Physical Geometry and the

1
Institute of Mathematical Physics Newport News,
Virginia, USA
Physical Geometry and the Fundamental Structure
of Space and Time Martha Harrell, Institute of
Mathematical Physics, Newport News, Virginia, USA
1. INTRODUCTION
6. FURTHER DEVELOPMENTS
Without consideration of what Einstein called
practically rigid bodies which he used in
treating the geometry and physics of General
Relativity, or some other form of matter, as
opposed to treating what Pauli called abstract
ideas and not experimental objects (1958, p.
148), it appears obvious that we do not know
whether any geometrical theorem of a mathematical
geometry is true of matter or, even if so,
whether a geometry applies to what we may call
our spacetime, or world, either globally,
regionally, or locally. This obvious
consideration is usually ignored, however I will
attempt to show that it is indeed beneficial, and
even necessary in certain regards, to separate
Mathematical Geometry from Physical Geometry.
Following Riemann, Einstein and Pauli appear to
be correct in holding that Geometry supplemented
in the ways they direct then becomes a branch of
Physics. Some of the concepts, objects, theorems
and related techniques of each kind of Physical
Geometry will differ from those of its
counterparts in Mathematical Geometry. Our main
interest here lies in differences in fundamentals
and their primary consequences. Essentially,
this new category of Geometry requires that the
fundamental concepts, such as line,
line-interval, point, plane, manifold,
and even space, as well as fundamental axioms
or postulates of any one type of Physical
Geometry, be reconsidered and modified as
necessary in regard to matter, as well as that
some concepts new to Geometry be introduced from
Physics, such as time, spacetime, and
motion. Though it may appear to some simply a
pedantic exercise in semantics or taxonomy to
distinguish Physical Geometry as a category of
Geometry, some fundamental results of General
Relativity and features of the model of Unified
Field Theory considered here already indicate
otherwise. Details concerning these matters will
be shown in sections 2 through 4. It is
historically interesting to compare the geometric
portion of the exercise in pure mathematical
logic embodied in the Whitehead-Russell Principia
Mathematica (1910-1913) to the formulation of
Physical Geometry for Theoretical Physics. The
fourth volume of the former, on Geometry, was
never published. Whiteheads extant manuscript
and related letters concerning what he did
produce for Volume IV indicate clear issues
arising from dramatic developments in progress
during his writing. Much of Whiteheads
difficulty in completing this work was due to the
impact of certain aspects of the early
twentieth-century revolution in physics,
especially the geometry of General Relativity
(see my (1984), (1988)). The Einstein (1961) and
Pauli views (1958) noted above began to shed new
light on Whiteheads issues for the author in
June 2006. See the authors 5.
_____________________________ The venue of this
poster is the September 4-8, 2006 Workshop
entitled Noncommutative Geometry and
Physics Fundamental Structure of Space and
Time at the Isaac Newton Institute
of Mathematical Sciences. _______________________
______ Please feel free to contact the
author at Instit_harrell_at_widomaker.com
. _________________________________

• A new category of geometry is needed,
  Physical Geometry. Physical Geometry is any
  geometry which adds to its mathematical
  structure, or theorems, at least a physical
  specification of a means of measurement of
  spatial and, usually, temporal intervals.
  Riemann 1, Einstein (1917) 2, and Pauli
  (1921) 3 recognize such a geometry as
  significant. Geometry usually presupposes some
  type of space, hence when it does the structure
  of space, time, and space-time in any physical
  context must be properly specified in order to
  judge the validity of a Physical Geometry. More
  recently, Sir Roger Penrose(2005)4 has offered
  abundant clarity in the detailed presentation of
  the many differences between mathematical and
  physical techniques, meanings, structures, and
  perhaps most importantly, criteria for choosing
  which, especially perhaps which Geometry, is
  likely to be found on the best road to reality.
  The object of finding this road is, in his work,
  to find a more satisfying fundamental theory of
  physical reality than we have or to reformulate
  an existing one to meet its major current
  challenges.
• My object at the Workshop is to place
  Non-Commutative Geometry in the context of
  Physical Geometry itself.
• The authors companion paper by the same title is
  available for those interested.

• A limited number of further constructions and
  related work are in progress, limited since the
  programme is naturally so demanding.
• Study of the potential formulation of Twistor PG
• 2. Study of the utility and potential for a
  formulation of a PG for aspects of the Big Bang
  scenario
• Analysis of the relation of Riemannian Physical
  Geometry to M-Theory which theory has enjoyed
  successes in deriving such physical results as
  quantities relating to Hawking radiation.
• The author would be very glad to
• have your suggestions for other
• investigations. You may contact me

3. NONOMMUTATIVE GEOMETRY AS A PG
5. THE DEVELOPMENT OF PHYSICAL GEOMETRY
2. PHYSICAL GEOMETRY

The following elements and sub- categories are in
analysis and review stages. 1. Physical
Geometry as a Branch of Physics 2. Is PG also
a Branch of Mathematics? 3. Hybrid Geometries 4.
Historical PGs 5. Following Penrose directions
for finding the best or a decent Road to
Reality 6. Potential fundamentals of Second-Level
UFT PG ? Fundamental objects Vector field
function, or matter vector , Christoffels G,
the modified Riemann tensor, e w tensor. ?
Consideration of the role of Dirac space (the
space of ket and bra, dual, vectors) ?
Determination of the geometry part of the action
integral, or Lagrangian ? singularities and
singularity theorems in GTR and their roles in
the UFT model ? absence of spacetime coordinates
x and t in Second-Level PG and physics
Physical Geometry is understood here to be a
higher-order category of Geometry including any
type of geometry treating matter in some form,
material objects we may say, as opposed to the
usual abstract or ideal objects treated in
Mathematical Geometry, in Pure Geometry as
Einstein termed it. E.g., we may need to develop
for certain purposes Riemannian Physical
Geometry, Projective Physical Geometry, Twistor
Physical Geometry or Minkowski Physical Geometry.
Following Riemann, Hilbert, Einstein, Pauli, and
others Physical Geometry may be founded on
differing modifications or additions to existing
definitions, postulates, or theorems of
Euclidean, Riemannian, and other geometries.
These alternatives will be studied and
developed in my continuing research, including
that in Field Theory, especially Unified Field
Theory. We should also recognize that there
already exist, logically and historically, a
third category of higher-order Geometry as well,
which may be called Hybrid Geometry. These
include treatment of both material and abstract
objects in some fundamental, irreducible way.
For example, Riemannian geometry may develop as a
Hybrid Geometry, explicitly or implicitly. In
papers in progress, Riemannian Physical and
Hybrid Geometries are formally considered in
relation to the co-variant differential UFT model
cited in section 3. below. It is well known that
early geometries were used for the practical
purpose of measuring land for tax collection
purposes without clear separation of practical
and pure geometry, hence a classic confusion of
Mathematical and Physical objects and related
propositions resulted. This kind of confusion
must be eliminated to avoid certain
inconsistencies and other issues in the twin use
of geometrical and physical description as
avoided so well by Einstein in his General Theory
of Relativity. Fundamentals of Riemannian and
other Geometries must be reconsidered and stated
in the terms and propositions of PG systems
including such fundamental concepts and
propositions as the general Riemannian manifold
, the Riemann metric given by the
infinitesimal line element and the
Riemann or Riemann-Christoffel curvature tensor
.
Noncommutative Differential Geometry in the
filed of NCG is of particular interest to the
author now in current investigations in
Geometry and geometrical methods of Quantum,
Relativistic, and especially Unified Field
Theories. The Penrose festschrift paper of
Alain Connes 6 is especially interesting in
its replacement of terms of distance measurement
from points to states . With the clear
basis in theoretical Physics of Connes new
concept of a geometrical space and its matter
States, as well as elements of differential
geometry, this geometry contains elements of a
Noncommutative Differential PG of use and
interest in modern Theoretical Physics. It
also gives a nod to Penroses Twistor Theory,
which will be most interesting to attempt
providing a kind of Emperors new clothes in
Twistor Physical Geometry form.
6. CONCLUSION
4. PHYSICAL GEOMETRY IN A UFT MODEL
While this poster is one of the first public
presentations of this new work in Physical
Geometry, the very early development appears to
provide a reasonable basis for attempting
further construction of Geometry as a branch of
Physics. This work appears suitable to pursue
for purposes of Unified Field Theory in
particular. Coupled with fundamental
theoretical physics of QFT and UFT, it should
help provide important testable physical
descriptions and predictions which have not
been forthcoming from other approaches. The
Fundamental Structure of Space Time The
Fundamental Structure of Spacetime
The model currently under study, developed by
Yaroslav Derbenev from late 2002, is a covariant
differential model5. It has been developed at
the Second Level to date. The author has
collaborated on the models geometrical approach
and Physical Geometry (PG). Some fundamental
features of the model and PG are as follows.
The Second-Level UFT Model introduces a
vector field function in an -
dimensional curved Riemannian manifold and
provides physical foundations for a Riemannian PG
for Quantum and GTR fields. Space-time as a field
is reserved for consideration in derivations from
the top-level model and its equations.
Prospects for derivations at the Second and First
Levels ? Preview of First-Level UFT Model
and its Physical Geometry (PG) ? Model
and PG of the Dirac equation and the
Electromagnetic field ?Physical quantities in
PG- ? Light velocity c, potential
basis of quantum and macro- measurement
? Matter-representation objects in n-
dimensional and n4 PG on Real and
Complex Riemannian manifolds ? PG of the
Unified Gravitational field ? PG of the
Electroweak field ? PG of the Strong and Weak
nuclear fields



REFERENCES
1 Riemann, B. (1854). On the Hypotheses which
Lie at the Foundations of Geometry. 2
Einstein, A. (1917) Uber die spezielle und die
allgemeine Relativitatstheorie. 3 Pauli, W.
(1921) Theory of Relativity. 4 Penrose, R.
(2005). The Road to Reality. QC20.P366 2005. 5
Harrell, M. (1988). Extension to Geometry of
Principia Mathematica and Related Systems II
140-160. Antinomies and Paradoxes, ed.
I.Winchester. McMaster U. Library Press. 6
Connes, A. (1998). Noncommutative Differential
Geometry and the Structure of Space-Time
49-80. The Geometric Universe, ed. S.Huggett et
al. 7 Derbenev. Y. A Covariant Approach to
Modeling Unified Field Theory. To be published.
Work supported by the Institute of Mathematical
Physics, Applied Research Center 243, Newport
News, VA, USA 23606