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From: Rock Brentwood on 29 Jul 2010 16:36
On Jul 20, 6:26 am, jmfbahciv <See.ab...(a)aol.com> wrote:
> >> > How do you define mass?  How do you measure it with a ruler?
> >> Mass is a measure of gravitational attraction...
> > making GR compatible with QM.
> You have claimed that everything can be described using only
> space and time.  So I've asked you to describe mass using
> only those two entities so that a mass can be measured in
> a lab.

Place a stationary shell around a spherical region of space and then
let it go. Its volume will contract with an acceleration given by 4 pi
G M, where M is the mass contained within the region. Once it gets
moving, the equation for the volume contraction will also involve the
first derivative of the volume contraction. Both cases are instances
of the Raychaudhuri equation. The Raychaudhuri equation is present in
both in relativistic and non-relativistic theory.

This is analogous to how charge is measured with a flux meter.

Mass can be defined solely in terms of the Lie group underlying the
symmetries of space-time. This is the case both relativistically and
non-relativistically.

In non-relativistic theory, each system has a state space that is a
representation of the Galilei group, this group describing the
symmetries of non-relativistic space-time. If the system is elementary
the representation is irreducible. The mass of such a system is the
central charge of the representation.

So, in non-relativistic theory, mass is defined solely in terms of the
transformation properties of the system under the action of the
Galilei group.

The notion of central charge was not widely known until the early to
mid 20th century. So, classical physicists never even had the
opportunity to recognize this important property, nor to make this
definition. This proves, by the way, that the field of classical
physics EVEN NOW is still in a state of continual evolution -- even if
retroactively -- as more and more gaps and oversights from "classical"
classical theory emerge. So we now have to distinguish "classical"
classical theory from "modern" classical theory (and even classical
non-classical theory from modern non-classical theory, since many of
the new insights also get inherited by non-relativistic theory).

The other two invariants of the Galilei group (for the generic
irreducible representation) are the one given by P^2 - 2mH (where m is
the central charge, P the generator of spatial translations and H the
generator of time translations), and W^2 where W = mJ + P x K (x
denoting cross product), where J is the generator of spatial
rotations, K the generator of Galilean boosts. These give you,
respectively, the internal energy and internal angular momentum (i.e.
spin) of the system.

In relativistic theory mass can also be defined solely in terms of the
behavior of a system under transformation by the underlying space-time
symmetry group.

There, the irreducible representations are classified as either
translation-invariant or not (as they also are in the non-relativistic
case). For the translation non-invariant systems, a further
classification into "tardion", "luxon" and "tachyon" exists (for the
non-relativistic case, luxon and tachyon combine into "synchron" --
something which was also absent from classical classical physics).

In all cases, the two invariants are P^2 - (1/c)^2 E^2 and W^2 - (1/
c)^2 W_0^2, where W_0 = P.J, where E is the generator of time
translations. The distinction between tardion, luxon and tachyon rests
solely on the sign of the first of these invariants.

Tardions have negative sign, so one can define the invariant m by m^2
= (1/c)^4 E^2 - (1/c)^2 P^2, taking the sign of m the same as the sign
of E. That defines the mass of the system.

For tachyons, the invariant is positive, so one can only define the
*impulse*, Pi by Pi^2 = P^2 - (1/c)^2 E^2. These systems represent an
"instantaneous" transfer of impulse Pi across space (where
"instantaneous" means, "instantaneous in at least one frame of
reference"). The "synchrons" in non-relativistic theory share this
feature with tachyons. So, the frame of reference for tachyons in
which the transfer is instantaneous might be called the "synchron
frame".

There is no meaningful definition for "mass" for tachyons. The
"impulse" takes over that role.

Luxons fall into two classes, based on whether W has components
perpendicular to P or not. If not, then W is parallel to P. I call
these the "helions". The photon falls into this class. In so, then
this leads to the representations known as the "continuous spin"
representations. There are no fundamental systems known that fall into
this class.

In both cases, there is no meaningful attribute "mass".
Conventionally, it's just taken to be 0, since the Luxon is the m -> 0
limit of the tardion. It's also the Pi -> 0 limit of the tachyon, so
its "impulse" can also be taken as 0.

Synchrons are the m -> 0 limit of tardions in the non-relativistic
case. So their mass is 0. There is also a frame of reference in which
H is 0. In non-relativistic theory, synchrons correspond to action-at-
a-distance forces.
From: Rock Brentwood on 29 Jul 2010 16:41
On Jul 22, 5:57 pm, Michael Gordge <mikegor...(a)xtra.co.nz> wrote:
> On Jul 22, 11:35 pm, bert <herbertglazie...(a)msn.com> wrote:
> > Spaceandtimeare two sides to the same coin. Einstein merged them.
> Only because he refused to accept / he rejected the law of identity.

It was not Einstein, but his teacher, Minkowski who "merged them". But
even this is wrong. It was Galileo who merged them -- what counts is
that the appearance of a mixed space-time symmetry. That's what
marries space and time. The marriage was already in place when the
extra "space-time" term in the transformation law appeared under
Lorentz. This did not effect the marriage, but merely consummated a
union that was already in place for around 300 years.

Before Galileo, the symmetries for space (3 degrees of translation
symmetry + 3 degrees of rotational symmetry) were separate from the
symmetries for time (1 degree of translation symmetry). After Galileo,
there were also the 3 extra degrees of symmetry: the boosts (3 degrees
of COMBINED space-time symmetry).

Hence, the union. A gunshot wedding that was kept quiet (even after
the union was consummated by Minkowski).

It took 300 years for the marriage to be consummated and 400 years for
the true age of the union to be recognized.
From: John Stafford on 29 Jul 2010 17:00
In article
<efe6a582-9984-46e0-8243-90c9797acbec(a)x21g2000yqa.googlegroups.com>,
Rock Brentwood <federation2005(a)netzero.com> wrote:

> On Jul 20, 6:26�am, jmfbahciv <See.ab...(a)aol.com> wrote:
> > >> > How do you define mass? �How do you measure it with a ruler?
> > >> Mass is a measure of gravitational attraction...
> > > making GR compatible with QM.
> > You have claimed that everything can be described using only
> > space and time. �So I've asked you to describe mass using
> > only those two entities so that a mass can be measured in
> > a lab.
>
> Place a stationary shell around a spherical region of space and then
> let it go.

Well, that is that. Impossible. Moving on, then...
From: Day Brown on 30 Jul 2010 20:26
On 07/28/2010 11:46 AM, Tim Golden BandTech.com wrote:

> Well, this realization frames the human situation perfectly. What are
> we overlooking?... this is the conundrum. When we do not come up with
> an answer to this puzzle we simply are going along on the existent
> assumptions. When we do find something that we've been overlooking
> then one of the most natural things to do is to return to the question
> again and ask what else we are overlooking. This path has led me to my
> current position. That said I do seek falsification.
>
> Another curious aspect in terms of theoretical construction is that
> fairly parallel theories may exist. This is not unlike thinking in
> analogies, which Hume argues is an inherent feature of the human
> learning process, and I pretty well agree with that. Within that
> context we know that an analogy is imperfect, yet can still be
> appropriate to discussing some quality that lays in parallel. Likewise
> in parallel theories the slight discrepancies or inversion of
> priorities as to what is fundamental sometimes have deep significance.
> The word parallel is a bit strict for these congruences, so I do wish
> to loosen that strict geometrical parallel to a rougher case, where
> more difference can be found. This in terms of construction can occur
> by shuffling the axioms.
....
> I think that the humans give themselves too much credit when they
> claim that they can think in 3D. If we could then when we looked at a
> tree we would likewise see the trees beyond the tree, and stars beyond
> those trees, and the stars beyond those stars. But this is not the
> ordinary human perception. Instead we receive a two dimensional
> projection with occlusion, and so our '3D' perception is in fact
> occluded and what we really have is a pair of 2D projections to work
> with. All the rest, and even some at that level, is image processing.
> Yes, our image processors do a fair job, but still the raw input
> prevents any true 3D observations. As to whether this all can be used
> to provide an argument for actual higher dimension than 3D space... I
> don't think so. When we do the math and have the freedom to set up
> solids within a region we truly are able to yield the 3D
> representation. It is also true that these solids have another two
> degrees of freedom so that a 6D representation can be claimed. Whether
> we grant the point position the six degrees, well, Euclidean geometry
> says no, but particle theory says yes, especially when spin is
> included. There have been numerous 6D theories presented, and it
> happens that polysign provides a 6D level just before the 10D level,
> and just after the 3D level:
>
> T1 : 0D : P1
>
> T2 : 1D : P1 P2
>
> T3 : 3D : P1 P2 P3
>
> T4 : 6D : P1 P2 P3 P4
>
> T5 : 10D : P1 P2 P3 P4 P5
>
> ... : ...
>
> This is the polysign progression. The above information is redundant
> per line and are three representation, the most fundamental being on
> the RHS. Travelling downwards this is merely a progression, and there
> is no reason yet known to end on T5. Regardless of how long the
> progression grows spacetime support will remain due to a behavioral
> breakpoint in P4+, where multiplication behaviors yield dimensionally
> degenerate cases, lacking in distance conservation, and preventing
> division.
>
> Best of all the polysign progression yields unidirectional time along
> with time's seeming paradoxes(from a real valued perspective). Time is
> zero dimensional, and until this flaw in the definition of dimension
> is addressed the human race is a bit of a farce. Still, I go on
> reusing the word dimension because it can sit right where it has been,
> yet the existinece of a zero dimensional entity challenges the
> cartesian notion of zero dimensional. Dimension is off by one. We are
> off by one.
Only by one? Seems a lotta minds are a little more off than that.
> The notion of duality as fundamental requires breakage. The duality of
> the real number denies its fundamental nature. This duality problem is
> one I think you may have some valuable contribution on so I bring it
> up. Almost certainly what you read historically will affirm duality,
> yet the existence of universality is a primary challenge to duality,
> followed by triality, whose practical usage seems to go unovserved in
> human language. The construction of a triality is a challenge. A
> balance will be present when a triality is taken in. Extending the
> mathematics I can even claim that this triality will have a semblance
> to two dualities which are orthogonal to one another. The triality
> will then be a simplification, and be more accurate, though obviously
> by the stance on human behavior we will fail to appreciate the new
> form. This is a new language that is not necessarily spoken yet.
Dr. Freud said neurotics cant tolerate ambiguity. Dualism, such as
good/evil, Heaven/Hell, blessed/cursed Conservative/Liberal chooses to
overlook the gray, never asking what is missed.

Arguably, the earliest hominid, 6 million years ago, was found in Chad,
in what had been at the time a closed drainage basin like the Okavango.
Thing is, favored fruit and nut trees didnt move, but the flash floods
every year often carved new channels, so the hominid had to calculate
new routes, and even learn to swim, to access resources. This entailed
the construction in the mind of a 3D landscape that existed only in the
mind. Pattern recognition and construction have been with us since.

I dont have the math skill to follow your logic very far, but readily
admit you may be substantially right about it. I dont need certainty.
But many others seem to.

From: Tim Golden BandTech.com on 31 Jul 2010 09:10
On Jul 30, 8:26 pm, Day Brown <dayhbr...(a)gmail.com> wrote:
> On 07/28/2010 11:46 AM, Tim Golden BandTech.com wrote:
>
>
>
> > Well, this realization frames the human situation perfectly. What are
> > we overlooking?... this is the conundrum. When we do not come up with
> > an answer to this puzzle we simply are going along on the existent
> > assumptions. When we do find something that we've been overlooking
> > then one of the most natural things to do is to return to the question
> > again and ask what else we are overlooking. This path has led me to my
> > current position. That said I do seek falsification.
>
> > Another curious aspect in terms of theoretical construction is that
> > fairly parallel theories may exist. This is not unlike thinking in
> > analogies, which Hume argues is an inherent feature of the human
> > learning process, and I pretty well agree with that. Within that
> > context we know that an analogy is imperfect, yet can still be
> > appropriate to discussing some quality that lays in parallel. Likewise
> > in parallel theories the slight discrepancies or inversion of
> > priorities as to what is fundamental sometimes have deep significance.
> > The word parallel is a bit strict for these congruences, so I do wish
> > to loosen that strict geometrical parallel to a rougher case, where
> > more difference can be found. This in terms of construction can occur
> > by shuffling the axioms.
> ...
> > I think that the humans give themselves too much credit when they
> > claim that they can think in 3D. If we could then when we looked at a
> > tree we would likewise see the trees beyond the tree, and stars beyond
> > those trees, and the stars beyond those stars. But this is not the
> > ordinary human perception. Instead we receive a two dimensional
> > projection with occlusion, and so our '3D' perception is in fact
> > occluded and what we really have is a pair of 2D projections to work
> > with. All the rest, and even some at that level, is image processing.
> > Yes, our image processors do a fair job, but still the raw input
> > prevents any true 3D observations. As to whether this all can be used
> > to provide an argument for actual higher dimension than 3D space... I
> > don't think so. When we do the math and have the freedom to set up
> > solids within a region we truly are able to yield the 3D
> > representation. It is also true that these solids have another two
> > degrees of freedom so that a 6D representation can be claimed. Whether
> > we grant the point position the six degrees, well, Euclidean geometry
> > says no, but particle theory says yes, especially when spin is
> > included. There have been numerous 6D theories presented, and it
> > happens that polysign provides a 6D level just before the 10D level,
> > and just after the 3D level:
>
> > T1 : 0D : P1
>
> > T2 : 1D : P1 P2
>
> > T3 : 3D : P1 P2 P3
>
> > T4 : 6D : P1 P2 P3 P4
>
> > T5 : 10D : P1 P2 P3 P4 P5
>
> > ... : ...
>
> > This is the polysign progression. The above information is redundant
> > per line and are three representation, the most fundamental being on
> > the RHS. Travelling downwards this is merely a progression, and there
> > is no reason yet known to end on T5. Regardless of how long the
> > progression grows spacetime support will remain due to a behavioral
> > breakpoint in P4+, where multiplication behaviors yield dimensionally
> > degenerate cases, lacking in distance conservation, and preventing
> > division.
>
> > Best of all the polysign progression yields unidirectional time along
> > with time's seeming paradoxes(from a real valued perspective). Time is
> > zero dimensional, and until this flaw in the definition of dimension
> > is addressed the human race is a bit of a farce. Still, I go on
> > reusing the word dimension because it can sit right where it has been,
> > yet the existinece of a zero dimensional entity challenges the
> > cartesian notion of zero dimensional. Dimension is off by one. We are
> > off by one.
>
> Only by one? Seems a lotta minds are a little more off than that.> The notion of duality as fundamental requires breakage. The duality of
> > the real number denies its fundamental nature. This duality problem is
> > one I think you may have some valuable contribution on so I bring it
> > up. Almost certainly what you read historically will affirm duality,
> > yet the existence of universality is a primary challenge to duality,
> > followed by triality, whose practical usage seems to go unovserved in
> > human language. The construction of a triality is a challenge. A
> > balance will be present when a triality is taken in. Extending the
> > mathematics I can even claim that this triality will have a semblance
> > to two dualities which are orthogonal to one another. The triality
> > will then be a simplification, and be more accurate, though obviously
> > by the stance on human behavior we will fail to appreciate the new
> > form. This is a new language that is not necessarily spoken yet.
>
> Dr. Freud said neurotics cant tolerate ambiguity. Dualism, such as
> good/evil, Heaven/Hell, blessed/cursed Conservative/Liberal chooses to
> overlook the gray, never asking what is missed.
>
> Arguably, the earliest hominid, 6 million years ago, was found in Chad,
> in what had been at the time a closed drainage basin like the Okavango.
> Thing is, favored fruit and nut trees didnt move, but the flash floods
> every year often carved new channels, so the hominid had to calculate
> new routes, and even learn to swim, to access resources. This entailed
> the construction in the mind of a 3D landscape that existed only in the
> mind. Pattern recognition and construction have been with us since.
>
> I dont have the math skill to follow your logic very far, but readily
> admit you may be substantially right about it. I dont need certainty.
> But many others seem to.


Good enough Day. I think you have a sharp mind, but accept that you're
not up to the math side of things. I believe that a grade school child
will take to polysign better than an educated adult, for we have had
the two-sigend real number drilled into our heads as fundamental ad
nauseum.

I always enjoy reading your anthropological posts and appreciate your
focus on the human condition, though I am about as versed in those
endeavors as you are in math. Whatever; sometimes fresh minds
contribute more than those versed in the lore, for they start with a
cleaner slate. So I don't mind entertaining any mathematical ideas you
might have, though I might bend them into a context that I find
useful. This is how we should all make our attempts and this is a
great medium for that endeavor.

Thanks, as usual, for contributing what you have. I agree that
navigational abilities may be one form of genetically coded spatial
analysis, but also that this tends to be a surface map, consistent
with the 2D projection of a 3D space, rather than a pure 3D form. We
do not typically look into a mountain and see the peak of another
lower mountain beyond, whether with our eyes or with our mind, so we
should perhaps surrender our claim of 3D visualization, just as most
give up on 4D visualization. I admit we do at least have some partial
3D ability, but that it is incomplete.

- Tim
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Prev: NP+complete-problem navigation, search In computational complexity theory, the complexity class NP-complete (abbreviated NP-C or NPC), is a class of problems having two properties: * It is in the set of NP (nondeterministic polynomial time) pr
Next: Continuity and Uncountability