m exist which meet the necessary
constraints. Indeed *any* real value meets the constraints even though
we have specific values for all other terms. The equation fails to
constrain the value of m under these conditions. Another equation,
relating the observed mass (m) to the frequency of the photon (as observed
in that frame of reference) fills the requirements of constraining m
to a single value.
Topher
|
| 629.2 | Probably Not... | TLE::JONAN | Into the Heart of the Sunrise | Wed Jan 20 1988 20:03 | 224 |
| Re: 629.1
>That refers to the mass property of the particle which is intrinsic to
>the particle and independent of the inertial frame of reference: to wit,
>the rest mass, which is as I said 0.
Since no frame of reference is prefered over another, none of the
considered properties is *intrinsic* or independent of the observer's
frame. Photons are light signals and as such must *always* have velocity
v = c in vacu and thus by the Lorentz and Mass transformations cannot
have mass in the ordinary sense. We must, as you state, consider their
inertial mass, which (in this case) consists soly of (kinetic) energy.
So, photons are not "material things" (there is no matter associated with
them - only energy. This is something the classical guys couldn't figure,
light had to be propogated in some material thing (the ether), it couldn't
have a reality unto itself...).
> That *assumption* has been proposed to explain why tachyons have not been
> observed, but, as far as I know, there is no a priori reason to expect
> it to be true.
That assumption is a direct consequence of the second postulate and its
friends the Lorentz transformations. Symmetry would soly state that
photons and the like interact with tachyons just as photons do interact
with ordinary objects. This in *noway* implies that there can be any
communication between the "two worlds".
> There have been a bunch. Most of them made further assumptions about
> the observed mass, the charge, etc. One I remember looked for Cherenkov
> radiation in a vacuum .....
Boy, now that you mention it, this stuff does ring a bell! The lack of
success is just further evidence in support of SR. Oh BTW, how could
anyone suggest that a cosmic ray could possibly have enough energy to
create a superluminal velocity particle???!!?!?! Perhaps some sense can
be made of this but I doubt it....
> > This is not where E = MC^2 comes from.
>
> OK, if you say so. It is a simple derivation which comes directly out
> of the mass transformation and the addition of velocities formula and
> the assumption of the conservation of "energy" (maintenance of the last
> being shown to require a broader interpretation of the term than
> classically used, QED).
Well now, *that's* more like it, but hardly QED! :-) I don't think that
the conservation laws enter into the derivation, and the conservation of
mass-energy is a consequence. Basically you just do what you do in
classical mechanics: you integrate F through a distance. But here, F
is no longer just ma it's (m(dv/dt) + v(dm/dt)). What you endup with is
E - E0 = (m - m0)c�. In classical mechanics you would take E0 = 0, but
here another analysis shows that we need to take E0 = m0c�. From which
the result finally follows.
> (I know of no other general ones as simple though, and it is frequently
> used in elementary expositions both popular and technical; including,
> if I remember correctly, Einstein's own popular book on the subject).
I don't either. But, because you really need to perform some (admittedly
straight forward) differentiation and integration to get it I don't think
it would be in a "popular book" on the subject. I believe that in his
popular book, Einstein only derived the Lorentz transformations.
> I don't think I have it confused and it is not even close to "the argument
> to support the first postulate of SR". The latter does not exist -- one
> does not support postulates one postulates them.
Well, the second comment is certainly a common belief, but in many cases
an erroneous one. As to the first, it *is* basically the simulteneity
paradox in different (SR *inconsistent* - see below) guise. And this
*has* been used to strengthen the case for the second postulate. Remember,
Lorentz had already devised his transformations *before* SR, but he added
the postulates that the length of objects contracts as a function of their
velocity in the direction of their velocity and that there were different
*apparent* times in different reference frames. Here you get the variation
that the velocity of light *appears* constant in different frames. In SR,
Einstein ties the definition of time (via simulteneity) to c and makes the
old "apparent" time "real" and junks the old postulates of absolute space
and time (Newtonian prefered reference frames). This resolves the
simulteneity paradox *and* accords with the experimental results for the
constancy of c.
This is not an issolated instance of a postulate being given support for
its acceptance - it is not always the case that axioms are intuitively
obvious (at least not in the beginning). Others: Newtonian absolute time
and space. These were "obviously" true because they accorded with everyday
experience; The (in)famous fifth postulate of Euclid. This one generated
2000 years of work trying to show that it *wasn't* an axiom until the
invention of model theory was used to *prove* it was an axiom; Cantor's
proof that the set of reals is strictly larger than the set of counting
numbers was the support for the postulates of set theory stating that
there were "closed" complete infinities which have the same ontological
status of more "ordinary" numbers; The simplicity, non-anthropocentric
elegance of the Copernicun system over the Ptolomeic (at the time *both*
could account for the observational evidence), the Continuom Hypothesis,
etc., etc., etc.
> Which postulate did I deny? ...
Basically the second (c is constant in all inertial frames). I say this
because the whole concept of simulteneity and therefore time itself
is, in SR, tied to the second postulate. Given this, in SR there is
*no place* for superluminal velocities - period. To postulate them is a
direct contradiction to the definition of simulteneity in SR. You seem
to want to call this a paradox for SR but clearly it isn't. If you
accept the basis for SR you *can't* with any consistency go around talking
about superluminal velocities.
> Perhaps, but that's really not a very big deal. Einstein's formulation
> is completely in terms of high-school algebra. Physicists generally use
> a more "efficient" for calculation form based on "tensor algebra" which
> is less familiar, but not actually too hard to learn. In any case it
> can all (I'm only talking SR here, not GR) be done with simple algebra.
To get at all of it you really need freshman calculus (and in some
places multivariable calculus would be nice). For example, to get the
Lorentz *velocity* transformations, you differentiate the Lorentz
transformations, to get E=Mc� you need to do a little differentiation
and integration, etc. As to this not being a "big deal", I would tend
to agree. But that in *noway* detracts from the claim that to really get
a clean grasp of it, you need to do the derivations yourself, ie., you
need to *understand* the mathematical description (not just use it as
a "cookbook" to solve some problems...)
>The hard part to learn is the different, less absolute, view of such
>seemingly concrete concepts as space, time, mass and velocity ....
Right. And you bring your intuition up to speed here by *deriving*
the results - not by just working a bunch of problems that assume and
use the results! As an analogy - from having taught much undergraduate
mathematics, I am quite willing to state (perhaps with some hyperbole)
that 99.9% of the people who know how to "do calculus" don't have any
understanding of what it's all about. Knowing how to differentiate
and integrate is the calculus equivalent of multiplying and dividing.
No matter how much of this you do, it doesn't increase your understanding
of the subject. For that you must understand the basic concepts and derive
(or at least prove on your own) (much of) the rest. Of course, if you're
a genius, this step can perhaps be skipped :-).
> Einstein claimed to have mentally seen the results and only then to have
> written the equations to fit the images.
Yeah, just like Mozart wrote Don Giovanni like he was taking dictation!
Like I said, genius...
> > In a nutshell, division = f(x,y)... This is *the* best way of
> > understanding this...
>
> It is certainly is a way, and for some purposes it may even be "*the* best"
> way, but it completely obscures the point I was trying to make, which is...
Look, division is a *function* f: R^2 -> R (confining ourselves to the
reals for explicitness), no matter what means you use to define it - any
number of rules, a graph, an extensional set definition of 3-tuples -
whatever. As such, division by zero is *undefined* period. If this isn't
adhered to we run into contradictions (as I clearly pointed out). Once
we have contradictions in our system, *anything* can be "derived" or
"proven". So, they are to be unconditionally avoided!
> You look at the equation as a *formula* ....
> I prefer to look at it as an *equation* ....
Come on. I stated that it was an equation, and that it is most useful
to view this equation as defining a function m(m0, v). It certainly
can also be used as a formula in the sense you give.
> m�(c� - v�) = m0�c�
This has no more useful information in it than the original (in fact,
since the negative root is now lurking in it, it's somewhat less
informative). And there's a problem lurking in this form. Lets apply
the algebra of limits to this guy and see what we get:
Lim m�0 = Lim m� Lim (c� - v�) = Lim m�(c� - v�) = Lim m0�c� = m0�c�
v -> c v -> c v -> c v -> c v -> c
You want to say that there is no problem in dropping the limit on m�
while I maintain there is a big problem with dropping it. At this point
we know that m depends on v!! Hence, even though m *looks* constant wrt
v here, we know that it isn't and thus that m�0 = m0�c� is *not* a correct
deduction. The fact that m could be *any* value is a sign that something
has gone *wrong*. Basically, since the limit is indeterminate, the equation
really just doesn't have anything to say about the case v=c. This is
true whether or not m0 = 0. And this is hardly surprising. For m0 > 0,
we know that v can never = c (as you point out) lest 0 /= 0! Next, we
know that the mass transformation is a statement about "material things"
as that's the context in which it is derived (the lorentz transformations).
Since m0 = 0 things are not "material" (not composed of matter) the
equation is silent about them. I think that you have to use the equation
m = E/c� (which could apply even when the mass transformation doesn't,
i.e., there could be a more general derivation of which the mass transfor-
mation one is a special case) and then the E = h*Nu equation for photons
(I think Nu is the Greek letter for frequency - it's been *years* since
I've thought about this stuff :-).). Then again, talking about the mass
of a photon is pretty misleading ...
One final note:
Relative but is it relevant?
I don't know, but probably not. Not that it isn't a very good thing for
people who are interested in "reality" to think about and toy about with
SR (and GR). But somehow, I think that most of the readers of this
conference just couldn't give a hoot about all this. In addition, our
yammerings have been l o n g and technical (and for most readers
probably incredibly boring (*sigh*)). It wouldn't surprise me a bit if
NO ONE else (but you) gets this far into this note...
/Jon (I doubt if I will respond anymore to this, even if I disagree
with any other responses...)
|
| 629.3 | right now, some of the above is inconclusive | INK::KALLIS | Has anybody lost a shoggoth? | Thu Jan 21 1988 10:03 | 44 |
| Re .2 (Jon):
> ............... This in *noway* implies that there can be any
>communication between the "two worlds".
If there's interaction, there's a potential interface; hence, it
in _no way_ implies that there _can't_. :-)
> ........................ Oh BTW, how could
>anyone suggest that a cosmic ray could possibly have enough energy to
>create a superluminal velocity particle???!!?!?!
Oh, I suppose fairly easily, if one suggests a "tunneling" mechanism
vaguely analogous to the mechanism present in a tunnel diode. If
something can reach one state from another witout "passing through"
intermediate states ....
>> Einstein claimed to have mentally seen the results and only then to have
>> written the equations to fit the images.
>
> Yeah, just like Mozart wrote Don Giovanni like he was taking dictation!
> Like I said, genius...
That one's a toughie, by the way: Fermat wrote that he saw (or wrote)
a simple proof _in the margin of a book page_ that for any integer
n greater than two, there were no numbers A, B, and C for which
A^n + B^n = C^n
That's Fermat's Last Theorem. Gauss thought it was hogwash (the
claim of seeing/writing a proof), and _he_ sure was a genius. But
to this day, Fermat's Theorem has never been disproven -- _or_
proven! Genius indeed ....
>..... But somehow, I think that most of the readers of this
>conference just couldn't give a hoot about all this. Inaddition, our
>yammerings have been l o n g and technical (and for most readers
>probably incredibly boring (*sigh*)). It wouldn't surprise me a bit if
>NO ONE else (but you) gets this far into this note...
Cheer up! You two are not alone. :-D
Steve Kallis, Jr.
|
| 629.4 | Tachyonic Miscellany | RSTS32::WAJENBERG | Celebrated ozone dweller | Thu Jan 21 1988 10:18 | 56 |
| Re .2
You remark that the responses have been too long and technical to interest
most of the DEJAVU readers. I suggest that (1) people not quote each other's
notes quite so extensively, and (2) people try to keep individual entries
under 100 lines. These guidelines work very well in the PHILOSOPHY
conference. It makes your comments likelier to get read instead of skipped
and is easier on the moderator(s).
I shall, accordingly, try to be brief in my quotes as I pick a few more nits.
You take Topher to task for describing the rest mass as "intrinsic to the
particle and independent of the inertial frame of reference" on the grounds
that "no frame of reference is preferred over another." Those grounds are
irrelevant, since the definition of "rest mass" is "mass as measured in the
rest frame of the body in question." Now, it is a nice mathematical point
whether or not photons can be said to have a rest frame at all, since they
must always move at c in any frame. But one could reasonably say that, if
they have no rest frame, they have no rest mass, so their rest mass is zero.
At the very least, physics literature routinely speaks of photons as "having
zero rest mass."
"Symmetry would soly state that photons and the like interact with
tachyons just as photons do interact with ordinary objects. This in
*noway* implies that there can be any communication between the `two
worlds'."
It may not imply it in the sense of strict logical implication, but it very
strongly suggests it, since the vast bulk of physical interaction we
experience in daily life is mediated by photons.
"Oh BTW, how could anyone suggest that a cosmic ray could possibly
have enough energy to create a superluminal velocity particle???!!?!"
Well, that's one of the amusing symmetries of tachyon theory. The faster
tachyons go, the more momentum they have, but the less energy. If a tachyon
made instantaneous transitions (in a given frame), it would have zero energy
and a maximum of momentum corresponding to the "rest mass" of a normal
particle. So there's no difficulty is supplying sufficient energy for the
production of tachyons.
"Given [the constant value of c], in SR there is *no place* for
superluminal velocities - period."
No. (See, anyone can be dogmatic. It's very easy.) Even without discussing
tachyons, we can find examples of superluminal velocities in relativity. It
is true that they are not the velocities of bodies, but they are still there,
they are permitted by the mathematics. For instance, there are phase
velocities, the velocities of interference patterns in a wave train. Phase
velocities often go higher than c.
And, as Topher has already explained, the mathematics of special relativity
continues to work fine past c, provided you are willing to cope with complex
numbers.
Earl Wajenberg
|
| 629.5 | surprise! | ULTRA::LARU | Let's get metaphysical | Thu Jan 21 1988 11:34 | 11 |
| � probably incredibly boring (*sigh*)). It wouldn't surprise me a bit if
� NO ONE else (but you) gets this far into this note...
Can anybody recommend efficient, intuitive books to help me
understand the mathematics of relativity??? I almost made
it through integral calculus 20 years ago, and have always
wanted to start up again...
/bruce
|
| 629.6 | Bibliography | RSTS32::WAJENBERG | Celebrated ozone dweller | Thu Jan 21 1988 11:41 | 18 |
| I recommend "Spacetime Physics" by Wheeler. It is probably available
as a large paperback in college bookstores.
Einstein himself is also quite lucid, I'm told, if you can find the
translations from German.
Martin Gardner wrote a book on relativity long ago. It was good, but
I've forgotten the title.
For actively entertaining instruction, try "Mr. Tompkins in Wonderland"
by George Gamov. His hero, Mr. Tompkins, is the son-in-law of a
physicist. His father-in-law comes over once a week and drones on
about relativity and quantum mechanics, etc. Mr. Tompkins invariably
falls asleep and has a surreal dream based on the subject under
discussion. After the dream, you get to read "the professor's lecture"
that inspired it (a short, lucid, non-fiction essay by Gamow).
Earl Wajenberg
|
| 629.7 | ...but what do you expect from one of the founders i of CSICOP? | INK::KALLIS | Has anybody lost a shoggoth? | Thu Jan 21 1988 11:57 | 9 |
| Re .6 (Earl):
>Martin Gardner wrote a book on relativity long ago. It was good, but
>I've forgotten the title.
It was called _Relativity for the Millions_. Not bad, but a bit
conceited about the press run on the part of Gardner. :-)
Steve Kallis, Jr.
|
| 629.8 | Novels | REGENT::BROOMHEAD | Don't panic -- yet. | Thu Jan 21 1988 12:05 | 7 |
| And if you want a *very* high entertainment to information work,
read _Tau_Zero_ by Poul Anderson (runner-up to _Ringworld_ for
the 1971 Hugo Award (sm)) and/or _Time_for_the_Stars_ by Robert
A. Heinlein (which has the further enticement of being about
telepathy).
Ann B.
|
| 629.9 | Never say "never" :-) | TLE::JONAN | Into the Heart of the Sunrise | Thu Jan 21 1988 14:11 | 72 |
|
Re: .3
> in _no way_ implies that there _can't_. :-)
Yup - completely agree.
>> Yeah, just like Mozart wrote Don Giovanni like he was taking dictation!
>> Like I said, genius...
I meant this tongue-in-cheek - should've put a smiley on the end
of it... :-) I never saw anything where Gauss said that. The only
thing I remember about it was that he said it was a waste of time
to work on it because he thought there wasn't anything interesting
about the answer - there wasn't anything "profound" about FLT said.
Then again, much (most?) of modern algebra can find its roots in attempts
to find the answer. Also, there has been some significant progress in
the search in the past year (it turns out that some aspects of elliptic
function theory bear directly on FTL. But as that isn't one of my areas
of specialization, I can't make head or tail out of most of it).
Re: .4
>the definition of "rest mass" is "mass as measured in the
>rest frame of the body in question."
My point was only that this is not intrinsic and is tied to the
mass in question's frame of reference.
> Now, it is a nice mathematical point whether or not photons can be said
> to have a rest frame at all, since they must always move at c in any frame.
My point exactly.
> But one could reasonably say that, if they have no rest frame, they have
> no rest mass,
Yes, but
> *so* their rest mass is zero. [[emphasis mine]]
I disagree. This is analogous to the ol' empty set arguments. You can't
have a rest mass of zero if you don't have a rest mass *at all*
> At the very least, physics literature routinely speaks of photons as "having
> zero rest mass."
No argument there.
> Well, that's one of the amusing symmetries of tachyon theory. The faster...
Fascinating. I suppose this is due to just accepting "imaginary" (a
terrible term, whose roots are completely historical) objects and applying
the mass transformation to them. As v increases without bound, their mass,
and thus their energy goes to zero. That'll learn me to confine my vision
to directly around the c=v "singularity"!!!
> No. (See, anyone can be dogmatic. It's very easy.) Even without discussing
Not dogmatic, just consistent within the confines of the theory. I'm
open to other possibilities and
> explained the mathematics of special relativity continues to work fine past
> c, provided you are willing to cope with complex numbers.
Was the point I originally made which started all of this!!
/Jon (Who's not feeling so down today :-) )
|
| 629.10 | digression | ERASER::KALLIS | Has anybody lost a shoggoth? | Thu Jan 21 1988 14:41 | 11 |
| Re .9 (Jon):
> ... I never saw anything where Gauss said that. ...
It was quoted in one of Eric Temple Bell's books. Either _Men of
Mathematics_ or _The Final Problem_ (a book using FLT as a theme).
Bell was a professional mathematician, and both books are entertaining
reads. Bell wrote a number of science fiction novels under the
pseudonym of John Taine.
Steve Kallis, Jr.
|
| 629.11 | Love it! | SDOGUS::DEUTMAN | I'd Rather be INSANE DIEGO | Thu Jan 21 1988 21:48 | 17 |
| Hey, love this stuff! I'm not too fluent on it myself, but I do
try to keep up with sub atomic particle physics...
I have a book (which I just dusted off) titled "Spacetime Physics"
by Edwin F Taylor (MIT) and John Wheeler (Princton U). It is
a "brief, readable exposition of modern relativity theory, illustrated
and amplified by a wealth of problems, puzzles and paradoxes, and
their detailed solutions".
Published by W H Freeman (San Francisco) C 1963, 1966 (which makes
it out of print probably, but maybe it can be found in those out
of the way book stores).
Larry @.@
o
|
| 629.12 | massless can mean zero mass. | PBSVAX::COOPER | Topher Cooper | Thu Mar 03 1988 12:34 | 51 |
| I was somewhat handicapped in this discussion by the fact that most
of my books are still in boxes after the move (we're doing a bit
of renovation, and until its over, we don't have room to set up
the shelves). However, the current Scientific American has provided
a place to cite.
The March, 1988 issue of Scientific American (Vol. 258, #3) contains
an article entitled "Gravity and Antimatter" by Terry Goldman,
Richard J. Hughes and Michael Marin Nieto. All are, according to
the "The Authors" section, theoretical physicists at the Los Alamos
National Laboratory.
At the top of page 52, last column they say "The force carrying
particles have a definite rest mass (zero in the case of photons)..."
This, of course, in *no way* can be taken as evidence that photons
have zero rest mass -- argument "from authority" is one of the
classic logical flaws.
It *does* though indicate that at least some mainstream theoretical
physicists dealing in areas closely related to those we are speaking
of, hold this view and are willing to present it in a widely read
publication in an offhand matter (which implies that they do not
consider it a controversial stand).
Personally I believe that neither interpretation of rest massless
(zero rest mass vs. without the property of mass) is correct or
incorrect in any absolute sense -- since it only a matter of how
we express our theories rather than the theories themselves. Both
work, BUT I think that zero rest mass leads to simpler descriptions.
For example, a with a zero rest mass, we can say that the mass
conversion formula applies to *all* particles, though it fails to
be sufficient to constrain the observed mass of those with zero
rest mass, such as the photon and the neutrino.
Another example, we can say, somewhat informally, that the range
of a force is inversely proportional to the mass of its mediating
particle. Thus the range of the electromagnetic force (as well
as the gravitational and chromatic forces) is infinite because
the photon (the mediating particle for electromagnetic forces) has
zero rest mass, while the "weak" force, mediated by the massive
W and Z particles, has a finite range.
It is useful, consistent and (with some additional work) rigorously
justified to say that the photon has a zero rest mass. Although
I can't prove this by quoting one or even a few individual sources,
I believe this is the standard interpretation of the term "massless"
and has been since Special Relativity was first published.
Topher
|