On Oct 28, 2004, at 9:32 AM, Jack Sarfatti wrote:


PS

Note that Tab is a 4x4 matrix.

There is a principal axis transformation that diagonalizes it. The
diagonal elements in that case are the eigenvalues.

Trace{Tab} is the sum of the eigenvalues. It is a frame invariant under
these kinds of transformations, which are not obviously GCT, but
something else? A different kind of group. What is the physical meaning
of this new group? In classical mechanics of coupled particles they are
"normal modes" of independent motions in a linear theory. What do they
mean here in classical local field theory?

Det{Tab} is the invariant product of the eigenvalues.

All of this is local in space-time.

There are also other sums of products of the eigenvalues forming the
coefficients of the secular polynomial of the matrix.

Question: can we generalize this formally to tensors and spinors of
rank > 2?

For example a third rank tensor is a lattice in a 3-space. Can we
diagonalize it?

There is a set of diagonals in these higher dimensional cases.

Similarly, a 4th rank tensor is a lattice in a 4-space and so on.


On Oct 28, 2004, at 8:29 AM, Jack Sarfatti wrote:

1.2 Roger Penrose "The Road to Reality" (2004) 19.5 The
Energy-Momentum Tensor

1.2.1 "[What is] the energy density of a field?, this density being
the source of gravity ... such as Maxwell's ... it does so via a
tensor quantity ... This is a symmetric [02]-valence tensor which
satisfies a [local] conservation equation [COVARIANT 4-divergence of
energy-momentum tensor vanishes] ... The quantity T^ab collects
together all the different densities and fluxes of the energy and
momentum in the fields and particles ... in a standard [flat]
Minkowski coordinate system, the covector T^0b defines the density of
4-momentum, and the three co-vectors T^1b, T^2b, T^3b, provide the
flux of 4-momentum in the three independent spatial directions ...
T00 measures the energy density, and T11, T22, T33 measure the
pressure, in the three directions of the spatial coordinate axes."

[Note by Jack on micro-quantum random zero point exotic vacuum
energy-momentum tensor (AKA stress-energy density tensor true for all
quantum fields of any spin;

Trace{Tab(zpf)} = T00 + T11 + T22 + T33

For real photons with w = +1/3 this is 2T00

At the boundary w = -1/3 between ordinary and exotic matter, the trace vanishes.

P(zpf) = T11 + T22 + T33

Energy Density (zpf) = T00

w(zpf) = P(zpf)/[Energy Density (zpf)] = -1 ]

For an isotropic case, no Casimir type plates for example, I should
have written

3P(zpf) = T11 + T22 + T33

T11 = T22 = T33 = P(zpf) = -T00 from Lorentz + General Coordinate
Covariance + Equivalence Principle

Therefore

Trace{Tab} = T00 + T11 + T22 + T33 = Too + 3P = Too(1 + 3w)

This is generally true for isotropic case for both real and virtual
stuff.

For virtual off-mass-shell vacuum zpf stuff

Too(zpf) + P(zpf) = 0

i.e. w = -1

Therefore,

Trace{Tab(zpf) = -2Too

For virtual photon ZPF, Too > 0 i.e. negative pressure.

Kip Thorne's criterion for "exotic matter" (1986) for the metric
engineering of warp, wormhole and weapon is

Trace{Tab(exotic matter)} < 0