Gravitomagnetism
Ziggi
indulge me if you can.
Ok, so I was thinking the other day: "Would it be possible to write down a
set of differential equations for some field that, in flat space, looks
kinda like EM, but it curved space has a gravity term/component? Sort of
like the way a magnetic field at zero velocity looks partially electric at 0
< v < c".
I know it's a bit of an odd question, but I was curious as to how one would
contruct such a theory and what it would look like. Answers on a postcard
:p
Thanks in advance
Ziggi
ps, spare no technicality in your response... I'm not exactly a "lay" person
:)
robert bristow-johnson
in article ckh4d2$6fb$1...@titan.btinternet.com, Ziggi at one_...@hotmail.com
wrote on 10/12/2004 14:45:
> This is going to sound like a very bizarre and possibly insane question, but
> indulge me if you can.
>
> Ok, so I was thinking the other day: "Would it be possible to write down a
> set of differential equations for some field that, in flat space, looks
> kinda like EM, but it curved space has a gravity term/component? Sort of
> like the way a magnetic field at zero velocity looks partially electric at 0
> < v < c".
>
> I know it's a bit of an odd question, but I was curious as to how one would
> contruct such a theory and what it would look like. Answers on a postcard
> :p
i don't see it as an odd question at all. i've been thinking about it
myself for as long as i understood (as best as a "lay" physiker can - i'm an
electrical engineer so that might give you an idea of the limits of my
physics expertise) how Electromagnetic forces could be derived from
Electrostatic forces with Special Relativity taken into consideration. i
have thought "Why not do the same for gravity? They are both inverse-square
forces and have a velocity of propagation of c, so why not?" folks on this
newsgroup haven't been too impressed and that's fine with me.
Anyway, there is a name for this theory and it's called
"Gravitoelectromagnetism" (GEM) and there isn't yet a Wiki page for it yet.
This GEM theory has counterparts to Maxwell's Equations that look just like
Maxwell's Equations (and the Lorentz force equations) with "q" replaced by
"m", 1/(4*pi*epsilon0) replaced by -G (just as it is in the Coulomb force
law to get to Newton's law of gravitation) except that the magnetic flux in
GEM is expressed as "B/2" instead of "B". There are at least two papers:
http://arxiv.org/PS_cache/gr-qc/pdf/9912/9912027.pdf
http://www.iop.org/EJ3-Links/26/B2PcnrMQ9Qr,dG8lppV,HA/q01911.pdf
that derive these GEM equations from GR (Einstein's Field Eq.) for flat
spacetime.
I haven't understood the B/2 scaling thingie (they say its because gravitons
are spin-2 particles) because it seems like, at velocities of c/2, the
gravito-magnetic forces completely counteract the gravito-static force and
that should not happen (from the p.o.v. of Special Relativity) until the
velocity is close to c. at least that's how this amateur looks at it. i
wish the experts here could give me an explanation of that seeming
contradiction.
> ps, spare no technicality in your response... I'm not exactly a "lay" person
> :)
but i am.
r b-j
Gregory L. Hansen
In article <ckh4d2$6fb$1...@titan.btinternet.com>,
Looks to me like you're describing general relativity. The differential
equations you're looking for would be
G = 8 pi T
Spin a mass and you get frame dragging, which is kinda like magnetism in
EM. Wiggle it and you get gravitational radiation, which is kinda like
electromagnetic radiation. Unlike electromagnetism, you can't get a
magnetic field with zero net charge.
--
"What are the possibilities of small but movable machines? They may or
may not be useful, but they surely would be fun to make."
-- Richard P. Feynman, 1959
Danny Ross Lunsford
"Ziggi" <one_...@hotmail.com> wrote in message news:<ckh4d2$6fb$1...@titan.btinternet.com>...
> This is going to sound like a very bizarre and possibly insane question, but
> indulge me if you can.
>
> Ok, so I was thinking the other day: "Would it be possible to write down a
> set of differential equations for some field that, in flat space, looks
> kinda like EM, but it curved space has a gravity term/component? Sort of
> like the way a magnetic field at zero velocity looks partially electric at 0
> < v < c".
>
> I know it's a bit of an odd question, but I was curious as to how one would
> contruct such a theory and what it would look like. Answers on a postcard
> :p
Yes, here:
Online for free here:
http://cdsweb.cern.ch/search.py?recid=688763&ln=en
Feel free to ask questions here or by email (note: email in paper was
spammed out of existence).
-drl
Ken S. Tucker
antima...@yahoo.com (Danny Ross Lunsford) wrote in message news:<2b93dd16.04101...@posting.google.com>...
Hi, I see you use as a ref Weyl's paper "Gravitation and Electricity".
I have a problem with it, (using Dover's PoR), Eq.(7) implies
phi_u
is a gradient. In Eq.(10) he takes the curl of phi_u, to define
the EM field tensor, which, I think, vanishes ie.
curl grad (scalar) =0
Do you see any problem with that?
I hold other comments on your paper pending your reply.
Regards
Ken S. Tucker
Danny Ross Lunsford
dyna...@vianet.on.ca (Ken S. Tucker) wrote in message news:<2202379a.04101...@posting.google.com>...
(by PoR I assume you mean "Principle of Relativity")
That is unfortunate notation - replace phi_m by A_m everywhere to make
it more transparent. He's simply saying the change in calibration is a
linear, homogeneous expression in the coordinate differentials under
an infinitesimal displacement. This is an assumption, not a
consequence. By no means is A_m (or phi_m in Weyl's notation) a
gradient. (If it devolves to a gradient, Weyl geometry reduces to
Riemannian geometry.)
Also note that Weyl has a footnote where he clarifies his notation.
-drl
Ken S. Tucker
antima...@yahoo.com (Danny Ross Lunsford) wrote in message news:<2b93dd16.04101...@posting.google.com>...
> dyna...@vianet.on.ca (Ken S. Tucker) wrote in message news:<2202379a.04101...@posting.google.com>...
> > antima...@yahoo.com (Danny Ross Lunsford) wrote in message news:<2b93dd16.04101...@posting.google.com>...
...
> > > http://cdsweb.cern.ch/search.py?recid=688763&ln=en
> > >
> > > Feel free to ask questions here or by email (note: email in paper was
> > > spammed out of existence).
> > >
> > > -drl
> >
> > Hi, I see you use as a ref Weyl's paper "Gravitation and Electricity".
> > I have a problem with it, (using Dover's PoR), Eq.(7) implies
> >
> > phi_u
> >
> > is a gradient. In Eq.(10) he takes the curl of phi_u, to define
> > the EM field tensor, which, I think, vanishes ie.
> >
> > curl grad (scalar) =0
> >
> > Do you see any problem with that?
> > Ken S. Tucker
>
> (by PoR I assume you mean "Principle of Relativity")
Yes, sorry I see you refd to Weyl's "SPACE TIME MATTER",
chp 35, more below.
> That is unfortunate notation - replace phi_m by A_m everywhere to make
> it more transparent. He's simply saying the change in calibration is a
> linear, homogeneous expression in the coordinate differentials under
> an infinitesimal displacement. This is an assumption, not a
> consequence. By no means is A_m (or phi_m in Weyl's notation) a
> gradient. (If it devolves to a gradient, Weyl geometry reduces to
> Riemannian geometry.)
>
> Also note that Weyl has a footnote where he clarifies his notation.
> -drl
Thank you. In your article, eq.(23) clarifies that much
better.
Weyl, in his SPACE TIME MATTER preface, page "v",
claims his approach, employed in chp 35, using gauge
invariance, does not connect EM potentials A_u with
gravitational potentials g_uv, but rather to the wave
field.
Would you agree with his assessment?
About your paper, I noted, following eq.(26) you've
determined a non-zero covariant derivative for the
4D metric, i.e. g_uv;w =/=0, (that's gutsy), for
example, one cannot do arbitrary associations like,
X_u = g_uv X^v
if g_uv;w =/=0. (Also, how does that affect the
covariant derivative of the Kronecker delta?).
I noted you did use a 4D association going from
eq.(39) to (40), is that what you meant to do?
I've discussed g_uv;w =/=0 with a mathematician
and he tell's me that's ok, but I could never
make them work, as they violate the Principle
of Equivalence.
Recall that the PoE allows for a CS where
g_uv;w = 0, and I certainly do understand you are
lifting the requirement of PoE in the presence of
EM fields such as A_u in your article.
Incidentally, as you use 6D does the covariant
derivative of the 6D metric vanish?
Regards
Ken S. Tucker
Danny Ross Lunsford
dyna...@vianet.on.ca (Ken S. Tucker) wrote
> Weyl, in his SPACE TIME MATTER preface, page "v",
> claims his approach, employed in chp 35, using gauge
> invariance, does not connect EM potentials A_u with
> gravitational potentials g_uv, but rather to the wave
> field.
> Would you agree with his assessment?
Of course, partially - the standard model works as phenomenology, so
whatever A is, it has to have a footprint in that world. The problem
is to show how A (and its Yang-Mills analogues) can exist in both
contexts, as geometry and as a field operator. All that is really
needed at this point is to generalize the idea of conformal weight so
that a non-Abelian "weight group" is allowed. The variational
principle can be taken over directly without any modification.
> About your paper, I noted, following eq.(26) you've
> determined a non-zero covariant derivative for the
> 4D metric, i.e. g_uv;w =/=0, (that's gutsy), for
> example, one cannot do arbitrary associations like,
>
> X_u = g_uv X^v
>
> if g_uv;w =/=0. (Also, how does that affect the
> covariant derivative of the Kronecker delta?).
That's just how it goes in Weyl space - intuitively, going from place
to place, both the direction cosines and the volume element
represented by g_mn are changing - in Riemannian geometry there is a
kind of reducibility of these two. This gets lifted by the free choice
of calibration. Stated another way, in Riemannian geometry, metricity
defined by parallel transport is only "partially" local (length is
integrable but direction is not), while in Weyl geometry it is
"purely" local (neither is generally integrable).
> I noted you did use a 4D association going from
> eq.(39) to (40), is that what you meant to do?
(39) and (40) are generally true regardless of dimension - pulling the
contravariant metric under ";" on the left is compensated via (27) by
adding the corresponding terms with A on the right.
> I've discussed g_uv;w =/=0 with a mathematician
> and he tell's me that's ok, but I could never
> make them work, as they violate the Principle
> of Equivalence.
> Recall that the PoE allows for a CS where
> g_uv;w = 0, and I certainly do understand you are
> lifting the requirement of PoE in the presence of
> EM fields such as A_u in your article.
In this theory the matter emerges directly from the 6d vacuum and is
not posited on the RHS of G_mn, so strictly speaking there *is* no
Principle of Equivalence until one has made the approximation leading
to Einstein-Maxwell. It is very interesting that the emergence of the
PoE is directly associated with the CC issue in the sense of this
approximation.
> Incidentally, as you use 6D does the covariant
> derivative of the 6D metric vanish?
The *conformal* covariant derivative vanishes in any dimension -
again, in Weyl space, one has the general rule
g_mn;a = -g_mn Aa g^mn;a = g^mn Aa
Conformally these are expressed as
Da g_mn = (;a + Aa) g_mn = 0
Da g^mn = (;a - Aa) g^mn = 0
that is, g_mn is weight 1 and g^mn is weight -1.
-drl
Danny Ross Lunsford
dyna...@vianet.on.ca (Ken S. Tucker) wrote in message
> if g_uv;w =/=0. (Also, how does that affect the
> covariant derivative of the Kronecker delta?).
This is a good question. Symbolically, it's very simple:
Da(g_mn g^np) = g^np Da g_mn + g_mn Da g^np = 0
Since the delta is of weight 0, Da reduces to da and so the simple
covariant derivative of the delta vanishes.
-drl
Ken S. Tucker
antima...@yahoo.com (Danny Ross Lunsford) wrote in message news:<2b93dd16.04102...@posting.google.com>...
> dyna...@vianet.on.ca (Ken S. Tucker) wrote
>
> > Weyl, in his SPACE TIME MATTER preface, page "v",
> > claims his approach, employed in chp 35, using gauge
> > invariance, does not connect EM potentials A_u with
> > gravitational potentials g_uv, but rather to the wave
> > field.
> > Would you agree with his assessment?
>
> Of course, partially - the standard model works as phenomenology, so
> whatever A is, it has to have a footprint in that world. The problem
> is to show how A (and its Yang-Mills analogues) can exist in both
> contexts, as geometry and as a field operator. All that is really
> needed at this point is to generalize the idea of conformal weight so
> that a non-Abelian "weight group" is allowed. The variational
> principle can be taken over directly without any modification.
Yes, but we need to understand that we are examining
the effect of "voltage" on a field. Subsequent to Weyl's
initial strike circa 1918, voltage was subsumed by more
complicated relations, as Weyl pointed out, i.e wave
equations.
However, as you say, the "weight group", if I understand
correctly, may stand in for the "wave equations", if that's
what you mean, that's an exciting thought.
> > About your paper, I noted, following eq.(26) you've
> > determined a non-zero covariant derivative for the
> > 4D metric, i.e. g_uv;w =/=0, (that's gutsy), for
> > example, one cannot do arbitrary associations like,
> >
> > X_u = g_uv X^v
> >
> > if g_uv;w =/=0. (Also, how does that affect the
> > covariant derivative of the Kronecker delta?).
>
> That's just how it goes in Weyl space - intuitively, going from place
> to place, both the direction cosines and the volume element
> represented by g_mn are changing - in Riemannian geometry there is a
> kind of reducibility of these two. This gets lifted by the free choice
> of calibration. Stated another way, in Riemannian geometry, metricity
> defined by parallel transport is only "partially" local (length is
> integrable but direction is not), while in Weyl geometry it is
> "purely" local (neither is generally integrable).
We have a great body of physics interconnected using
association in 4D, i.e. g_uv;w=0. I'm concerned that
body of effort will be rendered obsolete by g_uv;w=/=0.
Must that be done - in your opinion - to achieve a
unified field theory?
Speaking of weights, do you have a definition of
the covariant derivative of the determinant of
the covariant second rank metric, g = |g_uv|, ie,
g,w or g;w ?
Normally the covariant (g;w=0) and set's scale,
but I'm guessing you would have an addition there.
Regards
Ken S. Tucker
> -drl
Ken S. Tucker
antima...@yahoo.com (Danny Ross Lunsford) wrote in message news:<2b93dd16.04102...@posting.google.com>...
Yah, need something to zero in a geometry!
Very nice the way you did it, also about dealing
with the the weights |g_mn| = -|g^mn|, also neat.
I'm still concerned about losing 4D association,
but I presume you'll address that issue later.
Let's presume your 6D theory is totally consistent
mathematically, but I am an electronics technician,
who understands 4D voltage, (1 electrostatic, and
3 motion dependant, aka magnetic potentials), and
are curious about the other 2 dimensions assigned to
the 6D potential you call "A_m". Can you physically
justify the creation of those extra 2 dimensions,
in a measureable way. I was thinking about gravity,
but GR seems to subsume those potentials in the
electrical energy density. How can we physically
imagine/measure your 6D mathematics from a physical
stand-point?
Regards
Ken S. Tucker
Danny Ross Lunsford
dyna...@vianet.on.ca (Ken S. Tucker) wrote
> Let's presume your 6D theory is totally consistent
> mathematically, but I am an electronics technician,
> who understands 4D voltage, (1 electrostatic, and
> 3 motion dependant, aka magnetic potentials), and
> are curious about the other 2 dimensions assigned to
> the 6D potential you call "A_m". Can you physically
> justify the creation of those extra 2 dimensions,
> in a measureable way. I was thinking about gravity,
> but GR seems to subsume those potentials in the
> electrical energy density. How can we physically
> imagine/measure your 6D mathematics from a physical
> stand-point?
Once the approximation that leads to the Einstein-Maxwell action is
made, everything goes as before, only one is actually dealing with a
charge distribution in the true sense of the inhomogeneous Maxwell
equations, rather than something like {sum_over_e} e_k delta(x_k). One
can recover deltas by Fourier analysis if needed. That there are 2 new
dimensions simply corresponds to the fact that there is antimatter. If
one had a large bulk of antimatter and allowed it to interact with
matter in a less-than-catastrophic way - say, by placing it in a thin
gas of matter - then with ordinary EM we're stuck with an unsolvable
*classical* problem, to explain the copious radiation. If we didn't
already know about antimatter, then this theory could predict it, even
before quantization.
In another sense, the thing that needs to be demonstrated in any
physical theory is sufficiency to task and lack of ambiguous features.
That we see the world as the interaction of matter and fields in 4D,
is simply a consequence of the 40 orders of magnitude difference in
relative strength of g vs. A.
In a third sense, the mathematical structures that emerge are just
very interesting! So for example, to recover a *4D* vacuum, one ends
up with spacetime attached to a complex plane at every point. The
extra components in A now satisfy the Cauchy-Riemann equations. Such a
thing is IMO intrinsically interesting as a structure rich enough to
(one hopes) answer some real questions.
-drl
Ken S. Tucker
>
> > Let's presume your 6D theory is totally consistent
> > mathematically, but I am an electronics technician,
> > who understands 4D voltage, (1 electrostatic, and
> > 3 motion dependant, aka magnetic potentials), and
> > are curious about the other 2 dimensions assigned to
> > the 6D potential you call "A_m". Can you physically
> > justify the creation of those extra 2 dimensions,
> > in a measureable way. I was thinking about gravity,
> > but GR seems to subsume those potentials in the
> > electrical energy density. How can we physically
> > imagine/measure your 6D mathematics from a physical
> > stand-point?
>
> Once the approximation that leads to the Einstein-Maxwell action is
> made,
Just a formality, but action is an invariant, usually.
How would one approximate an invariant?
>everything goes as before, only one is actually dealing with a
> charge distribution in the true sense of the inhomogeneous Maxwell
> equations, rather than something like {sum_over_e} e_k delta(x_k). One
> can recover deltas by Fourier analysis if needed.
Yes, if I'm not mistaken those delta's may either be
the probabilty field of the classical charge location
or it may, in another context be the electric field of
said charge, that is sometimes described by virtual
particles(?).
>That there are 2 new
> dimensions simply corresponds to the fact that there is antimatter.
There's nothing "simply" here,(for me). If I understand you correctly,
you have generated a 6D field to account for antimatter, is
that what you meant?
Danny Ross Lunsford
> > made,
>
> Just a formality, but action is an invariant, usually.
> How would one approximate an invariant?
Well an invariant doesn't mean a constant!
The variational principle is based on a product of two coordinate
scalars and the volume element sqrt(det(g), that is a calibration
invariant. Now suppose R and W differ but little from some constant
values:
R = R0 + e R1 + O(2)...
W = W0 + e W1 + O(2)...
RW = R0W0 + e (W0 R1 + R0 W1) + O(2)...
so the variation reduces to the sum of variations of the first order
parts. So the approximation amounts to discarding higher order terms,
as usual.
> >everything goes as before, only one is actually dealing with a
> > charge distribution in the true sense of the inhomogeneous Maxwell
> > equations, rather than something like {sum_over_e} e_k delta(x_k). One
> > can recover deltas by Fourier analysis if needed.
>
> Yes, if I'm not mistaken those delta's may either be
> the probabilty field of the classical charge location
> or it may, in another context be the electric field of
> said charge, that is sometimes described by virtual
> particles(?).
Could you rephrase this question?
> >That there are 2 new
> > dimensions simply corresponds to the fact that there is antimatter.
>
> There's nothing "simply" here,(for me). If I understand you correctly,
> you have generated a 6D field to account for antimatter, is
> that what you meant?
Well this idea works for electrodynamics alone with one extra timelike
dimension, that is
div E = d/dx5 S4
(curl B - dE/dt)k = d/dx5 Sk etc.
where
Fm5 = Sm
You also get
T(Maxwell)mn,n = Fmn (d/dx5 Sn)
However there is no way to make this into a field theory for A and g
together because sqrt(det(g)) has non-integer weight. To get g in you
must go to six, and the current gets almost duplicated but not quite -
the EM laws in the limit are now
div E = (d/dx5 S4 + d/dx6 T4)
(curl B - dE/dt)k = (d/dx5 Sk + d/dx6 Tk)
T(Maxwell)mn,n = Fmn (d/dx5 Sn + d/dx6 Tn)
The current is still conserved but only the sum! So in the limit we
get inhomogeneous EM with creation/annihilation accounted.
-drl
Ken S. Tucker
> dyna...@vianet.on.ca (Ken S. Tucker) wrote
Studied your post, and your theory, are your
6D axes orthogonal, or do they require non-
orthogonality?
Incidentally, 6D is very new to me, so my
interpretations below are likely going to
sound dumb to you, but I'll risk it, be kind.
> > > Once the approximation that leads to the Einstein-Maxwell action is
> > > made,
> >
> > Just a formality, but action is an invariant, usually.
> > How would one approximate an invariant?
>
> Well an invariant doesn't mean a constant!
OOp's understood, I tangented...
> The variational principle is based on a product of two coordinate
> scalars and the volume element sqrt(det(g), that is a calibration
> invariant. Now suppose R and W differ but little from some constant
> values:
>
> R = R0 + e R1 + O(2)...
> W = W0 + e W1 + O(2)...
>
> RW = R0W0 + e (W0 R1 + R0 W1) + O(2)...
>
> so the variation reduces to the sum of variations of the first order
> parts. So the approximation amounts to discarding higher order terms,
> as usual.
Ok, what threw me was the thought that "action"
may not be invariant in 6D, i.e. possibly relative.
I'll presume action is invariant in 6D.
> > >everything goes as before, only one is actually dealing with a
> > > charge distribution in the true sense of the inhomogeneous Maxwell
> > > equations, rather than something like {sum_over_e} e_k delta(x_k). One
> > > can recover deltas by Fourier analysis if needed.
> >
> > Yes, if I'm not mistaken those delta's may either be
> > the probabilty field of the classical charge location
> > or it may, in another context be the electric field of
> > said charge, that is sometimes described by virtual
> > particles(?).
>
> Could you rephrase this question?
Well sir, it wasn't really a question. Only a
matter of my *semantic* interpretation, which
is crude at best, so the "(?)" was meant to mean
I didn't intend a declaration.
However if you want, I'll expand on what I wrote
in that paragraph.
> > >That there are 2 new
> > > dimensions simply corresponds to the fact that there is antimatter.
> >
> > There's nothing "simply" here,(for me). If I understand you correctly,
> > you have generated a 6D field to account for antimatter, is
> > that what you meant?
>
> Well this idea works for electrodynamics alone with one extra timelike
> dimension, that is
>
> div E = d/dx5 S4
> (curl B - dE/dt)k = d/dx5 Sk etc.
>
> where
>
> Fm5 = Sm
I think that parallels Kaluza's 5D, is that
where some of your ideas come from?
> You also get
>
> T(Maxwell)mn,n = Fmn (d/dx5 Sn)
If I read that correctly, "dx5" is invariant?
> However there is no way to make this into a field theory for A and g
> together because sqrt(det(g)) has non-integer weight.
Ok, you don't need to respond to this comment,
but I have no major problem with non-integer
weight if you allow non-integer dimensionality.
>To get g in you
> must go to six, and the current gets almost duplicated but not quite -
> the EM laws in the limit are now
>
> div E = (d/dx5 S4 + d/dx6 T4)
I think I see what you're saying, the last term
in the () bracket is a graviational component?
> (curl B - dE/dt)k = (d/dx5 Sk + d/dx6 Tk)
>
> T(Maxwell)mn,n = Fmn (d/dx5 Sn + d/dx6 Tn)
>
> The current is still conserved but only the sum! So in the limit we
> get inhomogeneous EM with creation/annihilation accounted.
Again allow my crude interpretation, but in your last
equation, it appears that's a description of photon
exchange (albiet in terms of Maxwell) is balanced.
And again please cut me a bit of slack here, in
your dimensions x5 and x6 we have a "virtual"
equation between matter in x6 (v<c) and photons in
x5 (v=c).
> -drl
Incidentally, I employed a 3D metric
g_uv = g*g_uv + A_u B_v
g = |g_uv| , A_u and B_v are EM potentials,
and showed how EM, GR and Strong Nuclear
force result. See,
Subject: Introduction to Unitivity. (kst)
Newsgroups: sci.physics
Date: 2004-10-17 17:13:59 PST
if you want. My approach may be helpful.
Regards
Ken S. Tucker
Danny Ross Lunsford
> Studied your post, and your theory, are your
> 6D axes orthogonal, or do they require non-
> orthogonality?
>
> Incidentally, 6D is very new to me, so my
> interpretations below are likely going to
> sound dumb to you, but I'll risk it, be kind.
There are 6 dimensions - how you set up coordinates is your business,
that is, one assumes 6d general covariance - and calibration
invariance.
> Ok, what threw me was the thought that "action"
> may not be invariant in 6D, i.e. possibly relative.
> I'll presume action is invariant in 6D.
The action is an "absolute" invariant, that is, the scalar R W is
weight -3. This is the whole point of Weyl's approach, an absolute
invariant leads to calibration invariant field equations.
> > > Yes, if I'm not mistaken those delta's may either be
> > > the probabilty field of the classical charge location
> > > or it may, in another context be the electric field of
> > > said charge, that is sometimes described by virtual
> > > particles(?).
> >
> > Could you rephrase this question?
>
> Well sir, it wasn't really a question. Only a
> matter of my *semantic* interpretation, which
> is crude at best, so the "(?)" was meant to mean
> I didn't intend a declaration.
> However if you want, I'll expand on what I wrote
> in that paragraph.
Yes.
> > Well this idea works for electrodynamics alone with one extra timelike
> > dimension, that is
> >
> > div E = d/dx5 S4
> > (curl B - dE/dt)k = d/dx5 Sk etc.
> >
> > where
> >
> > Fm5 = Sm
>
> I think that parallels Kaluza's 5D, is that
> where some of your ideas come from?
No. This is emphatically *not* a Kaluza approach - recall that in KK
theory the extra components in the metric are interpreted as related
to the gauge fields. Here, the metric in 6d has a self-standing role
and the gauge field Am has also a self-standing, independent role.
Also, there is no assumption at all about the global topology, nor is
there any restriction on the extra dimensions (curled up-ness) other
than the signature they carry.
> > You also get
> >
> > T(Maxwell)mn,n = Fmn (d/dx5 Sn)
>
> If I read that correctly, "dx5" is invariant?
The above is a statement in spacetime - so d/dx5or6 does not change
the Lorentz invariant property of whatever it acts on. Sn of course is
Fn5 (n=1..4), so in the 6d context it's part of a bivector, but from
the spacetime perspective it's a Lorentz vector - and so also is d/dx5
Sn.
> > However there is no way to make this into a field theory for A and g
> > together because sqrt(det(g)) has non-integer weight.
>
> Ok, you don't need to respond to this comment,
> but I have no major problem with non-integer
> weight if you allow non-integer dimensionality.
Non-integer? Don't understand this..
> >To get g in you
> > must go to six, and the current gets almost duplicated but not quite -
> > the EM laws in the limit are now
> >
> > div E = (d/dx5 S4 + d/dx6 T4)
>
> I think I see what you're saying, the last term
> in the () bracket is a graviational component?
No! this statement is effectively flat-space, or in an infinitesimal
region. In the paper, the first part concerns a model for
inhomogeneous electrodynamics as a projection of a 6d free EM field
onto spacetime, before gravity is added via the Weyl ansatz. The main
point of the current approach is that one can have a limit of
vanishing gravity in which *inhomogeneous* electrodynamics in
effectively flat space persists - this is in contrast to the Weyl
approach in 4d alone, where, even had it been possible to derive
equations for Rmn of the right order that were essentially coupled to
Maxwell-like equations for A[m,n], there would only be a limit to
*free* electrodynamics.
> > (curl B - dE/dt)k = (d/dx5 Sk + d/dx6 Tk)
> >
> > T(Maxwell)mn,n = Fmn (d/dx5 Sn + d/dx6 Tn)
> >
> > The current is still conserved but only the sum! So in the limit we
> > get inhomogeneous EM with creation/annihilation accounted.
>
> Again allow my crude interpretation, but in your last
> equation, it appears that's a description of photon
> exchange (albiet in terms of Maxwell) is balanced.
> And again please cut me a bit of slack here, in
> your dimensions x5 and x6 we have a "virtual"
> equation between matter in x6 (v<c) and photons in
> x5 (v=c).
Ontologically the 6d space is a vacuum! Only the splitting into 4+2
reveals interacting matter and field in spacetime.
> Incidentally, I employed a 3D metric
>
> g_uv = g*g_uv + A_u B_v
>
> g = |g_uv| , A_u and B_v are EM potentials,
>
> and showed how EM, GR and Strong Nuclear
> force result. See,
>
> Subject: Introduction to Unitivity. (kst)
> Newsgroups: sci.physics
> Date: 2004-10-17 17:13:59 PST
>
> if you want. My approach may be helpful.
>
> Regards
> Ken S. Tucker
I will look, but again, I feel it is necessary for a reasonable model
of gravity and light together to treat gauge and metric fields as
"separate but equal" in the sense that, in the passage from
Gauss-Ampere to Maxwell to Einstein, E and B are "separate but equal"
and together form a single whole. Here, the "single whole" is the
combination of g and A which have transformation properties under both
coordinate and calibration changes.
Oz
>In a third sense, the mathematical structures that emerge are just
>very interesting! So for example, to recover a *4D* vacuum, one ends
>up with spacetime attached to a complex plane at every point. The
>extra components in A now satisfy the Cauchy-Riemann equations. Such a
>thing is IMO intrinsically interesting as a structure rich enough to
>(one hopes) answer some real questions.
Is there any chance of a description for dummies here?
I am imagining, at every point in spacetime, the following:
3 spatial dimensions
1 time dimension
2 dimensions configured as one complex number.
You say this formulation specifically embodies the distinction between
matter and antimatter. Since your complex dimensions will allow a
rotation either clockwise or anticlockwise, is this how you distinguish
matter and antimatter?
That would be very intuitive, one can imagine that reversing the time
direction could reverse the direction and thus switch one to another. A
stationary particle would be a helix in your complex dimensions about
the time direction.
How does light work in this scenario? Does it spiral round a spatial
direction, or about the periphery of a light cone?
I am unclear why the two extra dimensions need to be complex.
It is an expression of a circular dimension, rather than an infinite
one? If so, what sets the size of that dimension? I ask this because
aesthetically I would like a long wavelength particle to live in a large
circular dimension, and a small one in a small one.
--
Oz
This post is worth absolutely nothing and is probably fallacious.
Use o...@farmeroz.port995.com [ozac...@despammed.com functions].
BTOPENWORLD address has ceased. DEMON address has ceased.
Ken S. Tucker
> There are 6 dimensions - how you set up coordinates is your business,
> that is, one assumes 6d general covariance - and calibration
> invariance.
...
> weight -3. This is the whole point of Weyl's approach, an absolute
> invariant leads to calibration invariant field equations.
...
> > > > Yes, if I'm not mistaken those delta's may either be
> > > > the probabilty field of the classical charge location
> > > > or it may, in another context be the electric field of
> > > > said charge, that is sometimes described by virtual
> > > > particles(?).
> > >
> > > Could you rephrase this question?
> >
> > Well sir, it wasn't really a question. Only a
> > matter of my *semantic* interpretation, which
> > is crude at best, so the "(?)" was meant to mean
> > I didn't intend a declaration.
> > However if you want, I'll expand on what I wrote
> > in that paragraph.
>
> Yes.
Well here's my understanding, let's begin with the
classical notion that an electron is a particle with
a *definite* location. We've modified that notion, by
inventing *wave mechanics* to one where the location of
a particle is described by a probability field as a
function over volume.
We add to that concept the idea that virtual photons
describe the potential (sometimes electric field is used
instead), and these virtual photons have an energy and
frequency corresponding to the electric potential of
the sample electron particle, that too is a function
over volume.
So I think the "wave mechanical probability field"
and the "virtual photon potential" are pretty much
the same thing.
To spur a bit of controversy, I also think the GR
energy density T_uv, is indistinguishable from the above
concepts. The reason I think that is because the units
are pretty much the same, so the differences result from
different perspectives.
> > > Well this idea works for electrodynamics alone with one extra timelike
> > > dimension, that is
> > >
> > > div E = d/dx5 S4
> > > (curl B - dE/dt)k = d/dx5 Sk etc.
> > >
> > > where
> > >
> > > Fm5 = Sm
> >
> > I think that parallels Kaluza's 5D, is that
> > where some of your ideas come from?
>
> No. This is emphatically *not* a Kaluza approach - recall that in KK
> theory the extra components in the metric are interpreted as related
> to the gauge fields. Here, the metric in 6d has a self-standing role
> and the gauge field Am has also a self-standing, independent role.
> Also, there is no assumption at all about the global topology, nor is
> there any restriction on the extra dimensions (curled up-ness) other
> than the signature they carry.
OK, that's clear, (curled up-ness, need to quote that :).
> > > You also get
> > >
> > > T(Maxwell)mn,n = Fmn (d/dx5 Sn)
> >
> > If I read that correctly, "dx5" is invariant?
>
> The above is a statement in spacetime - so d/dx5or6 does not change
> the Lorentz invariant property of whatever it acts on. Sn of course is
> Fn5 (n=1..4), so in the 6d context it's part of a bivector, but from
> the spacetime perspective it's a Lorentz vector - and so also is d/dx5
> Sn.
> > > However there is no way to make this into a field theory for A and g
> > > together because sqrt(det(g)) has non-integer weight.
> >
> > Ok, you don't need to respond to this comment,
> > but I have no major problem with non-integer
> > weight if you allow non-integer dimensionality.
>
> Non-integer? Don't understand this..
Well you know from SR as you move an object faster
and faster it's length contracts and it's time
slows down, and at "c" these dimensions vanish.
So in the intermediary, between v=0 and v=c,
can we say definitely that we have a fixed integer
dimensionality?
To put that on a more rational mathematical foundation
one can integrate, (no integration constants)
$ x dx = (1/2) x^2 (area)
$$ x dx dx = (1/6) x^3 (volume)
which shows how integration generates dimensions.
But now recall the Gamma function is continuous,
so write integration in a form,
$...$ 0 dx = kx^n/Gamma(n), (needs $ 0 dx = k)
n
please notice "n" is continuous. For example, you can
integrate using n = 2.5 , etc.
(likewise inter-dimensional differentiation follows).
If you want there is physical evidence of this effect,
by examining the perihelion shift.
> > >To get g in you
> > > must go to six, and the current gets almost duplicated but not quite -
> > > the EM laws in the limit are now
> > >
> > > div E = (d/dx5 S4 + d/dx6 T4)
> >
> > I think I see what you're saying, the last term
> > in the () bracket is a graviational component?
>
> No! this statement is effectively flat-space, or in an infinitesimal
> region. In the paper, the first part concerns a model for
> inhomogeneous electrodynamics as a projection of a 6d free EM field
> onto spacetime, before gravity is added via the Weyl ansatz. The main
> point of the current approach is that one can have a limit of
> vanishing gravity in which *inhomogeneous* electrodynamics in
> effectively flat space persists
GR is quite clear, G_uv = T_uv is where I'll begin.
If I understand you correctly, you state
"vanishing gravity" (implying G_uv =>0) and
"*inhomogeneous* electrodynamics"
(implying T(Maxwell)_uv =/=0
are compatible?
Yes, I'm trying to make the simplest possible metric
consistent with a nonorthogonal space, i.e.
g_uv = g*g_uv + A_u B_v
^ ^
calibration EM antisymmetry => A_u B_v = - A_v B_u
(Weyl=>gauge) (Einstein)
where det g = 1 - AB, (that's a bit crude, but close).
Regards
Ken S. Tucker
Danny Ross Lunsford
Oz <o...@farmeroz.port995.com> wrote
> I am imagining, at every point in spacetime, the following:
>
> 3 spatial dimensions
> 1 time dimension
> 2 dimensions configured as one complex number.
This is true in a certain limit, where the world separates into
spacetime and Matter.
> You say this formulation specifically embodies the distinction between
> matter and antimatter. Since your complex dimensions will allow a
> rotation either clockwise or anticlockwise, is this how you distinguish
> matter and antimatter?
Exactly - the distinction of matter vs. antimatter is purely
conventional. One can make a rotation in the 5-6 plane and change one
to the other. Because the "timespace" is 3D there is no
"forward-vs.backward in time" issue of order. There is simply Matter
with a capital M and that encompasses both anti- and koino-matter.
> I am unclear why the two extra dimensions need to be complex.
x5 and x6 (u and v) are real, but in a certain limit can be thought of
as a single complex value. In that limit and in flat space one has for
example a 6-d covariant gauge
div A + d/dt phi + d/du psi + d/dv chi = 0
that splits into 4+2 form
div A + d/t phi = 0
and
d/du psi + d/dv chi = 0
The condition that there be no creation/annihilation becomes
d/du chi - d/dv psi = 0
so (chi + i psi) can be thought of as an analytic funtion of (u + iv).
If we insist that the potential be bounded, then by Liouville's
theorem, chi and phi are constants. So the limit mentioned amounts to
a strict separation of Matter and 4-d spacetime. The 6-d vacuum
devolves to a 4-d one with light and no Matter.
-drl
Oz
>
>> I am imagining, at every point in spacetime, the following:
>>
>> 3 spatial dimensions
>> 1 time dimension
>> 2 dimensions configured as one complex number.
>
>This is true in a certain limit, where the world separates into
>spacetime and Matter.
OK, do I take it to mean that one *particular* slice/co-ordinate system
can give you this. There is then an implied statement that we should
observe mixing when viewed from a different frame.
>> You say this formulation specifically embodies the distinction between
>> matter and antimatter. Since your complex dimensions will allow a
>> rotation either clockwise or anticlockwise, is this how you distinguish
>> matter and antimatter?
>
>Exactly - the distinction of matter vs. antimatter is purely
>conventional. One can make a rotation in the 5-6 plane and change one
>to the other. Because the "timespace" is 3D there is no
>"forward-vs.backward in time" issue of order. There is simply Matter
>with a capital M and that encompasses both anti- and koino-matter.
Ok. So in effect a particle is (say) a clockwise spiral, and an
antiparticle an anticlockwise particle. Clearly reversing the time
direction will reverse the rotation so making the particles appear to
reverse direction. That's very neat.
Presumably there will be an energy in that rotation which I hope is
equivalent to the restmass. There should also be something that reflect
angular momentum (same thing, I guess). So producing a particle-
antiparticle pair conserves this momentum, IYSWIM. That's neat too.
As you say, it completely gets around the problem of 'two sorts of
time', that has niggled me for years. A time associated with a particle,
and a 'global time'. The helix sorts out the 'particle time' very neatly
(and obviously, dammit).
Oh, there's stuff that could just fall out in the wash, too....
Relativistic contraction will contract the spiral, presumably making the
particle look more massive. That should mean that the tightness of the
spiral is related to momentum (in the time direction). Er, that's not
quite right, but I expect you have it properly sorted. Anyway, there
should be a relationship between mass and some frequency.. now where
have I heard that before?
NB In the thread "The structure of electrons"
==========================
Alfred Einstead <whop...@csd.uwm.edu> writes
>At the semi-classical level, the Dirac fermion would be coiling
>in a helical worldline of about this radius (for electrons) at
>light speed.
>
>the picture that emerges is of a worldline with the following
>properties:
>
> (1) It's helical with mean free motion at a velocity
> v = p c^2/E (in a momentum eigenstate), parallel
> to p (opposite to p for negative energy states).
> (2) The velocity, counting the helical motion is just
> light speed.
> (3) The radius of the circular part of the motion is
> r = L (1 - (v/c)^2)
> where L is the Compton wavelength.
=============================
>> I am unclear why the two extra dimensions need to be complex.
>
>x5 and x6 (u and v) are real, but in a certain limit can be thought of
>as a single complex value.
OK. I'm happy to switch two real for one complex.
Of course I am not cognisant with any other implications this may have.
>In that limit and in flat space one has for
>example a 6-d covariant gauge
>
>div A + d/dt phi + d/du psi + d/dv chi = 0
<snip>
>The condition that there be no creation/annihilation becomes
>
>d/du chi - d/dv psi = 0
>
>so (chi + i psi) can be thought of as an analytic funtion of (u + iv).
>If we insist that the potential be bounded, then by Liouville's
>theorem, chi and phi are constants. So the limit mentioned amounts to
>a strict separation of Matter and 4-d spacetime. The 6-d vacuum
>devolves to a 4-d one with light and no Matter.
Obviously someone of my level doesn't follow this as he should. <cough>
However I would like to add two questions:
1) What are the implications of having a bounded potential and is the
size of the bound important?
2) One can rewrite (u + iv) as [n + v(1 + i)].
The implication seems to me that we have a real part and a circularly
rotating part. Does this mean anything?
Danny Ross Lunsford
Oz <o...@farmeroz.port995.com> wrote
> >> I am imagining, at every point in spacetime, the following:
> >>
> >> 3 spatial dimensions
> >> 1 time dimension
> >> 2 dimensions configured as one complex number.
> >
> >This is true in a certain limit, where the world separates into
> >spacetime and Matter.
>
> OK, do I take it to mean that one *particular* slice/co-ordinate system
> can give you this. There is then an implied statement that we should
> observe mixing when viewed from a different frame.
I am always impressed by your imagination :) This is true, but not in
the above context! In fact in flat-vacuo, one can start with no Matter
and free light, for which we have
div A + dt phi = 0 (Lorentz gauge)
d chi + d psi = 0
d psi - d chi = 0
(The former is a special 6-covariant gauge, akin to the "radiation"
gauge in 4-d
div A = 0
dt phi = 0 )
The 3rd equation expresses the absence of creation/annihilation (we
assumed no matter and no antimatter, that is, no Matter). We further
will now have
du S + dv T = 0
du T - dv S = 0
which express the absence of Matter (J = 0 and K = 0). So we are back
at 4-d empty space with free radiation.
> >> You say this formulation specifically embodies the distinction between
> >> matter and antimatter. Since your complex dimensions will allow a
> >> rotation either clockwise or anticlockwise, is this how you distinguish
> >> matter and antimatter?
> >
> >Exactly - the distinction of matter vs. antimatter is purely
> >conventional. One can make a rotation in the 5-6 plane and change one
> >to the other. Because the "timespace" is 3D there is no
> >"forward-vs.backward in time" issue of order. There is simply Matter
> >with a capital M and that encompasses both anti- and koino-matter.
>
> Ok. So in effect a particle is (say) a clockwise spiral, and an
> antiparticle an anticlockwise particle. Clearly reversing the time
> direction will reverse the rotation so making the particles appear to
> reverse direction. That's very neat.
This is a wonderful picture and might even be true, but I am not
making such assumptions. I'm simply saying that the extra potentials
in general mediate the presence of Matter, and that Matter is in a
sense associated with deviation from analyticity in the 5-6 (u-v)
plane of psi + i chi regarded as a complex function of u + iv. So, for
example, we might imagine poles in that plane as somehow corresponding
to Matter, with the sign of the residue giving matter vs. antimatter.
These things need a mathematician to sort out at some point.
> 1) What are the implications of having a bounded potential and is the
> size of the bound important?
Well this is just a way of ensuring a strictly local theory. We always
imagine we can draw a large bounding domain and that the potential
settles down "sufficiently fast" within it. This for example is
standard in both EM and GR (boundary conditions at infinity).
The most important points to keep in mind are
1) There is a strict, consistent definition of the vacuum (unlike
string theory) that is associated with a definite mathematical
criterion
2) The existing 4-d structures emerge completely intact and totally
naturally, in a suitable limit
3) The energy problem of GR is fixed
4) *Even in vacuo* in full curved 6-d, one has equations that look
like matter coupled to geometry, an in particular, this persists in
4-d. This is a remarkable circumstance and demands that someone take
on the characteristic theory of ultrahyperbolic equations.
-drl
strong_field
> So far, our inhouse theories can't make
> "frame dragging" work either,
> for similiar reasons, we'll need to wait and
> see what GP-b finds, and go from there.
Can you expand on this a little more? What kind of problems do you see
with frame dragging?
Thanks
Ken S. Tucker
Oh-oh, I was hoping it's because I'm stupid ):
Awhile ago I posted,
"Frame Dragging Gedanken Paradox?"
that's fairly low-brow.
Hope you'll be patient, as I reproduce it better.
In a laboratory of negligable gravity, let two
spheres "A" and "B" be measured by a balance
(ok a bit of gravity) to be have equal mass.
In "A" there is a rapidly spinning flywheel,
in "B" there is an equal amount of kinetic energy
due to temperature. The idea being, if each sphere
was cooled to absolute zero, and the flywheel
has no centrifugal force, they would balance.
So each of those sphere's has a kinetic energy
increment, that maintains an equal mass using
a balance.
I hope that scenario is clear because the "frame
dragging" hypothesis suggests we may perform
gravitational experiments to distinguish the
nature of the kinetic energy within sphere "A"
and "B". Specifically, sphere "A" is conjected
to have a different effect on the spacetime field
than "B", because "A"'s kinetic energy is due to
rotation, as opposed to "B"'s kinetic energy
content being random.
That said, let's survey the spacetime field near
the sphere's using light-rays and measuring the
deflections, see if we can predict a difference.
In my analysis, I can't see how, (I might be
missing something though).
I reason that all sub-light particles in either
sphere will have an equal relative speed wrt the
light-rays i.e. the speed of light "c", hence the
*velocity* (speed and direction) of the particles
in the spheres is the same relative to the light-ray
and will make no difference in the deflection.
To be a bit more sophisticated, let's look at Einstein's
Law,
G_uv = k*Tuv
Filling in the T_uv, sphere "A" uses stress in the
flywheel imposed by centrifugal force, while sphere
"B" uses the pressures of kinetic energy. No where
can I find how the pressure or stress components
in T_uv affects the LHS G_uv that predicts the g-field,
and that's my first problem.
I can't see anyway for the "stress" or "pressure"
components to show up differentially in G_uv, from T_uv
that is asymmetrical. Frame dragging is asymmetrical,
because reversing the direction of rotation reverses
the effect as I understand it.
My next problem involves the "Kerr metric" and I'll
use Weinberg's "Grav and Cosmo" pg. 240, as the definition
of what I mean by Kerr metric, whereon is a definition
of the metric g_0i.
Also, for good measure see Einstein's, "Meaning of
Relativity", Eq. (117)... for an excellent read about
"frame dragging".
In each case, the're g_0i's are symmetrical, so
passing asymmetrical information concerning direction
of rotation threw the symmetric g_0i , requires a
"leap of faith" I'm unable to underwrite, and that's
consistent with my problem above using G_uv=k*Tuv.
Regards
Ken S. Tucker
Oz
>making such assumptions. I'm simply saying that the extra potentials
>in general mediate the presence of Matter, and that Matter is in a
>sense associated with deviation from analyticity in the 5-6 (u-v)
>plane of psi + i chi regarded as a complex function of u + iv.
Obviously you are way past my level of competence.
I just try and see what you are saying in simple physical terms.
However, begging your indulgence, if I could postulate further and
perhaps get some idea of what your model says, whilst being mindful that
you haven't examined it in the required depth.
1) You have a 6D model that seems to express mass as related to two
extra dimensions.
2) These dimensions can be seen (if you take the appropriate viewpoint)
as a complex plane attached to each point in 4D spacetime.
So my questions (which you probably cannot answer) would be:
1) What is the condition for one or both of your 'mass dimensions' to
become real? Of course I have an event horizon or a singularity in mind
here.
2) If you were asked to include EM in this model, would you expect this
to generate another two (or perhaps one) extra dimensions?
3) Your model implies some oscillating something attached to each point
in spacetime. One is inevitably drawn to associating this with the
compton wavelength for a massive particle. That is, a massive particle
could be seen as a particle moving in and out of spacetime by moving in
and out (or rather within) your two extra dimensions.
>So, for
>example, we might imagine poles in that plane as somehow corresponding
>to Matter, with the sign of the residue giving matter vs. antimatter.
>These things need a mathematician to sort out at some point.
I much regret that I can be of absolutely no help in this area.
strong_field
> spheres "A" and "B" be measured by a balance
> (ok a bit of gravity) to be have equal mass.
>
> In "A" there is a rapidly spinning flywheel,
> in "B" there is an equal amount of kinetic energy
> due to temperature. The idea being, if each sphere
> was cooled to absolute zero, and the flywheel
> has no centrifugal force, they would balance.
I didn't understand the experiment well enough yet but I found this
equation which gives the precession of a gyroscope inside a hollow
sphere
omega / OMEGA = 4GM/3c^2 R
M = Mass of the sphere
R = radius of the sphere
omega = rotation of the sphere
OMEGA = the precession of a gyroscope inside the sphere
What would be the equation for the other sphere in your experiment?
Danny Ross Lunsford
> >This is a wonderful picture and might even be true, but I am not
> >making such assumptions. I'm simply saying that the extra potentials
> >in general mediate the presence of Matter, and that Matter is in a
> >sense associated with deviation from analyticity in the 5-6 (u-v)
> >plane of psi + i chi regarded as a complex function of u + iv.
>
> Obviously you are way past my level of competence.
> I just try and see what you are saying in simple physical terms.
>
> However, begging your indulgence, if I could postulate further and
> perhaps get some idea of what your model says, whilst being mindful that
> you haven't examined it in the required depth.
>
> 1) You have a 6D model that seems to express mass as related to two
> extra dimensions.
Well, I avoid the term "mass" at this stage, because of its
implications, and stick with "Matter".
> 2) These dimensions can be seen (if you take the appropriate viewpoint)
> as a complex plane attached to each point in 4D spacetime.
Yes, but the wholeness of the 6-d conformal space should be kept in
mind.
> So my questions (which you probably cannot answer) would be:
>
> 1) What is the condition for one or both of your 'mass dimensions' to
> become real? Of course I have an event horizon or a singularity in mind
> here.
They are always real. In a certain context they may be regarded as a
single complex value.
> 2) If you were asked to include EM in this model, would you expect this
> to generate another two (or perhaps one) extra dimensions?
E.M. is alrealy in the model:
A_m -> E.M. -> "length" curvature
g_mn -> gravity -> "direction" curvature
Connection = 1/2 { g_an,m + g_am,n - g_mn,a } + 1/2 { g_an A_m + g_am
A_n - g_mn A_a }
By replacing replacing F_mn by a general Yang-Mills field, one can
bring in other gauge fields in an analogous way.
> 3) Your model implies some oscillating something attached to each point
> in spacetime. One is inevitably drawn to associating this with the
> compton wavelength for a massive particle. That is, a massive particle
> could be seen as a particle moving in and out of spacetime by moving in
> and out (or rather within) your two extra dimensions.
Well this is a nice image and right in a limited way - e.g. free plane
waves for F_mn in flat 6-space show up as propagating Matter waves in
spacetime. But the essential viewpoint here is "pre-particle".
Particle behavior will have to come from quantization. If you wanted
something close to a particle equation, there is
D_m D_m W = 0
where D_m is the conformal covariant derivative (d_m - 2 A_m) and W is
the electromagnetic scalar (weight -2). Perhaps the extra cross-terms
coming from A_m can be thought of as generating mass.
-drl
Danny Ross Lunsford
> > > Ok, you don't need to respond to this comment,
> > > but I have no major problem with non-integer
> > > weight if you allow non-integer dimensionality.
> >
> > Non-integer? Don't understand this..
>
> Well you know from SR as you move an object faster
> and faster it's length contracts and it's time
> slows down, and at "c" these dimensions vanish.
> So in the intermediary, between v=0 and v=c,
> can we say definitely that we have a fixed integer
> dimensionality?
>
> To put that on a more rational mathematical foundation
> one can integrate, (no integration constants)
>
> $ x dx = (1/2) x^2 (area)
>
> $$ x dx dx = (1/6) x^3 (volume)
>
> which shows how integration generates dimensions.
Well conformal "weight" is a concept that is independent of
dimensionality, that is, a covariant in a Weyl conformal space has
properties under both coordinate and scale changes. The idea of weight
embodies the latter.
> GR is quite clear, G_uv = T_uv is where I'll begin.
> If I understand you correctly, you state
> "vanishing gravity" (implying G_uv =>0) and
> "*inhomogeneous* electrodynamics"
>
> (implying T(Maxwell)_uv =/=0
>
> are compatible?
Yes! That is what is so surprising. Nowhere is a current posited -
there *is* no "RHS" here. In GR, as in Maxwell-Lorentz, one has a
field that is driven by a *posited* current (energy tensor, charge
current resp.), and then that field acts back on the current. Here,
there is *no* posited current at all, rather, the assumption of
strictly local "metricity" requires both the metric and calibration
field, and these *jointly* assume roles in a manner that appears as
symmetric Ricci driven by energy-momentum, and Maxwell driven by
charge-current! Of course they are really 6-d *vacuum* equations:
Rmn - (2R/W) Tmn + (1/2W) (DmDn + DnDm) W = 0 (Rmn = symmetric part
of CCT)
1/S d/dxm ( S R Fmn ) - 5/4 (Dn W) = 0 (S = sqrt det g)
The new physics would be found in the "pure geometry" terms 1/2W
{Dm,Dn} W and (Dn W), which are respectively, energy-momentum and
charge current. These are absent in flat space
> Yes, I'm trying to make the simplest possible metric
> consistent with a nonorthogonal space, i.e.
>
> g_uv = g*g_uv + A_u B_v
> ^ ^
> calibration EM antisymmetry => A_u B_v = - A_v B_u
> (Weyl=>gauge) (Einstein)
>
> where det g = 1 - AB, (that's a bit crude, but close).
>
> Regards
> Ken S. Tucker
Could you perhaps send a paper? Thanks.
-drl
Ken S. Tucker
You may want to double check me, but I think
the OMEGA and omega are reversed??
((I'm looking at Weinberg's "Grav and Cosmo" pg 241)).
> What would be the equation for the other sphere in your experiment?
Hey, that's a neat question. Sphere "B" consists
of a shell with equal but random Kinetic Energy,
(equal to the KE possessed by rotating sphere A,
relative to an inertial frame), like temperature
would cause magnitudinally. I'm very sure temperature
would have no effect on the gyro within the "hot"
sphere B, (due to homogeneous isotropy).
As always, great care must be used in GR, so my
mention of an inertial frame above to measure
angular energy is colloquial, because I can choose
a rotating CS afixed to the center of sphere A
and *transform* away relative rotation and angular
energy. However I cannot transform away the stress
in the rotating sphere A, that results from
centrifugal force.
That stress, is radial and *independant* of the
*direction* of rotation, and that's a problem for
gravitomagnetism.
IMO, that stress in Tuv is a *magnitude* and
does not distiguish *direction* of rotation, and
because General Covariance, we are allowed to use
that CS when measuring Tuv.
Let's have a look at the GR logic map, from the
"Tuv" source, through to the effects.
(A) and (B) are the different spheres.
These Effects on these
Tuv(A) => Guv(A) => guv(A) => photon | gyro (GP-b)
Tuv(B) => Guv(B) => guv(B) => photon | gyro
Source => Field => potential => action/result
Sphere(A)'s spin vector, must end up as an
asymmetrical effect on the gyro, that's the
problem, how to do it.
We have another problem with the Minkowski limit
the Kerr metric and the Brill-Cohen metric are
based on, that may yield CS artifacts.
Regards
Ken S. Tucker
Danny Ross Lunsford
> 1) You have a 6D model that seems to express mass as related to two
> extra dimensions.
Well, "Matter" = matter + antimatter. Mass comes later (Dirac eqn)
> 2) These dimensions can be seen (if you take the appropriate viewpoint)
> as a complex plane attached to each point in 4D spacetime.
Yes. In regions where there is no Matter, and assuming infinitely weak
gravity, then A6 - i A5 is analytic in (u+iv), that is
dA5/du + dA6/dv = 0
dA5/dv - dA6/du = 0
> So my questions (which you probably cannot answer) would be:
>
> 1) What is the condition for one or both of your 'mass dimensions' to
> become real? Of course I have an event horizon or a singularity in mind
> here.
Again, they are always individually real.
> 2) If you were asked to include EM in this model, would you expect this
> to generate another two (or perhaps one) extra dimensions?
EM is *already* in the model, coming from the calibration field A.
This is the whole point, to get a set of essentially coupled eqns for
g and A that are of the right order and that reduce to
Einstein-Maxwell in the appropriate limit.
> 3) Your model implies some oscillating something attached to each point
> in spacetime. One is inevitably drawn to associating this with the
> compton wavelength for a massive particle. That is, a massive particle
> could be seen as a particle moving in and out of spacetime by moving in
> and out (or rather within) your two extra dimensions.
Well, wave-like solutions will clearly play a large role
(ultrahyperbolic eqns) and this may be true in some context, but it
must come later, as there *are* no particles at this level, just
densities.
-drl
strong_field
> Strong...@hotmail.com (strong_field) wrote in message news:<e929f229.04112...@posting.google.com>...
> the OMEGA and omega are reversed??
> ((I'm looking at Weinberg's "Grav and Cosmo" pg 241)).
Yes. you are right, omega is the precession of the gyroscope, OMEGA is
the rotation of the sphere.
But I realize that in that equation we can set all terms (including
4/3 !) to unity and we would get an equation saying that the
precession of the gyroscope inside a rotating hollow sphere is
inversely proportional to the radius of the sphere.
Two problems with this: According to this equation changing the
direction of rotation does not effect the direction of precession? And
I don't see why the mass of the sphere should be in the equation. The
precession, by symmetry at least, should be independent of the mass of
the sphere.
I imagine that the gyroscope inside the sphere is aligned with the
rotation axis of the sphere, so there is really no wobble, but all we
could observe is the motion of the nodes. I don't know if this agrees
with your model.
Then I would think that there should a constant like "the permeability
of spacetime," in order to find the actual number of the precession.
Ken S. Tucker
> dyna...@vianet.on.ca (Ken S. Tucker) wrote
>
>
> > > > Ok, you don't need to respond to this comment,
> > > > but I have no major problem with non-integer
> > > > weight if you allow non-integer dimensionality.
> > >
> > > Non-integer? Don't understand this..
> >
> > Well you know from SR as you move an object faster
> > and faster it's length contracts and it's time
> > slows down, and at "c" these dimensions vanish.
> > So in the intermediary, between v=0 and v=c,
> > can we say definitely that we have a fixed integer
> > dimensionality?
> >
> > To put that on a more rational mathematical foundation
> > one can integrate, (no integration constants)
> >
> > $ x dx = (1/2) x^2 (area)
> >
> > $$ x dx dx = (1/6) x^3 (volume)
> >
> > which shows how integration generates dimensions.
>
> Well conformal "weight" is a concept that is independent of
> dimensionality, that is, a covariant in a Weyl conformal space has
> properties under both coordinate and scale changes. The idea of weight
> embodies the latter.
Thanks, if weight happens to be an integer, independant
of dimensionality, that's big thanks. I'm going
to check that out, but I think you are right.
You know, moving from field to quantum physics
where are those integers.
> > GR is quite clear, G_uv = T_uv is where I'll begin.
> > If I understand you correctly, you state
> > "vanishing gravity" (implying G_uv =>0) and
> > "*inhomogeneous* electrodynamics"
> >
> > (implying T(Maxwell)_uv =/=0
> >
> > are compatible?
>
> Yes! That is what is so surprising. Nowhere is a current posited -
> there *is* no "RHS" here. In GR, as in Maxwell-Lorentz, one has a
> field that is driven by a *posited* current (energy tensor, charge
> current resp.), and then that field acts back on the current. Here,
> there is *no* posited current at all, rather, the assumption of
> strictly local "metricity" requires both the metric and calibration
> field, and these *jointly* assume roles in a manner that appears as
> symmetric Ricci driven by energy-momentum, and Maxwell driven by
> charge-current! Of course they are really 6-d *vacuum* equations:
>
> Rmn - (2R/W) Tmn + (1/2W) (DmDn + DnDm) W = 0 (Rmn = symmetric part
> of CCT)
I have a problem, I can read that in the usual 4d terms,
but when you uprate to 6d are your "mn" over 4 or 6,
ah...your pushin the envelope so please tell us your
encryption.
> 1/S d/dxm ( S R Fmn ) - 5/4 (Dn W) = 0 (S = sqrt det g)
>
> The new physics would be found in the "pure geometry" terms 1/2W
> {Dm,Dn} W and (Dn W), which are respectively, energy-momentum and
> charge current. These are absent in flat space
"absent in flat space", that's what Einstein said
when he doubled his prediction for the deflection
of light. Would you be able to provide how your
equations reduce to GR in the the weak field, that
way giving us ameans to connect GRist's with the
physicality of your 6D.
> > Yes, I'm trying to make the simplest possible metric
> > consistent with a nonorthogonal space, i.e.
> >
> > g_uv = g*g_uv + A_u B_v
> > ^ ^
> > calibration EM antisymmetry => A_u B_v = - A_v B_u
> > (Weyl=>gauge) (Einstein)
> >
> > where det g = 1 - AB, (that's a bit crude, but close).
> >
> > Regards
> > Ken S. Tucker
>
> Could you perhaps send a paper? Thanks.
What paper?
ken
> -drl
antima...@yahoo.com
get a spacetime interpretation, you need to separate everying 1-4 and
5-6, analogous to separating a 4-vector into a 3-vector and a time
component. 1-4,5-6 amounts to "spacetime" and "Matter" components.
Recall that in flat space we had a 6-energy law
Tmn,n = 0 (m,n=1..6)
that looked like
T_mu_nu(Maxwell),_nu = F_mu_nu J_nu (_mu,_nu=1..4)
when separated. You can do similar with the other eqns.
> > 1/S d/dxm ( S R Fmn ) - 5/4 (Dn W) = 0 (S = sqrt det g)
> >
> > The new physics would be found in the "pure geometry" terms 1/2W
> > {Dm,Dn} W and (Dn W), which are respectively, energy-momentum and
> > charge current. These are absent in flat space
>
> "absent in flat space", that's what Einstein said
> when he doubled his prediction for the deflection
> of light. Would you be able to provide how your
> equations reduce to GR in the the weak field, that
> way giving us ameans to connect GRist's with the
> physicality of your 6D.
Again, I can reproduce all of standard GR because that is the low-order
limit.
-drl