













Physical Geometry
© Gustavo R. González Martín 2007


...A unification
through geometry... The
book was born from a series of lectures over unified theories given at
Universidad Simón Bolívar. Really, it
is a coherent recollection of scattered publications on the geometric
unification of physics, including unpublished works. The objective is to
establish the foundations for this unification in order to give an answer to
the following question: Is there a Physical Geometry? The fundamental ideas
and some results are published in the references. It is recognized that the
action of matter defines certain concepts and their relations, all of them
capable of geometrical representation. The main aspects of the theory are the
following:
The results
indicate that gravitation and electromagnetism are unified in a non trivial manner. Multipole aproximations
determine the geodesic motion with the Lorentz force. If we restrict to the
gravitational part, we obtain the Einstein equation with cosmological
constant and a geometric
energy momentum tensor that indicates a geometric internal
solution. In vacuum, the known gravitational solutions are obtained. The
constant curvature parameter (geometric energy density) of a hyperbolic
symmetric solution may be related, in the newtonian
limit, to the gavitational constant. In general the gravitational parameter G
is variable under nonriemannian fields. This effect
may be interpreted as the presence of dark matter. Electromagnetism is
related to an SU(2)_{Q} subgroup. If we
restrict to a U(1) subgroup we obtain Maxwell's
field equations. The geometry has a canonical bracket operation for
generalized Jacobi vector fields and
determines fermionic and
bosonic operator fields end their rules of quantization (QED). In
fact, it appears that this geometry is the germ of quantum physics including its probabilistic aspects.
The geometric nature of Planck's constant h and light speed c
is determined by their respective relations to the connection and the metric.
The mass may be defined in an invariant manner in terms of energy, depending
on the connection and matter frames. The geometry shows a triple structure
that determines various physical triple structures in the classification of
particles. The geometric excitations have fundamental
quanta of charge e, action h/2 and
flux h/2e that may be used to explain the fractional
quantum Hall effect. The quotient of bare masses of
three stable particles may be calculated
and leads us to a surprising geometric expression, previously known but physically
unexplained, that gives the value 1836.1181 for the proton to electron mass
ratio. There are connection excitations whoose
masses correspond to the weak WZ boson masses and allow a
geometric interpretation of Weinberg's angle and represent a weak interaction
. The geometric equation of motion (a generalized Dirac equation) determines
the anomalous bare magnetic moment of both
proton and neuctron .
The first QED correction for the proton gives 2(2.7797) for the Landé gfactor. The "strong"
electromagnetic SU(2)_{Q} part, without the
help of any other force, generates short range atractive
nuclear potentials which are sufficiently strong to determine the binding energy of the
deuteron (2.20 Mev.), the alpha particle and other light
nuclides . The bare masses of the lepton families and mesons may
be calculated as topological excitations of the electron, giving 107.5927 Mev for the muon and 1802.7 Mev for the tau. The neutrino
mass energy and its oscillations are calculated from the geometric
curvature. The proton shows a triple structure. The combinations
of the three fundamental geometric excitations (associated to the proton,
the electron and the neutrino), forming other excitations, may be used to classify
particles and show a symmetry under the group SU(3)xSU(2)xU(1). The alpha coupling constant is a
geometric coefficient which is calculated to be 1/137.03608245. 

The Universe: inexorable geometric
action? 



Super Nova 1987a, Hubble Telescope, Space Telescope Science
Institute; www.aip.org (7/7/99).




Universe infrared radiation, COBE Spacecraft, Space
Telescope Science Institute; www.aip.org (7/7/99). 

…Mach felt that
there was something important about this concept of avoiding an inertial
system… Not yet so clear in Riemann's concept of space. The first to see
this clearly was LeviCivita: absolute parallelism
and a way to differentiate… …The
representation of matter by a tensor was only a fillin to make it possible
to do something temporarily, a wooden nose in a snowman… …For most
people, special relativity, electromagnetism and gravitation are unimportant,
to be added in at the end after everything else has been done. On the
contrary, we have to take them into account from the beginning… Albert Einstein
from Albert Einstein's Last
Lecture^{1}, Relativity Seminar, Room 307, Palmer Physical
Laboratory, Princeton University, April 14, 1954, according to notes taken by
J. A. Wheeler. 

_{1 J. A. Wheeler in: P. C. Eichelburg and R. U. Sexl
(Eds.), Albert Einstein (Friedr. Vieweg & Sohn, Braunschweig) p. 201, (1979).} 

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