Discussion with R. Kiehn
On Feb 9, 2005, at 2:43 AM, RKiehn2352@aol.com wrote:
Matrix Affine connections are transitive. All points move. There are no fixed points (such as an absolute origin).
Right. In Einstein's informal language the space of global Lorentz inertial frames (really flat spacetime everywhere) where each "point" is a possible inertial frame (set of 3 rapidities and 3 orientation angles, a rapidity is an imaginary angle), there is no preferred point. In Lorentz's there is. People like Cahill and Consoli claim experimental evidence for Lorentz's preferred point breaking the affine transitivity that I explain as a vacuum condensate effect analogous to ground state preferred magnetization in a ferromagnet.
In 4D, the affine group has 13 arbitrary functions.
If the affine group in 4D is constrained to have a "fixed direction",
then the number of arbitrary functions is reduced to 10. The group dimension is 10.
Precisely the dimension of the Lie Algebra of the Poincare group of global special relativity. Curious.
Such constrained matrices can be interpreted as a change of axes without shifting the direction of the origin.
(Three axes change, the fourth axis does not direction.)
No metric has been imposed.
If the affine group is constrained to be orthogonal, then the group dimension is 4.
In all cases, the determinant is NOT zero.
Hence all such matrices can be used as basis elements for a vector space.
As groups, all such basis elements have a well defined connection,
if the group matrix of functions is differentiable once.
Note that Projective connections have group dimension = 15.
Same as dimension of the conformal group in 4D used by Penrose in twistor theory.
The Poincare invariance of Maxwell equations is related to the 15 element group,
not the affine 13 (or less) sub group.
Yes, of course.
SOURCE = Turnbull "The theory of determinants, matrices and invariants" Dover 1960 p. 162-163
When a metric is imposed, then a key property of general relativity appears to be related to the question of
when and if the functional form of the metric goes to zero, or its associated Gauss curvature goes to infinity.
That's a singularity e.g. Big Bang initial singularity, black holes.
IF metric is an artifact of the presence of matter, then the question arises:
are there points or domains in the 4D space where the metric exhibits a singularity?
What are the states of matter in such disparate regions?
That was all done by Penrose & Hawking and you do not even need matter, i.e. the vacuum field equations
Ruv = 0 (neglecting exotic vacuum w = -1 "dark" zero point energy density (c^4/8piG)/\zpf of either positive or negative micro-quantum pressure) has singular black hole solutions even when Tuv = 0.
For example, in certain regions of 4D space can the metric change (can the states of matter change)
from a Eulidean to Lorentzian signature, or visa versa?
That's roughly Hawking's picture of inflation in quantum cosmology (Wick rotation) the Big Bang initial singularity is associated with that topological change in signature ++++ to -+++ of a Wick rotation "imaginary time".
Can the metric of 3D subspaces of matter change from Euclidean to Lorentzian?
What is the state of matter in such situations?
That would be a new one. No need for that one I think. Also "matter" takes a back seat, at most it's only 4%, the rest is zero point energy exotic vacua in two forms dark energy that anti-gravitates and dark matter that gravitates. Both are w = -1 but from afar dark matter of w = -1 with positive pressure mimics w = 0 CDM.
Wait that would be maybe t'Hooft-Susskind world hologram, i.e. 2D hologram screen from DeSitter local horizon that is 2+1 space-time rather than 3+1 space-time.
That's 3 translations x,y,t, (x-y) 1 space rotation, (x-t) (y-t) 2 Lorentz boosts, 3-conformal boosts, 1 dilation
3 + 1 + 2 + 3 + 1 = 10
in the de Sitter local Hubble bubble horizon (or is it anti-de Sitter? - depends on sign of /\ = cosmological constant)
I call your attention to arXiv:math.DG/0311330 v2 27 Jan 2005,
where the differential topologists investigate the concept of a 3D Minkowski space,
and the possible embeddings of zero mean curvature surfaces in such spaces.
They call such surfaces "Maximal Surfaces" (not "minimal surfaces).
Without such knowledge about 10 years ago I wrote a Maple program to solve for such oddities
and was surprised by the various (physically interesting) features of such objects.
The solutions to such problems are NON-AFFINE, and admit fixed points (centers
expansion, or rotational axes. Singularities that may be "black holes" )
Minimal surfaces (of zero mean curvature) in Euclidean do not admit such "fixed points".
From a physics point of view all of this appears to be weird stuff, do to the dogma of assuming 3D space must be Euclidean (at least for a space of particles).
HOWEVER, the Falaco Solitons in Fluids appear to be realizations of these Maximal Surfaces with fixed points.
I am investigating the idea that the interior of the Solar system is dominated by a Euclidean metric ( in aggreement with dogma) but at some orbit size, the metric becomes Minkowskian and will add an acceleration component inwardly directed to maintain the rotational features of a "MAximal: rather that a "Minimal" surface..
I have already explained that Pioneer anomaly with elementary battle-tested ideas. But go ahead by all means. However a ++++ metric in Solar Systems not possible because no light cones there - if that is what you mean? Do you mean in a small sphere at the Center of the Sun?