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From: Jarek Duda on 29 Oct 2009 14:53
While working on ellipsoid field theory (there is real symmetric 4*4
matrix preferring some eigenvalues in each point) instead of choosing
Lagrangian density, I thought that it would be more physical to start
with energy density - which integral is invariant while time
evolution. There is a delicate difference between these two entities:
for example in electrodynamics Lagrangian density uses (E*E-B*B),
while energy density uses (E*E+B*B).

The interesting observation is that if we take energy density which
doesn't emphasize any (time) dimension - we choose it to make
evolution along this axis. Using this choice we can transform it into
Lagrangian density - it occurs that it automatically gets Lorentz
invariant form - for arbitrary chosen arbitrary time dimension.
For example in ellipsoid model energy density
1/2* \sum_abc (\partial_a M_bc)^2 + V(M)
gives Lagrangian density
1/2* (\sum_bc (\partial_0 M_bc)^2 - \sum_ibc (\partial_i M_bc)^2 ) - V
(M)
Details can be found in 5th section of
http://arxiv.org/abs/0910.2724
The interesting about this ellipsoid field is for example that
rotational modes of three dimensions gives a bit generalized
electrodynamics and their forth axis creates gravitational
interaction.

Have you heard about such approaches in which time dimension can be
chosen completely arbitrarily?
From: xxein on 29 Oct 2009 22:49
On Oct 29, 2:53 pm, Jarek Duda <duda...(a)gmail.com> wrote:
> While working on ellipsoid field theory (there is real symmetric 4*4
> matrix preferring some eigenvalues in each point) instead of choosing
> Lagrangian density, I thought that it would be more physical to start
> with energy density - which integral is invariant while time
> evolution. There is a delicate difference between these two entities:
> for example in electrodynamics Lagrangian density uses (E*E-B*B),
> while energy density uses (E*E+B*B).
>
> The interesting observation is that if we take energy density which
> doesn't emphasize any (time) dimension - we choose it to make
> evolution along this axis. Using this choice we can transform it into
> Lagrangian density - it occurs that it automatically gets Lorentz
> invariant form - for arbitrary chosen arbitrary time dimension.
> For example in ellipsoid model energy density
> 1/2* \sum_abc  (\partial_a M_bc)^2 + V(M)
> gives Lagrangian density
> 1/2* (\sum_bc (\partial_0 M_bc)^2 - \sum_ibc (\partial_i M_bc)^2 ) - V
> (M)
> Details can be found in 5th section ofhttp://arxiv.org/abs/0910.2724
> The interesting about this ellipsoid field is for example that
> rotational modes of three dimensions gives a bit generalized
> electrodynamics and their forth axis creates gravitational
> interaction.
>
> Have you heard about such approaches in which time dimension can be
> chosen completely arbitrarily?

xxein: Time? Considered from within a subsystem (frame) or from
without (god's eye)? Things (events) happen in sequence/consequence
(causal). Stick with that and hopefuly you'll find your way out.

Eigen values? I'm not sure what you mean but any math construct can
have them. We can call them irreducible factors or 'pretty numbers'
along with various other coincidences artificially incurred. But if
you mean something like 2M or 3M regarding c around black holes or
that a relation of circular orbit velocity divided by escape velocity
at an R for an M is sqrt(2), then you have something more solid to
work with. Hardly anyone realizes that they have to use the 'god's
eye view' to make these relations though.

Try to understand that if a math does not describe the physic, 'that
math' cannot create a desription of the physic. While EM, gravity and
expansion may be similar in many respects, why and how are they
different?

Why didn't the first atomic bomb chain react and destroy the Earth?
Why won't a creation of a mini BH in the LHC devour the Earth? What
is real? A mathematical elipsoid field theory?
From: Jarek Duda on 30 Oct 2009 04:26
The interesting about these symmetric energy density theories is that
they are kind of totally Lorentz invariant - not only boosts, but even
while choosing orthogonal time direction, still propagations will
travel at constant speed.
And there is still place for e.g. time dilatation in this model -
gravitational curvature changes fundamental constants like electron
charge, magnetic momentum and masses a bit - in first approximation it
causes everything to 'rescale', for example reducing distances and
making everything happen faster.

I don't agree that math cannot describe physics...
My original motivation to work on ellipsoid field is that the
structure and behavior of its topological excitations corresponds well
with these from particle physics - with three generations of spin 1/2
fermions as the basic excitations, which can 'oscillate' one into
another and further ones corresponding to leptons, mesons, baryons,
nucleons with expected mass gradation and decay modes.
These ellipsoids are kind of generalization of quantum phase into
three dimensions, for example to be able to create traveling twist-
like wave to transmit angular momentum.
In this model magnetic flux is quantified not only in superconductors,
but everywhere as Aharomov-Bohm effect suggests: for example lines in
animation on
http://en.wikipedia.org/wiki/Magnetic_reconnection
would be physical and have energy density per length - additional
tendency to shorten - it could be missed effect why "This process is
not well understood: once started, it proceeds many orders of
magnitude faster than predicted by standard models." (from description
of the animation)
From: eric gisse on 30 Oct 2009 03:54
Jarek Duda wrote:

> While working on ellipsoid field theory (there is real symmetric 4*4
> matrix preferring some eigenvalues in each point) instead of choosing
> Lagrangian density, I thought that it would be more physical to start
> with energy density - which integral is invariant while time
> evolution. There is a delicate difference between these two entities:
> for example in electrodynamics Lagrangian density uses (E*E-B*B),
> while energy density uses (E*E+B*B).

The Hamiltonian [energy] and Lagrangian formalisms are equivalent.

>
> The interesting observation is that if we take energy density which
> doesn't emphasize any (time) dimension - we choose it to make
> evolution along this axis.

Energy is the conserved quantity / Noether current associated with
invariance under displacement in time. Hard for energy to be time dependent
here.

> Using this choice we can transform it into
> Lagrangian density - it occurs that it automatically gets Lorentz
> invariant form - for arbitrary chosen arbitrary time dimension.

And thus you have discovered the point transformation between a Lagrangian
and a Hamiltonian, and their associated densities.

> For example in ellipsoid model energy density
> 1/2* \sum_abc (\partial_a M_bc)^2 + V(M)
> gives Lagrangian density
> 1/2* (\sum_bc (\partial_0 M_bc)^2 - \sum_ibc (\partial_i M_bc)^2 ) - V
> (M)
> Details can be found in 5th section of
> http://arxiv.org/abs/0910.2724
> The interesting about this ellipsoid field is for example that
> rotational modes of three dimensions gives a bit generalized
> electrodynamics and their forth axis creates gravitational
> interaction.
>
> Have you heard about such approaches in which time dimension can be
> chosen completely arbitrarily?

You chose nothing but a different way to express the same thing under
certain conditions.
From: Jarek Duda on 30 Oct 2009 05:28
Yes it's called Hamiltonian ... in quantum mechanics. In classical
field theories it's usually ignored.
The interesting thing about it is that for Lorentz invariant theories,
this energy density doesn't emphasize any direction - time for
evolution is chosen arbitrary!
Now the transformation into Lagrangian makes kind of Wick's rotation
of this coordinate.
It looks that our flowing time is practically only the result of
entropy gradient...
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