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4 Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 74-17,764
CORUM, James Frederic, 1943- AN INVESTIGATION OF THE ANHOLONOMIC NATURE OF THE ROTATING LORENTZ TRANSFORMATION WITH APPLICATIONS TO ELECTRODYNAMICS.
The Ohio State University, Ph.D., 1974 Engineering, electrical
University Microfilms, A XEROX Company , Ann Arbor, Michigan
© 1974
JAMES EREDERIC CORUM
ALL RIGHTS RESERVED AN INVESTIGATION OF THE ANHOLONOMIC MATURE OF
THE ROTATING LORENTZ TRANSFORMATION WITH
APPLICATIONS TO ELECTRODYNAMICS
DISSERTATION
Presented in Partial Fulfillment of the
Requirements for the Degree Doctor of
Philosophy in the Graduate School o f The
Ohio State U niversity
by
James Frederic Corum, B.S., M.Sc.
********
The Ohio State U niversity
1974
Reading Committee: Approved By
C.V. Heer
U.H. Gerlach
H.C. Ko Adviser
Department of Electrical Engineering ACKNOWLEDGMENTS
The author wishes to express his sincere appreciation to the following people:
...... Professor Ulrich H. Gerlach, of the Department of Mathematics,
(who provided an introduction to Cartan calculus, and the vortex of
“Modern R elativity"), for his infinite office hours and for numerous discussions concerning those mystical derivations which are known ubique et ab omnibus*, and are so "triflin g " that no one bothers to exhibit them in the literature.
...... Professor C.V. Heer, o f the Department o f Physics, (whose lectures on Landau and Lifshitz provided an introduction to the structure of classical general relativity), for his stimulating and perceptive comments.
...... Professor H.C. Ko, of the Department of Electrical
Engineering, whose perceptive insight and penetrating inquisitiveness in the area of relativistic electrodynamics has been so inspiring over the last seven years.
...... Dr. L. Glaser o f B a tte l!e Memorial In s titu te - Columbus, for consultation on the relativistic expansion of elleptic integrals.
* everywhere and by everybody ...... Mr. Chiwei Chuang for several discussions and for his kind interest in this paper.
...... Dr. and Mrs. Fred T. Corum, fo r th e ir encouragement in th is academic venture.
And lastly, Linda - "Die du mir Jugend Und Freud' und Mut"
(Goethe) VITA
August 15, 1943 . . . Born - Natick, Massachusetts
1965 ...... B.S.E.E., Lov/ell Technological Institute, Lowell, Massachusetts
1966 ...... Electronic Engineer, National Security Agency, Washington, D.C.
1967 ...... M.Sc., The Ohio State University, Columbus, Ohio
1968 ...... Research A ssistant, The Ohio State U n ive rsity Radio Observatory
1970 - 1974 ...... Instructor, Department of Physics and Mathematics, Ohio In s titu te o f Technology, Columbus, Ohio
FIELDS OF STUDY
Major Field: Electrical Engineering
Studies in Radio Astronomy. Professors H.C. Ko and John D. Kraus
Studies in Electromagnetism. Professor R.G. Kouyoumjian
Studies in Antennas and Propagation. Professors C.H. Walter and J. Richmond
Studies in Communications. Professor W. Davis
Studies in Applied Mathematics. Professors H. Colson and S. Drobot
Studies in General R e la tiv ity . Professors C.V. Heer and U.H. Gerlach TABLE OF CONTENTS Page ACKNOWLEDGMENTS ...... i
VITA ...... H i
LIST OF FIGURES...... v ii
Chapter
I . INTRODUCTION...... 1
I I . MATHEMATICAL TOOLS ...... 8
] . A rithm etic space
2. Topological spaces
3. Manifolds
4. Curves on the manifold
5. Tangent vectors
6. Basis vectors
7. Differential forms
8. Anholonomic frames
9. Affine connection
10. Covariant d e riva tive
11. Interpretation of transformations
12. Anholonomic transform ations
13. Anholonomic coordinates
14. Lie derivatives
15. The object of anholonomity
iv page
I I I . ROTATIONAL WORLD LINES ...... 44
IV. CALCULATION OF THE ANHOLONOMIC OBJECT . . 53
1. Spherical polar coordinates
2. Cylindrical polar coordinates
V. INTERPRETATION OF THE TRANSFORMATION . . . 56
VI. THE SAGNAC EFFECT...... 65
V II. THOMAS PRECESSION...... 71
V III. ANHOLONOMIC FIELD THEORY ...... 75
IX. INVARIANT EQUATIONS ...... 80
1. Charge Invariance
2. Invariance of Maxwell's Equations
X. THE FIELDS ARISING FROM ROTATING SPHERICAL
CHARGE DISTRIBUTIONS...... 84
1. Charge Rotates
2. Observer Rotates
a. Method o f Anholonomic Frames
b. Method o f Galilean Frames
c. Method of "Instantaneous (M iller) Frames"
XI. SCHIFF'S PARADOX ...... 118
X II. A POINT CHARGE AND AN OBSERVER IN
RELATIVE ROTATION ...... 122
1. Observer a t Rest - Charge Rotates
2. Charge at Rest - Observer Rotates
X III. CONCLUSIONS...... 137 V APPENDIX Page
A. Summary of Cartan's Notation ...... 140
B. Summary of Classical Tensor Computations 141
C. Metrical Conventions ...... 142
D. Maxwell's Equations ...... 150
1. Cylindrical Coordinates
2. Spherical Coordinates
BIBLIOGRAPHY...... 153
Vi LIST OF ILLUSTRATIONS Page Fig. 1. A point in R ...... 9
Fig. 2. A manifold M ...... 12
Fig. 3. A differential curve in M ...... 14
Fig. 4. Tangent vectors and d iffe re n tia l forms. . 22
Fig. 5. Vectors and forms in polar coordinates. . 23
Fig. 6. Points on the manifold p, q* e M . . . . 26
Fig. 7. Geometrical interpretation of the Lie d e r iv a tiv e ...... 35
Fig. 8. An alternative interpretation of the anholonomic object ...... 38
Fig. 9. Basis vectors in polar coordinates . . . 41
Fig. 10. Differential forms in polar coordinates . 42
Fig. 11. Space-time diagram of a rotating observer 49
Fig. 12. World view of an inertial observer . . . 57
Fig. 13. Local tetrad frame of an inertial o b s e rv e r ...... 58
Fig. 14. World view of inertial observer .... 59
Fig. 15. Local tetrad frames fo r anholonomic o b s e rv e r ...... 60
Fig. 16. Local tetrads of galilean observer . . . 61
Fig. 17. Local tetrads of "instantaneous" observer ...... 62
Fig. 18. World line of a rotating observer . . . 66
Fig. 19. Thomas p re ce ssio n ...... 72
Fig. 20. A Charged s p h e r e ...... 84 v n* • page F1g. 21. Magnetic flux arising from rotation . . . 88
Fig. 22. A spherical charge distribtution . . . . 118
Fig. 23. Rotating o b se rve rs ...... 123
Fig. 24. D e fin itio n of w o r k ...... 124
Fig. 25. Calculations in special relativity . . . 138
Fig. 26. Calculations in anholonomic theory . . . . 139
v i i i I . INTRODUCTION
At the turn of the century, discussions of inertial motion led to the group of space-time transformations called the Lorentz Trans formations. Carried out in rectangular Minkowski space, they demon strate an intrinsic marriage of space and time. The possibility of global c o o rd in itiz a tio n and temporal synchronization in moving frames arises because o f the hoionom icity, or in te g ra b ility , o f transformed quantities. The success of the relativity transformations has demonstrated the impotency of the Galilean presuppositions of absolute space and universal absolute time.
Moreover, almost from the beginning, Ehrenfest (1909), Planck
(1910) and subsequently a surfeit of other authors continuing up to the present, have recognized a d ifficu lty concerning uniform rotation, or non-inertial motion. It would be unkind to maintain that several of the ensuing discussions generated more heat than light. However, like the twin paradox, the topic seems to reappear and be "re-resolved" in ten year intervals.
The purpose of this investigation shall be a new and original examination of the applicability of anon-rectangular rotational transformation to several classical problems. Our goal is a logically consistent explanation of the relativity of rotating electric fields in fla t space-time. V/hy is a new examination appropriate? The orthodox approach to problems of this nature has, in the past, been 1 2 a d ire c t ga lilea n ro ta tio n a l coordinate transform ation. However, such
an approach neglects (i.e ., supresses) all phenomena whose explanation
1s dependent upon the relativity of simultaneity. This is because
galilean transformations assume a universal absolute time. We w ill want to demonstrate the relatiorr between simultaneity and "anholono- mity" when rotation is present.
The formulation of the problem is most clearly executed within the framework of modern differential geometry. In particular, the concept of a field of orthonormal tetrad frames w ill provide a valu able tool for the clarification of the issues discussed, and for going to the heart of the galilean shortcoming.
Orthonormal tetrad frames are frames whose legs (basis vectors) are normal to each other and are o f u n it length. This orthonorm ality holds not only for the triad of spatial vectors, but is also appli cable to the vector pointinq into the time direction. In other words, each event of space-time is endowed with an orthonormal tetrad (a
Lorentz frame). A Lorentz frame is recognized by the fact that lengths and angles of vectors in such a frame are determined by the m etric
The choice of a Lorentz frame at an event is not unique. All frames at an event are related by a Lorentz transformation. At another event there is another set of frames real ted by a Lorentz transforma tion at that event. A choice of a frame at each event of space-time is a field of frames. 3
Measurements in physics are primarily performed with respect to orthonormal frames (often called "physical frames"). Which particular field of frames do observers in space-time choose? In fla t Minkowski space-time observers very often choose a global inertial field of orthonormal frames. This is the field of frames so fam iliar from special relativity. The metric (machinery for computing lengths and angles) has the form
ds^ = dx^ + dy^ + dz^ - c^dt^, which means that there is a global coordinate system (x, y, z, ct) whose tangent vectors
-y -y -y _ f 3 3 9 1 _3 Iex» ey» ez» e4' = i 3x, 3y, 3z, c 3t' at each event are orthonormal. Not only is this tetrad field tangent to coordinate curves, but each is normal to a stack of coordinate surfaces. (See Fig. 9). Such a field of tetrads constitutes a
"holonomic11 coordinate basis. That is, the coordinate differentials are integrable (i.e ., are exact differentials).
Now consider a field o f frames associated with rotating observers,
Their world lines are the helices discussed in Chapter III. Instead o f using the Cartesian coordinate system {x , y , z, c t} , use c y lin d r i cal polar coordinates (r, 0, z, ct} , so that the metric becomes
ds^ = dr^ + r^d9^ + dz^ - c^dt^, and the associated field of natural frames w ill be What is the field of frames associated with a set of observers
orbiting with an angular velocity + u> around a common center? The
coordinates for such observers are supposedly related to the coor
dinates of a set of inertial observers by
6' = 0 + wt
t* = t .
The above m etric becomes 12 2 ds^ = dr'2 + r'^ d9'^ + dz'^ + 2wr' d0* dt' - c^ [1 - ^ ^
If, however, we analogously try (in the historical tradition) to take tangents to the primed coordinate lines
e . * — r ' a r'
t at = 3 - e " 36'3 3 Gz' = 3 Z* ♦ 3 ef 3 3 t' as a field of natural frames, then we discover that these basis vectors are not orthogonal (?t , ® eQ, = 9t'e ' ^
Hov/ever, we obtain a field of orthogonal frames by letting the world lines of the ratating observers provide the new time-like direction and construct the spatial vectors as is done in Chapter III
The result is the field of frames: which is orthogonal and related to the inertial field of frames at
every space-time event by a Lorentz transformation. Nevertheless,
as desirable as these properties may be, they were obtained at a very
high price: these orthogonal basis vectors are not the tangents to
the coordinate axes of any coordinate system. As Pi rani points out,
" in general it is not possible to find a coordinate system
at large which incorporates the local coordinates used by different observers, because the latter, defined by the tetrads, w ill not in general be holonomic. It is the dichotomy between congruences of
curves, to which the tetrad vectors are tangent, and families of coordinate surfaces, which, geometrically speaking, lies at the root of the difficulties about measurements in general relativity theory."
(Pirani, 1956). This is always the state of affairs unless one uses
(and in the case of g ra v ity one is not even able to use) a global
in e r tia l coordinate system whose coordinate tangents y ie ld a fie ld
o f holonomic frames.
A derivation and discussion of the physical results of a number o f phenomena (1. Sagnac e ffe c t, 2. Thomas precession, 3. Heer's question, 4. Schiff's paradox) can be easily understood most directly in terms of quantities measured with respect to orthogonal frames. The reason why these phenomena seemed paradoxical in the 6 past v/as that the quantities (vectors, tensors, etc.) were determined with respect to a basis which was both non-orthogonal and non-rela- tivistic. Employing the orthonormal basis, derived in Chapter III, leads to a conceptually clear explanation of the above physical phenomena. Hov/ever, measurements w ill now be made w ith respect to a basis which is not a coordinate basis (i.e ., a nonholonomic basis).
This creates special problems when one wishes to compare vectors in two close by frames. Mot only do components o f a vector change because the vectors themselves differ (e.g., are not parallel), and because the basis vectors gradually turn from frame to frame, but also because the physical basis vectors have a certain "microscopic" twisting in them. This twisting makes them incapable of being derived from any coordinate lines, and thus incaoable of remaining normal to any coordinate surfaces. (See Fig. 13).
It is this special tv/isting effect, also known as anholonomity, which underlies any discussion of rotational relativity. The physical effects of twisted basis vectors is contained within a geometrical quantity known as the "object of anholonomity". This object, having its origin in the "rotating" (i.e ., ordinary rotation in three space plus pseudo-rotation of the space-time hyper-plane) Lorentz transformation, and therefore in the relativity of simultaneity, has been overlooked by previous investigators. In particular, those using a Lorentz type of transformation in electrodynamics completely C 3Ab aAa overlooked the appearance of ftab and proceeded to treat - j—g- as a tensor under the rotational transformation. The results of the present approach now clearify the physical content of rotational phe nomena in tnat geometrical effects are distinct from physical effects. 7 A word concerning notation is necessary. He have attempted to
use classical tensor calculus along with the concepts and notations
due to Schouten and to Cartan. This should simplify what Schrodinger
(1954, p.32) has called, "The bewildering dance of indices." If
mastered, Schouten's concepts permit one to be able to ". . . escape
the danger, pointed out be Hermann Weyl, to fa ll into 'Oriaien des
Forma1ismus'. . ." (Van Dantzig, 1954). And Cartan's beautiful and
concise method of differential forms avoids "... les debauches
d'indices." (Willmore, 1959, p. 261). In keeping with current
notational conventionalities, we let natural basis vectors be denoted
by e^. Physical basis vectors w ill have a circumflex over their
indices, and be denoted as e^. Thus, we w ill write
£ • e = g
and -* -*■ u v 'yv where the g are the components of the metric tensor and is the yv pv Lorentzian metric with signature (+, + , +, -). The next chapter, which is on mathematical tools, discusses these concepts in greater detail. This notation was dictated by the preprint edition of
Gravitation (Misner, Thorne and Wheeler, 1970) and is consistent with the Publication edition of that book, which appeared in fa ll o f 1973. I I . A BRIEF OUTLINE OF FUNDAMENTAL CONCEPTS
The purpose of this chapter is to review the basic presuppositions of tensor calculus on manifolds. It is not meant to be extensive and is by no means an elegant treatment. Hov/ever, it w ill provide a framework for the terms and concepts which are employed elsev/here in th is monograph. This discussion should serve as an in tro du ction to the nomenclature of modern tensor calculus for the engineer or s c ie n tis t, whose tensor background is usually stong in index ju gg lin g and deficient in geometrical conceptualization. Article one of
Misner, 1969, provides a similar introduction. The principle draw back of the engineering approach is that one is never informed of the properties of the space he is working in. Consequently, he never knows the range of validity of his analysis. This is because if one tries to coordinitize the points of the space globally, he is
"doomed to failure". It can be done only in a small patch, "but not fo r the whole space a t once. The best v/e can do is to cover the space w ith such regions, defining a coordinate system in each, and state how the coordinate systems are related in any two overlapping regions". (Whitney, 1937). The properties of the primitive concepts; set, order, number and dimension, are assumed.
1. Arithmetic Space
We start with the n-dimensional arithmetic (numerical, Cartesian) space Rn. A point in cartesian n-snace is an ordered n-tuple 8 (x-j, Xg* . . . , Xn ) w ith components x^eRn (Read: is an element of Rn). For example we may represent a point in R as the ordered pair (x-j, x2).
* ( * 1> x2^
------X t
Fig. 1 A point in R2.
The set of ordered pairs of elements from sets A and B written as
A x B = {(a,b) | a e A, b e B}
is termed the Cartesian product o f A and B. In p a rtic u la r, the
arithmetic space Rn is obtained from an n-fold iterated cartesian
product of R, the set of real numbers. (Wemust distinguish between
Cartesian and Euclidean space. They are not the same, as Euclidean
space has the additional metric structure. Furthermore, all points of a Euclidean space are equivalent - none is preferred above his
fellows. Hov/ever, in Rn the point (0, 0, . . . , 0) is definitely distinguishable from other points of Rn.)
Functions. Consider the sets A and B. A function from A into B, denoted f: A-+- B, is a mapping (correspondance) which assigns to each
a e A an element f(a) = b e B. If all B is covered by the mapping
(i.e., if for every value of b e B there exists at least one a e A
for which f(a ) = b), it is called a mapping onto B. Otherwise, into B. 10 2. Topological Spaces
Let X be any set (aggregate of elements). For example, an
unstructured "sea" of points. Further, let T be a collection of all
subsets of X such that
a) The empty set
b) If G-j, Gg» . . . , 5n e T then their intersections n belonq to T. n Gn e T. i= l c) The union of any collection Ga e T also belongs to T
(v/here a e I , and I is an a rb itra ry s e t). That is
if G e T for all a e I , then u e T. ael
When the above is fu lfille d , T is said to define a topology for X.
The set X together with the collection of subsets T is called a
topological space. The elements of T are called open sets of the
topological space. Any open set containing a point p e X is called a neighborhood of p.
Let X and Y be two topological spaces, and let f be a function
on X with values in Y. At every point p of X we have p e X ^f(p ) £ Y.
That is f: X ■+• Y. If f is a one-to-one mapping of X onto Y such that
f: X -► Y and f"^: Y -*• X are both continuous, then f is called a
homeomorphism and X and Y are homeomorphic (1-1 and in ve rta b le ).
Furthermore, if both f and f"^ are differentiable, f is called a
diffeomorphism. We shall le t L(X;Y) denote the space of maps from
X in to Y.
3. Manifolds
Let M be a topological space in which every point has a neigh
borhood homeomorphic to some open set in arithmetic n-space. M is 11 then said to be an n-dimensional manifold of points. (See: Steenrod,
1951, p. 21).
Let Ua be a collection of open neighborhoods covering M. (See
Fig. 2). For every a le t Qa be a homeomorphic mapping o f UQ in to the arith m etic space Rn. The homeomorphism Qa assigns to every point p e Ua the ordered n-tuple of real numbers (x^(p), x2(p), . . , xn(p)) which are the coordinates of Qa(p) in Rn. That is, for every p e U v/e have the coordinates at p,
Q(p) = (x^p), x2(p), . . . , xn(p)) which are real valued differentiable functions of p in the region where they are well defined, p is a point on the manifold M, Q is a coordinate map, U is a coordinate neighborhood, and the
A function f(p) on M assigns real numbers to points p e M. ’We w rite th is as f : M -► R. f can be represented in Rn space as a function of n-tuples. Such representations are called Coordinate representatives, f (p) = Frgp.fx^p). x2(p), . . . , xn(p)) where Frep: Rn*R and are d iffe re n t in each d iffe re n t system o f coordinates (x^p), x2(p), . . . , xn(p)), (yl (p), y2(p), . . . , yn(p)), etc. It is essential to keep clear the distinction between invariant!v defined points p e M and functions f e M, and their non-invariant representations as n-tuples Q(p) e Rn. (See: Misner, Thorne and Wheeler, 1970, p. 128). 12 n ,n R ■ Q jn Qa(P) Fig. 2. A manifold M, a point on the manifold p, two in te rse ctio n neighborhoods Ua, th e ir homeomorphisms Qa, the image of the point p in the local coordinates (Ua, Qa), and the coordinate trans formation Qjj ° Qa“ . 13 In Fig. 2, we are given a point p on the manifold M which belongs to the overlap of two local coordinate neighborhoods Ua n Ug. The point has different coordinates in the two local coordinate frames (Ua , Qa) and (Ug, Qg). These tv/o frames are Cr related i f the maps Qa o Og"^ (read: the composition of Qg"* followed by Qa) and Qg ° Qa“^ have continuous derivatives of order r. The mapping Og ° Qa“^ is called a coordinate transformation on Ua n Ug from the x1 coordinates into the y1 coordinates. Notice that a coordinate transformation is in the space Rn, not in M. M is said to be a differentiable manifold if (a) it is covered by a set of neighborhoods >Ja, each having the same number of coordi nates, and (b) different coordinate patches (Ua, Qa) in a common region related by a differentiable transformation. In general, no single Ua together with its homeomorphism Qa is able to cover all M. Consequently, the above coordinates of p are called the local coordinates of p in the local coordinate frame (UQ, Qa). The local coordinates, or coordinate patches, may be extended globally and cover the entire manifold only if some U = M. Otherwise global coordinates do not exist. This is the source of the geometrical ambiguity known as the Schwarzschild singularity, and also the source of difficu lty in discussing coordinates for rotating observers. Again, some words of caution concerning Fig. 2 and manifolds are in order. "When dealing with differentiable manifolds, one must always keep clear in one's mind the distinction between the space M and the space Rn to which i t is lo c a lly homeomorphic. The points 14 of the manifold are points of M, but the coordinates of the points o f M are points o f Rn. The space M is a topological space whose elements are of an unspecified nature, while the space Rn is just an ordinary arith m etic space whose elements are n-tuples of real numbers. Between these tv/o spaces there is a local homeomorphism defined by the local coordinates o f the points o f M." (Trautman, 1964, p. 70) 4. Curves in M The meaning of a curve, or parametrized curve, on the manifold may now be understood as a differentiable mapping of the real line onto M: C: S0 e R p0 = p(S0) e M. This is portrayed in Fig. 3. A congruence of curves is a family possessing the property that exactly one curve goes through each point of some region. N Fig. 3. A differentiable curve in M. 5. Tangent Vectors Let F°°(p) be the collection of all C" functions (smooth) f : U ■+• R, where p e U. A tangent vector, u, to M at p e M is defined as a function (operator) u: F°°(p) + R Such th a t fo r every f , g e F°°(p) and a, b e R (a) u is linear: tT(af + bg) = a tT(f) + b u(g) (b) u satisfies Leibnetz's product rule: u(fg) = f(p) u(g) + g(p) u(f). Consequently, u assigns a real number u(f) to every smooth function f on M. The set of all tangent vectors at p, Tp, forms a vector space over R and is called the tangent space at p. Chevalley points out that (Chevalley, 1946, p. 76), given any vector u, at p e M, tangent to M and f e F°°(p), the number u(f) is "often called the derivative of f in the direction u." In the same place, Chevalley also proves the statement "if (x^, x^, . . . , xn) is any coordinate system at p, and u any tangent vector at p, v/e have, for all f e F°°(p), 5 (f) 3xT This . . . , shows that a tangent vector is uniquely determined by the values it assigns to the functions of a coordinate system." (Also see Flanders, 1963, p. 53) 6. Basis Vectors: (See Trautman, 1964, p. 76) Consider the set n linear independent maps of F°°(p) R defined by 16 e,(f) - 2f_ a = 1, 2, n a 3Xa x = x(p) where f e F°°(p) and p e M. Since the ea(f) satisfy the tv/o criteria for tangent vectors mentioned above, they form a set of tangent vectors at p. A vector field is an assignment of a tangent vector to every point of M, consequently any tangent vector u(f) may be written as 3 ( f) - ua ea(f ) u a ^ fo r a ll f c F"(p). The ua are called the coordinate components of u with respect to the local coordinate system xa. The subscript (a) of ea numbers the vector, not the component of ea. Also notice that the basis vectors = a t axa t are the tangents to the local coordinate curves xb = cb(s). If we have another local coordinate system valid in the neigh borhood of p, xu, then the tangent vector u(f) can be written in terms of the basis and also in terms of -~r 3xa 9xu 3(f) = u“ 3 (f) = uu 51- . uu 2*1 3f = ua u 3xy 3xa 9xa 17 As a consequence o f the la s t e q u a lity, we can w rite the transform ation law for contravariant components of a vector as or ua = ha UP u Furthermore, at every point p on M ’ ha V As p runs over M, the frames £a(p) consitute a fibre bundle with M as the base space. (See: Bishop and Goldberg, 1968, p. 118). We w ill now consider the space of elements which is dual to the tangent space Tp. The elements of this cotangent space, T*, are called differential forms. 7. Differential Forms The relation between tangent vectors and cotangent vectors w ill now be examined to see how differential forms are bu ilt out of a field of cotangent vectors. In the definition given on page 15, a tangent vector was regarded as an operator which mapped real valued differentiable functions at p, F°°(p), into a real number. The set of a ll tangent vectors at p e M was called the tangent space Tp e M. In the local tangent space of a point p e M, one may introduce a set of basis vectors e^ which are defined with respect to the local coordinate variables Q(p) = (x^p), x2(p), . . . , xy(p)) by 18 The vectors - 2tt are tangent to the local coordinate curves on fl 3xM at p and form a basis for Tp. Besides the tangent space Tp at a point on the manifold we may consider the vector space L(Tp;R). This special case is denoted by Tp and is called the dual space of Tp. That is, at every point p e M, in addition to the set of basis vectors ep(p) we may consider the space o f vectors which is dual to the tangent space. Any element of the cotangent space Tp can be expressed in terms of basis covectors dual to the tangent vectors ep in the tangent space Tp. These covectors are uniquely determined by: - < if, > - < • The elements of Tp map any linear combination of Tp, e.g., the vector -* a -*■ u = ua ea, linearly into the real numbers. Consequently, the contravariant components of the tangent vector u are the result of letting the basis covectors operate on u: < ca, u > = < aa , ub eb > h 3, ">■ K *3 = uD c 0 , e5> = uD sb . Hence < ^ a , u > = ua . Suppose we wish to specify a whole fie ld of tangent vectors over M. It is generally impossible to describe a vector field over M by giving its components with respect to a single coordinate 19 system ‘in some neighborhood, Ua e M. " I f components are to be used at a ll, they have to be given with respect to the coordinates of the several distinct coordinate patches. In each patch there is great freedom of choice about the coordinate system." (Misner and Wheeler, 1952, p. 260) The concept o f a vector is in tr in s ic or coordinate independent. The basis vectors of the tangent space used above, e^ = ^ 7 , were tangent to the local coordinates at p e M = x1 , X2 , . . . , x^. Any covector ^can be expanded in terms of the dual basis A covector wis determined by its effect on any element of Tp= < ^ , u> = a number. However ^ is already quite well defined by specifying how it operates on a basis {ea } o f Tp. I f we le t , ea> = then the expansion of ^in terms of the dual basis ^r3} is £■ We now desire a physical picture of these differential forms. If a scalar field f s F°°(p) is given, then the equations f(p) = const, represent a system of (n-1) dimensional surfaces in M, called isotimicsurfaces (from the Greek: isotimos, meaning of equal value). The gradient of a scalar field, in a local coordinate system, is the field of covectors (Schouten and van der Kulk, 1949, p. 48) I f a t every point a vector has an (n-1)-direction [(n-1) dimensional blade] which is tangent to an M^n ^ of a system o f ‘Jl( n - l ) 's ’ t ^ien t ^le vec1:or 1S said to be j-forming. How can one visualize a field of co tangent vectors? “Consider the hypersurfaces f = f0, f = ffl + e, f = f0 + 2e, etc. If e is sufficiently small, to every point there belongs a system o f two 'in fin ite s m a l1 p a ra lle l hyperplanes in the 'infinitesmal neighborhood* of this point, one through the contact point and one on the side of f increasing. This figure represents the covariant vector (covector) e 3xa at this point. Hence a gradient field is characterized by the property that not only its (n-1)-directions fit together and form a system of but th a t fo r e ■> 0 the two 'in fin ite s m a l* hyper planes f i t together and form a system o f p a irs ." (Schouten and van der Kulk, 1949, p. 49) (in Fioure 4, — dxa = 3dxa. e 3xa That is there are three (n-1)-dimensional blades per unit coordinate interval of the vector space.) Consequently we draw the following two conclusions: 1.) Tangent vectors and differential forms may be pictorially represented as in Figure 5. That is, the determine the edges of an n-dimensional parallelooiped, and the aa 'Vr the system (n -1)-dimensional faces. 2 .) A neighborhood Ua e M can be covered by a system o f coordinates provided the congruence of curves to which the basis vectors are tangent have (n-1)- directions to which the families of coordinate surfaces are tangent, and these !i(n_-|) pairs coalesce to f ill the whole region without gaps. 8. More on Basis Vectors: Anholonomic Frames At any point p e fl one has a local tangent space Tp, in which one may introduce the basis vectors ea, and a dual space Tp with covectors ^a. The ea and aa can be defined with respect to the coordinate variables xa = x^, x^, . . . , xn, as * 3xa and £a = ^xa, i.e ., as tangents and gradients of the local coordi nates of Ua e Mn. The ea build up on M n congruences of curves and the £a bu ild up n systems o f ^ ' s which coalesce to f i l l the whole Ua e M without gaps. Hov/ever, one may arbitrarily introduce basis vectors ea(p), independent of the xy of U e M, to form a "" Uv basis for Tp. From these one can uniquely derive a set of covectors wa from the quality relation One then has a new coordinate system y^ = y \ y ^, . . . , yn w ith ea(p) as a basis, in Tp - but not necessarily in M. Consequently one is able to introduce a system of 'local1coordinates into the local Tp, defined by means of a vector d x y" ay 3e ax Figure 4 Pictorial representation of tanoent vectors and differential forms. (See Misner, Thorne and Wheeler, 1970, p. 147; Schouten, 1924, p. 22; Schouten, 1935, p. 6; Schouten and van der Kulk, 1949, p. 43; Schouten, 1951, p. 33; Schouten, 1954, p. 23; van Dantzig, 1954, p. 76) 23 r Figure 5 Pictorial representation of tangent vectors and differential forms in a cylindrical polar grid. 24 y = yD eb which has the contravariant components < y > = < wa, y beb > = y b with respect to coordinates which exist in Tp but are independent o f the x*1 e M. "The %{?) then b u ild up in the Mn n congruences o f curves, but the ua do not necessarily bu ild up n systems o f °°^ % M(n_u's." (Struik, 1934, p. 11) That is, these eb(p) are not necessarily M^n_-j j-forming. "It is the dichotomy between congruences of curves, to which the tetrad vectors are tangent, and families of coordinate surfaces, which, geometrically speaking, lies at the root of the difficulties about measurements in general relativity theory." (P ira n i, 1956, p. 391) In M the reference frame formed by the ea is called an anholonomic frame in contradistinction to the reference frame formed by the ep , which is called a natural coordinate frame. (Schouten, 1928; Schouten, 1929; Schouten, 1935, p. 67; Schouten and van der Kulk, 1949, p. 51; Schouten , 1951, p. 81; Schouten, 1954, p. 99, 164-177; Schouten and van Kamoen, 1930; S tru ik , 1929; S tru ik , 1934; Cartan, 1945; Cartanand Schouten, 1926) 9. Affine Connections (Gerlach, 1972) Corresponding to every element p e M there exists an n-dimen- sional tangent space Tp. Consequently two distinct points p and q e M have isomorphic tangent spaces if a connection is provided on M. (Two spaces are isomorphic i f there exists a homeomorphism Q ; Tp -*• Tq such that Q and its inverse Q"1 are differentiable 25 mappings. That is, in terms of their vector space properties TD and Tq are indistinguishable even though ea(p) and ea(q) may be construc ted differently). Let C(s) be an arbitrary differentiable curve on M connecting p and q. The isomorphism valid for all points in the region con sidered and for unspecified directions in these points is called a displacement and between Tp and T^ i t may be specified in the follov/ing manner. Each tangent vector ea(p) e Tp is mapped into Tq where it s image has the corresponding components aP i 9 j = 1, 2, . . . , n. That is, the image vector is ea^) + d^a^) = ®a + G V These image components are dependent upon: a.) the basis ea(p) of Tp b.) the basis ea(q) of Tq c.) the tangent vector of the curve connecting p and q e M. If the local coordinates at p e M are x1(p) and the coordinates of • • *S the point q are x n(p) + dx1, then the components wJ may be expanded a in the coordinate bases as J = T\ dx1 + r J’ dx2 + . . . + Ti dx11. a al ad an Then d ?, = h; dxk a ak ,] The p* are n^ arbitrary functions of the (p) and are usually 9 K called the components of the affine connection of the manifold. Figure 6 Points on the manifold p, q e £asis vectors ea(q) e Tq and ea(p) e Tp w ith it s imaqe ®a + d£a G Tq, and connecting curve on M. They provide a mapping of the elements of Tp into the elements of Tq th a t is , a mapping of the tangent space o f p in to the tangent space o f q. Under the above conditions d ea(q) has the following properties a.) d ea(q) =0 if q = p (i.e., if dx^ = 0) b.) d ea(q) = 0 if ea(p) = 0 where d ea = (oa eb. Thus ea e Tp and ea + ej e Tq are related by a displacement along any arbitrary curve passing through an points p, q e M. An element o f Tp and the displaced (or image) element in Tq are said to be p a ra lle l or p a ra lle l displaced. (Levi Civeta, 1917; Schouten, 1918) If on M we have a vector fie ld which at p assigns a vector Vp and a t q assigns Vq, then in Tq v/e have If we displace Vp along any curve to q g M, then in Tq we have it s image vector Vq + dVq where dV - d(va ea) = [dva + vb] % . S b a The firs t term in the brackets on the right describes the effect of translation upon the components of the vector (which are functions), while the second term t e lls how the basis changes as one moves from point to point on M. The quantity in brackets is called the covariant differential. Geometrically, it is the difference between the vector field evaluated at q e M and the value of the vector field that has been displaced parallel from p to q. A vector moves by parallel displacement if d V = 0, that is if d va + vb = 0. 10, Covariant D erivative An operator called a covariant derivative, may now be defined, which assigns to every pair of vector fields U, v a new vector field V with the following properties (Misner, Thorne, Wheeler, 1970, p. 157; Schouten, 1954, p. 124); I. Distributivity II. Linearity III. Leibnitz product rule V-(fV) = (V-ff) V + f VjjV. To denote the application of this operator along a basis vector e^ the following notation may be employed and the covariant differential may be reexpressed as a derivative 29 The covariant differential of an arbitrary vector may be reexpressed as a covariant derivative V. va = + r a vb k v 3xk kb where the components va are fun ction s. Under p a ra lle l transport along any curve w ith tangent u v j i = 0 the curve is called a geodesic. 11• Interpretation of Transformations There are two different types of transformations that have been discussed so far. Their distinction must be clearly understood, a.) Coordinate Transformation: (Passive Transformation) A coordinate mesh over a set of elements Ua e M is a correspondence between elements of Ua and the points of the arith m etic space Rn. (See Fig. 2 on page 12) A transformation of coordinates on Ug means passing to another one to one correspondence between the same elements of Ua and the points of another region of Rn. A coordinate transformation on M corresponds to a point transform ation in Rn. (See Schouten, 1954, p. 63; Veblen and Whitehead, 1932, p. 23) P hysically, we think of this as "fixed" events in space-time "described by many d iffe re n t systems o f coordinates." (Synge, STR, 1955. p. 75) 30 In contradistinction to a "coordinate transformation" on Ua e M a t p e M, we may also have a transformation of the elements of M. That is, we may pass from point p e M (with TD) to another point q e M (with its Tq). b.) Point Transformation (in M): (Active Transformation) A transformation of the elements of M p *> q, where p, q e M, is called a point transformation. Bearing the above in mind, the following two properties of the covariant differential are apparent: (1.) With respect to point transformation on M, the covariant differential is_ the difference between the vector field on M evaluated at q e M and the value of the vector field that has been displaced parallel from p to q. (2.) With respect to coordinate transformations on some neighborhood Ua, the covariant differential provides us with a coordinate invariant expression for the difference between the vector field evaluated at a point in Ua, and the value of the vector which has been displaced to the same point. (Just as a vector is a coordinate invariant quantity.) The transformation law for the affine connection can be determined by employing the tensor character of the covariant derivative. 31 . ■*. 12. Anholonomic Transformations The coordinate transformations discussed on page 29 concerned transformations on !Ja n from a natural coordinate system xa to another natural coordinate representation, ya, of the same points of M. Specifically, the coordinate variables are functionally related as yv = h(xv ) along with a unique non-singluar inverse transformation. Again, this all takes place in the space Rn, not in M. By differentiation one obtains d y^ =. h^ dxv where, in this case, = —rr . Consequently at every ooint in the v 3y region considered there exists a homogeneous linear transformation of the coordinate differentials. The above differentials are the gradient fields of the natural, coordinates y^1 and xy, and as such the gradient fie ld s possess ^ pairs which coalesce to f i l l the region considered. Hov/ever, at any point p e M one could select as basis vectors fo r Tp a set o f vectors, ?a, which possess some desirable property but are not the natural frames of any system of coordinates on M. Anholonomic frames do not belong to a system of natural coordinates because the are not the gradient vectors of natural coordinates. _ y . The ea so introduced "... form a non-holonomic n-hedral at each point of Mn, and hence define a coordinate system (coordinate net); hov/ever the transformation given by ya = h(xa) cannot be defined over a domain which cannot be reduced to a point." (Petrov, 1969, p. 43) The differentials d ya have significance at every point p c M and may be related to the natural coordinate derivatives by some specified transformation ha (in this case d ya = h^J dxu). Furthermore dxu = h £ d y a. However, on M the ^ do not build up n systems o f ^ coordinate surfaces which coalesce to f ill the region of Mn without gaps. 13. Anholonomic Frames If one is working in the tangent space of some point and has a set of differentials d ya related to a system of natural coordinates with differentials d xy, the question now arises; "How can one te ll whether the d ya correspond to holonomic coordinates or to anholonomic coordinates?" The necessary and sufficient condition for the existence of a holonomic (natural) system is that the covectors are gradients (i.e ., are irrotational in classical terminology) and consequently the curl of the covector must vanish. That is 33 must vanish if the ya are holonimic. If the ya are holonomic then the h® are given by 3ya 3xv and we recognize that these equations are the conditions for inteqrability and express the fact that the wa = d ya would then be exact differentials. Shortly, we w ill have another name for the expression on the right hand side of the equation above. There exist alternative criteria for the existence of holonomic coordinates which may be applied to the tangent vectors themselves. - y That is, one can te ll from the ea whether or not they are the tangent basis vectors of a natural coordinate system. This is discussed in the next section. 14. Lie Derivatives Vie now introduce the Lie derivative because it w ill serve as a powerful tool in distinguishing anholonomic frames from natural frames. [Historic Note: the name "Lie derivative" was firs t used by van Dantziq in 1932. I t appears th a t the e a rlie s t extensive discussion was be Schouten and Van Kampen (1933). I t was Schouten who proposed the notation£instead of D or <5 for the Lie derivative (Schouten, 1950). The Lie derivative was firs t derived in terms of integral curves and local one parameter families of transformations by Yano (1945). Prior to the development of the Lie derivative, Eddington had discussed the non-closure of the pentagon (1921),] 34 Consider a vector field $ on M. An integral curve of V is a smooth curve C: (a,b) -*■ M such that the tangent vector to C at each point along the curve is the value of the vector field, V, evaluated at that point: C(s) = V[C(s)] s e (a,b). The Lie derivative can be geom etrically interpreted in the fo llo w in g manner: Let § and u be smooth vector fie ld s on M, and let^Pp e M. Suppose we move along the integral curve of V through p0 until the parameter has moved from o to /s'; then move along an integral curve of u from o to /s'; then move back along an integral curve of V, the parameter now varying from o to and finally move back along an integral curve of u from o to -/s, as in Fig. 7. »Afe w ill not in general return to our starting point. As s -*■ o, our end poigt w ill trace out a curve through p0. The bracket [V, u](p0) is precisely the tangent vector to this curve. (Singer and Thoroe, 1967, p. 127; also see Schouten, 1954, pp. 127-129; Schouten, 1928; Eddington, 1929) In any natural (holonomic) frame e, = -^77 are the duals to ------U 3Xm gradient fields = jjx^, and so [Sa. ?b] - + 2 n cab ?c t 0. The Lie derivative then distinguishes between natural coordinate bases and other kinds of bases. As such it constitutes a criterion for determining whether or not a given set of basis vectors are the tangent vectors of some natural coordinate system. A congruence of curves is ‘^(n_-|) normal i f and only i f (See: Misner, 1963, p. 900) 35 |u.o) Cp = integral curve of V through p Bp = integral curve of ft through p Figure 7 Geometrical interpretation of the Lie Derivative. (See: Singer and Thorpe, 1967, p. 128; N ijenhuis, 1951, p. 210) 36 [ea , eb] = 0. As an example, on the surface of the two-sphere one usually employs the physical basis t * - 1 _A = 1 e 0 r 30 r 9 as ^ 3 = 1 £ The gap or "defect" vector is then £$• ^ ^ v - “se In a natural basis 0 30 t - 3 Ce0. = 0. There is an alternative interpretation of which is due to Cartan. Thus far, we have only displaced the vectors ea and not the contact points of the tangent spaces. (See Schouten: 1954, p. 127- 129, 375; 1935, p. 80-81) That is, we have related quantities in holonomic and anholonomic frames at a single point on the manifold by means of It is also possible to displace the contact point along an arbitrary curve on Mn. Following Schouten, 1954, p. 128, le t us suppose th a t there is a closed curve on Mn, starting with p, going through points q and 37 back to p' which is p.. The points of the curve may be considered as a point field each with a local tangent space. As the contact point P (we use P to indicate the contact point o f the moving Tp) moves along the curve, the moving tangent space Tp coincides at every instant with one of the tangent spaces of the individual points of the curve. At every moment the position of any such point can be identified in the moving Tp, and the entire collection gives an image curve in Tp. Schouten a ttrib u te s to Cartan the proof o f a theorem stating that if the original curve on Mn is a closed curve, its image in the moving Tp is closed i f and only i f the a ffin e connection is symmetric. On a Riemannian manifold th is means tha t ” nbc 15 non-vanishing, displacement of a contact point P about a closed path on Mn does not, in the moving Tp, bring p' back to the starting point, p. In the moving Tp, p' and p are now separated by a gap or defect which is given by ^ wc = 2 waA See Fig. 8. 15. The Object of Anholonomity In discussing the properties of the anholonomic object, let us consider three distinct situations. 1.) In a natural coordinate frame, the natural components of the affine connection r a “ < aa , V e > y v ^ y v are symmetric in their lower indices j.a _ r a yv vy and are generally non-zero. (In a Riemnannian Manifold r a reduces to {“ } in holonomic natural yv yv 38 P P Figure 8 An alternative interpretation of a may be made in terms of the displacement of the image of a manifold point in a "moving" tangent space. frames.) Consequently, the symmetry of these symbols im plies the condition ^p ev ep ^ey* ev3 “ ^ and the coordinates x11, to v/hich the ey are tangent cover some neighborhood of M, (See Fig. 9a) with out gaps. In a Riemannian Manifold the added structure -m e tric II II O exists and the physical frames, e*, are-p ’ orthonormal P v 'pv where ^ = f o t = f S 3 p p a y 9xa w ith dual covector $ = f£ ca = f £ d xa ^ a ^ ot a> but they are not the tangent vectors o f some system of natural coordinate curves. In such a frame, the physical components of the affine connection A A = < wa, Va e^ > = {/s5f} pv ** p v pv may be symmetric in p and v. If this is the case, (as it is in spherical polar coordinates, for example), 40 The Lie bracket is non-zero because the components o f the physical basis vectors are functions of the coordi nate variables. (This was the case in the previous example o f the tv/o-sphere.) Physical frames, which possess the convenient property of orthonormality, generate n-hedrals at every point on M but the coordinate surfaces do not mesh without gaps. Fig. 9b is an example of this. Here the Lie derivative is a measure o f the gap defect. (See: N ijenhuis, 1951, p. 210) 3.) Let us start with a natural coordinate system that has a symmetric r“v = . At every point in the region considered introduce, by means of an anholonomic trans formation , a system o f basis vectors given by eae = na hy ey e = na hy ' ( Ty f a ea; e ) * Since the affine connection components transform as (Schouten, 1954, p. 124) Tkbe * a b c hr yvr", - beh'h h? ^ or r a ,a ,P ,v r a a 3hc be ‘ ha hb hc r w + h j ^ they v/i 11, in general, not be symmetric in the new frame r bc * Lcb even though l yvJ lvy-f 41 (a) e = 2- ® de t r =&- a r (b) _ 1 a e* P A = ------e 0 r de -* a e * = — r a r Figure 9 Basis vectors in Polar Coordinates; (a) natural basis vectors, (b) physical basis vectors and Lie differential. 42 (a) de d r (b) u 1 = d r A OJ - rd e Figure TO Differential forms in nolar coordinates. (a) natural differential forms, (b) physical differential forms. 43 (Note: This is not torsion which is a property of the manifold and not coordinate transformations - see Schouten, 1954, p. 169) Consequently ? r a = r r a - r a i = ha ( — — ~ --1— ) [be] 1 be rcb] hU 3xb 3xc is nonzero. Then 7a*b - Vb?a = < J ’ + = + 2 nab ?c • In this case the object of anholonomity is made up of tv/o separate components. The f ir s t term on the rig h t hand side arises because, once again, the bases vectors are functions of the coordinates. The second term occurs as a result of the anholonomic transformation. If the local anholonomic frame consisted of mutually orthoqonal constant vectors then the firs t term would vanish, and in this case «ab * - i Crab - r y . Perhaps we are being overly pedantic here, but we must c le a rly distinguish between anholonomic frames with symmetric affin itie s (for example polar coordinate physical frames) and bases with a ffin itie s which also depend upon an anholonomic transformation and can be unsymmetric. It is this distinction to which we now turn our attention. I I I . ROTATIONAL WORLD LIMES In this section, we shall obtain a transformation between natural inertial tetrad frames and a set of orthogonal tetrad frames which may be employed by a rotating observer. That is, at every event on the space-time manifold we w ill introduce an orthogonal tetrad frame which, for the rotating observer, w ill span the local tangent Minkowski space. The ro ta tin g observer may then make measurements with respect to his local instantaneous rest frame. The results of these measurements w ill then be ju s t the natural components o f the corresponding tensors, that is, the scalar products of these tensors w ith te tra d vectors. (See: P ira n i, 1957, p. 143} By analyzing the helical world line of the rotating observer in terms of the Frenet-Serret formulae (Synge, GTR, p. 10) we w ill arrive at a set of vectors which are tangent and orthonormal to his world lin e . (Irv in e , 1964, p. 1165) Vie have l r = b B l 5Bl = c C1 + b^A1 6C1 = d D' - c B1 6s = -d c 1 Ss where fy 45 The A1 = are the components of the unit tangent to a time-like • « j curve r in space-time, and B1, C1, D' are the components of the firs t, second and third normals to r. The scalars b, c, d are the firs t, second, and third curvatures. In cylindrical polar coordinates in Minkowski space-time v/e may w rite ds^ = - dr^ - r Zd Further { > = " r 2 2 1 - t 21} -F Consequently A dx a r " Y . Consider a rotating observer with coordinates seen from the inertial (non-rotating frame) as r = R0 z = zo x4 = c t d d) dx^ yco where u = • Then d i~ = A1 = (0, 0, y). Now SAl = |A l + { 1 } A'1 ±XJ 6 s^ FT 1 j k ' ds 46 so 6A1 _ r 1 , a2 dxk 3s~ 1 2k ' ds“ Thus 5A^ _ y^ru? 6s " _ Consequently B1 = (1, 0, 0, 0) vn th y-rgj2 b = - C2 Next r i i R1 dx^ S s ~ “ 1 \ 2 } B _ds“ Thus 5B2 yw 6s ~ rc . Nov/ c c ’ + b A 1 So th a t we have c C2 + b A 2 = ^ I V c C4 + b A 4 = 0. Consequently C1 = (0, X , 0, m ) and y 2 c = 1 “ c • 47 Next 6 c l* = / 1 x zl\ # ■ <-> ■ - 4“ Now | 1 - d D< - c B1 OS so th a t Hence d - 0. * A " * Nov/ A1, B , C1, and D1 are orthonormal and thus D1 = (0, 0, 1, 0). In summary, we have A1 = (0, y~, 0, Y) B1’ = ( 1 , 0 , 0, 0) c 1 = ( 0, 1 , 0, ) D1 = (0, 0, 1, 0) w ith ,2^2 b = - E E “ ‘ 2 c = I3> c d = 0. 48 * 1 1 1 - - — The components A1, B', C1, D' provide an orthonormal set of basis vectors (physical frame) at any event alonq the world line r. Hence they may be employed by a ro ta tin g observer to span his local tangent Minkowski space. In the inertial coordinate frame we may write a field of ortho- normal tetrads (physcial frames) as: ->■ 3 = e e f1 = 3r r to = 1 3 = I C r 36 r rolco I = £ * = n 3 p ^ - 1 3 „ i 4 " c 3 t c These are the fam iliar physical basis vectors of Minkowski space. The index (n) labels the n^h basis vector. At a given point, the tetrads of the rotating observer are related to the tetrads of the inertial observer above by a certain transformation. We shall now find the transformation which transforms the inertial natural frames into the rotating tetrad frames. Let us expand the field of physical frames for the rotating observer as: * 2 ■ C ■ (}) «9 ♦ «4 ■ * v % ■ (y f ) % * W V 49 I ct \ Figure 11 Space-Time Diagram o f Rotating Observer. 50 The physical basis vectors in the rotating frame (let us denote this as the primed frame, for the moment) are: and so, the "natural" basis vectors employed by the rotating observer are related to those of the inertial observer as 2 Z e 4 ' = (Y c ) e0 + (y) e4. The transformation relating these two sets may be found from k a = a e y where here (and fo r the re st o f th is work) we w ill le t the Greek subscripts denote components in the inertial frame, and Latin subscripts denote components with respect to the rotating (anholo nomic) frame. The transformation may then be written as: / I 0 0 0 \ 0 Y 0 C 0 0 110 0 0 Y j 51 with the inverse 1 0 0 ° \ 0 Y 0 +y| 0 0 1 0 +/ 2w 0 0 J In a sim ilar manner, we may w rite down the coordinate basis vectors of the inertial coordinate system in spherical polar coordinates as: -> - 3 = el § r er 3 * 1 ->■ 30 r e0 _ 1 3 1 ej 3 r s in 0 3 The Frenet-Serret orthonormal tetrad family is then: *> -> el = er ' = er _ 1 _ 1 e2 - r e0' = f e9 % = ■ (r^) % + 0® ^ ej = e4. • m.) + (y ) ?4 . Consequently h^ and h& are given by: t,a _ 52 and T O O 0 0 1 0 0 0 0 y \ o 0 + r& gisl:.'?. Y / In general, it w ill not be possible to find a natural system of coordinates which incorporates the above tetrad vectors employed by the rotating observer. In the inertial frame, the polar coordinate tetrads are anholonomic in the sense o f case 2 on page 39. That is the rjja =' } are symmetric. However, the transform ation, h®, which we have chosen to employ is an anholonomic transformation and in doing physics referred to the new tetrad frame we must realize we are dealing with case 3 of pages 40 to 43. Consequently, the r®c are unsymmetric in b and c and the delicate distinction mentioned on page 43 must be observed. I t is th is d is tin c tio n which Irvine, and others that have used a Lorentz rotation, have neglected. We shall now employ the Cartan Calculus to calculate the object of anholonomity for the transformation in both cylindrical and spherical polar coordinates for the rotating reference frame. IV. CALCULATION OF THE ANHOLONOMIC OBJECT We shall now proceed to calculate the components of the object of anholonomity in spherical and in cylindrical coordinates (in fla t space) by employing Cartan differential forms. 1. Spherical Polar Coordinates We write the set of dual basis vectors (differential forms) corresponding to a natural (coordinate) frame: a* = dr o2 = d0 = d = cd t. We perform a transformationji;; or ^ = ha = ay ,',a and obtain a nev/ set of differential forms: J = dr a)2 = de to3 = _ -y(cid) - w dt) J = C Y(dt - r?2L |in !e d*) c and dcf> = (y) o? + (y^t) < /. 53 dw3 = ( g £ -s.™29Y 2 ) w1 A w3 + ( r_2^ sine cos9y 2 c2 c2 04 _ ( -rt^sin 28y2 j wl A + ^-r 2cj2sin9 cos By 2 c2 c 2 - ( 2rai sin29v2 ) W1 a w3 - ( 2r2(JljSin9 cos8y2 j c c Now dwa = 2 fl? wb A u)c DC where we have defined n?c * - \ Crbc - Icb ] then 1 2 ru£ s in 20 Ri Y ' 2 c 2 1 2 r^a r sin9 cose Y 2 c2 7 9 . 1 ra r sin^Q Ga 2 Y2 c2 1 r 2w2 sine cose 2 Y2 c2 55 2. Cylindrical Polar Coordinates lie write the set of natural differential forms: = dxy a* = dr a3 = d0 03 = dz 04 = cdt. We perform a transformation (dx)a = ha dx ^1 and obtain a new set of differential forms: wa = ha dxy U w* = dr o? = y (d 0 - a) dt) o? = dz w4 = c y(d t - de) and note that dQ = (y) w 2 + (y^) w4. Then doj^ = dw2 = 0 dw2 = ( y^ ) cJ A dbft - y 2 ) W A O)4 + (-2 y 2) J A a)2. Hence 2 _ 1 2 ru£ ft 12 ‘ 2 Y c2 12 Y c SJ? . - i y2 ru ? 14 2 Y ^ 7 V . INTERPRETATION OF THE TRANSFORMATION The in e rtia l observer possesses global coordinates ( r , 9, z, c t) and at every point on the space-time manifold, M, he has a natural tetrad frame given by: 2 = 3__ 1 9r t29 = -996 - = f— 3 9z _ 1 e4 = C at . These are tannent to his coordinate curves and the dual set cr *1 lie in the coordinate surfaces, which mesh together and f ill the whole manifold without gaps. (See Fig. 12, 13) This observer describes the rotating observer's world line as the previously discussed helix, and the Frenet-Serret formulas have provided the rotating observer with an orthonormal tetrad which may be employed at every event along his world line. (See Fig. 14, 15) The inertial tetrad frame and the rotating observer's tetrad frame are related by the anholonomic transformation which v/as derived above. That is •> , u ea = ha ep- Since the transformation is anholonomic there w ill exist no coordinate system in M to which the ea w ill c o n s titu te a basis. 56 57 c t world lin e \ Figure 12 The world view of the inertial observer with several of the tetrad frames and his world line. Note that £4 is the tangent vector to his world line. 58 A -ds2 = d r2 ■f r 2d 4? + dz2 -■ c2d t2 ■ ^0 © ie v a 1 O) = dr e f = 3 - %/s 9r 1 = 1 - rd4> e lig h t cone \ cdt Figure 13 The local tetrad frame employed hy the inertial observer, and his differential forms. 59 c t world line of inertial observer world line of rotating observer Figure 14 World view o f the in e r tia l observer with the rotating observer's world line and several of his orthogonal tetrad frames. 60 A A -ds^ = dr^ + r^d d = dr * f ■ d r - 'vv(dct> - wdt) e 2 = X ( L . + ^ M r v3tj) T 9 S + / J - dz 'Vi ^A _ 3 3 ' 57 of = ^c(dt - d$) c X ( L . 0) ) *4 = c D *V 6b eA lig h t cone A. Figure 15 The local tetrad frame of an inertial observer^ , the local tetrad frame of a rotating observer 1aaH and the rotating observer's differential forms wa. 61 -ds2 = d r2 + r 2d02 + dz2 + 2 d9dx 4 - (1 - ) c2 d t2 c c2 = dr fT = 9_ ei 3r J = rd9 + m dx4 •yj c 1 i _ A e £ = rrr 89 H? = dz 9_ = w4 = dx4 3 9z = 8 __ to 8 _ 8x4 C 90 . a _ Pa A » A f Figure 16 The local tetrad frame of a rest observer e , the local tetrad frame of an observer in a "galilean rotating frame", and tfye galilean observer's differential forms wa . Figure 17 The local tetrad frame of a rest observer , the local tetrad frame of an observer in a frame "instantaneously" at rest with respect to some rotating observer (with the constraint that this instantaneous observer has chosen the skewed metric and oblique basis vectors given above), and th is,instantaneous observer's differential forms wa . These are called "M iller frames". The tetrad frames of the accelerated observer provide only an infinitesmal coordinate frame, i.e ., valid only at a point on M. They are, in fa c t, the frames employed by a comoving in e rtia l observer with respect to which the accelerated observer is moment arily at rest — their world lines are tangent at this one point on M. 1. Properties of the Transformation Before proceeding too much further we should mention that the transformation derived above preserves the form of the metric tensor. That is, given -ds^ = g dxy dxv where 10 0 0 0 r 2 0 0 0 0 10 0 0 0 -1 then 9ab * ha hb % where 64 gives 1 1 0 0 0 0 r 2 0 0 0 0 1 0 ^ 0 0 0 -1 where -ds2 = dxa dx^. Consequeptly ^ab^ = That is, the line element is invariant, and the transformation h^ a is a conformal transformation, with unity scale. (Schouten, 1924, p. 170) Conformal transformations have the property th a t they leave the angle between two vectors unaltered. Furthermore, we have an interesting relation between the e^ of the inertial observer and the £4 of the rotating observer. The world line of a rotating observer w ill be a curve whose tangent (£4) makes a constant angle with a fixed line specified by £ 4. This is the definition of a helix. V I. THE SAGNAC EFFECT If one follows along a closed path on a rotating disc, one finds, upon returning to the starting point, that his clock w ill differ from an inertial clock by an amount At. We shall calculate, an expression for this difference by comparing the world line of an inertial (rest) observer v/ith that of a rotating observer. Our choice of e^ = A1"^ from among the Frenet-Serret set was motivated by the desire to have the rotating observer always be at rest in this frame. That is we have let his velocity four-vector specify a time-like direction for him. Furthermore, his helical world line, to which e^ = is tangent, is an integral curve of e^. The proper time, (which is the time between any two events occuring at the same spatial point), that elapses in the inertial frame while the rotating observer is away on his trip , is found by setting dx^ = dx 2 = dx^ = 0. Consequently, as the inertial observer floats up his time axis +ds2 = c2 d tj and his world line is of length where T is the period o f the ro ta tio n . (See Fig. 18) As the rotating observer follows his world line, the inertial 1 observer assigns dx‘ = dx = 0 and measures the rotating observers world line length as the helical arc length 65 66 c t X Figure 18 World lines of an inertial observer, Sj, and a rotating observer, S^, with respect to the coordinates of the inertial observer. 67 / IZ T ~ ds = / l - e V c d t i . Hence If the rotational velocity is much less than the speed of light ( < < 1), then SR '/I0 0 "T2 Tc‘ * cdtl' Consequently, the path difference is given by AS = S j - or, as measured by the inertial observer AS = — c where A is the projected area of the helix onto the spatial plane. In terms of the proper time of the inertial observer, this may be w ritte n as ^ cjjA ATI = 7 ■ A similar result is obtained by a different technique in Miller. (1952, p. 262) The equation above which described the incrimental path length of the rotating observer in terms of the inertial time incriments, TIT ds = / 1 - c d tj , c1- may be expressed in terms of the non-inertial time incriments. That is, under the anholonomic transformation hg , 68 d tj = y(d tr + ^ d0r ) (where + refers to rotation in the + azimuthal direction) so that or ds = c (d tr + d 0r ). Now this is an inexact differential and reflects the anholo- nomicity of the rotating observer's frame. That is, there exists no function s(r, 0, z, ct) which has this expression as a differential, since it is non-integrable. In the rotating frame, proper time unrolls differently at r than at r + dr. Consequently global synchronization of clocks, in the rotating frame, is impossible. Note that the congruence of helicies with r = constant w ill all unroll the same arc length ds^ in the same inertial interval d tj. However, the helical world lines at distances r and r + dr w ill unroll different arc lengths, during the interval d tj, and w ill consequently measure different proper time intervals. Confining ourselves to a fixed distance from the origin, let us measure the arc length (and hence the proper time) which unrolls during a given circuit in the rotating frame so th a t (See: Langevin, 1937, p. 306) SR = c T R t M 69 where the sign (+) or (-) refers to following the circuit with or opposed to the direction of rotation. Sagnac (1913) caused two beams o f monochromatic lig h t to interfere after passing a closed circular path in opposite directions on a rotating disc. From the above, the time difference for the' propagation is + _ 4wA " "TZ where A is the area enclosed by the circuit. The form of this result has previously been attained by other types of derivations. Landau and Lifschitz (1962, p. 297) obtain the same result by considering the off diagonal terms, gQa, of a rotating galilean metric. These terms, however, w ill create problems when interpreting the electromagnetic force density f 1 = p FiJ* J . c j (Synge: GTR, 1966, p. 359; Irv in e : 1964). The ga lilea n ro ta tin q metric leads to electromagnetic forces proportional to the particle velocity, even when the particle moves in a purely electric field (F0^ = 0, Fa^ = Ew). This is because, fo r example, - 1 p14 „ ,4 .1 f14 n ,2 f ~ c F g44 J c F g42 J or f r = PEr + Er ( ) c which is proportional to rw. The galilean terms must be carefully interpreted in order to make sense. (See: Chapter X, Section B. 2) This problem does not arise in the anholonomic formulation. 70 The form of AtR appears to have firs t been attained by Sagnac (1913), and also by Langevin (1921) and Gordon (1923). For an in te re s tin g review see Post (1967). Trocheris (1949) does not employ a simple galilean frame in finding AtR. r VII. THOMAS PRECESSION If one carries a defined axis in a given frame of reference along a spatially closed curve with an acceleration not parallel to his velocity, then as seen by an inertial observer, the reference axis w ill have rotated through a small angle, da. See Fig. 19. This e ffe c t is called Thomas Precession (Thomas, 1S26; Thomas, 1927). In 1926, Uhlenbeck and Goudsmit introduced the concept o f quantized electron spin angular momentum. They were successful in explaining anomalous Zemann e ffe c t, however the s p in -o rb it in te ra c tio n they obtained was twice too large. (See: Jackson, 1963, p. 366) Thomas' discovery of a purely kinematic precession, independent of the cause of acceleration, reduced the spin-orbit splitting. The Thomas precession arises because under successive Lorentz transformations a compass o rie n ta tio n changes by an amount equivalent to th a t obtained by a Lorentz transformation plus a rotation. ...im agine a set o f in e r tia l frames moving so th a t the spatial track of each origin is tangent to a circle of radius R, and in such a way that we meet these origins at zero relative velocity as we go round the circle at constant angular speed w; if any two successive such frames consider their axes to be oriented without relative rotation, then, as we complete the circle, the axes of the last frame are nevertheless rotated relative to those of the firs t by an angle it R2 4 • c^ (Rindler: 1969, p. 172; note that Rindler sets c = 1 on p. 168) 72 / d d Figure 19 Thomas precession of a reference axis after ro ta tio n . 73 To see that Thomas Precession is contained within the anholonomic transformation, le t us examine the precession of a reference axis after rotation. In particular, if one follows along a circular path instead of the pentagon o f Section 11.14 (See also; Schouten, 1954, pp. 127-129), he vdll find that, after proceeding the angular distance 2ir, his reference axis is not at the starting position. The gap or defect being given by [er , eQ] = + 2 ec . That is, the angular difference is given by the two-form dto2 = 2 ft2. wa A ofi. ab So th a t dw2 = y 2 J Aw2 zr or, to order •!=• c d w2 • c Consequently, after one complete circu it, the angular difference is given by (to this order) da = J ' J ' dr A d0. 0 So that the rotation produces an angular defect of order da = ir r 2 ^ z r • 74 This result is, of course, a relativistic effect and not attainable v/ith a galilean rotation. V I I I . ANHOLONOMIC FIELD THEORY As a consequence o f experimental data concerning the invariance of electric charge, we assert that the electric four current is a legitimate four-vector. Consequently we write the equation of charge conservation in an arbitrary coordinate frame as v„ JW = o. Furthermore, the invariant action principle leads to Vb Fab = + 41 ja where, by definition ^ab “ va vb ^a* All our electromagnetic definitions and sign conventions w ill follow Landau and L ifs h itz , 1962. (A th ird e d itio n of th is book appeared in 1971, with a different metrical signature - and consequently a different sign convention.) These are tabulated in appendices B, C and D for reference. Now, in the anholonomic frame rbc t r ?b* and so F_h * + 2 Ac . ab 3xa 3xb ab As Schouten points out, "If any expression with respect to holonomic coordinates is transformed with respect to an anholonomic system, correction terms appear, all containing the object of anholonomity". 75 (Schouten, 1954, p. 100) Suppose v/e have a vector q u a n tity, By, which in some holonomic frame, satisfies the equation F - n R o R - 3B, _ Fwv - vii Bv -’ v Bu - 3XU ' axv - Upon transformation to some anholonomic frame, we have fab = ’A -V a= +2^bBc . That is Fab * "I »b V For example, suppose we have some arbitrary bona fide vector with components in a holonomic frame given by By = (0, 0, 0, K). In the holonomic frame we form the curl h, v = vu bv - vv bw = ( 2 - ! £ ) =0. The components of the vector transform to the rotating anholonomic frame as Ba - ha Bu * (°> r - f - > °» vK). Again forming the curl, we have (3b2 3 Bj_ ) • ^c u 3 # ro k 3X 3 X ‘ o *>3 3 - Y3 k t 0. O'3 Further 3 2 «12 Ba = Y — 3 - k - 2Y3 ^ k C'3 = + Y3^ k ^ 0 so tha t H,h = - i B_i ) + 2 fiC. B = 0. 3b 3xa 3xb ab c That is , with the above definition for Hab we really have Hab * hH hb V However, if we had only defined as Hab - (!!L.- !!* ) ab 3xa 3xb we would have qotten the inconsistant result Now, the method of anholonomic frames, employed above is s t r ic t ly valid at one and only one space-time point. How then can one rigor ously re la te q u an tities at one space-time p o in t to a second near by point? The problem we are facing is how to relate the tangent space Tp to the tangent space at a near by point, Tq. This problem has already been discuessed in Section 1.9. A mapping between the elements of Tp and Tq is provided by the affine connection r ^ . The method of instantaneous frames (M iller's method) permits one to map Tp onto Tq but, it requires one to use oblique (i.e., non-ortho- gonal) tetrad frames at p and q. Then the instantaneous frame's m etric w ill be non-diagnonal. Describing the universe from a system of oblique frames leads to several queer conclusions: There w ill be a component of electromagnetic force proportional to a particle's velocity, even when a particle moves in a purely 'electric' field. There w ill be ficticious charges and currents described by observers using these oblique frames. Space-time w ill be non-Euclidean - even though it may be fla t. Maxwell's equations w ill be different when w ritte n e n tire ly in terms o f covariant or contravariant components. There w ill be confusion with respect to the definitions of IT and Ef in the rotating system. "These diverse definitions lead, if con sistently applied, to the same physically observable results. None the less, the several alternate formulations ..., form a perplexing 79 picture for the physicist trying to apply the results to a particular problem." (Irvine, 1964) However, the method o f anholonomic frames provides us w ith not only a technique for the erection of instantaneous orthonormal frames at all points on Mn, but also smoothly connects them together. By doing this we have not lost "pieces" of Fa^. Further, in moving from p ■+ q, the moving tangent space Tp is spanned by an orthonormal basis to provide a local Minkowski metric. A consistant use of the anholo nomic affine connection has supplied us with information concerning the continuous changes in Fa^ with position, and permits a consistant tensor calculus. All definitions of field variables, and the tech niques for handling them, are now exactly the same as the covariant definition in inertial frames. The interpretation of physical results is clearly much less complicated when all results are referred to orthogonal reference frames. We w ill soon turn to a specific situation which explicitly illustrates the differences and sim ilarities between the "method o f g a lile a n frames", the "method o f instantaneous frames" (M iller's method), and the "method of anholonomic frames". IX . INVARIANT EQUATIONS 1. Charge Invariance One may demonstrate that charge is an invariant quantity, that is, charge does not depend upon the choice of reference frame, even in rotation. First v/rite dov/n the four-current density in the charge's re s t frame as (Landau and L ifs h itz , 1962, p. 80) ■= (0, 0, 0, C Po ) . Transforming to a rotating frame gives Ja = h j or 0a = (0, -Ywpo, 0, cYP0). In the charge's rest frame we must integrate over all of three dimensional- space, dVQ. That is, we must integrate over the entire hyperplane = const. It w ill be convenient to use the four vector dSy, whose "magnitude" is equal to the "area" of an element of hyper surface and is in the direction normal to this element of hypersurface. It is clear that dSy = rdrd9dz = dV0 . Consequently, in the charge's rest frame dSy = (0, 0, 0, dV0). 80 81 Thus, we have the following expression for the total charge 'c f * V % * / podvo - In the rotating frame, we must integrate over the transformed element of hyperspace, that is, over the same events on the space time manifold. Now dSa * ha dV That is : dSa = (0, + r - f 4 dV0, 0, YdV0). Consequently dSa = / ( - Y2 £ £ ) Po«o + / y 2 P0dV0. c * ^ ) W = Q. An thus, we have the following expression of charge invariance, under the transformation hjj. 'c f jV As one might expect, a continuity equation may be written in the form Va Ja = 0. 82 2. Invariance of Maxwell's Source Equation We shall demonstrate that Maxwell's source equation V Fvu = + Jv M C is form invariant under the rotational transformation h!J. To see that this is so, one considers the inertial equation - % jV) ds^ . 0 and shows that this also vanishes in the rotating frame. For simplv city, let us consider an electrostatic distribution = (0, 0, 0, c p ) so that = - Er is the only non-vanishing component of Fvii. Now dSv = (0 , 0, 0 , dV0). And, in the in e rtia l frame we have f \ F'’1' - M . Now dSb = hb dS v so tha t dSb = (o, dV0, 0, ydV0) 83 and 7. Fba = hb 7„ ?w . a v P _ (for the moment v/e are using a bar to denote the computation in the anholonomic frame). V1 F41 = y V1 F41 V F^" = -y - V-, F41. 1 c 1 Then y Va Fba dSb = j y 2 V, F41 dV0 - f ( y | V, F41)41' y r “ dV0 :41 = / 'v 1 F‘H dV0 = 4ttQ . Notice that Fab = ha hb Fvy v y and Aa = ha AV. X. THE FIELDS ARISING FROM ROTATING SPHERICAL CHARGE DISTRIBUTIONS In this chapter we desire an expression for the vector potential arising from a spherical charge distribution which is in relative rotation with respect to a frame of reference. We shall consider the problem by employing tv/o separate approaches: A) A charged sphere rotates (non-relativistically) with respect to an inertial observer at rest. B) A r e la t iv is tic a lly moving observer revolves about a charged sphere which is at rest. We shall find that there exists no electromagnetic symmetry between cases (A) and (B). Case A A sphere of radius a carries a uniform charge distribution on its surface. The sphere is rotated about a diameter with constant angular velocity w. Our problem is to find the vector potential and magnetic flux density both inside and outside the sphere. z X Figure 20 A charged sphere. 84 The current density in the rest frame may be written as T (x ') = P0v = PQ to x a or J(x') = sin e' 6(r* - a) In Cartesian coordinates, the vector potential is given by _ 1 f . M ' L . d3x' In order to determine the vector potential, we choose the observation point in the x-z plane for calculational convenience. Then Jy = cos We are le ft with only the y-component of the vector potential, which is A<}>(x). t {—\ - ^ f sin 9 1 cos ft1 5 (r1 - a) r 1^ dr'dfl1 A<|>lx ' = 4-nac J |x - x ' | where |x - x ' } = [ r * + r ,d - 2rr'(cos9 cos 9' + sin 9 sin 9' cos This integral can be expressed in terms of the complete ellip tic integrals K and E, but the results are not particularly illuminating. A neater expression re su lts i f we employ Tessera! harmonics. The term -----3------may be expanded in spherical harmonics as 86 1 A = 4ir t I ynx -^r Y,m ^0l’ Yim ^ |JT- ^ I 1=0 m=-i 21,1 r> lm lm where r> (r,,) is the greater (lesser) of r and a. Further, the angular factor in the integration may be written as sin e' cos Consequently, the orthogonality of the .spherical harmonics Jf Y,«m« l m (0> ^ Y im (0> % (r» 0) = *lc IF" sin 0. The magnetic flux density is found from F = Vx I giving (in cylindrical coordinates) B(r < a) = z 3ac F(r > a) = ■ (3 cos 9 r - z), The magnetic field is uniform within thesphereand that of a dipole outside. 87 In spherical coordinates ?T(r, 0, r - 1 3(A* sin 9) r " r sin 0' a 0 gives Br ■ ( |) £ cos 9. And since 0 r 3 r we obtain Be(r > a) = sin 0 3 r3c and B (r < a) = - sin 9. Vie shall now proceed to case B. CASE B 1. Method o f Anholonomic Frames We now desire to calculate an expression for the magnetic field seen by an observer who is rotating about a fixed spherical charge distribution. l*/e distinguish the frame of the rotating observer jj!xa and the coordinates of the charges 1 rest frame <£x^. The covariant components of the four potentials in the rest frame are A» = (0, 0, 0, - *) 88 Figure 21 Magnetic flux density arising from a rotating uniformly charged spherical shell. 89 where 0 a r < a $ = < 1 r > a r The transformation from the (holonomic) rest frame of the charges to the (anholonomic) rotating frame is given by ^dxa = ha y dxy where 0 0 /’ 0 \ o 1 0 0 ha = y o o o v r^ s in ^ e / c The potential four-vector transforms as Aa - ba Av • So That is njQ sin^ 9 - Y r > a A3 * < . ^ f t r 5y _ y3 ro)2 s in 2 9 ar 90 and 9y 3 r 2q)2 s in e cos 9 80 s y 2 c The radial component of the magnetic flux density is found from F23 = r2 sin 6 Br = |*1 where Aa = (Ar , rAe, r sinQ A^; - $) so th a t A3 = - yQ^ —- r sin2 9. 6 c a Now = - 2y^r ~ r sin 0 cos 0 - y3 r3 sin30cos 6 c and _4 ? r 2ai sin 0 cos 9 23 = - Y ------e ------ ^3 _ 1 .2 r 2w2 sin9 cos 9 ij23 2 Y “ 2 so th a t, fo r r> a r2 sino Br = -2y ^ - 5i p 0-s- 6 (1 - y2) -2 y3 Qy3^3 s in 3 9 cos 9 c3 and fo r r < a 2 r2 sin 0 Br = -2 y~ sin 0 cos 9(1 - y2) -2 y 3 s in 3 9 cos 9. acJ 91 Consequently Br = 0. Next, the theta component o f W is found from F-io = - r sin 0 B0o = 3r + 2 13 A. a . No v/ ( vQcj sin2 6 _ y^Qr^u)3 sin^ 9 (r > aj 3A3 C‘ 9r - 2 y ^ . sin*2 0 0 - _ YJ „3 Q\ Lr T 3w3 ^ _ P,.„4sin* 9 (r < a) ac ac-5 and si4 = . ,2 ro> s in 2 0 13 3 1 2 rco2 s in 2 0 Si 13 " 2 Y ------ For r > a oj s in 2 9 -r sin 0 B0 = Y Q and fo r r < a - r sin 0 Bq = 0. Consequently f Ooj sin 0 - y-1 —----- r > a < 0 r < a. 92 Furthermore, since A . 1 2 )2 sin 2 Y 41 " 1 c2 4 1 2 Y *42 ' 2 Vie also have an electric field. The radial component is found from _ _ 3A, 4 . 14 " r *14 1/1 A4‘ -Y - r > a o r < 2. r < a ? 9 3v _ 3 ruT sinc 6 9r so 3A 4 Q 3 Qco2 s in 2 e H r = Y ^ and ( y r > a = < r c Er - V\ r K 0 r < a If any further electric field is present, it is found from 9A4 9A2 ^24 = rEe = 90 9x + 2S224 - V Nov/ 3y _ 3 r 2y2 sin e cos e 99 ~ y t i------93 and so rFn s - v3 rto2 sine cos9 Q rw2 sin 0 cos e_Q 0 I O ” ” • I o C Hence E0 - 0. Our conclusion is that, except along the axis of rotation (where the magnetic field vanishes in both frames), the fields measured by an observer rotating about a fixed charged shell are different from those measured by an inertial observer in the field .of a rotating charged shell. 2. Method o f "G alilean Frames" Before turning the S c h lff's paradox, le t us examine the "method of galilean frames", and explicitly see how results are obtained by that technique. The inertial frame metric is given by -ds2 = dr'^ + r'^de'2 + r '2 s in 2 0 d The positional relation between inertial and rotating observers is given by r' = r dr' = dr 9' = 0 de1 = do t* = t d t' = dt . Putting these into the above ds2 gives, for the rotating observers 94 -ds** = dr2 + r2d02 + r2s in 2 0 d^ + 2r2w s in 2 0 d /, r 2a)2 s in 2 0 x i Aj? ; a , h _ (I _ ------j c dtc = gab dxadx . 0 That is 0 0 ^ab .2 s1n20 A j.s ln L .e 0 r ^ 51in 20 -Y -2 where x^ = ct, and y = (1 - Psin- 9 ) ^ Now g = || dgb 11 = - r 4 sin 2 0, and so 0 0 r 0 0 0 ,ab _ k 0 0 1 w/c ■y2r 2sin 2 0 w/c -1 One should compare this oblique frame rotation metric with the orthogonal frame rotating metric obtained on page 64. Also see Fig. 16. It is apparent that the "method of qaiilean frames" is simply a rotation of spatial axes. Let us assume Maxwell's equations in the form (1) IfHi + JIfli = o 3xCT 3x^ 3xv (2) T^r fv ( ^9 F^) - j". /ng 3xv c These are covariant under transformations of the form xa - xa 01 s 4, ', ct') a ~ 1, 2, 3 t = t* since the transformation is integrable, and all vanish. In the rotating frame, v/e may define Fab as rB^ -r sin 0 Bg Er \ 0 r 2 sin 0 Bf - r V rE9 Fab = r sin9 Be, -r 2s in 0 Br 0 r sin 0 E -Er -rEe -r sin 0 E^ Then, in the ro ta tin g frame, the f i r s t o f the above Maxwell equations is satisfied. Now, with respect to the second Maxwell equations Fcd = gca gdb F ab .ab . where g is the metric given above for the rotating frame. Let us explicitly display the calculation of each separate component of F cd 96 :34 _ 933 g44 F34 + g34 g43 F43 - r sin 9 .13 g11 g33 Fi3 + g11 g34 Fi4 !_ _ 2d rw sin 0 c ^ r sin e B c rJ :12 = g11 g22 f 12 p23 _ g22 g33 + g22 g34 p^ - ^r^ 4 sin n r e r ^ ' 2 We display Fab in m atrix form on the next page. Note: The method of galilean frames leads to the disheartenina result that Bq 0 Bb/r r sine -Er -B 4>/r 0 Br _ Ee r Fab 1 r 2 sin 9 Be Br 0 . r sin e r^ sin e r sin 0 Fr E0/r 0 r sin 0 which is_ valid in the inertial frame. This is the nrice one pays for employing a skewed frame of basis vectors for reference directions. - 2o rw sin 0 c \ . . 0 B, W ------£------brJ _ rto sin 6 Be) r sin 0 / -2« + ru) sin e (Y &r — ------Ee) (Eg- * 4 ^ Br) o r * sin e •ab _ , - 9- , r E<|) (Er + OLSilUL b0 ) (BS - Bp) r sin 9 I Contravariant Components of the galilean Field Tensor ■*•*1VO In the inertial frame, the current four vector is given by ^8 ' jp - p c J " *o c ds and = (Jr, £i , cP). r r r sin 6 Further Jrr = ayvg dv so th a t Ju = (dr , rJg, r sin 0 - cp). In the rotating frame, since we are remaining with the definition of the covariant components F ^, we w ill be consistent and retain the definition of the covariant components of the four current, i.e., Oa = (dr , rJQ, r sin 9 - cp). Now I 1 0 0 0 \ 0 0 1 -o)/c \ 0 0 0 1 j so da = Ga dp = (0, 0, 0, - PqC). The contravariant components of the four-current are found as da * gab Jb = G* Jy so 99 ■»3 _ 33 , . 34 , 9 ^ 3 9 ^4 = - P0ix> and J4 - g44 J4 + g43 J3 = Poc ' Let us apply this analysis to the situation of an observer rotating about a single charged spherical shell of radius a. Then o = Q3 (r - a) 0 " 4ua2 and Ja = (0, 0, - PqW, P0c ). Let us explicitly write out the second of Maxwell's equations in the ro ta tin g frame ■2-k ( Fab) - % /^ g - 3xb c a = 1 0 n « —2d no sin 9 r 59 [r2 sin 9 (5i ) ] + | j [r2 sine (!. BP ~ c_____ ^ )] au r -r sin e 1 a r ? . . _ _ no sin 0 _ % „ + c n ^ sin 9 r ~ Be) ] - 0 TOO a = 2 •n o r, _ -?n . sin 9 _ jL. [r2 sin 0 (- _!)] + 2_ [r2 sin g(Y Br + c E9 )1 3 -2„ rto sin 9 r v -2d j. ru sin 9 + | r [ r 2 sin e (T Br c E* ) ] - r 2 sin 9 + jr f r [ r 2 sin 9 ( ) ] = ~ r 2 sin 9 J3 c r sin 9 c a = 4 | p [ r 2 sin 9 (Er + 9 B0)] F(1 rc» sin 9 + |g [ r 2 S in 9 ( E° ~ c r ) ] + [ r 2 s1n 0 ( _ £ l _ ) ] = *1 r2 sin 0 j4. 3(P r sin 9 c A solution that satisfies the boundary conditions (no singularities and vanishing at infinity) is: 2? r > a Tc 2r = 0 r < a 101 0 r > a Bf 0 r < a which are not the same as the fields found by the method of anholo- nomic frames. With that technique, however, there was no ambiguity concerning the interpretation of the components of Fa^. As a check, let us insert the above into the equations for a = 1, 2, 3, 4, and integrate. The only interesting cases are for a = 3 and 4. Writing these out explicitly gives: r^inJL ( ^ ) (1 - 3_ r 2 sin 0 3r r sin 0 + a(o)+i3 lo i : r2 sin 9 a3 + ae c at c a_ 'rC 9t|i!!_e (-l)]n I % x2 sin 0(- V )- 3r To check, le t us integrate both sides across the thin shell [ -Q to sin 9j _ |-- Q to sin 0^ L .L' 2 • 1 r r ru sin 8 „ , n sin 6 [Ef + ------BqJ + 0 3r + o = r2 sin e c |_ (Q 9)2 4„ r2 sin e po . 102 Integrating both sides across the thin shell gives Q sin 9 = Q sin 9 It is seen that the above choice of E" and Ef fields and four current Ja are a solution of the Maxwell field equations in the form given above. The appearance o f "fic tic io u s charges" and currents, and the ambiguity in interpreting components of the field tensor, are not the only disagreeable aspects o f the "method o f galilean frames". There also appear electromagnetic forces proportional to charge velocities, even when the fields are purely electrostatic. Consider the situation where Jb = (°» °» r sin 9 J4>‘> - c^)* Suppose we have a purely radial electrostatic field in the rotating frame. Then F-j^ = Er is the only non-vanishing component of Fa^. Mow F3^ w ill have the expression 0) £ i 0 0 m c r 0 0 0 0 :3b _ “ r 0 c br 0 0 E r 0 0 0 The force on a charge density moving with a velocity R0 Q fa - 1 Fab Jb. So that the separate components are given by 103 and f2 - f3 - f4 - o. This leads one to make the interpretation that there exists a force of magnitude rw sin 0 f1 = p[Er + Er Ro a ] There exists a component of force proportional to the charge's velocity, ft, even though the particle moves in a purely electrostatic field. This curious result, and all these seemingly arbitrary ambiguities must of course, be interpreted in light of the fact that the reference tetrads are not orthogonal, i.e ., 934 f 0. In using the "method of galilean frames", one has committed himself to describing the universe from an oblique frame of reference. In this respect, the "method of anholonomic frames" provides one w ith a simpler set of tetrads for reference. 3. Method of Instantaneous Frames Instead of the "method of galilean frames", which corresponds to a simple rotation of the spatial axes, some authors employ the "method o f instantaneous frames" (M ille r's method). (See Fig. 17) This widely used technique is often misinterpreted, and confused w ith the "method o f g a lilea n frames". One should no tice, however, that the reference tetrads are different (compare Fig. 16 and Fig. 17). Where the time coordinate is absolute in the "method of galilean frames", the t' coordinate is not a physical or clock time in the "method o f instantaneous frames", but provides a measurement o f "coordinate interval". Under a galilean rotation ^ = dx^ and is integrable so that t = t ‘. All anholonomic objects vanish. With the "method o f instantaneous frames" the re la tio n between in e rtia l time,dt, and local "coordinate interval time" is dt'= (1 --■^ ) ^ d t3„ qC ^ x is * This is not an exact differential. The problem with such a technique is that, "Not only does time flow at different rates at different points, but the very concept of synchronization does not exist." (Zeldovich and Novikov, 1971, p. 11) Problems o f th is so rt in the "method of instantaneous frames" are discussed in this and other references (e .g ., M ille r , 1966, pp. 222-6, 240-50, 262; Adler, Bazin and S c h iffe r, 1965, pp. 111-121; Davis, 1970, p. 240-421) Vie also note that this method provides one with an oblique local referenpeTf^ame since 934 ? 0. Bearing all this in mind, we will soon demonstrate that this unwieldy technique, if properly used, w ill lead to the same Er and B0 as previously found by using anholo nomic frames. The closing paragraphs of the present chapter discuss this curious result. A fu rth e r (and previously unnoticed) aspect of the "method of instantaneous frames" is that it is anholonomic in addition to being oblique. That is, in the "M iller frame" v/e have c since the objects ft ^ are non-vanishing. Nov/ the interval -ds2 = dr2 + r 2de2 + r 2 sin 20 dcj>2. + 2 r2w s in 2 e d ^ = dr g = de jj3 = yd -ds2 = g , oa (x) crb. ab % 'Xi The "instantaneous frame" anholonomic object is found from 4 i f ■ 2 nab • The non-vanishing components of of interestare: 3 _ 1 „2 rw2 s in 2 0 «13 " 2* Y------^ ---- ^4 _ ..2 rw s in 2 e 13 “ Y c 4 1i . «-2 rw2 i tu sin o 2 e “ l4 = ’ 2 Y _ 72c* The transformations relating inertial and "instantaneous frame differential forms and basis vectors are where 0 0 0 1 0 0 r = 0 Y 0 1 0 A i s in 2e -Y Y and r 0 0 \ 0 1 0 0 i r = 1 0 0 0 yr^cosin2 Y 2 e 0 0 Y J c These are neither Lorentz nor Galilean transformations* but they do relate vector and tensor components on inertial bases vectors e^ to the components along the "instantaneous frame" basis vectors at every point in sapce-time. For example, in the inertial frame, the four- po te n tia l may be expanded as: The same space-time four vector, expanded in the instantaneous frame, 107 f \ - a3 s a • So tha t now Aa = I a o r Aa , = T*1 a A y * As a s p e c ific case where the anholonomic aspect o f the method of "instantaneous frames" is important, consider a static charged sphere of radius a, in an inertial frame. Let us write Av , compute Aa and then Fab. Secondly, let us compute and then transform to fin d Fab. In order to obtain a consistent re s u lt, one must employ the object of anholonomity, even when using the "method of instanta neous frames". The calculations which we are about to perform may be illustrated as the following: v ^3[yAv] V Fab In the inertial frame Av = (0, 0, 0; -$0) 108 where 2 r > a r *0 = S. a r < a Then Aa ■ la A. so that 2,.. cin2 Aa = (o, 0, - Y r- - ^1n e- * -Y Next Fab = U ' Vb Aa so that 2 £)c. A, . ab ab c \3xa 3 X1 Then, in the rotating frame: f 13 = - f 31 " 3r + 2 n13 A3 + 2 413 A4 Y 0 tosin^ 6 c r > a r < a F14 = - f 41 = + 2 A4 r > a r < a 109 A ll other components vanish. Let us return to the in e rtia l frame where A = (0, 0, 0; - *Q). Then 3AV gAy F. yv 3xP 3xv given only F = [ %■ r > a h14 <( r * 0 r < a . Next, the relation between inertial and rotating field tensors is Fab = *«i *b Fpv • Consequently, since = y and = Y r — -■s-——- a _ F14 = -F41 = ye ; = $ r r | 0 r < a f.I 9 ? . 9 „ i y Q usings r^to sm^ 8 i — ^ ------r > a f 13 = “ f 31 = Y c Er = , r < a i 0 Which was to be shown. Thus, i f one desires to employ the "method of instantaneous frames" he should not ill treat the anholonomic object in calculations where it becomes necessary for internal con sistency. I t hardly seems necessary to remark th a t the "method o f n o anholonomic frames" provides us with a technique which is not only simpler to apply, but also conceptually much clearer. In a previous section, we explicitly calculated Fa^ and demon- strated th a t, in the ro ta tin g frame, the Maxwell equations are s a tis fied if one applies the "method of galilean frames" consistently. We w ill now demonstrate that fille r's "method of instantaneous frames", if consistently applied, w ill lead to consistent results. We start with the inertial metrical line element -ds^ = dr^ + r^de^ + r^ sin2 e d Now so that as in the "method of galilean frames". But now, instead of the gali- lean metric, since gab = Ia 1^ gyv» we have 0 0 0 r,2 0 0 9ab r^sin^O .2r\£sin^e .y^uisin^e “ 7 “ c2 ™ C m I a I b gvvwe have y v a Y 1 0 0 0 \ 1 ^ 1 / r 2 0 0 ,ab _ 0 Y2 _ y 2 w 0 ” 2 s in 2 0 C 1, - 2 2 r 2w2 sin 2 0/ 0 _y2w -Y +Y - \ 0 c 12 / We note that these are different from either the components of the galilean rotating metric or "the anholonomic metric". If we consider the static charged sphere, in the inertial frame we have Jy - (0, 0, 0; -CPQ) and Jy = 9yVJv = (0, 0, 0; CP0), Nowthe components in the "instantaneous frame" are Ja = I a = (0 , 0 , Os ) and Ja = la Jy = (°» - ^ 2« sin20 PQ; - YCP0). Furthermore, the above metrical components provide the consistent "instantaneous frame" relations: Ja = gab Jb and J b * 9ba 0 112 That is *b Jv Jv Jb (0,0,0;CPo) (0,0,-Y r2wsin2QPo; A - y CP0) g ^ J 9a b jb or 9baja i8 Jy (0,0,0;-CPo) . (O.O.O;— ) as in the inertial frame, we may define 0 rB -rsin e B0 4> Er \ -rB r2sin0 Br rEg $ ‘ab rsine Bfi -r2sine B* rsin e EA ! * i -E, -rE, -rs in 0 E (We could also have defined Fa^ and proceeded to calculate Fab> but since v/e followed the above procedure in the galilean case le t us pursue it again here.) Mow 113 - B9 r sin 0 - B Br r 2 sin 9 •ab _ g3C gbd Bcd r sine 0 ' 2 sinG r sine Ee E* 0 / r r sinG That is F14 • g11 g44 F„ ♦ g11 g « f13 = (- y-2 + y2 r2^ sin2 6 j ^ + 2| r sin 9 Bj) F24 - g22 g « F24 + g22 g43 F = (_ y ~2 + y2 r 2to2^sin 2 9 ^ E y - + Y- j—) i9_ - y2 £ sin 9 Bv c C j* c « p34 _ 33 44 P . 34 43 F I- ~ 9 g F34 + c? g f43 = r2sin20 + y4 4" ) r Sln 8 E4> + y4 ^ (-»* sin 9 E^) F13 = g11 g33 F „ + g " g2 4 F 13 14 r t i n ^ (~ r sin 0 Be> - T 2 Fc E r ^12 - g11 g22 F-j 2 = h 114 F23 = g22 g33 F23 + g22 g34 F 24 2 w a Tr H X-~2~" s in 6 9 s1n 8 Br) ' y* Jr- r 6c (r Ee) We then have: la lb F «v_ Fab yv gac gbd F ab or g ° V vF. yv cd 9ac 9bd f V jc jC FaS V pa3 a 0 pcd Let us apply this to the specific case of the charged sphere and demonstrate that the previously found Er and Bq are solutions to the Maxwell equations in the "instantaneous frame" form. That is, le t us show th a t r - Q Er ' Y ~ z R = - y ^ Sin 9 8 ' r c are solutions of ab 4jr_ VF » where F and Ja are as specified above by the "method o f in sta n taneous frames". The only non-vanishing component of Fab is rl4 _ 0 A ..3 Q«2 sin 2 e 3 0u2 s in 2 e _ Q r ------9- + V 5 ------y 5----- = ------ since F13 = . 0 r^c r^c also notice that we have the equivalency relations: F14 ■ 5 9le 94d FCd ■ F13 ■ I f '1 Fpv s 9ic 93d FCd * Y Now * grab V. f = — fr + r a c^b a. r c b 3xb rbc F + r bc F and (Schouten, 1951, p. 103) Fbc ={ bc} “ + gbd ga0 2ce + gcd ^ nbe * Consequently r c = _ L 3( be v^g 3xb Whether the local frame is holonomic or anholonomic. Further, a 1 - r3 , r ab _ pba Then, the only interesting component of Maxwell's equation is for a = 4: a = 4 F41) 4 41 04 F14 + n I $ 1 ,4 3r - n14 F - n41 F + 0 - c J /-g Let us write out this last equation: 1 9 r ? . n 0 n nf 1 .9 rw2 sin2 0 w Q x 1 4n7PnC) 7 2 1 ^ * [r'sine-V]^-^ — -2------> ( ^ > - c V Then 0 ( r < a) v 4b ■ t _ L _ _ L- [ H M ] + T92.2..4lnz 8 = 0 (r>a)- r2 sine 3r y r c£ Integrating across the thin shell gives a+ _Q v 4m J4 dr = z ^ Z / Yaa‘ Yaa‘ a- Again, this result requires the use of the "instantaneous frame" anholonomic object. Without its inclusion, this technique would not give consistent results. 117 Conclusions We obtained the same result (Er , BG) v/hether we used M iller's "method of instantaneous frames" or the "method of anholonomic frames" because the pertinent components of the associated transformations I BiC and hg v/ere the same. This is not true in general. Which method should one employ in any given calculation? I believe one should use that technique which is the simplest to interpret physically. Conceptual sim plicity, orthogonality of reference, and sim ilarity to special relativity, weigh heavily in favor of the "method of anholo nomic frames". Only w ith th is method do the components o f the fie ld tensor retain a unique meaning in all frames of reference. Clearly, it is the logical extension of relativity theory to non-inertia! systems. Let us now apply the "method o f anholonomic frames" to Schiff's paradox. XI. SCHIFF'S PARADOX In 1939, L.I. Schiff attempted to explain the vanishing of the electromagnetic field tensor in the frame of an observer rotating externally about a charged spherical capacitor, in terms of Mach's Principle. Consider a negative point charf^e -Q surrounded by a spherical charge distribution of net charge +Q. + Figure 22 A Spherical Charge Distribution The following three situations present themselves: A. Shell and external observer at rest. B. External observer at rest, shell in relative uniform rotation. C. Shell at rest, external observer in relative rotation with respect to the rest frame of the shell. Schiff's paradox runs along these lines: 1. In s itu a tio n A, by Gauss' Law, there is no external fie ld . Hence the field tensor F^v vanishes in both the frame of the 118 119 due to the cancellation of the actual physical current and the qravi- ta tio n a lly induced, or Machian current. Our answer to the above question is also no, but fo r less philosoph ical reasons. First, we want to see v/hy the field tensor is different fo r the th ird case. We shall employ the anholonomic form ulation and obtain the fields seen by a rotating observer. Now, for the negative point charge E = - H n r = .Mil (r> a) 0 nc v ' and for the positive shell E n r Bft = + ^ S-M ( r > a) 0 rc Hence, by superposition, the fields do vanish external to the shell. (The same result may be obtained by the "method of instantaneous frames" however, one must introduce certain ficticious charges and currents when interpreting the contravariant equations, as outlined in the previous cha pte r). Secondly, we notice that if one examines the reference frames that Schiff employed, he finds that they are oblique and arise as a result of a local coordinate rotation. Consequently, his metric (£ *e =q ) u v pv should not be diagonal. However, i f one employs a r e la t iv is tic Lorentz ro ta tio n , he obtains an orthogonal fie ld o f anholonomic reference frames. The electromagnetic fie ld tensor, produced by the charged capacitor, observer and in that of the shell: 2. In situation B, the observer at rest sees currents and therefore he observes a vector potential given by A and hence B"(r, 9) = (“ if3-- cos 0 ) r + — sin 0) e . 3rJc 3rJc the field tensor is thus nonzero for the observer. 3. In situation C, however, the observer circling about sees no magnetic field since, if the tensor Fal) vanishes, then so must the tensor Fuv. The paradox may be stated as a question. "Since the relative motion between spheres and observer is the same in both step 2 and step 3, should not the observer have the same field tensor in both cases?" S c h iff answered no as follow s. In the observer's frame, the distant stars are rotating and this v/arps his metric tensor (intro duces off-diagnal or non-Euclidean terms). If one defines the co v a ria n t components o f the fie ld tensor, and - by means o f th is v/arped metric - obtains the contravariant components of the field tensor, then Maxwell's source equation w ill have extra terms. Schiff inter preted these as the physcial manifestation of an extra current, induced by the action of the rotating distant masses on the electro magnetic field. The vanishing of the external field tensor is then 121 vanishes with respect to these frames simply by the principle of linear Superposition. The paradox is then resolved without the use of absolute space or d is ta n t masses, (see D eS itter, 1916). The significance of rotational electromagnetic phenomena, like Schiff's Paradox, is that observer rotation is a physically different operation from source rotation. X II. A POINT CHARGE AND AN OBSERVER IN RELATIVE ROTATION As a further application of the anholonomic approach, le t us consider the following example (which was firs t posed as a question by Professor C. V. Heer). Suppose there exists a charge Q on the end of a shaft of length Rq. (See Fig. 23). The shaft is assumed to be a perfect insulator. Observer #1 rotates with the shaft and computes the amount of work necessary to move a test charge from the origin out to r=a along a radial path. We call this W' . Observer #2 is assumed to be spatially fixed in an inertial (laboratory) frame of reference. He computes the amount o f work done on a te s t charge to move i t along the same path specified above, from r=0 to r=a. We call this 11 . Ro»a , is presented to guide us through the important physical aspects of the problem. A Digression on the Definition of Work We must, before answering the above question, e xp licitly specify what observers #1 and #2 measure. Consider the follow ing s itu a tio n . 122 123 z X (jj — Figure 23. Observer £1 rotates with the charge Q. Observer #2 is in an inertial (non-rotating frame). Is the work done by #1 on a test charge q in moving it from r=0 to r=a the same as the work done by #2 in moving the test charge over the same path from r=0 to r=a? (assume the test charge has zero mass). 124 Electrostatics: F - QE , E = - V$ A Fiqure 24. Concerning the definition of work. We want to obtain an expression for the work done (by an external mechanical force) on the charge in moving i t from A to B in the fie ld of force, F. Consider the system shown above, where a te s t charge 0 is moved from A to B. In electrostatics, the field of force is given bv Coulomb's Law: F = QF . A mechanical force Fm which is equal and opposite in direction to the field force must be applied to the charge in order to overcome the field of force F . When this is done, the total net force on the charge Q is zero, and the charge may be moved the incremental distance dT . The incremental v/ork dW done by_ the external mechanical force oi^the charge is dW = Fm • dl Since the mechanical force is equal and opposite to the field force, we may w rite dW = - F • dT Hence, the total work done by an external mechanical force on the charge in moving i t from A to B is Again, the minus sign appears because we are computing the work done on the charge (b^Fm) against the action of the field. Further, by definition, if.F m has a component along the direction of motion, the work done by F"m on Q is positive. Consequently, if Q is moved in a direction opposite to F, the work is positive. On the other hand, if Q is moved in the same direction as F, the work done by F is m negative. We also want to relate work and electrical potential difference. In a conservative mechanical system the change in potential energy is equal to the work done in moving from A to B. Analogously, the difference in electrical potential energy of a charge Q between points A and B is equal to the work required to move the charge from A to B. Letting U denote electrical potential energy W A-B Then Note that if the work done, , is positive, the potential energy at B is greater than at A. We may define an electric field potential V as the potential energy per unit charge, V = ^ . The electrical potential difference is then written as AV = V - V = - / B F • dT . b a a (End Digression) 126 1 . Observer at Rest - Charge rotates A. Determination of W (Inertial Frame) - In the inertial frame o+a______we may write dov/n the electrical potential as (r) = -2— - 2_ (r«Rj . V Ro2 0 The magnitude radial component of the electric field is then E„ = - - -2y (r«R ) r (Ro- r )Z Ro2 0 and is directed away from +Q. We may compute the magnetic fie ld by specifying the current distribution (seen in this frame), calculating the vector potential, and then F. (This is the path followed in the more detailed discussion later). Alternatively, to this approximation, we may write (for positive rotation) B = - -7 x E (r _ A where v is the v e lo c ity o f the charge: Vcharge = . Then, near the axis we have only B = 2 V w ith A = 9 ^ 1 . 2cR0 (By the v/ay, for a continuous ring of current we have, near the axis, A -H E . cRo and We now ask, "How much work is done on a charge q to move i t along the specified path dr from 0 to a?" a Mo-a V rm ° & * - V (F + v x B) • dr 0 — A A' and so, since vq = vr r + (rw) Q, W = - q f [ - + ^-2- ] dr. M J 0 R0 V 2 That is V a = « Cp-- m r 1 B. Determination of W by firs t employing classical techniques and then using the anholonomic approach. (a) Classical Techniques. The situation presented now is that of a static point charge. Consequently F = 0. By the way, an observer on axis at r = 0 is common to both rotating and non-rotating frames. The classical approach presents a paradox here, for at r = 0 it gives two different values for F. V/e note that the charge is non-inertial. Further, the electrical potential is 128 And the ra d ia l component of E is Er = T(R0- ^ \ r Z ) 2 - ’ R£ ( r « RQ) The work done by the classical observer on the test charge is a W' * - q f Er dr , o-»-a J r o So th a t (b) Anholonomic Approach In the rotating frame we have (for positive rotation) „2 . 1 2 n ? " R12 ' 2 Y c 2 4 2 rw n12 Y c a4 = . 1 Y 2 n s i 14 2 Y c2 . We may find the vector potential from Aa * "a so th a t a4 - y( - J - V ) + (Y I ) ) 129 Now “ £• = y3 ^ 2 and so c = . JT2 . Y3rm2 + v%2r 3 r (R o -r )2 (R0. r )C2 R0C2 3 Q rV 1 Y 2R„C4 Further _ yQwr2 y r 2wQ 2 = 2CR0 ' (R0-r)C and so 9A2 = yOcur + y3Qr3a)3 3r CRo 2 C \ 2yr(oQ _ y3r 3u)30 _ y r2toQ " (R0-r)C (R.0-r)C3 “ (R0-r)2C From th is we may w rite _ _ _ _ 9A4 3Ar „ F-.* - Er ------+ 2 14 ar 3x4 And so, for r « RQ 2 yQ + yOroj Er = " (R„-r)2 R„C2 130 Furthermore ^12yc = r2z L a ar—~ - 39—— + 2 ^ 2 Ac and so, r - yro/) ,« rg/) _ yr^gfl z ' «t0 (Ro r)C +2 ^^7)0 (R0-r)2c so a ll we have is o „ yQw y r wQ Bz - CR0 ■ (R0- r )2C . We now ask v/hat the work done on a charge q is, to move it along the same path as specified before. r * - ra - i - V * =J Fm ° ^ = -q J [E + c v x B] o dr . 0 Since there is no component of 1 v x B along dr in this frame, we have f * a Q - , Or - or.2 Wo^ = -q J [ - R2 RoC2 ] d r and the work done by Fm on q against the field of force is hi - „ r Ql - a2to2Q n o+a ~ R§ 2R0C2 ^ = o-*a If we attempt to remain consistent with the definition of electrical potential difference as the work expended per 131 u n it charge in moving the charge between two po in ts, we conclude that the electrical potential difference between (o) and (a) is the same in the anholonomic frame as in the inertial frame. 2. Charge at Rest - Observer Rotates In conjunction with the previous example, let us examine the similar (but physically different) problem, in which #2 rotates, and the charge frame is at rest. Here there is no^non-inertial motion of the charge. For computational sim plicity, we w ill use only the near axis approximation. A. In # l‘s frame (the inertial or rest frame this time). By inspection B = 0 and since the work required to move a test charge q from 0 to a is a o then Qa W,o-*a = q R§ 132 B. In #2*s frame (the rotating frame, assumed rotating in the positive sense). (a) By Galilean Relativity Ja = G* / = (0, -up, 0, CP) $ r V r “ (R0- r )2 F = 0 . And the te s t charge q is (w ith respect to #2) moving w ith v e lo c ity A A v = vr r + (-ru ) e . Consequently, the work required to move the charge from (0) to (a) given by a W = - q r [E + - vn x B ] o dr o-^a M J c q o is Qa W = n2 o-*a Ko that is W' „ and V/ „ are the same. But v/e are faced with o-*a o-»a the fact that B" vanishes everywhere in this frame. 133 (b) By the Anholonomic Formulation (again in the positively rotating frame). F irs t Aa = ha A^, where \ = (°» °» °» ~ R0- r J* Then, in the ro ta tin g frame 2 (R0-r)C . Furthermore A* - - -TP ■■ 4 Ro 'r and since i l - 3 rm2 ar - V c2 we have 3A4 _ yQ y3grb£ " “ (R0-r)'2 “ (R0-r)C2 Now _ 3A4 4 14 = Er = I f + 2 fi14 A4 so that for r « RQ •r " /n „\2 134 In the anholonomic frame we also have F12 = rBz = " a f + 2 n*2 Aa and so the anholonomic z component of the magnetic fie ld is . TfQar Bz ' ' (Rn-r)2c . Hence, Bz vanishes for observers along the r = 0 axis. The work done by F™ against the field of force to move a test charge q from r = 0 to r = a is then a a a "i* = / ^ dr = - / Fl dx’ * - / I Fla ja d x l- 0 o 0 Now Jy = (0, 0, 0, cP) where p = q Ja = h® = (0, -ycop, 0, yep). Then, in the anholonomic frame fl ■ c Flb jb gives f l * c F14 j4 4 F12 j2 or fl = YErP - b rB z 135 consequently y 2 Q P + y2Qr2to2P 1 = ' (Ro-r)2 (R0-r)2C2 . The viork is then calculated from so that, as in the inertial frame = ^ = Va In this case we notice that the work is the same as in the inertial frame. Furthermore, the magnetic field paradox is resolved. That is Ef = 0 for observers along the z-axis in both the inertial and non-inertial frames, when there is no actual charge motion. The conclusions v/e draw are as follow s 1. If the charge rotates a) IT j1 0 in both frames a t r = 0. b) VT = W ' o->a o+a 2. If the observer rotates a) F = 0 in both frames at r = 0 b) o+a a = W' o-+a 3. Source rotation and observer rotation are physical ly different operations. One produces a magnetic field at r = 0. One does not. The anholonomic object is an essential intrinisic ingredient in rotationally covariant electro dynamics. X I I I . CONCLUSION The essence of the foregoing analysis is that the Lorentz trans formation, expressed in non-rectangular coordinates, is an anholonomic transform ation. (More s p e c ific a lly : anholonomic case 3). Consequently there exists no rotating global t', corresponding to an inertial t. It is argued that the holonomic form of electrodynamics (See Fig. 25) is not a satisfactory formulation for relativistic rotation. The funda mental difference being that the anholonomic formulation provides an orthogonal reference frame while the holonomic approach employs an oblique frame of reference. The price one pays is that now the object o f anholonomity must be employed. That is , properties o f the a ffin e connection may not be neglected in such anholonomic frames. This problem did not arise in special relativity since the rectilinear Lorentz transformations are holonomic and the transformed a ffin ity remained symmetric. This is not the case for transformations like hy . a Post (1952, p.63) points out that "anholonomic constraints have been barely considered in field theory." I believe that this is the firs t time an anholonomic approach has ever been applied to rotating frames of reference. Specifically, I have never seen the object of anholonomity, as defined above, employed in any electromagnetic calculation. The anholonomic formulation of electrodynamics. (See Fig. 26) is proposed as a tenable explanation of rotational relativity in fla t space-time. 137 138 J * ^ a JU F}IV j " ft T a T b Jf i T V -ab -5» PKFab=A£ j a A -3s» Fab F F ab A A r - 5 > A c ab ab AZL ,a ■ £ > F c J Figure 25. Under the holonomic Lorentz transformation, Tlj , a ll paths lead to the same expression fo r Fa5 for rectilinear motion. But this program leads to ambiguities when applied to rotation. 139 « a Cb * 0 HP > a ' F A V ab ab _ 4 re - 5 > A - > ^ b F where In e rtia l Frame: F,.,. = V A_.-V.A. = + (r®r r ^ )A A yv y v v y vy yv a v 3XU 3X 3 A, 3 A p Anholonomic Frame: F . = V A.-V, A e —b - _ a.+ A ab abba jX 4 9Xb ab C Figure 26. The anholonomic electromagnetic program. Under the anholonomic Lorentz transformation, hj , any of the indicated paths lead to a consistant expression for F . . Furthermore, the rotational paradoxes noa longer arise, and the final results satisfy the principle of general covariance. Appendix: A. Sunmary of Cartan's Notation in Riemannian Space. (Schouten: 1935, p.139) dS2 *= g dxy dxv yv -*■ e = 9 y 9X^ “V t * e = g V v yv ii dxv ■£ y a ^ A a v = dx^dxv - ux ux 'X/ d e = ra dxv t a e * p y v a y a [6 . e ] = + 2 fia y v pv i a V V cS ll i i y v A r a Aa<* 'y % vofv _ 1 a A B II da)W + ■■ RU a Aa va$ <°-c v a v ~ 2 140 Appendix: B. Summary of Classical Tensor Computations (See: Schouten, 1951, p.103) -d 2 _ g dxa dxb 3ab V A1b - 3AL + r b Ac Q ac 3X V At 3Ab a b ” r abI A c 3xa 3Ah 3Aa c F ■ 1 ad / \ 2 cd,b db,c bc,d (r - rv ) s -r, c 2 |jC cb [be] - 1 (r1 - r1 ) r [ jk ] 21 ' jk M 1 f i _ 1_ 31 ng T 3 i \ j 1/ 2 3xJ 1 1 Rh »\| » • k ji 29[krj ] i +2r[k |l,^ + % Fli R* kj kjh 141 Appendix: C. Metrical Conventions Throughout this work, we have le t = dx11 = inertial •u n, wa = dx = anholonomic 'Xi and dxa = h9 dxy . 4 Furthermore, we have le t x = ct, and so -dS^ * g dxy dxv . pv Then, in cylindrical coordinates, ! 1 % 0 0 \ 0 r 2 0 0 \ 0 1 0 0 0 -1/ or, in spherical coordinates, 1 0 0 0 \ 0 r o o 0 1 0 0 r 2Sin CQ 0 1 0 0 0 -1' We also have /I 0 0 0 \ gvv= /0 l/r2 0 0 j ( 0 0 1 0 I \0 0 0 -1/ or 0 0 0 1 /r 2 0 0 0 1 _ 0 I r W e j 0 0 - 1 / I 42 depending upon whether we were employing cylindrical or spherical polar coordinates. The C h ris to ffe l symbols are then e ith e r 1 1 22 i 1 2 1 i 2 12: r or 1 ) 22? - - r f 2 I 9 1 i 112; = 21' = r { . y = -r Sin 0 = - Sine Cose (3 : ;'3 i I 113 = 1 3 1 ’- = r T3 3 ' 123 = ',3 2 , = Cote The Lorentz transformations then become (for counter-clockwise rotation): 1 0 0 ° \ 0 0 1 0 J 0 _Yr jw 0 c r / 143 and n 0 0 a) 0 0 Y Yc 0 0 1 0 r^w 0 Y----- Y i \ ° c or 1 0 o o o o h = 0 1 0) 0 0 Y “Y - 9 9 C A A r wSin 0 / 0 0 "Y------Y/ and 1 0 0 0 ' 0 1 0 0 0 0 03 Y Y L.c 2 2 r wSin 8 0 0 Y , Y c The components o f the anholonomic object are computed from a a b. c dw = 2fi w Aw r\j\> be ^ ^ and are either n2 -1 2 ra/ “ 12 ?.y 2 c 4 = _ - .2Y rw 12 q4 = _ 1 y 2 rw. 14 2 Y 2 c 144 .,2 r 2o)2SineCose Y 2 c 1 _2 r w2Sin20 2 Y 2 C 1 2 r^w2Sin0Cos0 ? Y 2------c 2 r.coSin20 c 2 2 r Pi SinQCosQ Filmed as received without page(s) 1^6-149 . UNIVERSITY MICROFILMS. Appendix: 0. Maxwell's Equations We have adopted the notation of Landau and L ifsh itz, 1962, and w ritte n -dS2 = g dXvdXv . Hence, in the so-called Gaussian system o f u n its (Landau and L ifs h itz , 1962, pp. 77, 288) we write v pVV = / v c (where Av - V ^ ) , which in the standard Gibbsian notation is V*E = 4np 1 3j VXB - r — = — J c 3 t c (B = VXA and E = - V(J>- iK ) . 31 We also write f = 7 FUVJV y C V™ which is f = pF + ~ p7 xF 1. Cylindrical Coordinates We w rite . , u dxp _ dxy _ VG Furthermore / ° rBz -rB z 0 F = g 9 QFa8 = yv ^yorvB -rB B9 - i t K 2. 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