"The evolution of the electromagnetic field with time proceeds most simply when the (3D spacelike) metric does not change with the geodesically normal co-time ... No such simplicity reigns when the metric changes with time in a complicated way. Then the (EM field) normal modes themselves alter. Consequently the (warped Maxwell) dynamical equations acquire new terms ... 1) the emission of energy into the transverse (EM) field, or absorption of energy from the transverse field, by the Coulomb fields due to the motion of the charges or other changes in the metric 2) the pumping or scattering of energy into one transverse mode out of another or out of the metric itself, as a result of changes in the metric. It is true that one can define the local density of electromagnetic field energy in an unambiguous way. However, the same is not true of the integrated field energy. There is no common set of space and time coordinates with respect to which one could even hope to refer the components of a total energy-momentum four-vector." p. 294 Geometrodynamics

Noether's theorem says conservation of energy and momentum only when the universe's total action is invariant under global translation in time and space. OK inflation says universe is 3D flat on large scale, but it is not time-translation invariant. Indeed, the dark energy density is constant in this limiting scale, therefore the large total w = -1 dark energy is increasing with cosmic time in the standard metric. You cannot use special relativity globally only locally. Note that the total energy of w = + 1/3 radiation. It's not conserved because of the stretching of space. The total energy of pressureless w = 0 matter is conserved. Note that the parameter "a" can be thought of as the coordinate of a Bohm hidden variable. Let Psi be the vacuum ODLRO field, then the classical equations below have additional macro-quantum terms corresponding to Bohm's "quantum potential" Q.



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3rd draft
On Feb 19, 2007, at 10:17 PM, Jack Sarfatti wrote:

Wheeler and Misner (mid 1950's) give a manifestly global invariant way to deal with the problem using antisymmetric Cartan's forms and deRahm period integrals for generalized Stoke's theorem with metric/signature dependent * Hodge duals. More on this anon. The multiple-connectivity of 3D spacelike slices (star gates) is key, i.e. homology, Betti numbers. Also you have to know where to start, i.e. spinors and tetrads not the usual symmetric Levi-Civita connection and symmetric metric tensor. Torsion is key as is local gauge invariance of the initial non-gravity fields. Wheeler uses powerful results from the French school in the 1950's. The split between non-metric and metric-dependent sectors is transparent.

Localize T4 to get the compensating tetrad A^a 1-forms

e^a = I^a + A^a

Think of A^a like the U(1) potential up to a point.

F^a = De^a = de^a + S^ac/\e^c

The spin connection 1-form is S^a^b = - S^b^a

This torsion 2-form is the analog of the internal Yang-Mills field.

It is zero in the limit of Einstein's 1915 theory.

Therefore, in the 1915 limit

de^a + So^ac/\e^c = 0

d(I^a + A^a) + So^ac/\(I^c + A^c) = 0

I^a are the globally flat Einstein-Cartan tetrads. Subscript o is for the zero torsion limit of 1915 GR.

For geodesic observers dI^a = 0, therefore this equation is true for all observers.

i.e.

dA^a + So^ac/\(I^c + A^c) = 0

d^2A^a + d(So^ac/\(I^c + A^c)) = 0

d^2A^a = 0

d(So^ac/\(I^c + A^c)) = (dSo^ac)/\(I^c + A^c) - So^ac/\dA^c = 0

(dSo^ac)/\(I^c + A^c) = So^ac/\dA^c = 0


You need to locally gauge entire T4xO(1,3) to get

F^a =/= 0

as a covariant closed form

DF^a = 0


and the period integral of its Hodge dual *F is like "electric charge" for the torsion field in multiply connected 3D space with non-trivial p = 2 homology, i.e. non-bounding 2-cycles (wormhole mouths).

Note that G(mass density) ~ c^2/\zpf

G(mass density)r^3/c^2 ~ /\zpfr^3 ~ meters (Skinwalker Ranch data point)

Therefore /\zpf can, in principle, induce strong "Sakharov" (anti) gravity, /\zpf is a local "torsion" field.

Note that the uniform large scale dark energy is just enough to induce Newton's G.

i.e. where /\zpf limits to the cosmological constant.

/\zpf(13.7 billion years)^3 ~ (13.7 billion years) as an observed fact.

1//\zpf ~ 10^122 BITS area of observer causal dark energy future de Sitter horizon.

The symmetric invariant is

ds^2 = e^aea

The usual geometrodynamic connections are from

|~ ^wvu = ev^a(d/dx^u)ea^w

Curvature 2-form is

R^a^b = D^S^a^b = dS^a^b + S^ac/\S^c^b

Beside the topological metric independent d there is a metric dependent & with warped d'Alembertian d& + &d

Einstein's "vacuum" field equation is basically

(d& + &d)e^aea = 0

There is also a Dirac spinor equation here.

e^a = Z^CC'psi^a^CC'

C & C' are qubit spinor indices

psi^a^CC' = quantum computer Bell basis pair-entangled 2-spinor state.

Generalized global Stoke's theorem frame for "conservation laws" is for globally invariant de Rahm integrals

(n-1)dim Boundary integral of A/\*B = n dim{Interior integral of B/\*dA - Interior integral of (&B)/\*A}

A = p - 1 form

B = p form

*B is an n - p form

this is signature dependent

More anon.

n = 4 in ordinary spacetime with signature s = 1

Jack Sarfatti
sarfatti@pacbell.net
"If we knew what it was we were doing, it would not be called research, would it?"
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