On discrete geometrodynamical theories in physics.

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Order Number 8816322

On discrete geometrodynamicaI theories in physics

Towe, Joe Patrick, Ph.D.

The University of Arjzona, 1988

Copyright @1988 by Towe, Joe Patrick. All rights reserved.

"-U·M·I 300 N. Zeeb Rd. Ann Arbor, MI 48106

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It

ON DISCRETE GEOMETRODYNAMICAL THEORIES IN PHYSICS

by

Joe Patrick Towe

copyright © Joe Patrick Towe 1988

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF PHILOSOPHY

In Partial Fulfillment of the Requirements For the Degree

DOCTOR OF PHILOSOPHY

In the Graduate College

THE UNIVERSITY OF ARIZONA

198 8

_._. __ ._- -------------------------------------- .... _ ....

THE UNIVERSiTY OF ARIZONA GRADUATE COLLEGE

As members of the Final Examination Committee, we certify that we have read

the dissertation prepared by __ ~J~o~e~P~a~t~rwi~cuk~T~o~w~e~ ________________________ _

entitled On Discrete Geometrodynamical Theories in Physics

and recommend that it be accepted as fulfilling the dissertation requirement

for the Degree of Doctor of Philosophy

Dec. 14. 1987 Professor Joseph L. Gowan

7ft '0V!l1 L Date

Dec. 14, 1987 Date

Dec. 14, 1987 Date

Dec. 14. 1987 o essor J. D. Garcia Date

~{J))u (fu.JL- Dec. 14. 1987 Professor John D. McCullen Date

Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College.

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.

~~JW.I~ Dec. 14. 1987 Date

STATEMENT BY AUTHOR

This dissertation has been submitted in partial ful­fillment of requirements for an advanced degree at The Uni­versity of Arizona and is deposited in the University Li­brary to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allow­able witho~t special permission, provided that accurate ac­knowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the copyright holder.

SIGNED:

Hl

ACKNOWLEDGMENTS

I want to thank the members of my committee for their

patience and assistance in this endeavor. I am especially

indebted to Dr. Joseph Cowan and Dr. J. D. Garcia who have

answered many questions for me and offered very helpful sug­

gestions. In addition to my committee members, I am also

indebted to Dr. Richard Young and Dr. Willis Lamb, from whom

I studied quantum mechanics, and to Dr. David Lovelock and

Dr. Hanno Rund of the Mathematics Department, from whom I

studied relativity and gauge fields.

I also wish to thank Dr. Barrett O'Neill, and Dr. Ernst

G. Strauss of the University of California Los Angeles

(UCLA), from whom I studied differential geometry and

relativity, and Dr. J. G. Miller, then a visiting professor

at UCLA, from whom I studied black hole dynamics.

I am also indebted to Alan Leighton, then a graduate

assistant at Caltech, for helping me when I was a beginning

student, and in particular for making available to me the

complete set of notes which later became the book entitled,

Gravitation, by Misner, Thorne, and Wheeler. I further ex­

tend my gratitude to Dr. John A. Wheeler for patiently

listening to my ideas and for the helpful suggestions he

made.

Notwithstanding that which lowe to all the above, my

greatest debt is to my mother, Toncie, for supporting me

iii

Il

iv

through so many years of school and for her encouragement

and inspiration; and to my father, Roy, for his example and

his continual admonitions. My Aunt and Uncle Milliken were

also instrumental in the development of my academic inter­

ests, through our many discussions of relativity and the

other theoretical foundations of physics. Thirdly, my Aunt

and Uncle Bowling were very supportive financially.

Fourthly, my wife and family have supported me strongly dur­

ing my recent work, this is especially true of my son,

James.

Finally, I wish to thank Julie Noon, who typed this

dissertation, as well as several other papers which collec­

tively formed the basis for my dissertation.

TABLE OF CONTENTS

Page

LIST OF FIGURES vi

ABSTRACT • • • vii

1. INTRODUCTION • 1

2. A REVIEW OF GENERAL RELATIVITY AND THE MAXWELL THEORY • • • • • • • • • • • • • • • • • • 11

3. A REVIEW OF THE RAINICH-MISNER-WHEELER THEORY • • • • • • • • • • • • • • • • • • 53

4. A SUGGESTIVE TOPOLOGICAL-GEOMETRICAL FORMALISM • • • • • • • • • • •

5. BOHR QUANTIZATION AS A CONSEQUENCE OF THE SL(4,R)®U(1)-INVARIANT REALIZATION

60

OF THE PROPOSED FORMALISM • 71

6. GENERAL QUANTIZATION • • • • • • • • • • 75

7. SPIN ANGULAR MOMENTA WHICH ARE SUGGESTED BY THE SL(2,R)®SU(3)-SYMMETRIC REALIZATION OF THE PROPOSED FORMALISM • • • • • • 80

8. THE SUPERSYMMETRY THEORY AS A CONSEQUENCE OF GEOMETRY • • • • • • • • • • • • • • 85

9. ASTROPHYSICAL PREDICTIONS OF THE PROPOSED MONISTICALLY GEOMETRIC THEORY . • • • • 93

10. A CLASSICAL FOUNDATION FOR QUANTUM LOGIC. • 100

11. INTERIM CONCLUSIONS AND SUGESTIONS FOR FURTHER DEVELOPMENT • • • 104

SELECTED BIBLIOGRAPHY 108

v

It

LIST OF FIGURES

Figure

1.

2.

3.

4.

Spacetime Path of Inertial Frame • • •

Spacetime Path of Non-Inertial Frame •

Light Cone •• . . . . .

. . . .

The Theorem of Pythagoras in Special Relativity • • • • • • • • • • • . . . . .

5. Lorentz Transformations . . . . . . . . . 6. Parallel Displacement in a Flat Space

7. Parallel Displacement in a curvilinear Space • • • . • . . • • • • • • •

8. Gauss' Law of Electric Flux

9. Gauss' Law of Magnetic Flux . . . . . . . 10. The Faraday-Lenz Law . . . . . . . . . . . . . 11. The Cross Product . . . . . . 12. Maxwell's Generalization of Ampere's Law.

13. Wheeler's Handle Topology

14. ]R3 x ]R32-1 Admits a Cartesian Coordinate

System • • • • •

15. 2

A Discrete Metric on IR3 x IR3 -1

16. A Discrete Metric on IR3 and the simplest SL{2,R)®SU{3)-Symmetric Space •••

2 17. Triplets and Anti-Triplets on IR3 x IR3 -1

18A. Hexagons Associated with the Additional Particles which are Required by the Commutator-Ant i-Commutator Algebra

Page

14

15

18

20

21

33

33

42

43

43

44

45

55

81

82

83

87

88

18B. Triplets Involving the Additional Particles. • 88

vi

It

LIST OF FIGURES (continued)

Figure

19. Quark-Squark Interaction • · . . . . . . . . . 20. Quark-Quark Interaction . . . . . . · . . 21-

22.

23.

Lepton-Lepton Interaction · . . . . . . . Lepton-Slepton Interaction • . . · . . Gravitational Interaction · . . . . . . · . .

vii

Page

90

90

91

91

91

------- --------------------------------

ABSTRACT

The authors of the Rainich-Misner-Wheeler theory no

longer believe that everything physical can be accounted

for in terms of the topological-geometrical structure of

ordinary spacetime. However, many p~ysicists and philoso­

phers entertain the possibility that a geometrodynamics

(a theory which accounts for sources as well as fields in

terms of topological-geometrical structure) may be

feasible in the context of a more general topology. In

this dissertation I consider two topological-geometrical

models (based upon a single suggestive formalism) in which

a geometrodynamics is both feasible and pedagogically

advantageous. Specifically I consider the topology which

is constituted by the real domains of the two broad clas­

ses of rotation groups: those characterized by the com­

mutator and anti-commutator algebras. I then adopt a

Riemannian geometric structure and show that the monisti-

cally geometric interpretation of this formalism restricts

displacements on the proposed manifold to integral mul­

tiples of a universal constant. Secondly I demonstrate

that in the context under consideration, this constraint

affects a very interesting ontological reduction: the

unification of quantum mechanics with a discrete, multi­

dimensional extension of general relativity. A particularly

viii

--- -_.-._-------------------------------------

ix

interesting feature of this unification is that it includes

and (for the world which is eharacterized b~ energy levels

which range in magnitude from low to intermediately high)

requires the choice cf an SL(2,R)~SU(3)-5yrnmetric reali­

zation of the proposed, generic formalism which is a

lattice of spins ~ and ~/2. (This is in the context of the

same universally constant scale factor as that which yields

the quantization conditions described above.) If the

vertices of this lattice are associated with the fundamen­

tal particles, then·the resulting theory predicts and pre­

cludes the same interactions as the standard supersymmetry

theory.

In addition to the ontological reduction which is pro­

vided, and the restriction to supersymmetry, the proposed

theory may also represent a scientifically useful extension

of conventional theory in that it suggests a means of under­

standing the apparently large energy productions of the

quasars and relates Planck's constant to the size of the

universe.

CHAPTER ONE

INTRODUCTION

In the nineteenth century, W. K. Clifford (1876, pp.

157-158) suggested that all aspects of every physical inter-

action can be accounted for in terms of the geometry of

space. Clifford's view was never widely accepted, but in

recent years John A. Wheeler has entertained a generaliza­

tion of Clifford's theory, suggesting that there may indeed

be a topological-geometrical context in which everything

physical can be accounted for in terms of the mathematical

structure of the formalism.

Wheeler's view emerged from his association with, and

generalization of the Rainich-Misner-Wheeler theory, which

reduces the combined field equations of Einstein and Maxwell

in the context of a point charge source to a set of equa-

tions which involves nothing more than the Ricci tensor and

its derivatives. Wheeler attempted to extend this theory to

contexts where point charge sources are replaced by general

4-distributions of charge by interpreting field sources as

areas where handles attach to the 4-manifold. Lines of

force were thought of as entering and issuing from these ar-

eas via the surfaces of the handles, so that it was no

longer necessary to describe field sources in terms of

singularities (points where the mathematical formalism is

not well defined).

1

In 1960 Wheeler described the questions which his re-

search was attsmpting to address with the following words:

Is curved empty spacetime a kind of magic building material out of which everything in the physical world is made: (1) slow curvature in one region of space describes a gravitational field; (2) a rippled geometry with a different kind of curva­ture somewhere else describes an electromagnetic field; (3) a knotted-up region of high curvature describes a concentration of charge and mass­energy that moves like a particle? Are fields and particles foreign entities immersed in geometry, or are they nothing but geometry?

2

For some years the prospects for a realization of the mo­

nistically geometric ontology which is suggested here seemed

encouraging. But ultimately it became clear (first to

Wheeler) that 'geometrodynamics,' as Wheeler's program was

known, confronted a problem in the microdomain. In

Wheeler's (1974) words:

In all the difficult investigations that led. • • to some understanding of the dynamics of geometry, the most difficult point was also the simplest: The dynamical object is not spacetime. It is space. The geometric configuration of space changes with time. But it is space that does the changing ••• (and) in the real world of quantum physics one cannot give both a dynamical variable and its time rate of change.

Wheeler therefore concluded that quantum uncertainty pre-

cludes a quantum scale knowledge of spacetime and spacetime

geometry; i.e., precludes a quantum scale geometrodynamics

in the context of the topology which is represented by ordi­

nary spacetime.

fl

3

Adolph Grunbaum (1974, p. 467) has also objected to

geometrodynamics as it was initially formulated on grounds

that a metric which is unique, up to a universally constant

scale factor cannot be established on spacetime without ref­

erence to a rod-clock, combination, field source or other

material device, which introduces circularity if one is try­

ing to account for field sources and material objects gener­

ally in terms of Riemannian curvature (i.e., in terms of

4-metrical properties).

I agree with Wheeler and Grunbaum that the geometri­

zation of mechanics which is represented by the Rainich­

Misner-Wheeler theory, is incompatible with a monistically

geometric ontology. On the other hand, I tend to believe as

Wheeler does (Misner, Thorne and Wheeler, 1974, p. 1180-

1183), that a monistically geometric ontology may be fea­

sible in the context of some more general topology. This

view is based upon my recent investigation of a very sugges­

tive formalism, which I will now describe.

I first adopt the most general physically relevant to­

pology of which I am aware. In particular I adopt the ge­

neric symmetry principle GL(4,R)®SU(n), which involves

both the commutator and anti-commutator algebras. The real

topology, which is associated with this symmetry principle

is, of course, spacetime and the (n2-1)-dimensional param-

eter space of the SU(n) group. Secondly, I adopt the most

----------- -------------------------------------

1/

general class of metrical geometry: the Riemannian geom­

etry.

4

The suggestive aspects of this formalism are as fol­

lows: The equations of motion which result from this

topological-geometrical formalism replace the mathematical

construct which is usually interpreted as representing the

electromagnetic field with the construct which is associated

with the generic Yang-Mills field. Secondly, the Einstein­

type field equations which correspond to this formalism are

apriori if one is committed to the monistically geometric

interpretation of the proposed formalism, in that the 4-

distribution of mass-energy must, in this context, be

equated to the 4-curvature of the world. Thirdly, the

Einstein-type equations which result from a monistically

geometric interpretation of the proposed formalism break

down into ordinary Einstein equations which describe the

4-distribution of mass-energy in terms of the stress-energy

tensor, which is in terms of the generic Yang-Mills field

and Maxwell-type equations which involve the generic

Yang-Mills field, provided that a certain sum of products

of connection coefficients is associated with the 4-current

of the Yang-Mills field. The association of this sum of

connection coefficients with the 4-current of the Yang-Mills

field is, of course, a necessary condition for the monis­

tically geometric interpretation of the formalism. (This

5

requirement is motivated by the same considerations as those

which motivated Wheeler's handle topology.)

An additional condition which is necessary for the mo­

nistically geometric interpretation of the proposed formal­

ism is that the model be intrinsic to the manifold (this is

to avoid the kind of logical circulatory discussed by

Grunbaum).

Since, according to Riemann's famous inaugural lecture

(1886), continuous manifolds do not have intrinsic metrics,

the manifold which I ultimately adopt must be a discon­

tinuous analog of the (4+n2-1)-dimensional real manifold de­

scribed above. Let us consider a Riemannian metri~ ds=nK,

where n is a real number and K is a universally constant

scale factor. If I permit that n be any real number, then

the manifold is continuous which, according to Riemann, pre­

cludes an intrinsic metric. Thus, for positive displace­

ments, n must be restricted to the set of positive rational

numbers, or the set of positive integers. If n is only re­

stricted to the set of positive rationals however, one en­

counters the same problem as when n is permitted to be any

real number; since there is no smallest positive rational

number, there is no natural unit of measurement (i.e., no

metric which is intrinsic to the manifold). I conclude,

therefore, that if the metric is to be intrinsic to the

(4+n2-1)-dimensional manifold (if one is to avoid logical

circularity in giving a monistically geometric

6

interpretation of the proposed formalism), then the most

• general form the metric ds can assume is ds=nK, where n is a

positive integer.

To demonstrate that interesting physical consequences

emerge from this second necessary condition I conjoin this

condition with the formalism introduced above and adopt the

SL(4,R)®U(1)-symmetric realization of this generic

theory (which is for a protonic charge equivalent to a dis­

crete version of the Rainich-Misner-Wheeler theory) in the

construction of a radially symmetric model of the hydrogenic

electromagnetic interaction. Up to an appropriately chosen

constant I obtain the Bohr radii. I also show that general

quantization conditions follow from this discrete theory.

This result is important for two reasons. First it repre­

sents a monistically geometric theory which avoids both the

Grunbaum-type circularity and the problem of quantum uncer­

tainty discussed by Wheeler (in that quantization conditions

automatically emerge from geometric considerations). Sec­

ondly, it indicates that Einstein was probably right when he

insisted that quantum mechanics not be regarded as the onto­

logical foundation of physics.

To show that additional consequences emerge from

physical interpretations of the proposed generic formalism,

I also consider the SL(2,R)®SU(3)-symrnetric realization of

this formalism (that with the invariance properties of the

supersymrnetry theory). Specifically, I consider the simplest

7

topological configuration which is compatible with the nec­

essary discrete metric and is invariant under SL(2,R)®SU(3).

This topology consists of hexagonal subsets of the cartesian

2 product IR3 x IR3 -1

For consistency, I adopt the same metrical constant as

that which yields the Bohr atom and general quantization

conditions in the context of the SL(4,R)®U(1)-invariant

theory. I find that the resulting model is immediately in­

teresting in that it restricts action to a configuration of

discrete packages which collectively suggest the spectrum of

integral and half-integral spins. [The three spacetime ~3

in which rotations SL(2,R) occur, does not permit propaga­

tion of fields, so that the magnitudes of action just de-

scribed cannot be related to translational motion. More-

over, in the context of the constant scale factor which gave

the Bohr atom in the SL(4,R)®U(1)-invariant theory, the

magnitudes of action which constitute the above described

world consist of integral and half-integral multiples of h.]

If rotations of coordinate systems in the above de­

scribed topology are associated with SL(2,R)®SU(3)-

invariant interactions, as rotations in ordinary spacetime

are associated with gravitational interactions, and if the

packages of action which constitute the above described

world are associated with the total spin angular momenta of

initial and final states of interacting systems of

-------- --------------------------------

8

particles, then these initial and final states are reduced,

in the resulting theory, to angular positions in a qt,antized

spectrum of orientations which reference frames are permit-

2 ted to assume in m3 xm3 -1. And the replacement of ini-

tial particle states with final particle states through the

mediation of a field carrier is reduced to the replacement

of one angular position by another through the mediation of

2 a quantized rotation in ~3 x~3 -1 (where quantization

emerges strictly as a consequence of the discrete metrical

condition which is logically necessary for the monistically

geometric status of the proposed theory). The permitted ro­

tations just described turn out, in the proposed theory, to

be restricted (by nothing more than the geometric consider­

at:ions I have mentioned) to spin angular momenta which cor-

respond to the spins of the usual spin-1 bosons and their

supersymmetric partners, and the interactions which are rep-

resented very strongly resemble the interactions predicted

by the conventional supersymmetry theory.

The results I have described, involving quantization

and the interactions of spin states are encouraging because

they seem to vindicate a theory which does for particle in­

teractions what general relativity does for large-scale

gravitational interactions (reducing these interactions to

rotations in a relevant topological context). Since we

already have quantization and a supersymmetry theory, the

Ii'

9

above described results do not represent a scientific

advance. Nevertheless, they are philosophically interesting

in that they represent an important pedagogical reduction.

In the closing sentences of this dissertation I point

out that other aspects of the proposed monistically geomet­

ric theory may be useful from the scientific standpoint.

Specifically I show that the proposed theory suggests a

means of understanding quasar energy production, and that it

relates Planck's constant to the size of the universe. with

regard to the first matter, there is a geometric interpreta­

tion of mass-energy density which emerges from the proposed

monistically geometric theory, and this interpretation seems

to indicate a large modification of proper time in the re­

gions where quasars are located. If this is taken into con­

sideration in evaluating the observations of the quasars,

the energy productions of these objects are what would be

expected of galaxies which are of comparable size. with re­

gard to the relation of Planck's constant to the size of the

universe, when the device which, in the proposed

monistically geometric theory, represents mass-energy den­

sity, is sUbstituted into the appropriate expression relat­

ing red shift to the distance between observer and observed

object, the order of magnitude of one unit of action can be

calculated from the order of magnitude of the universal ra­

dius. Given this relationship, I use the precisely estab­

lished value of Planck's constant to predict a precise upper

Ii'

10

bound for the universal radius. On the basis of this pre­

diction, the proposed theory seems to lend itself, in prin­

ciple, to falsification or confirmation.

Finally, by considering the current experiments which

are designed to test Bell's inequality, and by interpreting

these experiments in terms of the proposed theory, I suggest

a classical geometric foundation for quantum logic. Before

beginning the discussions outlined above, I will review gen­

eral relativity, Maxwell's theory and the Rainich-Misner­

Wheeler theory. Moreover, I will review the gauge field

concept in the process of discussing quantization. Some

knowledge of these is necessary for an understanding of the

proposed theory.

----.- -------------------------------

It

CHAPTER TWO

A REVIEW OF GENERAL RELATIVITY AND THE MAXWELL THEORY

Mechanics is the study of motion. This includes motion

in the absence and in the presence of force fields. Geo­

metrizations of mechanics are models of the world in which

particles always follow paths which are the straightest

available. If a theory involves a metric, or means of mea­

suring distance, then the paths followed are also the short­

est available, with respect to that metric. Such paths are

called geodesics. The equations which describe the geo­

desics of the topologies (or spaces or manifolds) which are

associated with geometrizations of mechanics are invariant

in form under transformations from one system of coordinates

to another, and since elements which are characterized by

this kind of invariance are called "geometric elements," the

models in which equations of motion are equations of geode­

sics are labeled "geometric." If a description of motion is

to be geometric, then both velocity and acceleration must be

geometric quantities (the reason for this requirement will

presently become clear). But, in spacetime contexts where

straight, mutually perpendicular coordinate axes cannot be

introduced (contexts like the surface of a sphere), accel­

eration is not, itself, geometric, unless certain terms are

added to this rate of change. A field of curvature is de­

scribed by the coefficients (called "connection

11

12

coefficients") of the terms which must be added to the rate

of change of velocity, so that this rate of change can be

geometric. In geometrizations of mechanics these fields of

curvature are identified with physical fields. Geometriza­

tions of mechanics are transformed into dynamical theories

when field equations are formulated in terms of the connec­

tion coefficients mentioned above, and solved for particular

field sources or distributions of mass-energy, and the re­

sulting connection coefficients are substituted into the

geodesic equations of motion.

By a geometrization of mechanics in the context of

gravity (the geometrization which constitutes the general

theory of relativity) one means a geometric model in which

all particles which interact with a gravitational field

follow geodesics with respect to the metric which is

determined (up to a universally constant scale factor) by

the gravitational field source. To understand in detail how

this works one first observes that a 3-dimensional manifold

is inadequate. This requires mathematics which has not yet

been introduced, but I will accompany the discussion with

two helpful references.

Consider a field with potential v(x,y,z), and a test

particle which is coupled to this field. There are several

paths through the field's domain which the test particle

could traverse with the same energy

------------------------------------------------

Ii'

(1) 2 E = (1/2)mv + v(X,y,z).

However, according to the principle of Maupertuis (Marion,

1969, p. 217), there is among these a unique path which

minimizes the action integral

P2

(2) mIvds; P1

P2

i.e. which satisfies the variational principle 6I vds = 0,

P1

or

P2

(3) 6I [(21m) (E-V)] 1/2dS = O. P1

13

since (21m) (E-V) is regarded as constant, the variational

principle (3) can also be interpreted as (Adler, Bazin and

Schiffer, 1975, p. 6)

where

(5) ds = V(2/m) (E-V)ds.

If we characterize a differential geometry by the metric ds,

then the Euler-Lagrange equations corresponding to (4)

--------------------------- _._-_ .....

fl

14

indeed represent a geodesic. But for a particle of differ­

ent mass and/or total energy, the metric ds would be a dif­

ferent one, which conflicts with our intuitive notions of

3-space.

Having now seen that a geometrization of mechanics in

the context of gravity cannot be associated with a 3-space

topology, we are in a better position to appreciate

Einstein's formulation of this geometrization in terms of a

spacetime topology. since he thought of spacetime as having

a metric (or means of measuring distance), Einstein con­

structed a model of those physical interactions which are

influenced by a gravitational field in terms of a Riemannian

4-geometry. The following is an account of this formula­

tion, beginning with the usual observations concerning the

"flat" or "zero-field" limit of the postulated topological­

geometrical system.

The path of an object which moves with constant veloc­

ity (with constant speed and in a straight, 3-dimensional

trajectory) is a straight line in spacetime:

S

t

Figure 1. Spacetime Path of Inertial Frame

-_ .. -... _--_ .... _._-----------------------------------

15

The slope AS/At of this line corresponds to the ve-

locity of the object. Objects which do not accelerate con­

stitute what are called inertial reference frames, and

spacetime contexts in which all frames are inertial are

called flat or non-curvilinear spacetimes.

The path of an accelerating object is clearly curvilin-

ear in spacetime,

------~--~~-----------+ t

Figure 2. Spacetime Path of Non-Inertial Frame.

and the velocity of the accelerating object is therefore de­

pendent upon the object's spacetime position. We say that

this velocity is a function of the object's spacetime posi­

tion. The instantaneous velocity of an accelerating object

is given by

(6) ds lim S(t+At) - set) dt = At-O At

The infinitesimal increment, ds, in the object's position,

with respect to an infinitesimal increment of time, dt, is

Ii'

16

referred to mathematically as the derivative of S with re­

spect to t. In the limit described by expression (6) the

chord through points P and Q of Figure 2 clearly approaches

the tangent to the curve at point P. Observe that the posi­

tion, s, of the object and the velocity, V, of the object

can also be dependent upon; i.e. be functions of the several

components xi : i=1,2,3,4 of spacetime position, so that a

"partial derivative"

(7) as --r-ax~

= lim S(Xi+AXi ) _ S(xi) AXi-+O Axl

i=1,2,3,4

of S (and an analogous derivative of V) can also be defined

for each 4-position component xi. In this context, ds/dt is

expressed as

(8) dS Cit =

Objects which accelerate constitute what are called

non-inertial reference frames, and a spacetime context in

which frames are non-inertial is called a curvilinear

spacetime.

Previous to his geometrization of mechanics in the con­

text of gravity, Einstein addressed the problem of choosing

the correct set of transformations from one inertial frame

to another. The Maxwell equations, which describe the elec­

tromagnetic interactions which occur in the macro-world, had

turned out to be invariant under a set of transformations

-------------------------------------------------

Ii

17

which came to be called the Lorentz transformations, while

the equations of motion of Newtonian Mechanics are invariant

under a different set of transformations proposed by

Gallileo. This inconsistency, of course, presented a prob­

lem for physicists. Einstein was impressed with the compre­

hensive nature of Maxwell's theory and he suspected that

Newtonian relativity (invariance in form of physical laws

under Gallilean transformations) would be replaced with

"Lorentz relativity" (at the time of Einstein's work the

Lorentz name was not widely associated with the Lorentz

transformations) if the principle of relativity (invariance

of the laws of physics under ~ transferal of reference

from one inertial frame to another) were conjoined with the

postulate that the speed of light is a universal constant.

Since Einstein inferred from the postulate that the

speed of light is a universal constant that there is no sig­

nal faster than light speed in a vacuum, Einstein's model of

the universe from the standpoint of a given inertial ob­

server was represented by a configuration which he called a

light cone:

t

S

Figure 3. Light Cone.

since, in general, xi=vit, the condition

refers to the entire shaded area (not including the

boundary) of Figure 3. Einstein labeled this region of

spacetime the "time-like connected" region. The regions

represented by the boundaries of this domain he labeled

18

"light-like connected," and the remaining regions he called

"space-like connected." Observe that the unshaded region on

the left (right) can know about the unshaded region on the

right (left) only if there are signals which move faster

than light speed.

The universe which is depicted by Figure 3, of course,

involves spacetime topology, and expression (9) indicates

the form a metric must assume on this topology. Since, for

light-like connected spacetime loci,

and since, according to expression (9)

-_._. ---_. ------------------------------

for the time-like connected region (the region in which

physics occurs), the 4-metric must assume a form

(lOa)

The signs of these terms (+---) are said to constitute the

"signature" of the postulated geometry. An alternative to

this convention is provided by adopting xO=ct, so that

(11)

The 4-metric (11) is now the more popular construction.

19

The above described conjunction of the relativity prin-

ciple with the postulated universally constant status of

light speed leads directly to the set of transformations

(from one inertial frame to another) which came to be called

the Lorentz transformations. To demonstrate this let us re-

call that if we are in a flat spacetime; i.e. if there are

straight world lines which can be adopted as axes, then we

can introduce rectangular coordinate axes:

20 S

In this rectangular, cartesian context, the theorem of Pythagoras can be applied.

Figure 4. The Theorem of Pythagoras in Special Relativity.

If we are in a flat spacetime, general spacetime displace­

ments (represented by the radius vector in Figure 4) are

clearly invariant under rotations of the coordinate axes.

What do such rotations mean physically? To answer this I

note (see Figure 5 below) that velocities (ax/at) of

observed particles are seen in the unprimed frame as faster

than in the primed, or rotated frame. The rotations of axes

depicted in Figure 5 therefore represent transferals of ref-

erence from one inertial frame to another inertial frame.

How do the coordinates themselves transform under rotations

of axes? Let us consider an arbitrary rotation a:

---_.-_ .. __ .- -----------------------------------

Il

I

X

x

I

ct

ct

Figure 5. Lorentz Transformations.

21

In the context of a signature +--- on spacetime, Figure

5 indicates that

(12) =

or that

or that

(14a)

The result (14a) tells us that the separation between in­

stants of time become very large, in the primed frame from

the vantage point of the unprimed frame, as vx-c, where

Vx denotes the relative speed in the x-direction of the

primed inertial observer with respect to the unprimed

------------- ---------------------------------

22

observer, and c denotes the speed of light. On the basis of

analogous considerations we can write

(14b)

Expression (14b) tells us that length (in the primed frame

from the vantage point of the unprimed frame) contracts in

the direc,tion of the primed frame's motion (with respect to

observers in the unprimed frame). The transformations de-

scribed by (14a) and (14b) are called Lorentz transforma-

tions, due to a certain historical consideration (Marion,

1969, p.9S). There are also Lorentz transformations on the

other components of 4-position which are due to the rota­

tions depicted in Figure 5. If motion is strictly in the

x1=direction (where x1=x, x2=y, x3=z and x4=ct), then

(Marion, 1969, p. 101)

x - vt (14c) xl

, 1 = J 1-v2/c2

, (14d) xl = x2

, (14e) x3 = x 3

(14f) t' = t-{VLc2}X1 J 1_v2/c2

These may also be expressed by

, (14g) xi = >. •• x.

l.J J

-------------------------------------------------

It

23

where

r : 0 0 iay

1 0 0 P = vic (14h) Aij =

0 1 0 I 0 'Y = 1 i

~1_p2 '-iay 0 0 y , L

, -J

Lorentz transformations (14h) act on the components of all

4-objects (objects with four components) and only those

objects which transform as invariants under (14h) are

thought of (in the Einstein theory just described) as objec­

tive descriptions of the physical world. The Lorentz trans­

formations, under which the Maxwell equations are invariant

in form, were therefore derived by Einstein as the correct

transformations for the context where an unspecified prin­

ciple of relativity is conjoined with the assumption that

light speed is a universal constant. This hypothesis, and

the resulting theoretical derivation of the Lorentz trans­

formations constitute what is called the "special theory of

relativity." When Newtonian mechanics was adapted so that

equations of motion were Lorentz invariant, the result was

called special relativistic mechanics. It was this adapta­

tion of Newtonian mechanics to special relativity which

produced Einstein's famous result E=mc2 •

The Lorentz transformations (14h) constitute what is

known as a transformation group, which means that:

It

•

24

1. The set of transformations is closed under a composition

of transformations (by "closed under a composition 0"

we mean that if a and b are elements of the set, then

aob is also an element).

2. The set is associative under this composition: If a, b

and' c are elements of the set then (aob)oc = ao(boc).

3. There is an identity transformation I such that loa =

aoI = a for all a in the set.

4. There is an inverse element a-1 for each element a in

the set such that a-loa = aoa-1 = I.

5. aob = boa, for all elements a,b in the set.

The Lorentz group is usually designated by SL(4,R),

which is read "special linear rotation group over four real

dimensions." "Special" refers to the fact that the coordi­

nate axes which are rotated are straight, mutually perpen­

dicular axes. "Linear" refers to certain mathematical

properties of the transformations:

1. ao(b+c) = aob+aoc: a, band c being transformations

(since there is a matrix for every linear transformation

"+,, can be thought of as matrix addition and the compo­

sition "0" can be thought of as matrix multiplication).

2. (b+c)oa = boa+coa: a, band c being transformations.

3. aao(b) = (aa)o(b) = ao(ab): a a real or complex constant.

If the group is non-commutative or non-abelian then 5.

is removed from the group axioms.

25

The rotations of curvilinear 4-axes in spacetime, which

I will now consider, constitute a second transformation

group designated GL(4,R) and read "general linear rotation

group over four real dimensions."

The action of the matrix (aji ) of a Lorentz transforma­

tion on the components of 4-position:

(15) 3

= E x)' a), i j=O

represents the constant projections of old axes onto new

axes. Thus, the a ji of the Lorentz transformation (15) can

be diagonalized (replaced by components of a matrix in which

nonzero components are restricted to the diagonal) by the

introduction of mutually perpendicular axes as the coordi­

nate axes which are associated with both primed and unprimed

coordinate systems. I now direct our attention to transfor­

mations which cannot be associated with diagonalizable ma-

trices. These are the transformations of coordinates from

an accelerating, or non-inertial frame to another such

frame. In this context straight, mutually perpendicular,

world lines do not exist (c.f. the surface of a sphere), so

that curvilinear coordinate axes (which cannot be perpen­

dicular; c.f. spherical coordinates) must be used. In

this context ordinary inner products, or constant

-------------------------------------------

oz

projections of i onto j do not exist.

must be replaced by

(16)

Thus, lJ i,j

where gij *6ij , and where the gij are functions of 4-

position (if the g .. were constant, then, clearly straight 1.)

26

axes xi,xj could be introduced; i.e. then diagonalizability

would exist).

In this context it is clearly necessary to distinguish

between entities with superscripts and those with sub­

scripts. Elements vi: i=O,1,2,3 which transform like

(17) -i v ax~ J. = --J v : ax i=O,1,2,3

under the group GL(4,R), of general rotations in spacetime

(rotations of curvilinear axes) are said to transform as the

components of "vectors," while elements Ai: i=O,1,2,3 which

transform like

(18) ax j = -i A

J. ax

are said to transform as the components of "co-vectors."

II

i1 .n More generally, entities v , ••• ,1 which transform under

general rotations in spacetime like

(19) .1 .n :v1 , ••• ,].

.1 . n ajC1 ai1

= -,..1\11 •• -. n ax) ax)

.1 . n v) , ••• ,)

are said to transform as the components of a rank (n,O)

tensor, while entities A. . which transform like 11 ,···,1n

(20) A. • l.1,···,1n =

axj1 jn .

ax A. . :1=1,2,3 n ) 'I' ••• ,) n -r- ... ~ .l.

ax1 1 ax1

are said to transform as the components of a rank (O,n)

tensor.

Vectors are clearly special cases of tensors. Two

.1 .n .1 .n tensors A1 , ••• ,1 and B7 '··"7 are equal if and only if

i 1 ,···,im 11 ,···,1m

all components with corresponding indices are equal. Sums

and differences are defined for tensors of the same rank.

"Tensor products" of tensors of rank (p,q) and tensors of

rank (r,s) are tensors of rank (p+r, q+s). Finally, divi­

sion of tensorial quantities is not defined.

Tensors are important in that equations involving

tensorial terms are invariant in form under transferals of

reference from one non-inertial frame to another. To see

this we write

27

-----------------------------------------------

Ii

(21) i 1 .n .1 in ¥ , ... ,1 = §1 , ••• ,

which implies that

1 n

. (22) .1 jn .1 jn

(T) , ••• , - s) , ••• , ) axi ""'axi --T"1 --.--ax) ax)

= 0,

which (since it must be assumed for tensors which have well

defined transformation properties: non-singular tensors,

.k .k that ax1 lax) *0) implies that

.1 .n .1 .n ( 23) T)' ••• , ) = s) , ••• , J •

Based upon this observation one concludes that a theory

which lends itself to description in terms of tensorial

equations is an objective theory.

(24)

Magnitudes of vectors (Vi) are defined as follows

2 i i j 3 i dr (v ) = gil' v v = ~ v vi'

i=o

since the gij are also assumed non-singular, the magnitude

of the co-vector is analogously defined:

(25) 3 i ~ A A ••

i=o 1

Inner products are generally defined by

(26)

-------------------------------------------------------------------------------

29

or by

(27 ) ij g AiBj •

As observed above, the formulation of geodesic equa­

tions of motion requires that the derivative of velocity be

tensorial. Let us now address this problem. Because the

entities (Ai+dAi ) and (Ai+SAi ) are vectors (SAi is the

change in the component Ai which results from the transport

of the vector (Ai) along a geodesic axis xk, at an angle

with xk which is constant. since xk is not generally a

straight axis, SAi is not generally zero), and because the

difference of vectors is a vector, the entity (dAi-SAi) is a

vector. Thus, if we express the change SAi, which is due to

the "parallel transport" of the vector, by

i i Ajdxk (28) SA = rjk (the negative sign is conventional),

we find that the increment

of the vector (Ai) is also a vector. The expression (29)

describes what is called the covariant, or tensorial de­

rivative of the vector (Ai). Covariant derivatives of

co-vectors and of all tensors are analogously defined. The

coefficients r~k are known as affine connection coefficients.

On the basis of the above discussion we see that if a

coordinate system is appropriately chosen (so that the axes

are as straight as the straightest world lines), then the

----------------------------------------- - ... ,., ..

I',' t

30

connection coefficients describe the curvature of the

• topological-geometrical context. This realization motivated

Einstein to observe that covariant equations of motion

(30)

can be formulated only at the expense of introducing into

those equations the field of intrinsic spacetime curvature

which is (in the above mentioned context) described by the

connection coefficients. He identified this field of curva-

ture with the gravitational field.

Before discussing the Riemannian curvature tensor and

the Einstein gravitational field equations, we express the

connection coefficients in terms of the metrical coeffi-

cients, gij' We first show that the metrical coefficients

are tensorial components and that the covariant derivative

of this tensor is zero. Specifically, since (Ai) is a

co-vector, and since (Aj) is a vector, the quotient theorem

(Marion, 1969, p. 29) requires from the expression

(31) A. = g .. Aj l. l.]

that (gij) also be a vector. Secondly, the relation

(32) DA. = g .. DAj l. l.]

holds for DAi , as for all tensorial components. But, from

(31) ,

-------_ .... ------------------------------------

It

31

so that

But, comparing this with (33), we see that

(35) Dgij = 0: i,j=O,1,2,3.

Now since (Thomas, 1972, p. 70)

(36)

(signs conventional), we obtain

(37)

Thus, permuting the indices i,k,l in the cyclic way indi-

cated below, we obtain

and

(39)

--------------------------------------- .- .... - ,,,.

Ii

or, adding (38) and (39) and subtracting (37) we obtain

(Landau & Lifschitz, 1971, p. 241)

(40) i gim (agmk agml

{kl} = (1/2) - +-axl axlt

32

Observe that if the gmk are constants, then the coefficients i {kl} are zero.

Given connection coefficients which were in terms of

the metrical coefficients, Einstein's strategy was to formu­

late field equations involving the {~l}' solve them for the

gij (which he interpreted as the components of the gravita­

tional 4-potential) and SUbstitute these values, for a given

field source, into the equations of motion

( 41)

i (the coefficients {jk} are called Christoffel connection

coefficients), thereby describing the dynamics of a par­

ticular gravitational field; i.e. the motion of a test par­

ticle in a particular gravitational field. Let us now dis-

cuss the Riemannian curvature tensor, and Einstein's

gravitational field equations.

Consider a rectangle in a flat plane.

Q=constant

dx~

Figure 6. Parallel Displacement in a Flat Space.

If a vector (Ai) is transported from point A to point C so

that the angle Q is held constant, then (Ai) will have

the same direction at C as at A, and the direction of (Ai)

at C will be the same regardless of whether the vector is

first transported along dxi and then along dxk or vice

versa. The same result clearly occurs if the rectangle is

drawn on the surface of a cylinder. However, let us now

33

draw the rectangle on a 2-sphere. In this case the sides of

the rectangle will not be straight lines but geodesics of

the sphere:

c

B

Figure 7. Parallel Displacement in a curvilinear Space.

Ii

34

Let us again transport a vector (Ai) from the point A to the

point C by introducing a reference angle Q which orients

(Ai) with respect to the tangent to the axis xi. As dis­

cussed above, the orientation of the vector (Ai) at C is not

the same as at A, and its orientation at C depends upon the

route of its transport from A to c. since the angle Q

which represents the orientation of (Ai) with respect to the

tangent to dxi and with respect to the line which is perpen­

dic~lar, or normal to dxk is held constant, the transport of

(Ai) which we have described is a parallel transport. It

is, of course, equivalent to the parallel transport which is

represented albegraically by expression (28).

Let us represent the components of the covariant de-i i rivative of (A ) by A Ilk. In the context of this nota-

tion the difference between (Ai+dAi ) which is indicated by

the vector field [Ai(Xj )] at C, and (Ai+SAi ), which is

indicated by the parallel transport of (Ai) from the point A

to the point C is

(42) Aj

lIi//k: i,j, k=O,1,2,3

if the path of transport is first dxi and then dxk. How­

ever, this difference is

(43) A j II k II i : j , i, k=O, 1,2, 3

if the path of transport is first dxk and then dxi. Thus,

the difference between (42) and (43) is

_.--_ ... - ....•. _----------------------------------

1/

(44) j j

A ~i~k - A ~k~i·

since the difference described by (44) is zero if and only

if the surface upon which the rectangle is drawn is flat,

the expression (44) is thought of as describing the curva­

ture of the surface. Since expressions (42) and (43) de­

scribetensorial components, and since the difference be­

tween two tensors is a tensor, expression (44) also

describes tensorial components.

35

Expression (44) can be written in a more familiar form

by recalling the following:

( 45)

so that

(46) t j - Aj - t j + {j }tr {s }tjs i II k - II i II k - ilk rk i - ik

or, substituting (45) into (46),

(47) A j Aj { j 1 Ar { j } Ar + II i Ilk = I ilk + ir'1 k + ir I k

Interchanging the indices i and k in expression (47) we ob­

tain

--_ .. _ .. _ .. _---_._-------------------------------

It'

Thus, the difference (44) is given in terms of the

Christoffel connection coefficients by (Adler, Bazin &

Schiffer, 1975, p. 148)

( 49)

The expression (Rjikl ) is called the Riemannian curvature

tensor.

36

The general theory of relativity began as a generaliza­

tion of special relativity; i.e. as a model of the world in

which the laws of physics are invariant under a transferal

of reference from one non-inertial (or accelerating) refer­

ence frame to another. In this kind of model the laws of

physics would, of course, be invariant under a transferal of

reference from one kind of non-inertial frame to any other

kind of non-inertial frame. This observation prompted

Einstein to consider a thought experiment in which ex­

periments are done in an elevator situated on the surface of

a gravitational field source where the acceleration due to

gravity is g, and also in a second elevator which is moving

upward through space with acceleration g. Einstein observed

that if general relativity is to hold, there should be no

37

experiment which one can do within the elevator which would

distinguish the two situations. His critics said that a

beam of light propagated in a direction parallel to the el-

eva tor floor would bend downward in the second case, but not

in the first; i.e. his critics maintained that general

relativity cannot hold. But Einstein expressed the opinion

that general relativity does hold, and that as a consequence

of this, light propagated parallel to the elevator floor

would be bent downward in the first situation as well as in

the second. This, of course, implied that the gravitational

field is equivalent to a curvature of spacetime; i.e. that

the equations of motion of energy (e.g. light) as well as

the equations of motion of massive objects are just equa­

tions of spacetime geodesics. It also implied that star

light would be bent in the gravitational field of the sun,

thus exposing general relativity to falsification. But this

prediction was confirmed by Eddington during a solar eclipse

in 1919 (Adler, Bazin & Schiffer, 1975, p. 4).

Due to the implications of general relativity for the

motions of test objects in gravitational field, Einstein re­

garded general relativity as a new theory of gravitation.

Since Einstein wanted this theory to be analogue of Newton's

theory, he invoked certain analogies which I will now ex­

plain. The Laplace equation

(50) 2 V r/> = -41rp

--_ ... __ .. _------------------------------------

It

38

(where ~ denotes the scales gravitational field potential,

where

(51)

and where p denotes the density of mass in 3-space) pro­

vides the field equation for Newton's gravitational theory.

Thus, Einstein sought a tensor representing 4-curvature

which could be regarded as proportional to the

energy-momentum tensor, which describes the 4-distribution

of mass and energy (Landau & Lifschitz, 1971, p. 268).

[Recall that mass and energy are equivalent in the context

of relativity (Landau & Lifschitz, 1971, p. 9)J. Since the

energy-momentum tensor is a symmetric, rank-2 tensor [which

means either rank-(2,O) or rank-(O,2)J tensor, with zero di­

vergence (Quigg, 1984, p. 33), Einstein sought a tensor de­

scribing 4-curvature which was also characterized by these

mathematical properties.

We have seen that the tensor (gij)' which is employed

to raise and lower indices, is necessary for the definition

of the inner product in 4-space. We now employ the gij once

again to introduce an operation called tensor contraction.

If we multiply the components of the Riemann curvature

tensor by components gij as follows

(52) R = grk R • jl rjkl'

--~--~-----. --------------------------------------

Ii'

39

we obtain a rank-(0,2) tensor, describing 4-curvature, which

is symmetric in its indices. However, this tensor is not

characterized by zero divergence, which is, mathematically,

to say that

(53)

However, since the divergence of (Ril) is given by (Adler,

Bazin & Schiffer, 1975, p. 170)

(54) jl 1 jl • R ~l = (~g R) ~l· R = gi Rj

j i'

Einstein observed that

and adopted the tensor

(56) j,l = 0,1,2,3

as that which would be regarded as proportional to the

energy-momentum tensor. The tensor (56) is known as the

Einstein tensor. The Einstein gravitational field equations

are therefore given by

where C is a universal constant. In a vacuum, the equations

(57) reduce to:

-----------------------------------------

(58) Rjl - ~ gjlR = 0: j,l = 0,1,2,3,

which are known as the Einstein vacuum equations.

The equations (58) were solved by Schwarzschild, who

found that (Adler, Bazin & schiffer, 1975, pp. 185-199)

(59)

40

where m is proportional to the mass of the gravitational

field source and r is proportional to radial position. If

the data characterizing the sun and the positions of the

planets and asteroids about the sun are substituted into the

Schwarzschild solution, and if this specific solution of the

Einstein vacuum equations is substituted into the equations

of motion (41), then the 4-trajectories of these masses, in­

cluding the shifts of'the perihellia can be accurately cal­

culated (Adler, Bazin & Schiffer, 1975, pp. 199-209).

Newton's gravitational theory had not predicted the shifts

of the perihellia, and since the perihellion of Mercury is

measurable, even with the instruments which were available

in the nineteenth century, the shifts of the perihellia pre­

sented a problem in the context of the Newtonian theory.

That general relativity correctly predicts the shifts of the

perihellia is regarded as one of the outstanding accomplish­

ments of this theory.

-------- -------------------------------------

The "black hole" phenomenon is also predicted by gen­

eral relativity. Note on the basis of the expression

(60) ~~,= (1- 2~),

41

which derives from the Schwarzschild solution, that if a

frame of reference is observed approaching the radial posi­

tion r=2m from a distance, then the ratio dt/dt' of the time

interval dt', characterizing the observer's frame, to the

time interval, dt', characterizing the observed frame, from

the vantage point of the observer's frame will approach zero

as the observed frame approaches r=2m, indicating that

dt'_oo in the observer's frame. Because dt' character-

izes photon frames approaching the radial position r=2m from

within this radius, light coming from within is prevented

from reaching this radius, from the vantage points of dis-

tant observers (Misner, Thorne & .fueeler, 1974, pp.

884-887).

The Maxwell theory describes all of the electromagnetic

interactions which occur in the macro-world; i.e. in the

world of everyday experience. The Gauss law of electric

flux is

~~

(61) VoE = p

where

42

is the electric field vector [the word "vector" here simply

• refers to an entity with magnitude and direction, but the

mathematical object consisting of the components of both the

electric and magnetic fields does transform under Lorentz

transformations as a tensor (Landau & Lifschitz, 1971, pp.

21, 62)], where p denotes the density of electrical

charge within some closed surface:

.. E

Figure 8. Gauss' Law of Electric Flux.

and where

" (1,0,0) 1.1 = -l A " (63) V = Il a. t.j: 1.2 = (0,1,0)

j J { = (0,0,1) 3

denotes "flux" or increment in the direction normal to the

surface. The inner product "0" is defined in terms of

the angle orienting A with respect to B as follows:

..... ..a. ~ ~ -I. .....lo

(64) AoB = IAIIBlcosa: IAI is the magnitude of A.

43

The equation (61) states that total electric flux (change in

electric field strength) across the surface is proportional

to the charge density inside.

The Gauss law of magnetic flux is

...... .,. (65) VoH = 0

meaning that total magnetic flux across a surface encompass­

ing a magnet is zero:

Figure 9. Gauss' Law of Magnetic Flux.

The Faraday-Lenz law which describes the counterclock­

wise curl or rotation of the electric field (looking down)

due to the downward motion of a magnet:

Figure 10. The Faraday-Lenz Law.

------------------------------------------------

Il

or the production of magnetic flux downward by a counter­

clockwise (looking down) current in the coil is

-"

(66) ~ -" -1 8H VxE = -6 at'

where the cross product "x" is defined by

(67)

where the angle ~ represents the orientation of A with

respect to B:

Figure 11.

-'"' B

2 x

The Cross Product.

Ampere's law, which describes the curl or rotation,

VxH, of the magnetic field vector, as pictured below,

44

due to the current, j, through a straight conductor was gen­

eralized by Maxwell because he observed that although there

is no actual current across the gap separating capacitors:

Ii'

capacitors

7 " ( ~~ 0 ~ e (k< H /H / H

"" curling magnetic . field

45

capacitors are objects which store electric potential energy or voltage

Figure 12. Maxwell's Generalization of Ampere's Law.

there is a curling magnetic field about the displacement

separating the capacitors. Due to this phenomenon Maxwell

postulated a "virtual current;" i.e. an entity yielding the

consequences of current across the displacement separating

the capacitors. He called this phenomenon a "displacement

current." Ampere's law as generalized by Maxwell is:

~ ~

(68) ....:..... l' 1 8E VxH = cJ + C §to

We have given each of Maxwell's equations: (61), (65), (66)

and (68) in Lorentz-Heaviside units.

The Maxwell equations: (61), (65), (66) and (68) can be

expressed in a form which is invariant under the Lorentz

group of rotations in flat or ("Minkowskian") spacetime by

adopting a notation

(69)

where

(70)

k -" (A ) = (V,A)

~

E = 8A '_ at ~

VV

Il

46

and

" A

'1 '2 '3 ,.io ~~

(71) H = VxA = a1 a2 a

3

Al A2 A3

To accomplish this we combine (70) and (71) to form the ma­

trix

0 El E2 E3

(72) Fkj = ajAk-akAj = -E 0 H3 -H 1 2 -E 2 -H 3 0 Hl

-E 3 H2 -H 1 0

We then show that (Fkj ) is GL(4,R)-tensorial (GL(4,R)­

covariant) by choosing an arbitrary 4-vector (Aj) and ob­

serving that the curl ajAk-akAj of (Aj ) is

GL(4,R)-tensorial. Finally we show that certain Lorentz­

tensorial equations in terms of (Fkj) are equivalent to the

Maxwell equations. choosing an arbitrary 4-vector (Ai):

i=0,1,2,3, let us observe that the GL(4-R)-covariant de­

rivative of (Ai) is

(73) Ai Ilk = Ai I k - <fk }Ar:

k,i = 0,1,2,3,

which implies that the curl

-------" ----------------------------------

Il

of (Ak) is a GL(4-R)-covariant entity. Clearly the matrix

(72) is invariant under the additions of arbitrary scalers

(GL(4,R)-invariant quantities) to the potential components

47

Aj (such additions are called gauge transformations of the

first kind--subsequently we will encounter gauge transforma­

tions of the second kind). Thus, the Aj of expression (72)

are arbitrary. In the above discussion we have also assumed

that the Aj of (72) are the components of a 4-vector. A

curl (Aj /i_Ai/j) can be obtained from a curl

iajb E = Eiajb (called the Levi-Cevita symbol) is equal to 1

for even permutations of the indices, and -1 for odd

permutations.

It can be verified by a straightforward calculation

that the GL(4,R)-covariant derivative of the skew-symmetric,

rank-2 tensor (F .. ), and the "cyclic" permutation of indices ~J

indicated in the following expression yields zero:

This result is due strictly to the skew-symmetry of the

tensor (Fki ). It can also be directly verified (by substi­

tuting the scaler potential V for AO and the vector poten­

tial A for (A1 ,A2 ,A3» that the equations (75) are

equivalent to the Maxwell equations (65) and (66).

It

If we assume that the entity J=(p,j), representing

what is called 4-current, is a 4-vector (transforms under

GL(4,R) as a vector), then the equations

48

are GL(4,R)-invariant [since the left side of equation (76)

is the covariant derivative of (Fjk)]. Thus, in the limit

represented by flat (or Minkowskian) spacetime, where the

Christoffel symbols are zero, the equations

(77) F jk - J j . )'-0 1 ~ 3 Ik - . -, ,&.,

are Lorentz-invariant. We will now demonstrate that

equations (77) are equivalent to the Maxwell equations (61)

and (68) by substituting particular values for the indices

into (77) and referring to expression (72) (representing

what is called the electromagnetic field tensor). Spe­

cifically, for j=O we obtain

(78)

or, referring to (72), we obtain

(79)

which is precisely the Maxwell equation (61). Secondly, for

j=l we obtain

(80)

or

(81) 1 .

= -Ji C

49

which is the xl component of the Maxwell equations (68).

The conservation of 4-current (which is the relativis­

tic analogue of charge conservation) can be derived from

equation (76): the GL(4,R)-invariant analogue of the Max­

well equations (77). To obtain this result we observe that

This is because the components Fab are skew-symmetric in the

indices a and b, and the derivative operators a2/axaaxb

are symmetric in a and b.

To clarify notation which will appear in Chapter Three,

I observe the following: The terms <tk}Fak of expression

(76) sum to zero (because the connection coefficients are

symmetric in a and k and the components Fak a~e skew­

symmetric in these indices). Secondly, the contracted con­

nection coefficients

(83 )

can be written in terms of the determinant of the metric

tensor:

(84) gijllh = gijlh -k

{hi }gkj k

- {hj }gik = 0,

which implies that

(85) ij i - {j } 0 g gijlh- (hi) hj =

or that

(86)

Or, since g = E~ikg. for any given value of i (where ~ik k l.k

ik is the co-factor of gik) 8g/8gik = ~ (Bronson, 1970,

pp. 23-27).

Thus,

(87)

so that

(88)

Thus, since g<O,

50

-----.... --------------------------------

It

In the context of these considerations then, equation (76)

can be expressed by

(90) ab ab 1.r:: ak F lib = F Ib + -(V-g) IkF

a = J •

Thus, if we write

(91) (j4ab = Fab{_g

and

I"\'a = a.r: (92) u J V-g ,

o

we can write expression (76) as

(93) '2jab _'Ta o· Ib -u •

51

Entities such as (91) and (92) have transformation

properties slightly different from those which characterize

tensors. Since

(94)

we have, taking the determinant of both sides,

(95)

so that

-----.... - ---------------------------------

(96) 1 2 a(x ,x , ••• )

-1 -2 I, a(x ,x , •.• )

1 2 a(x ,x , ••• ) where I I is the inverse of the absolute

-1 -2 a(x ,x , ••• )

value of the Jacobian of the coordinate transformation

(o'Neill, 1983, p. 10). Thus, the expression (91) trans­

forms under GL(4,R) like

(97) -1 -a(x ,x, ••• )

1 2 a(x,x, ... )

52

(this is assuming a positive Jacobian), and the entity (92)

transforms like

-1 -2 a(x ,x , ••• ) (98) 1 2 a(x ,x , ••• )

Objects which transform like (97) and (98) are called tensor

densities, and objects which transform like (96) are called

scaler densities. In the context of these remarks the

Einstein-Maxwell equation (100) of Chapter Three will be in

telligible, as will the discussion of Chapter Three.

CHAPTER THREE

A REVIEW OF THE RAINICH-MISNER-WHEELER THEORY

The Rainich-Misner-Wheeler theory is based upon the

Einstein-Maxwell equations for a context where the electro­

magnetic field is due to a point charge (i.e., for a context

which is characterized by a virtual absence of charged mat-

ter):

(99)

(100)

(101)

Rkj = CTkj

k' ~ J Ij = 0

k,j = 0,1,2,3

k = 0,1,2,3

k = 0,1,2,3.

Since, in this context, (Adler, et al., 1983, pp.

80-85) the electromagnetic field is due to a point charge,

the energy-Momentum tensor

(102)

reduces to the stress-energy tensor

(103)

(Because the mass-energy distribution consists of nothing

more than a point charge, the terms involving pare

53

.. , - .. 'I

Ii

negligible.) Thus (due to the skew-symmetry of the

electromagnetic field tensor) the trace Tkk of (Tkj) is

zero.

(104)

i.e.,

(105)

Consequently, since

since

k R = cT k'

the 4-curvature scaler R is zero for contexts where the

54

electromagnetic field is due to a point charge. It is for

this reason that the gravitational field equations (57) as-

sume, for the context we are considering, the form (99).

By employing extensive mathematical machinery (Adler,

et al., 1975) one can transform the Einstein-Maxwell equa-

tions (99), (100) and (101) into equations which are

strictly in terms of the Ricci tensor and its derivatives:

(106a)

(106b)

(106C)

(106d)

R Rj = kj a

(in real coordinates)

---------------------------------------------

55

These equations are called the Rainich Misner-Wheeler equa­

tions. These are non-linear differential equations, and

therefore difficult to solve. However, if one introduces a

skew-symmetric, rank-2 tensor which satisfies the Maxwell

equations, then the equations (106) reduce to the Einstein­

Maxwell equations (99), (100) and (101). This skew­

symmetric, rank-2 tensor replaces the electromagnetic field

tensor but it is interpreted as having no physical reality

[i.e., this tensor is interpreted as nothing more than a

convenient mathematical construct in a purely geometric

theory of gravitation and classic (non-quantum) electro­

dynamics] (Misner and Wheeler, 1957, pp. 525-603).

Since the Rainich-Misner-Wheeler equations hold only in

a context which is devoid of charged matter, Wheeler at­

tempted to extend the Rainich-Misner-Wheeler theory to the

general domain by replacing charged field sources with areas

where "handles" attach to the 4-topology which is associated

with the proposed theory:

4-sphere

Figure 13. Wheeler's Handle Topology

------- -----------------------------------------------------------------------

1/

56

Lines of force are thought of as entering and issuing from

these areas via routes along the handles, so that there is

no need to represent field sources in terms of singularities

(loci where the mathematical formalism is not well-defined).

In this context the energy-momentum tensor can, in general,

be described in terms of the stress-energy tensor (103)

i.e., in this context gravitational field sources as well as

electromagnetic field sources can apparently be reduced to a

mathematical construct having no physical reality: the

skew-symmetric, rank-2 tensor introduced above, and its de­

rivatives. It is on the basis of these considerations that

the theory of Rainich, Misner and Wheeler is interpreted as

a monistically geometric theory.

To emphasize the distinction between a geometrization

of mechanics such as that which constitutes general relativ­

ity and the monistically geometric (i.e., "geometrody­

namical") concept discussed here let us briefly digress to

recall the geometric substance of the general theory of

relativity.

The most directly geometric expression of Einstein's

hypothesis is that the geodesics of the spacetime manifold

determine the world lines of test particles in gravitational

fields. To demonstrate this we again employ the principle

of least action which results in the Euler-Lagrange equa­

tions (Marion, 1969, p. 197-201). Since action has the

------------------------------------------

Ii'

dimensions of the product mc2dr, where mc2 represents the

energy of a test particle and where dr is an increment of

proper time, we can introduce a metrical variational prin-

ciple

r (107) 6J dr = 0

ro

where

(108) dr gjkxjxk dr 'k dxk vk = x =JT =

We therefore obtain the Euler-Lagrange equations

(109) :

(L is called the Lagrangian of the problem). Thus, since

(110) Ilvll = II (Vk

) II

57

(This derivative is based upon the fact that = Ilvll and upon the chain rule for derivatives) (Thomas, 1972, p.

86); and since

(111)

Ii

58

so that

(112)

we obtain

or

(114)

or finally

(115)

which are equivalent to the equations of motion (41).

According to general relativity, however, (see page 28)

these geodesic equations are determined for specific 4-

regions by the distributions of mass-energy which character­

ize those 4-regions (or for vacuous regions by location and

orientation of motion with respect to gravitational field

sources, and by the characteristics of those field sources).

*Since L= II v II (i. e., since we are seeking the optimal II v II , IIvll is regarded as constant here).

---_ ... - ... _-----------------------------------

IX

59

Thus, general relativity bases the curvature field upon the

• 4-distribution of mass-energy.

Let us now contrast this account with the aspirations

of Clifford, who envisioned a complete reduction of physics

to geometry. Clifford (1876, p. 157-158) stated: "There is

nothing in the world except empty, curved space. Matter,

charge and electromagnetism and other fields are only

manifestations of the bending of space. Physics is geom­

etry." As we have seen, it is this kind of account of na­

ture which Wheeler is pursuing in his interpretation of the

Rainich-Misner-Wheeler theory.

It may be, of course, that Clifford would not approve

of Wheeler's proposal of a 4-dimensional theory as a pos­

sible realization of the Clifford ideal. However, as we

discussed in Chapter One, a 3-dimensional context for phys­

ics precludes even a geometrization of mechanics, which is

clearly a necessary precursor of a monistic geometrization.

---------------------------------------------

fl

CHAPTER FOUR

A SUGGESTIVE TOPOLOGICAL-GEOMETRICAL FORMALISM

I will now consider a topological-geometrical formalism

which is similar to the Rainich-Misner-Wheeler theory in

that it yields general Einstein-Maxwell type equations

apriori if I insist upon a monistically geometric interpre­

tation of the formalism.

Consider the Langragian:

(116) 3 k 0 n2-1

( E g oX xJ + E k,j=O kJ a=l

3 ~ oa .. k) 1/2 u9kxx :

k=O a

where

by which vector magnitudes are measured on the manifold

(117)

The terms 3 E gakdXadXk represent the projections onto

k=O

the spacetime manifold of displacements dxa on the parameter

space

(118)

60

of the group SU(n), and the metrical coefficients gak:

a=1, ••• ,n2-1 k=0,1,2,3 are associated with the components

of 4-potential of the SU(n)-covariant field, or generic

Yang-Mills field [derived from considerations analagous to

those which yield the U(l)-covariant field: the ordinary

derivatives of wave functions ¢ do not transform co­a

61

variantly under SU(n) so that an SU(n)-covariant derivative

must be introduced in terms of connection coefficients

Aa • 2 k. a=l, ••• ,n -1; k=0,1,2,3. These connection coefficients

constitute the 4-potential of the SU(n)-covariant field].

Displacement on the manifold (117) is given by

(119) 3 'k 3 n

2-1

cdr =c ( ,E gJ'kdxJdX + E E gk",dxkdxa) 1/2. .. J ,k=O k=O a=l u;

The invariance of the inner product (164) under

GL(4,R)®SU(n) can be easily verified. In particular, by the

invariance under GL(4,R) of the first group of terms

3 k ' E gk'x xJ of (116), one obtains (summing over repeated

k, j=O J

Latin indices from 0 through 3 and over repeated Greek indi­

ces from 1 through n2-1):

(120)

62

(Recall that the SU(n) aspect of the transformation as well

·k· as the GL(4,R) aspect leaves the terms x xk untouched).

Secondly, since the action of SU(n) on qph is

(121)

dx6 and on ar- is ..

(122)

Thus,

(123)

* (g h + ic' g hX~)

Q Q~ ,

=

or, expanding the right side of equation (122) I obtain

(124)

*where the Cr are the Cartan structural constants. Q~

... 'I

Finally, exchanging dummy indices Q and v in the second

term, the second and third terms of (123) cancel, and

recalling that we are considering only first order (in the

parameters xQ) gauge theory, the fourth term of (123)

also vanishes. I therefore obtain

(125)

63

I will now derive the Euler-Lagrange equations which corre-

spond to the metric Lagrangian (116). [Observe that this

amounts to deriving equations describing the projections

4 n2-1 , onto spacetime of the geodesics ofm xm , w1th respect

to the geometry (118).] Again, I adopt the usual summation

. convention: pairs (one upper, one lower, both the same let-

ter) of Latin indices in a term are summed from 0 through 3,

and pairs of Greek indices from 1 through n2-1. In this

context (116) becomes

(126)

In formulating the Euler-Lagrange equations I consider the

cylindricity condition (the condition agij/axQ = agpk/ax

Q

= axk/axQ = ax··k/awQ = 0.' ~ )'-0 1 2 3'~-1 n2 1 ... , - , , , , ..... - , ... , -,

64

which avoids modification of general relativity) which,

broadly interpreted, requires that

(127) a (g .. xixj + gPixPxj ) 0 - =

aXQ 1)

and that

(128) a (g .. XiXj iI',j 0: ,k dxk - + gpi X) = x = dr aX

Q 1) .. It is assumed that the same is also true of the derivatives of

gijXixj + gPixPxi and gijXixj + gPiil'xi with respect

to xQ. Given these considerations I take the derivatives of

the metrical Lagrangian with respect to variables xi and xi

only.

(129)

and

(130)

and

(131)

Thus, assuming that L=1, I obtain

d aL dTal .. x

Dr

65

Now since (dxQ/dr) is assumed covariantly [under SU(n)] con-.. stant, I obtain D/dr' (xa) = 0, or ..

(132)

where the

(133)

Q

d (~) CiT dr .. Q/J dx 1. k = C g"'Q arx , 1 fifJ ..

represent the connection coefficients which characterize the

SU(n)-covariant derivative, D/~: thus observing that

(134)

and that

(135)

I obtain

(136) d aL al ;; axl - axl

1 ag 1 + - (~

2 agk

1 ag 1 = (~ 2 axk

where the "A" in the last term denotes the skew-symmetry of

the indices 1 and k, or, multiplying (136) by glj, I obtain

------------------------------------------------

IT

(137) ql.jxl + qlj [ik,lj,;,!;« + !Ii- (a::~ - ::l:~ ::";« #1

lj ,8 dx'" .. k + g c~ gal 1\ gpk a;:-x = 0, . ,

or, interchanging indices a and,., in the last term of (137)

I obtain

(138)

where

k,l=0,1,2,3 (139)

2 a=1, •• ,n -1

66

are the components of the generic Yang-Mills field (Yang­

Mills, 1951). Thus, since the components of the covariantly

constant SU(n)-tensorial object (dxOi/dr) are interpreted as

the coupling parameters eOi (which couple the test particle

to the Yang-Mills field), I can express (138) as

(140)

67

Observe that if the gik: i,k=0,1,2,3 are constants,

then equations (140) reduce to the equtaions which were ob­

tained by S. K. Wong which describe, for the classical

limit, the motion of a test particle which is coupled to an

SU(n)-covariant field (Wong, 1970, pp. 689-694). I there-

fore regard equations (140) as a generalization, incorporat­

ing classical gravitation, of the equations which were ob-

tained by Wong.

Writing the eQ as components of (dxQ/dr), one might

also express equation (140) as

i=0,1,2,3 (141)

2 Q=1, ••• ,n -1

interpreting these GL(4,R)®SU(n)-covariant equations as a

generalization of the equations of motion which were ob­

tained by Kaluza and Klein (Bell, 1987) in their unified

theory of gravitation and electrodynamics.

If we wish to extend a monistically geometric interpre­

tation to the above formalism, then it is clearly necessary

that the Yang-Mills 4-current be equated to 4-curvature. In

this context Einstein type field equations are aprior: We

have

(142)

It

where Rjk=Rajak(Rajbk: a,j,b,k=0,1,2,3, ••• ,3+n2-1 are the

components of the Riemann curvature tensor in this 5-

dimensional context) and where C is a universal constant.

These equations can also be separately expressed as the

Einstein equations of general relativity:

(143)

plus the equations

(144)

where j=0,1,2,3; o=1, •• ,n2-1 and where Rjo=O for all

68

values of j,o because Tjo = 0 for all values of j,o (Clearly

R = 0 identically). Now since equation (144) can also 00

be written

.'

(145) Rjo = (lny:g) lolj-{~j} Ir+{~j}{~o}-(lny:g) In{jo} = 0,

or, since (In -g) lolj = 0, due to the cylindricity condi­tion,

(146) { r. } +.....L (.G) {~} oJ 1r ~-g V -g I k JO

But in the proposed theory {r.}: r,i=0,1,2,3; are the o~

components of the generic Yang-Mills field. Thus, the left

sides of equations (146) are the same as the left sides of

equations (90); i.e., the left sides of equation (146)

----.--- -------------------------------

constitute the GL(4,R)-covariant derivative of the generic

Yang-Mills field tensor. Thus, if I assume that

i=0,1,2,3 (147)

2 OI=l, ••• ,n -1

69

where (J .) denotes the generic Yang-Mills current, then the Oil.

sets of equations (143) and (144) become

(148)

(149)

where

(150)

is the stress-energy tensor in the context of the generic

Yang-Mills field.

(151)

The additional set of Maxwell-type field equations

{F"lk}=O, OIl.J

follow, of course, from the skew-symmetry of (Fij ) in i and

j. We are assuming that g = detgik for every value of 01.

70

Thus, as a consequence of its monisitcally geometric

interpretation, the proposed formalism becomes a classical,

unified field theory, combining the gravitational and

generic Yang-Mills fields. We know, of course, that there

is at least one additional requirement for this interpreta­

tion: that the metric on the manifold we associate with a

physical model be an integral multiple of a universal con­

stant. Let us now consider a consequence of this require­

ment.

Il

CHAPTER FIVE

BOHR QUANTIZATION AS A CONSEQUENCE OF THE

SL(4,R)®U(1)-INVARIANT REALIZATION OF

THE PROPOSED FORMALISM

If we introduce the realization of the proposed formal­

ism which is invariant under the symmetry principle

SL(4,R)®U(1) [where SL(4,R) is the Lorentz group in 4-

spacetime and U(l) is the group of position-dependent phase

transformations), then the set of field equations (148),

(149), (150), and (151) become

(152)

(153)

(154)

and

(155)

....... ir r i d"' I r = V-g J

These are the combined equations of Einstein and Maxwell.

For a point charge source such as a proton, equations (152),

(153) and (154) and (155) reduce to the equations (99),

(100), (101) and (103).

71

Ii

72

In this section I consider the radically symmetric so­

lution of this set of equations, and after conjoining this

formalism with the discrete metrical condition which is nec-

essary for a monistically geometric interpretation of the

formalism, I apply the modified theory as a model of the

hydrogenic electromagnetic interaction. As a result of this

application I obtain the Bohr model of hydrogen.

The radically symmetric solution of the set of equa-

tions (99), (100), (101) and (102) is

(156)

where

(157) = (1- ~ + 9.:2

2)

r r

where M is proportional to the mass of the field source,

where Q represents the electrical charge on the point which

provides the field source and where r denotes radical dis-

tance from the field source. This solution was produced by

Reissner and Nordstrom in 1918 (Misner, et al., 1974).

Thus, in adopting the discrete metrical condition

(158) C~T = nK: n=1,2,3, ••• ,

I obtain the result

73

T T

(159) ar = J dr = J goo dt = nk/c: n=1,2,3, ••• , o 0

where T represents the period of the electron in radial po­

sition r. Now in the virtual absence of gravity (which

characterizes the low-energy, micro-domain under consider­

ation), the metrical coefficient goo' which is given by ex­

pression (157), is essentially equal to Qr/r2, so that ex-

pression (159) reduces (essentially) to

T g~ (160) J dt = nK/c: n=1,2,3, ••• , 0 r2

or simply to

T (161) J (Q/r) cdt = nK: n=1,2,3, •••

0

Now, since the position coordinate r is fixed (has been

assigned a particular value) expression (161) can also be

written

(162) T J (Q/r) cdt = Q/r cT = nK: n=1,2,3, ••• o

Thus, if Q represents the electronic charge, and if T de­

notes the period of the electron (in radial position r) in

the modeled hydrogenic interaction, and if I substitute into

(162) in terms of the definition of period:

(163) T = 21rr/v

----_. __ ._-------------------------------------

Ii

(where r represents the orbital speed of the electron and

the condition of orbital stability) and the condition for

orbital stability:

(164) !uy2

r

(m is the electronic mass), I obtain

(165) n=1,2,3, ••• ,

74

or, if K2 = h2c 2/Q2, where h is Planck's constant, I obtain

the Bohr radii. Thus, I have shown that the quantization

ds=nK: n=1,2,3, ••• of the 4-metric is a necessary condition

for the monistically geometric interpretation of the pro­

posed formalism, and that this discrete metrical condition

leads apriori to the "old quantum theory" (the Bohr model of

hydrogen), provided that one chooses the appropriate scale

factor K. I will now show that the modern quantum theory

(the replacement of classical energy and momentum by the

usual operators) can also be accounted for as one of the al­

ternative conditions necessary to accomodate the s~me metri-

cal quantization and value of K.

Ii'

CHAPTER SIX

GENERAL QUANTIZATION

Bohr's quantization of angular momentum is given by

(166) mvdr =~: n=1,2,3, •• ,

where dr is an infinitesimal increment in the radial direc­

tion. (According to (165) r = n2~2/Q2m and according to

(164), Q2 = mv2r, so that mvr = ~.) But this is equivalent

to

(167) mvds 2;r' = nfi: n=1,2,3, •• ,

where ds represents displacement along a circular orbital.

Expression (167) can also be written as

(168) .... ..JIo

p.ds = 211'nif: n=l, 2 , 3 , •••

where p and ds respectively designate vectors of ordinary

3-momentum and 3-displacement. But the stationary state

(168) of the deBrogle wave ~ = eXP[-¥(P.dS)] entails that

(169) Dt/J = 0

or that

3 (170) d~ = i Il Ak ~dxk

k=l

75

It

3 (where 0 = E (BkdXk-iAkdXk) is the derivative which is

k=l

covariant under the group U(l) = (eio ) of local phase

transformations). That the phase-invariant derivative of

76

¢ is zero corresponds to the requirement that all nodes get

transformed equally under U(l)=(eio (Xk

)}o This precludes

increments of relative positions of nodes; i.e. precludes

increments of wave number.

But the condition

-i ........ -pods (171) D(e~ ) = 0

implies that

(172) -i .... S) o ( ..j1pods = 0,

or if coordinates are chosen so that p is strictly in the

xk-direction, (171) implies that

(173)

3 k J.' k But since the operation of i E Akdx upon,fiPkdx

k=l

is equal to the product of these expressions, an alternative

condition, which is equivalent to (171), entails that

------ ---------------------------------------------

Il

(174)

and that

(175) ...... p.ds = .n:

Thus, the replacement of ordinary 3-momentum Pk =

k=1,2,3 by the operator (-H/i)ak : k=1,2,3 is (in the

context of the scale factor which yields the Bohr atom)

just an alternative way of expressing the stationary state

or parallel displacement (169) of the deBrogle wave ~ =

exp(-~/i) p.ds; i.e. an alternative way of expressing

77

the discrete metrical condition (158) which is necessary for

the monistically geometric interpretation of the formalism

proposed in Chapter Four.

One is also motivated to replace classical energy with

the usual operator by observing that an alternative inter-

pretation of (165) gives

(176) !Edt

e~ = ei2~n.. n 1 2 3 =, , , ..• ,

which entails that D~ = 0, where

(177)

78

In this context, ~ = 0 implies that

(178) D(AEdt) = 0,

or that

(179) d(~Edt) = i E Akdxk clEdt) • k=l

Again, however, since the operation of iEAkdxk upon ~Edt is k

just the product of these expressions, an alternative condi­

tion which is equivalent to ~ = 0: ~ = exp ~dt entails

that

(180) .n d I dt = E,

and that

(181) Edt = 11

(The quantization conditions (174) and (180) are based

upon the space time signature + - - -.) Thus, the replace-

ments of classical momentum and energy with the operators,

which are the foundations of the Schrodinger and Dirac equa­

tions, can be accounted for as an alternative way of ex­

pressing the discrete metrical condition which entails the

result (165); i.e. as a way of expressing the metrical

It

79

condition which is necessary for the monistically geometric

interpretation of the proposed theory.

Chapter Seven discusses the existence of hexagonal lat­

tices as realizations of the proposed theory, but not the

uniqueness of these lattices. In the theory which is pro­

p~sed the lattice structure of a world consisting of m met­

rical levels (~r=nK: n=l, •.. ,m) is different from that of

a world constituted by q metrical levels: mlq. This is be­

cause physics is equated with geometry, so that geometry

(e.g. possibilities for increments of scale) must behave the

same under spacetime translations as under rotations in 2

spacetime X Rn -1 which involve no spacetime translation.

Thus, for the world which consists of two metrical levels.

(in physical language, the world of low to intermediate ener­

gy), a hexagonal lattice structure is requir~d (this argument

will become clear in. the context of Chapter Seven). This

uniqueness argument is important because it restricts our

choice of symmetries, for a.world of low to intermediately

high energies, to that which yields the spectrum of known

particles plus the standard supersymmetry theory. Specifical­

ly, in the context of the constant scale factor which yields

the quantization conditions described above, the vertices of

the above described lattice structure are characterized by

actions ~ and~/2. Moreover, in the context of the

SL(2,R)8SU(3)-invariance of the hexagonal lattice structure,

these actions, or angular momenta are unrelated to transla-

tional motion and are, therefore, identified as spins.

Il

CHAPTER SEVEN

THE SPIN ANGULAR MOMENTA WHICH ARE SUGGESTED BY THE

SL(2,R)®SU(3)-SYMMETRIC VERSION OF THE PROPOSED

FORMALISM

In the monistically geometric theory which I am consid­

ering, it is required (by the monistically geometric charac­

ter of the theory) that the associated metric be an integral

multiple of a universal constant. This metric is not there­

fore determined by the field equations (148), (149), (150)

and (151). How then are the metrical fields (gak) and (gjk)

to be determined? Due to the discrete nature of the postulated

theory, these fields will appear in spacetime as discrete pack­

ages of action which suggest spin angular momenta. Let us see

how this comes about.

IR3 x 32-1 32-1 The metric on the manifold IR (where m is the parameter space of SU(3» is given by

(182)

+

while the metric on spacetime, including projected displace-

ment, is

80

Ii

(183)

since the coefficients gap of the metric (182) are the

components of the cartan metric Cap = c c5a 'Y Cpc5 '

32-1 the sub-manifold ffi is flat. Thus, the manifold

2 ~3x ffi3 -1 is flat (in the same sense as a right-circular

81

cylinder), so that the world in which displacement is given

by the metric (182) can be diagramed in the context of the

following rectangular, cartesian coordinate system: E l'

E !

E J 1]32-1g xaxP

a,p=l ap 2

Figure 14. m3 x m3 -1 Admits a cartesian Coordinate System.

where E denotes the available energy of the world.

To permit a monistically geometric interpretation of

the proposed theory, and to permit consistency with Chapters

Five and Six (in the limit where the radius vector coincides

with the vertical axis), I adopt the condition of discrete-

ness

--------------------------------------- .......... .

Il

82

(184) Er = nfl: n=l, 2 , 3 , ••• ,..,

Due to the condition (183), the world, as modeled, reduces

to a series of concentric rings: E,!

Figure 15. 3 3 2-1 A Discrete Metric on m x m

Now when the radius vector Er coincides with the verti-

cal axis of Figure 14, the general metric (182) coincides

with the spacetime metric (183). In general, however, the

metric (182) projects displacements Er onto the vertical

(spacetime) axis which depend, in their magnitudes, upon the

orientation of the radius vector Er. Thus, since the

orientation of this radius vector is conventional in the

context I am considering, I am again confronted with a con­

tinuous (and therefore conventional) metric on spacetime.

So, since this conventional metric would contradict my ear-

lier assumption that the spacetime metric is intrinsic (the

hypothesis which made possible the results of Chapters Five

------ --------------------------------------

Ii'

83

and Six}, I must, for the sake of consistency, require that

the angle a, which designates the orientation in

2 m4 xmn -1 of the radius vector Er, be restricted to

integral multiples of some constant angular value a.

This angle is constrained by the requirement that the

world of Figure 13 reduce to a topological configuration

which is invariant under SU(3)®SL(2,R}. Since hexagonal

configurations are the simplest such topologies, I consider

the configuration:

E !

E T

E

Figure 16. A Discrete Metric on m3, and the Simplest SL(2,R}®SU(J)-Symmetric Space.

This configuration is interesting in that projections Er

onto the vertical axis of Figure 14 are given by

(18S) E! = E,!sinO': a = 30°,90°, ••• ,

which, by condition (184), is equivalent to

n=1,2, •• (186) Er = rrlrsinc:t:

c:t=30°, 90°, ••• ,

which yields the values

(187) 1'l ii 31i 211.' etc. 2' , 2 '

84

since fields cannot propagate in the 3-spacetime, lR3 , which

we are considering, the projections (187) of action onto

spacetime are not related to spacetime translation. Due to

this consideration and to the nature of the spectrum (187),

I identify the packages of action which constitute the

E! axis of Figure 16 as prototypes of particle systems

(possibly consisting of single particles) which are charac­

terized by spin angular momenta.

In Chapter Eight, I will interpret the SL(2,R)®SU(3)-

invariant rotations of the configuration which is depicted

by Figure 16 as interactions of the postulated particles

with the quantized field of curvature gak: 2 a=1, ••• ,3 -1;

k=0,1,2,3, which projects them into spacetime.

It

CHAPTER EIGHT

THE SUPERSYMMETRY THEORY AS A CONSEQUENCE

OF GEOMETRY

As explained in the introduction, if I regard the pack­

ages of action which are depicted by Figure 16 as initial and

final states of particle interactions (as indicated by the

nature of the values (187), and the fact that no propagation

of fields can occur in the 3-spacetime m3), then these ini­

tial and final states are reduced to angular positions in a

quantized spectrum of orientations which reference frames 2

are permitted to assume in m3 xm3 -1. And particle inter-

actions (the replacement of initial states by final states

through the mediation of a field carrier) are reduced to the

replacement of one angular position by another through a

quantized rotation of the refrence frame which is depicted

by Figure 16.

In this chapter I will show that the above interpretation

of the proposed formalism appears to be justified in that

the rotations which the formalism permits are characterized

by spin increments which correspond to the spins of bosons

and their supersymmetric partners; and in that the initial

and final spins which are associated with the permitted an­

gular increments correspond, in the proposed context, to the

initial and final spins which are associated with supersym­

metric interactions. Moreover (in addition to the

85

It

86

discussion regarding spin) I will show that the proposed

model contains a mechanism which provides a natural separa­

tion of strongly interacting systems, and leptonic, or

non-strongly interacting systems.

Finally, I will recall that the commutator and

anticommutator algebras which are associated with the group

SL(2,R)®SU(3) requires that for every particle, there is

a second particle which differs in its spin from the first

by n/2 (Wess and Bagger, 1983, p. 12).

with regard to the mechanism which separates baryonic

and leptonic interactions I observe that there are two cat­

egories of rotation which leave the configuration of Figure

16 fixed in orientation and shape. One of these consists of

multiples of 60° rotations of the entire hexagonal con­

figuration. The second consists of secondary rotations

(multiples of 120°) of spin-n/2 triplets such as those

connected by bold lines anG bold dashed lines below:

s=~

s=-11/2

Figure 17.

s=fi/2

\ I ,I .if",~ s=-!'i/2

, =;:;' , ,;.:.

, -- (negative signs indicate negative parity)

2 Triplets and Anti-Triplets on 1R3 x IR3 ~1

87

The additional particles which are required by the com­

mutator and anti-commutator algebras can be related,to the

particles of Figure 16 as follows:

--_ .. _-------------------------------------

Ii'

E

s=O --~-------+----~~~4-~E

I I

d».. ......

...... ......

......

Figure l8A. Hexagons Associated with the Additional Particles which are Required by the Commutator-Ant i-Commutator Algebra.

anti-squark triplet

s=-fl/2

quark triplet

88

Figure l8B. Triplets Involving the Additional Particles

89

In the context represented by Figures 18A and 18B,

• there are additional rotations which leave the configuration

of hexagons and triangles unchanged in shape and orienta­

tion. One of these is the rotation indicated by the dashed

arrow in Figure 18A. Another is exemplified by the rotation

of the dashed triplet of Figure 18B into the position of the

triplet, which is connected by solid lines, accompanied by

the rotation of the solid triangle into the position of the

dashed-line triangle.

The triplets of Figures 17 and 18B seem to associate

themselves with baryons, and the rotations of these triplets

(and of combinations of these triplets) seem to associate

themselves with strong interactions (Glashow-Gell-mann,

1961, pp. 45-46). Specifically, suppose I associate the

solid triangle of Figure 18B with a quark triplet, and the

dashed line triangle with an anti-squark triplet (as the

model seems to indicate).

Since the spin increments which are associated with an­

gular increments, or rotations are, in the proposed theory,

associated with the spins of mediators or field carriers,

the rotation indicated by the arrow of Figure 18B depicts

the interaction between a squark and a quark, mediated by a

gluino. In terms of a Feynman diagram, this is:

90

s=1l/2

~ s=o

Figure 19. Quark-Squark Interaction.

Similarly, suppose I associate the solid triangle of Figure

17 with a quark triplet, and the dashed triangle with an

anti-quark triplet (again, the model seems to indicate

this). Then the rotation depicted by the curved arrow of

Figure 17 represents the interaction of two quarks mediated

by a gluon. In terms of a Feynman diagram:

Figure 20. Quark-Squark Interaction.

Thirdly, suppose I associate rotations other than the

rotations among triplets with interactions other than the

strong interaction; i.e. with lcptonic interactions. In

this context the rotation indicates by the solid arrow of

Figure 18A depicts the interaction of two leptons, mediated

by a boson, and the dashed arrow of Figure 18A depicts the

interaction of a lepton and a slepton, mediated by a

photino, or wino. In terms of Feynman diagrams, these are

respectively:

and

s=If/2 }-vv s=~/2

Figure 21. Lepton-Lepton Interaction.

s=11:/2 ~~~ 040>

s=O

Figure 22. Lepton-Slepton Interaction.

Finally, I consider a higher energy level. Note that

if I attempt to depict the interaction of a massive boson

with another such boson in terms of the kind of model pro-

posed above:

spin-if

---f---t---F'---t--t-t~ E ' Ir-E-3-2-_-1g--X-a-x!'~' Va,p=! ap

Figure 23. Gravitational Interaction.

I require a graviton as the mediator.

91

Even this very cursory discussion of the proposed

theory strongly suggests the supersymmetry theory and its

conservation law. If I assign a number B=1/3 to each vertex

It

92

when that vertex participates in a rotation of triplets, and

if I assign a number L=l to each vertex when that vertex

participates in a rotation of the 1st energy level hexagon,

or of the hexagons immediately above and below, which are

associated with the additional particles (those regained by

the commutator and anti-commutator algebras), then all rota­

tions of verticals which can be represented as entirely for­

ward or backward in time conserve the evenness or oddness of

(188) R = L + 25 + 3B

(where S denotes spin, L denotes number of leptons and B de­

notes number of baryons). But this is the conservation law

which underlies the conventional supersymmetry theory. (Re­

call that B is associated with the verticles of triplets and

anti-triplets, while L is associated with the vertices of

hexagons.)

This concludes my discussion of supersymmetry. The re­

sult I have obtained is philosophically interesting in that

it seems to provide additinal confirmation (assuming the va­

lidity of the supersymmetry theory) of the discrete

geometrodynamics which I have proposed.

In the next chapter I will observe that this discrete

geometrodynamics may also be useful from the scientific

viewpoint in that it provides certain novel explanations and

predictions in the astro physical domain.

CHAPTER NINE

ASTROPHYSICAL PREDICTIONS OF THE PROPOSED

MONISTICALLY GEOMETRIC THEORY

The material of this chapter provides an explanation of

the apparently large energy productions of the quasars in

terms of the proposed theory. This chapter also relates

Planck's constant to the size of the universe, and thereby

provides a precise prediction regarding the radius of the

observable universe.

The theory which I have proposed is first and foremost

a representation of mass-energy in terms of spacetime, i.e.

a representation of mass-energy density in terms of 4-

curvature, which, in a radially symmetrical context, amounts

to a dialation, or wrinkling-up of spacetime. In this con­

text mass is proportional to 4-volume, and gravitational

scaler potential ~ = m/r is proportional to r4/r = r3:

i.e. in this context the scaler component goo of metrical

gravitational potential is given by

(189) 3 goo - r •

I now recall that luminosity, or electromagentic en­

ergy, E, production per unit time is, if I do not involve

93

It

94

time dialat.i.on, inversely proportional to the square of the

distance, L, separating radiation source and observer:

(190) 1 E = - . L2

Thus, if I wish to express a ratio of what I will call ac­

tual luminosity Eo: for example, that observed when the ra­

diation source is LO = 1 light-year from the source, and

apparent luminosity E: that observed when the observer is a

very large distance, L, from the source, I can write

(191)

Now, if I wish to apply expression (191) to radiation

from distant sources, in the context of the proposed theory,

then I must, due to the important dependence of goo upon

distance in this theory, modify (191) in terms of a time

dialation. For a spherically syn~etric universe this

modification takes the form

(192) dr = C~goo dt,

where, once again, goo is proportional to the cube of the

distance separating an observer on earth from the distant

radiation source. To do this modification I write

(193)

which yields

(194)

Now replacing r with the notation, L, I substitute into

(191) to obtain

(195) E

If I substitute into expression (195) the distance from

earth to the most distant quasar: Approximately 1010

c-years, I obtain

(196)

95

This ratio differs appreciably from that obtained by

excluding the effect of radial dialation as it occurs in the

context of the proposed theory. To demonstrate this I omit

the considerations (193) through (195), and assume that the

world lines connecting the present day earth with the

light-like event horizon are linear. In this context I ob­

tain the ratio

---------------------------------_ ........... .

96

(197) =

Now, due to their red shifts (3.5 or so), one can calculate

the distance, L, of the quasars from the earth, by utilizing

the equations which relate red shift to distance (Adler,

Bazin & Schiffer, 1975, p.139). The most distant quasar is

about 1.5x 1010 light-years away, and its observed luminos­

ity is about E = 1025 ergs/second, which means that, accord­

ing to the relation (197), the actual luminosity or energy

production is about 1045 ergs/second, an energy production

too large to be accounted for in terms of known processes

(this energy production is from one to one hundred times

that of a typical galaxy). However, if I take into consid­

eration the time dialation which occurs in the context of

the proposed model, then the actual luminosity of this qua-

sar is

(198) 1025 = 1033 ergs/so

~ o

Now a typical quasar is about one light month in diameter,

or about 10-1 light-years, whereas a typical galaxy is 103

or 104 light-years in diameter. If I multiply the value of

EO given by (198) by the size factor 105 (a galaxy which is

100,000 light-years across is about 105 times as large as a

typical quasar), then I obtain an actual energy production

of about 1038 ergs/second, which is typical of a galaxy of

104 light-years in diameter.

97

The interpretation of mass-energy density as a

dialation of spacetime; i.e. the interpretation of mass as

4-volume, which emerges from the proposed, monistically geo­

metric theory therefore provides one explanation of the ap­

parently large energy productions of the quasars, and

predicts an actual energy production which coincides with

what normal processes would lead one to expect.

This interpretation of the density of mass also enables

one to relate Planck's constant to the size of the universe.

I consider the equation

(199)

which relates red shift to the distance separating observer

and radiation source, and which does not depend upon the

principle of equivalence (a local principle: Adler, Bazin &

Schiffer, 1975, p. 139). ~V denotes the shift in

frequency of the radiation along the world line which

connects observer and radiation source. Vo represents

the frequency which is measured at a distance Lo from the

source (in the neighborhood of the source), ~~ denotes

the potential difference through which the radiation passes

from source to observer.

---_._-_._----------------------------------- ,- .. 'I

I now multiply the numerator and denominator of equa­

tion (199) by Planck's constant, h, to obtain

(200)

Finally, I calculate a~ in the context of the proposed

theory by recalling that ~ is proportional to r3. Inte­

grating over the distance L, from the earth to the

light-like event horizon (which is the edge of the observ­

able universe) I obtain

(201)

r=2x1010

A~ = J r 3dr = o

light-years 10

r41~X10 light-years

4

98

which yields about 4x1040 (light-years) 4. substituting into

(200) I obtain

AE 4X1040 (light-years) 4. (202)

since 4x1040 light-years is not in terms of the same system

of units as the denominator of the right side of (202), I

convert the numerator of the right side to gaussian units.

There are about 3X107 light seconds per light-year. Thus,

(202) becomes

,< - ", 'I

99

(203) h = 0.19 x 10-27 6E V~·

I therefore conclude that for a unit increment of action

AE/vO' h is 0.19x 10-27 erg-seconds. Thus, the in­

terpretation of mass density in terms of spacetime density,

permits that I derive a roughly correct value for Planck's

constant as a consequence the rough, currently accepted up­

per bound of the radius of the observable universe.

since Planck's constant has been measured accurately, I

utilize the propoosed theory t9 predict a new value for the

upper bound of the universal radius. According to the

theory I propose, the predicted distance to the light-like

event horizon is about

(204) 2.11 x 1010 light-years.

I now observe that the postulated geometrodynamical

theory may provide a classical foundation for quantum logic.

It

CHAPTER TEN

A CLASSICAL FOUNDATION FOR QUANTm~ LOGIC

In addition to the consequences just discussed, the

proposed theory may provide an explanation, in terms of

classical geometric considerations, regarding the outcome of

the current experiments which are designed to test Bell's

inequality (these experiments are refinements of the ex­

periment done by Alan Aspect and his associates at the Uni­

versity of Paris: Robinson, 1982, p. 432). This

explanation arises from a comparison of the proposed geomet­

ric model of the world, and the world as interpreted in

terms of quantum mechanics.

Let us consider a quantum mechanical world in which ev­

erything is measured. specifically let us assume that all

members of a particle ensemble are measured to determine

which of two mutually exclusive properties characterize each

particle. If the measurements are separated in space, and

in all frames of reference by a time interval small enough

to preclude the time-like connection of the measurements,

then (due to the special relativistic restriction which ex­

cludes faster-than-light signals) the events represented by

the space-separated measurements are truly independent.

In this context assume that each particle is character­

ized by (A,A), (A,B) or (B,B), and let (A,B,i) denote the

number of particles with properties A,B and i. If i must

100

--------------------------------------------

•

101

either be n+" or "-," (indicating the particles parity) then

according to common sense, "either - or" logic,

(205) n(A,B) = n(A,B,+) + n(A,B,-),

or since n(A,B,+) s n(A,+) (n(A,+) = n(A,A,+) + n(A,B,+»,

arid since n(A,B,-) S n(B,-),

(206) n(A,B) S n(A,+) + n(B,-).

This is essentially Bell's inequality.

Let us now consider the context in which it is not al-

ways determined whether a particle is characterized by "+"

or by 11_." In this case (according to Young's experiment),

each measurement influences or interferes with the other, so

that the probability

(207) P(A,B) = n(A,B,+)

n(A, B) +

n(A,B,-)

n(A,B)

which is given quantum mechanically by

(208)

(~ = c1~1 + c2~2) must, due to the above men­

tioned interference, be replaced by the description

-------- - --- -----------------------------

It

102

(209)

Now, in this case, it is clear that

(210) n(A,B) > n(A,B,+) + n(A,B,-)

so that one can no longer derive the inequality (206).

Thus, if the results of an experiment are to satisfy Bell's

inequality, then evidently the "+" or II_II nat.ure of each and

every particle must be accounted for. But why does it hap­

pen that expression (208) gives way to (209) if all par­

ticles are not accounted for as either "+" or "_?" Is there

an explanation in terms of classical physics? Until now we

have thought not, but in the context of the geometric model

I have proposed, the absence of the ability, in a given ob­

servation, to measure the spins (or parities "+" and II_II) of

particles is equivalent to the absence of an ability to

appropriate the postulated metric (if one cannot, ascertain

particle parity, then by the Figure 16 model, one cannot ap­

propriate the metric dr for that observation), so that,

if one accepts instrumentalism, there is no metric for that

observation. But, if there is no metric, then the spacetime

manifold is either continuous or rational in its topology,

and if the manifold is continuous or rational then there is

no separation between events. (Even in the rational case

there is another event between every two events.) And this

means that events are not independent, so that one

103

measurement can influence, or interfere with another without

a violation of special relativity. I submit that this clas­

sical consideration provides one means of understanding why

expression (208) must be replaced by (209) (why events

propagate like waves, involving interference) if all par­

ticles are not accounted for.

Let me make this argument explicit: The separation of

events indicated by the condition (198) entails an intrinsic

metric on spacetime; but according to instrumentalism, if an

observer cannot apply this metric, then, for this observer

there is no metric, so that, by modus tollens, there is no

separation of events.

---------------------------------------_ ...... - " ..

CHAPTER ELEVEN

INTERIM CONCLUSIONS AND SUGGESTIONS FOR

FURTHER DEVELOPMENT

Initially my purpose in the research, which led to the

foregoing dissertation, was to discover a geometric theory

which would avoid the problems confronted by the Rainich-

Misner-Wheeler Theory, and which would describe a larger do-

main than the RMW. such a theory was suggested to me by a

formalism which is based upon two considerations:

1) The broadest possible basis for tensorality, which

is a composition of symmetry principles involving

both co~"utator and anti-commutator algebra and

2) A discrete metrical condition to avoid logical cir­

cularity in the representation of field sources in

terms of geometry.

The first of these two characteristics is provided by a

4 n2 -1 4 real, topology 1R x 1R , where 1R is spacetime

2 and mn -1 is the parameter space of the SU(n) group. The

second is provided by a Riemannian geometry which is

modified by an integral discreteness condition.

When employed in the construction of a spherically sym­

metric hydrogenic model, this formalism yields the Bohr

atom, and when employed in the general representation of the

hydrogen atom, this discrete formalism yields general

104

------------------------------------------------

It

105

quantization conditions (providing at least a partial real­

ization of Einstein's hope that it would become unnecessary

to regard quantulll mechanics as the ontological foundation of

physics).

Thirdly, the SL(2,R)®SU(3)-covariant realization of

the proposed generic formalism yields a strictly geometric

reconstruction of the supersymmetry theory. Fourthly, the

interpretation of mass-energy density, which emerges from

the proposed theory, yields an explanation of the apparently

large energy productions of the quasars, and relates

Planck's constant to the size of the universe, thereby pre­

dicting a precise upper ground for the radius of the

observable universe. Finally, the proposed, monistically

geometric theory suggests a classical explanation of the

outcome of the experiments which were designed to test

Bell's inequality.

Due to the nature of these results (obtained strictly

from a monistically geometric interpretation of the proposed

formalism), my conclusion is that monistically geometric

theories can be feasible and productive in appropriate topo­

logical contexts. Also, I feel that the specific theory I

have proposed may represent a realization of the rationalism

which was pursued by Poincare and Descartes.

As stated earlier, if this discussion does represent a

vindication of rationalism, it does not recommend that we do

science by induction. It merely reconstructs a theoretical

__ ~_ 0---- _____________________________________ _

1/

106

structure apriori which has already been ascertained through

the conventional scientific method; thereby lending a philo­

sophically interesting "has to be" {;~l1aracter to the world it

predicts.

In my view the result I have obtained hints at a sort

of "genetic code" for the physics of the universe, this

structure represented by an intrinsic geometry on the to­

pologies which are determined by the aspects of the world

one wishes to consider. It is particularly interesting that

even the quantum mechanical world, which has usually been

regarded as constrained only by probability, can be derived

from the proposed geometric considerations (from a condition

which is logically necessary for the monistically geometric

interpretation of the proposed formalism).

possible lines along which this discussion might be ex­

panded are as follows:

1) A philosophical discussion of the apriori nature of

the proposed theory might be given, with particular

emphasis on the support it lends to rationalism and

Descartes.

2) A more extensive discussion of the metaphysical na­

ture of the theory would be interesting. The basic

question is about an ontological reduction--the

claim being that the furniture of the world can be

reduced to an intrinsically metrical spacetime,

provided one admits a larger world in which the

1/

•

107

furniture is also reduced to a topological-­

geometrical structure. What is most interesting,

of course (returning to the rationalistic aspect of

the theory), is that this apparently apriori

theory--a theory which seems very metaphysical--can

make predictions; i.e. can lend itself to empirical

falsification.

--.. -.-- ._--------------------------

1/

SELECTED BIBLIOGRAPHY

Adler, R., Bazin, M. & Schiffer, M. 1975. Introduction to General Relativity. New York: McGraw-Hill.

Bell, J. S. 1987. "Gravity." Fundamental Symmetries. Edited by P. Bloch, P. Pavlopoulos, and R. Klapisch. New York: Plenum Press, pp. 19-20.

Bronson, R. 1970. Matrix Methods. New York: Academic Press.

Clifford, W. K. 1876. "On the Space Theory of Matter." Proc Cambridge Phil Soc, 1.

Cohen-Tannoudji, C., Diu, B. and Laloe, F. 1977. Quantum Mechanics. New York: Wiley & Sons.

Glashow, W. L., and Gell-Mann, M. 1961. "Gauge Theories of vector Particles." Ann. Physics (U.S.A.), Vol. 15, No. 3.

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