Conformal Group and General Relativity
- From: Jack Sarfatti <sarfatti@xxxxxxxxxxx>
- Date: Tue, 26 Jul 2005 06:59:57 +0000 (UTC)
See also Roger Penrose "The Road to Reality" on conformal invariance.
"'Internal symmetries' are ... approximate ... an even more interesting
and amazing symmetry ... which prevails when all velocities are very
high, close to light velocities ... is the ... conformal symmetry
[which] ... implies, first of all, SCALE INVARIANCE [also seen in
fractal self-similarity and at critical point phase transitions with
infinite coherence length] ... This means that if one expands everything
in both space and time, nothing significantly changes."
Note that this implies the limit of zero rest mass m = 0, i.e. E/mc^2 ~
(1 - (v/c)^1/2)^-1/2 -> infinity, or "real quanta" physics literally on
the light cone.
"Thus, the angular distribution resulting from a collision will become,
at very hight energies ... independent of that energy ... Under ordinary
conditions, of course, scale invariance does not apply"
Rest mass breaks dilation symmetry, because it gives a gravity length
scale rs = Gm/c^2 that is zero when m = 0. Note that the quantum length
scale is h/mc that is infinite when m = 0. The virtual
electron-positron cloud surrounding a bare electron extends out to h/mc
and the classical electron radius is 1/137 of that.
"There are four others ... successions of 'inversions by reciprocal
radii', i.e. ...
xi -> x'i = (area)/(xi - ai)"
This suggests the dualities of string theory and the mappings inside to
out and back of the unit circle as well as the stereographic projective
mappings Penrose uses to relate tensors to spinors (and twistors in the
complexification of space time bringing infinities into finite position
on the "Penrose diagram").
"The whole conformal group has 15 parameters in contrast with the 10
parameters of the symmetry of special relativity..."
General relativity comes from the local gauging of the 4-parameter
translation sub-group T4 of the symmetry group of special relativity
giving the GCT Diff(4) group with compensating tetrad gauge potential
Bu^a that I relate to (argVacuumODLRO) i.e. spontaneous breakdown of
U(1) QED symmetry in the conformal group false vacuum dominated at long
distances by the mc^2 = 0 Dirac Sea. However all ZPF fields are included
in the final theory. My present model is only a first approximation to
that final theory. The fact that /\ ~ (Hubble Radius)^-2 is proof
positive that such a final theory exists. Indeed, we need only take the
semi-phenomenological Landau-Ginzburg theory to see what the final
effective c-number ODLRO theory will look like. Similarly locally
gauging the Lorentz group gives a new independent torsion field at the
geometrodynamic level bi-linear in the tetrads. Why stop there? Locally
gauging the 4 conformal "boost" inversions and the dilation will give
even more "gauge field potentials". We have 6 + 5 = 11 "charges" to
locally gauge beyond T4 to extend 1915 GR. We need to look at the
complete Lie algebra of the 15-parameter conformal group the way
Schwinger does the 10-parameter Poincare group to get a new idea here.
On Jul 24, 2005, at 2:32 PM, Jack Sarfatti wrote:
"Covariance" is derivative from "symmetry and invariance".
Invariance of the dynamical Action S of some physical system with
generalized coordinates q, q,t under some [continuous] group G demands,
in the extremal action principle "critical point" &S = 0 that the
equations of motion (Euler-Lagrange equations) are tensor (or spinor)
equations relative to the symmetry group G.
Invariance means "frame invariant" under some transformation of
"perspective" or "point of view".
The "passive" view is a fixed physical process observed from two
different frames.
The "active" view is a fixed frame of reference looking at an ongoing
physical process at a new place-time at least for a space-time symmetry.
The active view is strictly limited and problematical in General
Relativity that is essentially a local theory. Locality of macro
space-time physics comes from the ODLRO emergence of gravity from the
non-local micro-quantum substratum (pre-inflationary unstable false
vacuum). That is, the Einstein-Cartan tetrad in a convenient geodesic
basis is
eu^a = Iu^a + Bu^a
Iu^a is the "aligned" 4x4 identity matrix for the geodesic basis, either
in globally flat or variably curved space-time. Note that Bu^a = 0 in
globally flat spacetime.
Bu^a = BuLp(argVacuumODLRO)^,a
,a is ordinary partial derivative in tangent space
Bu = LpBu^a&a
Consistency constraint is the Planck Mass Shell on the Goldstone Phase
of the post-inflation Higgs field, i.e. the flat wave equation
(argVacuumODLRO)^,a,a = 1/Lp^2 = c^3/hG = /\
where /\ is the bare cosmological constant in the absence of Vacuum
ODLRO as it is in the pre-inflation false-vacuum dominated by the
conformal QED "Dirac Sea" with mc^2 = 0.
Einstein's geometrodynamic field is then given by the equivalence
principle EEP as
guv = eu^a(Minkowski)abev^b
*This essentially completes the derivation of Einstein's 1915 General
Theory of Relativity as an emergent macro-quantum theory in the sense of
Andrei Sakharov and P.W. Anderson's "More is different" both in 1967.
Sakharov's idea was incomplete because he did not have the idea of "More
is different", i.e. the cohering of the zero point vacuum fluctuations
as the essential ingredient. Also inflation was not around back then.
The symmetry group of the Einstein-Hilbert action with density
(-detguv)^1/2R is the local gauging of T4 -> Diff(4) with compensating
gauge potential
Bu = Bu^a&a
The elements of Diff(4) are Xu'^u(P) where
x^u(P) -> x'^u(P) = x^u(P) + L^u(x^u'(P))
Xu'^u(P) = L^u,u'(P)
P is a coincidence where non-geodesic LNIFs u & u' "intersect" in
Einstein's "local coincidences" i.e. "Physics is simple when it is
local." Wheeler.
Note the orthonormality
Xu'^u(P)Xu^u"(P) = Iu'^u" = 4x4 Identity Matrix
This does not demand orthogonal guv.
The indices a in the tetrads are LIFs intersecting the LNIFs at same P.
Any non-physical "relabelings" form a normal subgroup H of Diff(4) and
are factored out. We only deal with Diff(4)/H(relabelings).
"Covariance" is, therefore, relative to the given group G and refers to
the tensor/spinor properties of the equations of motion (classical
N-particle and classical field for now).
Tensors and spinors transform multi-linearly under the group G. Let g be
an element of G, with a faithful (homomorphic image) matrix
representation X(g). Let T be a tensor of rank n with n' and n" upper
and lower indices, then
T -> T' = X...X X^-1 ...X^-1T
n' factors of X and n" factors of X^-1
Note that a tensor component representative of the form Xu can be made
into a frame- invariant via
T = Tudx^u
where {dx^u} is a basis (local frame) of Cartan 1-forms
sum over repeated pairs of upper and lower indices
A tensor component of the form T^u can be made into an invariant
T = T^u&u
where {&u} is a basis of tangent vector 1-coforms.
"Newton's equations were not formulated in any special coordinate system
and thus left all directions and all points in space equivalent. They
were invariant under rotations and displacements ... the same applies
to his gravitational law. ... As to the conservation laws, the energy
law was useful ... The momentum and angular momentun conservation
theorems in their full generality were not very useful ... Most books on
mechanics written at the turn of the [19th-20th] century and even later,
do not mention the general theorem of the conservation of angular
momentum ... This situation changed radically ... as a result of
Einstein's theories ... Having lived in those days, I know ... Hamel, as
early as 1904, established the connection between the conservation laws
and the fundamental symmetries of space and time ... the concept of
symmetry and invariance has been extended into a new area ... much less
close to direct experience."
On Jul 23, 2005, at 9:50 PM, Jack Sarfatti wrote:
"Objectivity" = "Invariance" = Stillness in the movement among points of
view, changes of perspective all looking at the same process.
"One can distinguish between two types of invariance: the older ones
which found their perfect, and perhaps final, formulation in the special
theory of relativity, and the new one, yet incompletely understood,
which the general theory of relativity brought us. The older theories of
invariance postulate, in addition to the irrelevance of absolute
position and time of an event, the irrelevance of its orientation and
finally the irrelevance of its state of motion, as long as this remains
uniform, free of rotation, and on a straight line." Eugene Paul Wigner
"Invariance in Physical Theory" (1967)
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