On Apr 26, 2006, at 11:04 AM, Jack Sarfatti wrote:

On Apr 25, 2006, at 10:43 AM, Waldyr A. Rodrigues Jr. wrote:

1) You got the idea...In a de Sitter world you can exchange angular momentum and what appears to be "linear momentum", the Phi_mu, which is indeed angular momentum.

2) Take notice: the formula for the Phi_mu that I wrote is valid only instereographical coordinates. It is important to emphasize this point. In other coordinate systems you (of, course) will find different formulas.

On Apr 26, 2006, at 9:33 AM, RKiehn2352@aol.com wrote:

Hi

I hate to get embroiled in this dd -> 0 argument,

but I would like you both to know my opinion on a point, which I think is important.

Thanks Robert.

I think Waldyr is saying that d^2 = 0 always, consistent with it's non-metrical topological origin that you emphasize.

However, the metrical Pu from the 10 Killing vector field isotropies in deSitter O(4,1) have the form (non-rigorously heuristically here)

D = dx^uPu ~ topological pre-metrical Cartan d + metrical stuff proportional to /\zpf

/\zpf = 1/R^2 ~ dark energy density of our pocket universe

R is the 5D constant curvature parameter.

In Susskind's theory R^2/4Lp^2 is the Holographic Universe BIT content of our pocket universe confined to the observer-dependent de-Sitter horizon on the inflation bubble.

In any case I made the analogy of

D = dx^uPu ~ topological pre-metrical Cartan d + de Sitter metrical stuff proportional to /\zpf

to

D = d + A/\ in EM & Yang-Mills

&

D = d + W/\ in GR with W as the spin-connection

Now this has important ramifications.

The existence of positive dark energy density /\zpf > 0 on cosmic scale means we formally live in a 5D deSitter space-time. Since the dark energy density is small this closely approximates 4D space-time's Poincare group.

When we locally gauge all of the 10-parameter deSitter group O(4,1) we get NEW compensating gauge field potentials and this allows /\zpf to be a local scalar field rather than Einstein's cosmological constant from 1915 GR which is an approximation to what we are talking about now that includes torsion fields of course.

Then, it's a long story, I think we can locally amplify /\zpf "harness cosmic energy" on a small scale both positive and negative for propellantless geodesic propulsion (Alcubierre type) like we see in the "alien ET flying saucers."

This sounds crazy, but is it crazy enough to be true? (Bohr)

Some flaky URLS (disinfo also has some true info)

http://www.ancientx.com/nm/anmviewer.asp?a=65&z=1

http://www.indiadaily.com/editorial/tech_default.asp

On Apr 25, 2006, at 10:43 AM, Waldyr A. Rodrigues Jr. wrote:

1)You got the idea...In a de Sitter world you can exchange angular momentum and what appears to be "linear momentum", the Phi_mu, which is indeed angular momentum.

2)Take notice: the formula for the Phi_mu that I wrote is valid only instereographical coordinates. It is important to emphasize this point. In other coordinate systems you (of, course) will find different formulas.

-----Mensagem original-----

De: Jack Sarfatti [mailto:sarfatti@pacbell.net]

Enviada em: terĂ§a-feira, 25 de abril de 2006 14:26

Para: Waldyr A. Rodrigues Jr.

Assunto: Re: OK now I understand Waldyr's statement & ZPF anyonic propellantless propulsion

Pa = -i{[1 - (r^2 - t^2)/R^2]^-1&a + (x^bMab/R^2)]}

= -i{[1 - /\zpfs^2]^-1&a + /\zpf(x^bMab)]}

where

s^2 = r^2 - t^2

G = h = c = 1

/\zpf = 1/R^2 > 0 ~ dark vacuum energy density

take the weak field linear approximation

[1 - /\zpfs^2]^-1 ~ 1 + /\zpfs^2

-i&a is the MECHANICAL KINETIC MOMENTUM OPERATOR pa in quantum mechanics

The analogy is to EM

P = p + (e/c)A

P is the canonical momentum, (e/c)A is the EM field momentum, e.g.

Feynman Vol III

So in the linear approximation

Pa ~ (1 + /\zpfs^2)pa - i/\zpf(x^bMab)]

In NR quantum theory, the Hamiltonian for a neutral test particle of

mass m moving in this weak dark energy field is

H ~ pa^2/2m ~ (Pa - /\zpfs^2pa - i/\zpf(x^bMab))^2/2m

With curious NEW nonlinear couplings between the deSitter Lie algebra

"charges" (Killing isometries) translations and space-time rotations

that are mediated/catalysed by the dark energy field - suggesting

propellantless propulsion. Again this is a linear approximation for

temporary convenience.

This is analogous to a charge in an external EM field

H ~ (P - (e/c)A)^2/2m

Now if you confine the test particle to a nanolayer 2D film, then you

may get ANYON fractional quantum statistics in which /\zpf is

manipulated to act like the normal magnetic field.

Also one wants to do strong field case with large localized /\zpf

vacuum zero point energy density.

On Apr 24, 2006, at 5:52 PM, Jack Sarfatti wrote:

> Waldyr says that for the deSitter O(4,1) group

>

> Pa = -i{[1 - (r^2 - t^2)/R^2]^-1&a + (x^bMab/R^2)]}

>

> Therefore, lim Pa as R -> infinity is i&a.

>

> /\zpf = 1/R^2

>

> OK, so what I meant was

>

> D = dx^a(Pa/i)

>

> as the GENERALIZED d for O(4,1)

>

> This D^2 =/= 0 when /\zpf =/= 0

>

> Limit of D when /\zpf ---> 0 is the Cartan d

>

> Therefore, this D is analogous to a covariant exterior derivative

> with the extra stuff as a connection.

On Apr 26, 2006, at 9:33 AM, RKiehn2352@aol.com wrote:

Hi

I hate to get embroiled in this dd -> 0 argument,

but I would like you both to know my opinion on a point, which I think is important.

IMO the operator d is a differential concept, not a derivative concept.

*

My topology base is self -taught, and most of it - that which I truely understand - comes from point set topology.

I was impressed by Kuratowski, especially his ideas on a "Closure Operator = K" as being

something which would allow the construction of a topology, for then my imagination led

me to believe that the {Identity-union-the-exterior-differential-d} = K= {I + d}, appeared to satisfy the Kuratowski Closure axioms.

It was then exciting to comprehend that the exterior differential acting on p-forms created the limit sets of the p-form.

*

From early physics training, this fit in well, for from the postulate that dJ = 0, one can dervice the Maxwell-Ampere PDE's.

dG = J

which implies that

div D = rho.

Hence, as taught in EM 2.01, the D field lines orignate and end on charges.

These endpoints ( the charges) are the "limit points" of the field lines.

*

IN an abstract manner, it then became apparent to me that the exterior differential was a generalized limit point generator, indeed.

*

This idea was later justified by the development of Cartan Topological Structure for an arbitrary 1-form.

It is possible to construct (what I called) a Cartan topology form any 1-form of Action,

and in 4D, show that the exterior differential indeed generated the limit sets of all sets in the Cartan topology.

(See pdf attachment.)

*****************************************************************************

Before 1991, when the ideas of a Cartan topological structure was presented at Santa Barbara, and for several years afterwards,

I used the words exterior derivative, ( and even in the article math-ph/0101033),

but it slowly became apparent to me that this was a philosophical error that should be corrected. Since 2002 or so I have tried to be careful and use

the words "exterior differential" ( not exterior derivative) which do not explicity fix the definition of a limit.

***************************************************************************************************************************

The words exterior derivative imply that a limit process has been chosen such that

dx/dt = V.

I have called this topological constraint: "Kinematic Perfection"

Written as an exterior differential form, dx -V(x,y,z,t)dt = w,

if w = 0, the statement is in effect a severe topological constraint formulated by the differential system, dx -V(x,y,z,t)dt = 0 .

it would follow that

dw = -dV^dt

and

w^dw = 0^dw = 0,

which implies that the system is integrable (in the sense of Frobenius).

In fact, V =V(t) not V(x,y,z,t).

**

But if w is not zero, then

w^dw = dx^dV^dt,

and to be integrable, V must be a function of x and/or t alone.

For otherwise, dx^ (Vy dy^dt+Vz dz^dt) = w^dw is not zero and

there does not exist a unique integral equivalent. (Frobenius theorem)

*

HOWEVER IMO

The exterior differential is much more general than a derivative concept.

The exterior differential does not require Kinematic Perfection, and permits a

formulation of topological fluctuations about the guiding lines of Kinematic Perfection, which are possible

integrable solutions assuming NO topological fluctuations.

*

**

That is in 4D, for example,

dx - V(x,y,z,t) dt = a topological fluctuation 1-form = w = w(x,y,z,t,dx,dy,dz,dt)

**

Cartan was well aware of this idea, for he often would substitute the 1 - form (dx - V(x,y,z,t) dt) into expressions for dx, in order to prolong his original expressions.

For example, if

dx - U(x,y,z,t) dt =0,

dy - V(x,y,z,t) dt = 0,

dz - W(x,y,z,t) dt = 0,

was substituted into the expression for a 3D differential volume,

then

d(3Vol) = dx^dy^dz = Udt^Vdt^Wdt which is zero !!!

In other words, Kinematic perfection is not compatable with a differential volume element.

**

I now use the algebraic definitions of the exterior derivative and the Lie derivative as being the dominant ideas.

The limits of Kinematic perfection are treated as special sub cases,

that do not apply to problems of systems far from equilibrium,

interactions that involve cubic curvature (such as pressure and temperature),

and to processes that exhibit thermodynamic irreversibility.

**

By the Way, Just where are the cubic interaction curvatures in EGR?

regards

RMK

## Wednesday, April 26, 2006

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