einstein's apple - homogeneous einstein fields (gnv64)

9333_9789814630078_tp.indd 1 23/12/14 9:34 am

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World Scientific

9333_9789814630078_tp.indd 2 23/12/14 9:34 am

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Published by

World Scientific Publishing Co. Pte. Ltd.

5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

EINSTEINS APPLEHomogeneous Einstein Fields

Copyright 2015 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-4630-07-8

In-house Editor: Christopher Teo

Typeset by Stallion PressEmail: [email protected]

Printed in Singapore

Christopher - Einstein's Apple.indd 1 19/12/2014 3:54:39 PM

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January 6, 2015 11:37 Einsteins Apple 9in x 6in b2002 page -5

This work is dedicated to

Alex Harvey

who inspired our eort to put all the pieces together.

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May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory PSTws

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PREFACE

The role of homogeneous gravitational elds in the formulation of theequivalence principle and in the foundation of Einsteins theory of gravita-tion is well known. However, the original treatment of these concepts wasdone in terms of Newtonian gravity and for small velocities. We believe,therefore, that it is necessary to treat homogeneous elds relativistically inEinsteins theory of gravitation. In this book we discuss how this can bedone for manifolds with simply transitive isometry groups and we mentionpossible applications.

Since the results presented here are far from complete, we are aware ofthe preliminary character of our investigation. What we have tried to do isto study the concept of homogeneous elds in Riemannian manifolds fromdierent points of view in an exploratory spirit. This leads to a certainamount of repetition in the dierent chapters which we hope the readerwill excuse. Included are some straightforward calculations that we wouldexpect to readily nd in the literature but appear to be conned to papersin obscure forms or are not in the common bibliographies and texts.

We would like to acknowledge the assistance and support that we havehave received from various individuals during the gestation of this work.We would especially like to thank Alex Harvey, Friedrich Hehl, MalcolmMacCallum, Istvan Ozsvath, and Andrzej Trautman. Jie Zhaos contribu-tions, published and unpublished, have been gratefully appreciated. PeterBergmann should be mentioned for his lifelong eorts to clarify the contentof Einsteins Theory of Gravitation. New York University provided thefacilities where most of the work was carried out.

vii

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CONTENTS

Preface vii

Table of Contents ix

List of Figures xi

0. The Happiest Thought of My Life 1

1. Accelerated Frames 13

2. Torsion and Telemotion 28

3. Inertial and Gravitational Fields in MinkowskiSpacetime 38

4. The Notion of Torsion 47

5. Homogeneous Fields on Two-dimensionalRiemannian Manifolds 60

6. Homogeneous Vector Fields in N-dimensions 79

7. Homogeneous Fields on Three-dimensionalSpacetimes: Elementary Cases 94

8. Proper Lorentz Transformations 111

9. Limits of Spacetimes 136

10. Homogeneous Fields in Minkowski Spacetimes 162

11. Euclidean Three-dimensional Spaces 182

12. Homogeneous Fields in ArbitraryDimension 208

13. Summary 225

Appendix A. Basic Concepts 229

Appendix B. A Non-trivial Global Frame Bundle 240

Appendix C. Geodesics of the PoincareHalf-Plane 244

ix

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x EINSTEINS APPLE

Appendix D. Determination of HomogeneousFields in Two-dimensionalRiemannian Spaces 253

Appendix E. Space Expansion 256

Appendix F. The Reissner-Nordstrom IsotropicField 258

Appendix G. The Cremona Transformation 262

Appendix H. Hessenbergs Vectorial Foundationof Dierential Geometry 265

Appendix K. Gravitation Is Torsion 269

Appendix R. References 274

Appendix X. Index 284

Appendix N. Notations and Conventions 300

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FIGURES

1.A.1 Born Motion 15

1.C.1 A Homogeneous Gravitational Fieldin Minkowski Plane 22

1.C.2 A Homogeneous Gravitational Fieldin Minkowski Plane 24

1.D.1 The Newtonian Gravitational Field 26

2.A.1 Cartans Torsion 29

2.B.1 The Gravitational Frequency Shift 32

2.C.1 Fermis Torsion 35

4.C.1 Einsteins Torsion 51

4.D.1 The Notion of Torsion (1) 54




5.F.1 Two-Dimensional Space of ConstantCurvature Embedding 68

5.I.1 Hyperboloid of the Steady State Model 75

8.D.1 Pseudosphere Cross-section 120

8.D.2 Two Circles in Velocity Space 122

8.D.3 Case II: The Vector Is Spacelike 124

8.D.4 Klein-Beltrami Map I 125

8.D.5 Klein-Beltrami Map II 126

8.E.1 Lorentz Transformation Null Rotations 130

8.F.1 Observable Transverse Velocity 133

8.F.2 Apparent Transverse Velocity 134

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xii EINSTEINS APPLE

C.1 Geodesics of the Poincare Plane 246

C.2 Pencil of Geodesics in the Poincare Plane 247

C.3 Smith Chart 251

G.1 The Cremona Transformation 263

G.2 Bianchi Class A Degeneracy Pattern 264

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January 6, 2015 11:37 Einsteins Apple 9in x 6in b2002 page 1

CHAPTER 0

THE HAPPIEST THOUGHT OF MY LIFE

A. The Principle of Equivalence

In a speech given in Kyoto on December 14, 1922, Albert Einstein re-membered:

I was sitting on a chair in my patent oce in Bern. Suddenlya thought struck me: If a man falls freely, he would not feelhis weight. I was taken aback. The simple thought experimentmade a deep impression on me. It was what led me to the theoryof gravity.

This epiphany, that he once termed der glucklichste Gedanke meinesLebens [the happiest thought of my life], was an unusual vision in 1907.In the history of science it is referred to as Einsteins rst principle ofequivalence and we call it Einsteins Apple.

Einsteins apple was not envisioned after watching the antics of orbitingastronauts on television, sky-diving clubs did not yet exist, and platformdiving was not yet a sports category of the freshly revived Olympic Games.How could this thought have struck him? Had he just been dealing withpatent applications covering the safety of elevators?

Three years earlier, in 1904, the Otis Elevator Company installed inChicago, Illinois, the rst gearless traction electric elevator apparatus, thatwas of the direct drive type, known as the 1:1 elevator. This rst modernelectric elevator made its way to Europe where, on Zurichs Bahnhofstrasseand elsewhere in Switzerland, buildings went up that needed elevators. Itwould have been natural for Director Friedrich Haller at the Swiss PatentOce in Bern to put applications involving electro-mechanical machineryon the desk of Einstein, his expert (second class) with expertise in electro-magnetism. It would not be surprising to nd Einsteins signature approv-ing (or disapproving) a patent application for elevators in 1907.

However, only one patent application with Einsteins comments fromhis years as a Swiss patent examiner has survived. It does not concern

1

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2 EINSTEINS APPLE

elevators [Fluckinger, 1974]. The comments of the patent examiners inHallers les have not been preserved by the Swiss bureaucracy. Einsteinsexpert opinions on patents were destroyed eighteen years after the les wereclosed. Well probably never know how Einstein got his inspiration.

In his review The Relativity Principle and the Conclusions drawn fromit[Einstein, 1907], Einstein formulated his principle of equivalence for therst time. He wrote:

We consider two systems 1 and 2 in motion. Let 1 beaccelerated in the direction of its X-axis, and let be the (tem-porally constant) magnitude of that acceleration. 2 shall be atrest, but it shall be located in a homogeneous gravitational eldthat imparts to all objects an acceleration in the directionof the X-axis.

The next sentence contains the principle of equivalence:

As far as we know, the physical laws with respect to 1 do notdier from those with respect to 2; this is based on the factthat all bodies are equally accelerated in the gravitational eld.

It was this last fact that had inspired Einstein and prompted Sir HermannBondi, Master of Churchill College in Cambridge, England, to the observa-tion:

If a bird watching physicist falls o a cli, he doesnt worryabout his binoculars, they fall with him.

Although nowhere stated in the Principia, one may assume that IsaacNewton was already familiar with the principle of equivalence. In Propo-sition 6 of Book 3 of his Principia [Newton, 1687], Newton describes hisprecise pendulum experiments with gold, silver, lead, glass, sand, commonsalt, wood, water and wheat testing equalityas we would say nowofinertial and passive gravitational mass. His treatment in Proposition 26 ofBook 3 in the Principia dealing with the perturbation by the Sun of thelunar orbit around the Earth, discovered by Tycho Brahe and called theVariation, leaves no doubt that Newton knew how to transform away ahomogeneous gravitational eld.

The apparent enigmatic equality of inertial and passive gravitationalmass was also still a prize question at the beginning of the twentieth century.The Academy of Sciences in Gottingen, Germany, had oered the BenekePrize in 1906 for proving this equality by experiment and theory. TheBaron Roland Eotvos won three-fourths of this prize (3,400 of 4,500 Marks);only three-fourths, because he had only done the experiments and had notattempted a theoretical explanation [Runge, 1909].

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CHAPTER 0 THE HAPPIEST THOUGHT OF MY LIFE 3

The principle of equivalence was not new in Newtons theory of gravita-tion. New was Einsteins extension to all of physics. He wrote:

At our present state of experience we have thus no reason toassume that the systems 1 and 2 dier from each other inany respect, and in the discussion that follows, we shall thereforeassume the complete physical equivalence of a gravitational eldand a corresponding acceleration of the reference system.

B. Where the Principle of Equivalence Leads

This basic observation guided Einstein in the formulation of his theoryof gravitation. The principle of equivalence says, roughly, that all bodies,independent of their nature, experience the same acceleration in a gravita-tional eld of given strength. Newton checked this principle by experimentswith pendulums which were later rened by F. W. Bessel.

In his famous elevator gedanken experiment Einstein saw that this ex-perimental fact of Newtonian gravitation could also be described as theequivalence between motion in a homogeneous gravitational eld and mo-tion in an accelerated frame of reference. In his 1908 paper [Einstein, 1908]he used this equivalence to show that special relativity then demands theexistence of a gravitational redshift for clocks in a gravitational eld. Hisderivation assumes velocities small compared to c (the speed of light) andaccelerations g such that gL/c2


4 EINSTEINS APPLE

the gravitational potential survived in Einsteins theory as certain space-time components of the Riemann tensor.

Interest in homogeneous elds has a number of roots, some quite old.Even before the advent of the theory of general relativity, the concept wasintroduced by Einstein in his initial explorations of the principle of equiv-alence [Einstein, 1911]. After the introduction of general relativity it wasdiscussed in detail by Levi-Civita [Levi-Civita, 1917] [Schucking, 1985A].

For years the homogeneous gravitational eld has been central to thestudy of radiation by a charged particle undergoing uniform acceleration.Also, the concept has long since been implicitly utilized in the realm ofNewtonian mechanics when discussing so-called ctitious elds. It is char-acteristic of these ctitious elds that they can be transformed away. Ifone wants to give a precise meaning to the notion that a homogeneousgravitational eld can be transformed away, one needs to know what ahomogeneous eld is in Einsteins theory.

When T. Levi-Civita treated this concept in terms of general relativityin [Levi-Civita, 1917], it was as a special coordinate system in Minkowskispace, now widely known as Rindler coordinates [Rindler, 1966]. Physicalphenomena in frames adapted to these coordinates have been widely dis-cussed in connection with the interpretation of Hawking radiation [Hawk-ing, 1975] [Lee, 1986] [Unruh, 1976]. One has to point out, however, thatthe Levi-Civita eld is by no means homogeneous in the customary sensethat eld strength is independent of position.

Our objective is to establish a generic formalism for various specic ap-plications. For instance, it is well-known that, in certain areas of the studyof spacetimes, the tangent space provides a rst order local approxima-tion. We are interested in the utilization of homogeneous spaces to providethe next higher order local approximation. Secondly, homogeneous eldsprovide a means of understanding and treating, in a comprehensive fash-ion, ctitious elds encountered in Newtonian dynamics. Thirdly, in someinstances, homogeneous elds may be considered as limits of physically sig-nicant gravitational elds. Fourthly, homogeneous elds may provide auseful tool in analyzing gauge elds of the Kaluza type.

The pursuit of our objective has led us to implement a programme sug-gested by Elie Cartan in lectures presented at the Sorbonne during 192627[Cartan, 1927]. A consequence has been to expose an unobvious relation-ship between homogeneous elds and torsion. The ctitious elds are eldsof acceleration induced torsion by teleparallelism. [See Appendix H forsome of Hessenbergs initial development along this line.]

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To develop his theory of gravitation, Einstein had equated the action ofgravity with the eects of acceleration:

gravitational attraction acceleration . (0.B.0)

As his ideas developed, the geometrical idea of gravity as curvature becamedominant. Subsequently it has become a paradigm in its strength on theanalysis of astrophysical phenomena. The right hand side of the equivalenceequation has been neglected. Early on, Max Planck had pointed out thatEinsteins constructions were limited to low velocities; in fact, they arelimited to local events in space and time.

A new denition for the notion of a homogeneous eld is proposed herewhich applies to at and homogeneous spacetimes in Einsteins theory ofgravitation. It incorporates a principle of equivalence which holds for ex-tended spacetime regions. The new denition runs as follows: Let themetric of a pseudo-Riemannian spacetime be

ds2 =[0]2 [1]2 [2]2 [3]2 jk j k (0.B.1)

with dierential formsj(xk)

(0.B.2)

where Latin indices j, k 0, 1, 2, 3. These dierential forms uniquely deneconnection forms

jk(xl)

(0.B.3)

by means of the equations [Flanders, 1963] [DeWitt-Morette, 1977]

dj jk k = 0, jk + kj = 0, jk jl lk . (0.B.4)

The symbol denotes the skew-symmetric product of dierential forms.Development of the connection forms (0.B.3) in terms of the one-forms(0.B.2) gives

jk = gjkl

l . (0.B.5)

The coecients gjkl, which specify the contribution of each basis dierentialone-form to the decomposition, are termed the physical components of thegravitational eld. One now denes:

A spacetime allows a homogeneous gravitational eld if themanifold allows a frame l in which the coecients gjkl are allconstants, independent of position in spacetime.

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6 EINSTEINS APPLE

The coecients are 24 independent scalar functions with respect to co-ordinate transformations. These scalars satisfy

gjkl = gkjl . (0.B.6)

Under a constant homogeneous Lorentz transformation of the frames,

j(xk) Lj l l (xk) (0.B.7)

where the transformation satises

Lj l jk Lkm = lm (0.B.8)

and in whichLjl = constant , (0.B.9)

the coecients gjkl transform as components of a third rank tensor. Suchhomogeneous elds exist if and only if the spacetime metric is invariantunder a simply transitive group of isometries. This includes Minkowski-space. Examples of such homogeneous elds in Minkowski-space will bestudied.

We rst discuss the general formalism and apply it to a rigidly rotatingsystem followed by specialization to two-dimensional spaces. We then dis-cuss three-dimensional spaces for which the standard approach is by meansof the well-understood Bianchi symmetries [Ellis, 1969]. Though this ap-proach is inadequate for the study of higher dimensional cases some of theconcepts and techniques employed can be utilized.

We focus on the connection coecients as the essential descriptive ele-ments of the various possible spaces and their role as eld strengths . Theviewpoint here is that because the Riemann tensor is quadratic in the con-nection coecients the latter may be considered square roots of the com-ponents of the former. They are, from both physical and mathematicalpoints of view, simpler quantities and their study should lead to a betterunderstanding of the space. One can tell more about certain aspects ofthe dynamical behavior of a spacetime by examining the geodesics, thatis, connection coecients, than by studying the geodesic deviation, that is,components of the Riemann tensor.

Among the more interesting areas of the investigation are at spaceswhich a fortiori are homogeneous. The existence of such non-vanishingeld strengths in Minkowski space is extremely interestingnot least be-cause this lack of uniqueness is not yet understood. It indicates the exis-tence of homogeneous gauge transformations by which source-free ctitious

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gravitational elds can be transformed away. One has in this way a coordi-nate independent description of the homogeneous gravitational elds whichentered into the early formulation of Einsteins theory of gravitation.

A further study of such elds appears useful for several reasons. Firstly,an understanding of the reasons for the lack of uniqueness would facilitatethe use of ctitious elds on a systematic basis. Secondly, the existence ofnon-vanishing eld strengths in the case of Minkowski space suggests thatthe traditional usage of rotating frames in Newtonian mechanics for study-ing Coriolis and centrifugal forces might be extended to special relativisticmechanics. Thirdly, and possibly the most interesting, the techniques be-ing developed might help interpret gauge theories of the Kaluza type. Suchtheories may be set in a ve-dimensional Minkowski space. The expecta-tion is that one might be able to show that constant electric and magneticelds may be transformed away in a manner similar to homogeneous grav-itational elds in the lower dimensional case. The analog of the Einsteinelevator becomes here the Kaluza elevator that is built from matter ofthe same e/m ratio as that of the particles one studies in it.

It will be shown that this procedure can also give homogeneous elds de-ned as above in Einsteins theory [Geroch, 1969] [Schucking, 1985A]. Sincegravitational elds induced by masses and inertial elds induced by accel-erations are held to be physically equivalent, we call both Einstein elds.

Before we go on with the discussion, we have to say more about thereference system.

C. The Reference System

Einsteins reference system was based on identical clocks and rigid bod-ies. Through the work of Gustav Herglotz, Max Born, and Max von Laue, itwas soon realized that rigid bodies do not form suitable reference systems.The ideas of Hermann Minkowski and Henri Poincare allow us to describethe reference systems of special relativity more clearly. The Minkowskispacetime with a metric given by the line element

ds2 = dxdx , , 0, 1, 2, 3 , (0.C.1)

with

=

1 0 0 00 1 0 00 0 1 00 0 0 1

(0.C.2)has a set of distinguished coordinates x. Under a Poincare transformation

x = x + a , a = constant (0.C.3)

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8 EINSTEINS APPLE

with constant and

= (0.C.4)

the x again form such a set of distinguished coordinates. We call themlinear orthonormal coordinates.

To keep physics and mathematics clearly dened, we introduce four con-stant orthonormal vector elds e such that

e x

(0.C.5)

for a set of distinguished coordinates x. The orthonormal vectors alongthe coordinate axes just introduced by the operators /x are subject to

dx(

x

)= . (0.C.6)

The invariant characterization of a vector tangent to a manifold as adirectional dierentiation operator on the functions living on this mani-fold goes back to Sophus Lie. He used that concept in his Theory ofTransformation Groups, where vector elds became innitesimal trans-formations. Dening dierentials like dx as the duals of vectors like

/x emerged from the work of Elie Cartan and its interpretation byErich Kaehler and Nicholas Bourbaki.

The vectors in (0.C.6) are all parallel to each other and dened on all ofMinkowski spacetime. The components V (x) of any vector eld V

V = V (x)

(

x

), (0.C.7)

also known as the physical components, have a metrical meaning, that isto say, their numerical values are results of physical measurements in termsof meters or seconds. If one introduces new coordinates y that are non-linear and/or non-orthonormal functions of the distinguished coordinatesx

y = f(x) , det

[y

x

]= 0 , (0.C.8)

the basis vector elds /y will no longer be orthonormal and the com-ponents V (y) of

V = V (y)

(

y

)(0.C.9)

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will no longer be physical components. The metric tensor g will no longerbe , but instead we have

ds2 = g(y)y

xy

xdx dx = dx

dx . (0.C.10)

If we take the mathematicians point of view that coordinates are anarbitrary means for naming events in Minkowski spacetime, we cannot as-sign a direct physical meaning to the components of vector elds given inthose coordinates either. However, this was not a point of view of taken byEinstein originally and became the root of misunderstandings.

If we want to know the physical components V (x) in arbitrary coordi-nates y(x), we only have to remember the chain rule to see that

V (x)

x= V (y)

y= V (x)

y(x)

x

y. (0.C.11)

This gives the relation

V (y) = V (x)y(x)

x(0.C.12)

for the transformation of vector components from one coordinate system toanother.

The use of arbitrary coordinates in Minkowski spacetime is just a matterof convenience for adapting the coordinates to the symmetry of a given sit-uation. We should now stress that the identication of the constant vectoreld with the e, as dened in (0.C.10), is invariant under the Poincaregroup. The translations evidently do nothing and constant Lorentz trans-formations give

e = e

x=

x . (0.C.13)

We call the set of four constant orthonormal vectors, e , in each event ofMinkowski spacetime a frame.

In the nineteenth century physicists were already using more generalframes in 3-dimensional space. For instance, in problems with axial orspherical symmetry it was useful to let the frame vectors i, j,k be tangent tothe orthogonal lines of constant coordinate pairs. In this way one could de-ne, for example, the radial component of an electrical eld strength. Withthe advent of Minkowskis spacetime in 1908, such adaptable orthonormalframes also appeared in four dimensions.

We shall dene our reference system abstractly as an orthonormal framein every point of the spacetime manifold. Nowadays, one calls this a sectionof the frame bundle.

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10 EINSTEINS APPLE

D. The Physical Principle of Equivalence

We now return to the question: What is the physical meaning of Ein-steins rst principle of equivalence?

In his monograph Relativity, The General Theory John Synge con-fesses in his introduction

. . . I have never been able to understand this Principle.

and goes on to write:

Does it mean that the eects of a gravitational eld are indis-tinguishable from the eects of an observers acceleration? If so,it is false. In Einsteins theory, either there is a gravitationaleld or there is none, according as the Riemann tensor does ordoes not vanish. This is an absolute property; it has nothingto do with any observers worldline. Space-time is either at orcurved, and in several places of the book I have been at consid-erable pains to separate truly gravitational eects due to curva-ture of space-time from those due to curvature of the observersworldline (in most ordinary cases the latter predominate). ThePrinciple of Equivalence performed the essential oce of mid-wife at the birth of general relativity, but, as Einstein remarked,the infant would never have got beyond its long-clothes had itnot been for Minkowskis concept. I suggest that the midwife benow buried with appropriate honours and the facts of absolutespace-time faced. [Synge, 1960]

Are we beginning a chapter of forensic physics if we investigate thecorpse of a principle? It is easy to agree with Synge. If one admits a met-ric, one also buys into the Levi-Civita connection and its Riemann tensorand this connection makes itself felt. But that was not all. From the begin-ning of relativity, already in the special theory, there was the question of thereference body. A reference body can be dened mathematically througha section of the frame bundle of the spacetime manifold, with Fermis con-struction being an approximation for physics in some cases. So, it is themathematical reference body, the generalization of the constant parallel or-thonormal 4-vectors in Minkowski space-time that now show teleparallelismand torsion. Mathematically, we can formulate our principle as

= d = . (0.D.1)

Its meaning is that we enlarge the set of admissible frames for the descrip-tion of physical phenomena in spacetime.

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Einstein tried to understand gravitation through acceleration and thatmade it necessary to introduce accelerated frames, that is, frames withtorsion.

Although Gregorio Ricci-Curbastro had already introduced frames intomanifolds in 1895, their use appeared optional until the advent of PaulDiracs equation for spin 1/2 particles. Most physicists approached theinterpretation of general relativity from the particle point of view. But eldsare more important than particlesat least quantum elds. Fermi was anexception. In his study of Fermi transport, he worried about the descriptionof an electromagnetic eld in an accelerated reference system. How wouldthe energy density be aected by the acceleration? This raises the generalquestion of how elds are aected by torsion. The prime example for this isthe Unruh eect where the eld is the vacuum [Unruh, 1976]. It is clear thata mathematical section of the frame bundle does not give rise to physicaleects like energy densities, etcetera. But as soon as one attaches physicalobjects to the frames, one has reference bodies and for the Unruh eect[Wald, 1994], what one calls particle detectors. But, we have alreadyseen that where light is emitted by a source at rest and absorbed by areceiver also at rest, we get redshifts in accelerated systems as measured byPound and Rebka.

E. Newton, Mach, and Einstein

Einsteins apple is, like Newtons apple, a seminal thought of great pen-etrating power. It has been claimed that the story of Newtons apple wasapocryphal; but a book containing the recollections of William Stukeley[Stukeley, 1936] appears to conrm it. Stukeley, a doctor, from Lincolnshirelike Newton, became a close friend to Newton in Sir Isaacs last years.Stukeley describes the Summer evening when Newton, then in his eighties,recalled his thoughts from sixty years before:

After dinner, the weather being warm, we went into the gardenand drank tea, under the shade of some apple trees, only he andmyself. Amidst other discourse, he told me, he was just in thesame situation, as when formerly, the notion of gravitation cameinto his mind. It was occasiond by the fall of an apple, as hesat in a contemplative mood. Why should that apple alwaysdescend perpendicularly to the ground, thought he to himself.Why should it not go sideways or upwards, but constantly to theearths centre? Assuredly the reason is, that the earth draws it.There must be a drawing power in matter: and the sum of thedrawing power must be in the earths centre, not in any side

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12 EINSTEINS APPLE

of the earth. Therefore does the apple fall perpendicularly, ortowards the centre. If matter thus draws matter, it must be inproportion of its quantity. Therefore the apple draws the earth,as well as the earth draws the apple. That there is a power, likethat we here call gravity, which extends itself into the universe.

What Newton had done was to look at gravity from a new frame whoseorigin was in the center of the Earth.

In 1907 Einstein did not show the equivalence of acceleration and gravi-tation described by spacetime curvature. He did not show either the equiv-alence of geodesics and non-geodesics or the equivalence of rotating andnon-rotating systems.

What he did, we now can see more clearly, was the introduction of ac-celerated reference systems exhibiting torsion through distant parallelism.There were physical consequences that needed to be checked for these sys-tems, like the constancy of the speed of light independent of acceleration,no inuence of acceleration on the rate of clocks and the length of stan-dards. As far as these assumptions have been tested, they appear to be inorder.

In 1911 Einstein formulated an equivalence principle that involved rela-tive acceleration in an attempt to introduce ideas of Ernst Mach into histheory [Einstein, 1911]. This did not prove to be a happy idea since this no-tion makes mathematical sense only for bodies having the same 4-velocityand, from a physical point of view, accelerations are absolute. These ideasgave the theory its name.

However, as John Stachel [Stachel, 1980] pointed out, it was the olderidea of 1907 that guided him through Ehrenfests paradox of the rotatingdisc to Riemannian geometry. We can now see that going from Levi-Civitasconnection in Minkowskis spacetime to teleparallelism opens the door togoing back from torsion to a Levi-Civita connection with curvature.

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CHAPTER 1

ACCELERATED FRAMES

A. Born Motion

Borns motion stands in analogy to an observers motion on the circle

x2 + y2 = 2 (1.A.1)

of radius in the (x-y)-plane. If the motion is at constant speed alongthe circle, an observer experiences a constant acceleration in the directionopposite from the center of motion. In Born motion, we require an observerto feel constant acceleration in Minkowski spacetime.

We introduce into the (x-t)-plane (with c = 1) Born-polar coordinates

t = sinh , x = cosh , (1.A.2)

and they yield the line element in the form

ds2 = dt2 dx2 = 2 d2 d2 , (1.A.3)

since the dierentials of equations (1.A.2) are related by

dt = d sinh + cosh d ,

dx = d cosh + sinh d . (1.A.4)

The vector form of the line element is

s

s=

1

2

+

= e0 e0 + e1 e1 (1.A.5)

13

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14 EINSTEINS APPLE

where the vector elds

e0 1

, e1

(1.A.6)

are orthonormal. The coordinate lines = constant are the hyperbolae

x2 t2 = 2 (1.A.7)

of Born motion. Its time-like worldlines, with proper time s as parameter,

=s

, = constant (1.A.8)

have the unit tangent vector(dt

ds,dx

ds

)=

(cosh

s

, sinh

s

) e0 . (1.A.9)

The acceleration vector is given by(d2t

ds2,d2x

ds2

)=

1

(sinh

s

, cosh

s

) 1

e1 , (1.A.10)

with the aid of (1.A.4). This shows that 1/ is the magnitude of the acceler-ation and the hyperbolae (1.A.7) are the worldlines of constant acceleration.

For Born motion, acceleration becomes the analog of the curvature ofcircular motion and the hyperbolic functions replace the trigonometric onesof circular motion.

It is easy to construct a picture of the vector elds e0 and e1 of (1.A.3)that show, in a region of Minkowski spacetime, the local four-velocity andfour acceleration. [See Figure 1.A.1.]

It is clear that the frame eld is invariant under motions in the new timecoordinate but not under motions in the -direction because the size ofthe acceleration is 1/.

Here was a problem that Einstein apparently had not noticed in his 1907paper on the equivalence principle. In Newtons theory the acceleration,, could be a vector in the x-direction, constant in space and time, inde-pendent of the velocity of a body moving in the x-direction. Not so inMinkowski spacetime. There the acceleration vector has to be orthogonalto the 4-velocity; it would appear that homogeneity of the acceleration eldin spacetime could no longer be achieved.

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CHAPTER 1 ACCELERATED FRAMES 15

Figure 1.A.1 Born Motion Vectors

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16 EINSTEINS APPLE

Apparently Max Planck had noticed the problem. It was four months af-ter Einstein had mailed his paper that he sent a correction to the Jahrbuch.It began:

A letter by Mr. Planck induced me to add the following supple-mentary remark so as to prevent a misunderstanding that couldarise easily: In the section Principle of relativity and gravita-tion, a reference system at rest situated in a temporally con-stant, homogeneous gravitational eld is treated as physicallyequivalent to a uniformly accelerated, gravitation-free referencesystem. The concept uniformly accelerated needs further clar-ication.[Einstein, 1908]

Einstein then pointed out that the equivalence was to be restricted to a bodywith zero velocity in the accelerated system. In a linear approximation, heconcluded, this was sucient because only linear terms had to be takeninto account.

Einsteins retreat raises the question whether it is impossible to nd ahomogeneous uniformly accelerated reference system, or, assuming exactvalidity of his principle of equivalence, a homogeneous gravitational eld.

B. A Homogeneous Gravitational Field

A denition of homogeneity in an n-dimensional manifold involves theexistence of n linearly independent dierential one-forms

(x) = (x) dx , det [ ] = 0 (1.B.1)

and n independent functions

x = f(x) , det[f

x

]= 0 . (1.B.2)

The coordinate transformation (1.B.2) of the dierential forms (1.B.1), theso-called pull-back,

(x) = (x) dx = [ f(x) ]f(x)

xdx (1.B.3)

denes n new dierential forms (x) by

[ f(x) ]f(x)

xdx (x) dx (x) . (1.B.4)

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If the and , are the same functions of their arguments

(x) = (x) , (1.B.5)

we say the dierential forms are invariant.A simple example for (1.B.5) and (1.B.2) is the translation isomorphism

of Rn given by

(x) = dx , (x) = dx , x = x + c , (1.B.6)

where c = constant, which immediately yields

(x) = = (x) . (1.B.7)

The invariance, (1.B.5), of the dierential forms gives

(x)

f(x)

xdx = (x

) dx . (1.B.8)

Introducing the inverse of gives

(x)(x) = (1.B.9)

wherex

x=

f(x)

x= (x

)(x) . (1.B.10)

These are the equations of Lies rst fundamental theorem for Lie groups.Dierentiation of the gives the Maurer-Cartan equations

d +1

2C

= 0 (1.B.11)

and the invariance of the leads to constancy of the structure constants C . Further dierentiation makes the rst term of the left-hand side of(1.B.11) zero and creates thus the conditions for Lies third fundamentaltheorem of Lie groups by the integrability conditions

C d C d = 0 . (1.B.12)

With the Maurer-Cartan equations, we have

C C

= 0 . (1.B.13)

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18 EINSTEINS APPLE

This gives the dening conditions for a Lie algebra

C C + C

C

+ C

C

= 0 . (1.B.14)

By introducing Lie groups through invariant dierential forms, Elie Cartanwas able to simplify Lies derivations of his fundamental theorems.

A Riemannian manifold is called homogeneous if its metric is invariantunder a transitive group of motions. Mathematicians call a transformationgroup transitive if its action maps any point of the manifold into any otherpoint. The S2, for example, with its standard metric inherited from beingrigidly embedded into the 3-dimensional Euclidean space, is homogeneousunder the action of the rotation group O(3).

The homogeneous manifolds we need for gravitational elds are of a spe-cial nature. For them, the group of motions has to be simply transitive,meaning that there is only one group element that moves a point of themanifold into another given point of the manifold. In this case, a Riemann-ian manifold is homogeneous if the metric

ds2 = , det [ ] = 0 , = = constant , (1.B.15)

is given in terms of invariant dierential one-forms (x) for a simplytransitive group. These one-forms dene in each point of the manifold anorthonormal n-leg of vectors, also known as a frame by

(e) = . (1.B.16)

In coordinate components, these vectors are then given by (1.B.9) as

e . (1.B.17)

The Levi-Civita connection is dened by dierential one-forms , theconnection forms, that represent the gravitational forces given by

d = , = . (1.B.18)

These are Elie Cartans rst structural equations for a Riemannian spacewith vanishing torsion. Some authors dene the connection forms withthe opposite sign from our convention.

The components of the gravitational eld, g, are given by

= g

, g + g = 0 . (1.B.19)

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We have thusd = g

. (1.B.20)Comparing this equation with the Maurer-Cartan equation with the indiceslowered by gives

d =1

2(g g) = 1

2C

. (1.B.21)

We obtain, therefore, with even permutations,

C = g g , (1.B.22a)

C = g g , (1.B.22b)C = g g . (1.B.22c)

Because of the skew-symmetry of g in its rst two indices, according to(1.B.19), adding (1.B.22a) and (1.B.22b) and subtracting (1.B.22c) gives

2 g = C + C C . (1.B.23)

This shows that the physical components of a gravitational elds strengthare constant in a homogeneous gravitational eld.

For the further development it is useful to discuss an example of such ahomogeneous gravitational eld.

C. A Homogeneous Gravitational Field in the Minkowski Plane

It is easy to nd invariant dierential forms if one writes the metric forthe Minkowski plane as

ds2 = dx dx = 2 du dv , =

(0 11 0

)(1.C.1)

where we have used the null coordinates

u =12( t x ) , v = 1

2( t + x ) , (1.C.2)

and the associated vectors

u=

12

(

t

x

),

v=

12

(

t+

x

). (1.C.3)

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20 EINSTEINS APPLE

The one-forms

0 =du

u, 1 = u dv , (1.C.4a)

ds2 = 201 , = constant = 0 , (1.C.4b)

are independent and give us constant structure coecients. We have

d0 = C001 0 1 = 0 , (1.C.5)

d1 = du dv = 0 1 = C101 0 1 (1.C.6)which gives

C001 = 0 , C101 = . (1.C.7)

We then have, from (1.B.23), for the gravitational eld strength(s)

g100 = , g101 = 0 . (1.C.8)

The frame vectors are given by (1.B.16); in the current situation, they are

e0 = u

u, e1 =

1

u

v. (1.C.9)

A general velocity vector in the Minkowski plane is subject to

2V 0 V 1 = 1 (1.C.10)

and is thus time-like. It is directed into the future if V 0 > 0.The acceleration vector g has the covariant components gj equal to

gj = gjkl Vk V l , (1.C.11)

which is the geodesic acceleration adapted to our frame. In our case, weobtain

g0 = g010 V1 V 0 = V 1 V 0 , (1.C.12a)

g1 = g100 V0 V 0 = (V 0)2 . (1.C.12b)

Evidently, this expression for the acceleration fullls the requirement thatit be orthogonal to the 4-velocity; that is,

gj Vj = 0 . (1.C.13)

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If we choose the velocity as a constant, that is, V 0 = constant > 0, itfollows from (1.C.10) that V 1 is also constant and the velocity vector eld

V = V e = V0 u

u+

1

2V 0 u

v(1.C.14)

is invariant in the Minkowski plane. To better visualize the eld, we cal-culate the worldlines that have the vectors of the eld as tangents. Wehave

du

ds= V 0 u ,

dv

ds=

1

2V 0 u(1.C.15)

or, with integration constants u0, v0, we get

u = u0 eV 0 s , v v0 = e

V 0 s

2 u0 (V 0 )2. (1.C.16)

By eliminating u0 and s, we obtain the one-parameter set of curves

v v0 = 12 u (V 0 )2

. (1.C.17)

This is a set of identical hyperbolae that are obtained from the hyperbola

u v +1

2 (V 0 )2= 0 (1.C.18)

by translation in the v-direction. With (1.C.2), this hyperbola can also bewritten as

t2 x2 + 1(V 0 )2

= 0 . (1.C.19)

[See Figure 1.C.1; left and right hyperbolae.]Next, we calculate the acceleration vector eld. We have from (1.C.12)

g0 = (V 0)2 , g1 = V 1 V 0 (1.C.20)

and thus

g = g e = 2 (V 0)2 u u

+V 1 V 0

u

v. (1.C.21)

This vector eld is tangent to the curves

du

d= ( V 0)2 u , dv

d=

V 1 V 0

u(1.C.22)

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22 EINSTEINS APPLE

Figure 1.C.1A Homogeneous Gravitational Field in the Minkowski Plane

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described by the parameter . We obtain

u = u1 e(V 0)2 , v v1 = V

1 V 0

( V 0 )2 u1e(V

0)2 (1.C.23)

with integration constants u1, v1. By eliminating u1 and v1 together with(1.C.10), we obtain

v v1 = 12 u ( V 0 )2

. (1.C.24)

These curves are obtained from the hyperbola

u v =1

2 ( V 0 )2(1.C.25)

by translation in the v-direction.With (1.C.2), this hyperbola can also be written

t2 x2 = 1( V 0 )2

. (1.C.26)

[See Figure 1.C.1; upper and lower hyperbolae.]

By shifting the lower branch of the hyperbola by2/V 0 in the positive

v-direction, we intersect then the time-like hyperbola (1.C.18) at t = 0,x = 1/V 0. Tangent to the hyperbolae at their intersection are the twoframe vectors of the velocity eld and its acceleration. [See Figure 1.C.2.]Now, we wish to show that in the neighborhood of the intersection pointwe are approximating a Newtonian homogeneous gravitational eld.

D. The Newtonian Gravitational Field

We study the Newtonian eld with acceleration in the x-direction ofzero velocity and at t = 0. In the Minkowski (t-x)-plane, we take parallelworldlines x = constant that are geodesics. For t = 0 and x = 1/V 0,the parallel to the t-axis through this point is touched by the hyperbola(1.C.19). This hyperbola, parameterized by with proper time s,

t = sinhs

, x = cosh

s

, =

1

V 0 , (1.D.1)

is the worldline of an observer with constant intrinsic acceleration. Now,let this observer measure the distance to the straight line x = orthogonal

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24 EINSTEINS APPLE

Figure 1.C.2A Homogeneous Gravitational Field in the Minkowski Plane

The two hyperbolic branches intersecting at x = 1/(V 0) are the rightbranch of the hyperbola in Figure D.1 and lower branch of the hyperbolain Figure D.1 displaced in the v-direction. The two curves intersect orthog-onally and have as their unit tangent vectors e0 and e1 of a homogeneousgravitational eld.

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to his worldline. This distance is a function (s). From Figure 1.D.1, weread o immediately that

= cosh(s/)

= cosh(s/) 1cosh(s/)

=1

2s2 + . . . (1.D.2)

where higher terms are of fourth-order in time s. The acceleration is givenby

a =d2

ds2=

1

= V 0 . (1.D.3)

Since the velocity d/ds increases, the acceleration points into the negativex-direction.

It is for this special case: against absolute space, in the same event, andat the same velocity, an inertial observer and the accelerated one can inter-pret gravity as acceleration and vice versa. Here the Newtonian equivalencefor a homogeneous gravitational eld can be made relativistic in a point.This raises the question: Is it possible to save Einsteins equivalence for ahomogeneous gravitational eld.

E. Relativistic Equivalence

We have applied the calculus of Ricci Curbastro to Minkowski spacetimefor the description of a homogeneous gravitational eld in Einsteins theory.It turns out that such elds actually exist, without any spacetime curvature,in a at world. The existence of such a non-vanishing eld came as asurprise to us because its analogs on the S2 or the Euclidean plane do notexist. The reason for its existence on a at spacetime is that the Poincaregroup in (1,1)-dimensions has a simply transitive 2-dimensional subgroup.This is not the case for SO(3). The Euclidean motions in the plane, E(2),only have the trivial transformations.

In terms of the light-like coordinates u and v, we can write the Poincaregroup of the Minkowski plane as

u = u + a , v =1

v + b , = 0 (1.E.1)

with constants a, b, . In matrix form, this isuv1

= 0 a0 1 b

0 0 1

uv1

. (1.E.2)

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26 EINSTEINS APPLE

Figure 1.D.1 The Newtonian Gravitational Field

The worldline described by the parametric equations

x = coshs

, t = sinh

s

, =

1

V 0

has acceleration 1/. At t = 0, it osculates the geodesic = constant(the vertical dashed line). The vector determines the distance from thatgeodesic to the hyperbola in the rest system of the accelerated observer,orthogonal to his worldline.

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Clearly, the matrices with vanishing a 0 00 1 b0 0 1

(1.E.3)form a subgroup of the Poincare group depending on the two parameters and b. For events with u = 0, a unique event with coordinates (u, v) existsand is obtained from the inverse matrix of (1.E.3). This demonstrates thatthe subgroup acts simply transitively in the two demi-mondes u = 0. Theexistence of this subgroup is the source of the homogeneous eld. But it isalso the source of what is known as teleparallelism.

If we take an orthonormal frame in one original point with say u < 0,the motions of the subgroup move it to all other points with u < 0 ina unique fashion. When we assign to a vector V in the original pointthe components V 0 and V 1, we can now assign these same components tovectors with respect to the frames that were transported and now denethem as parallel to each other. This is clearly what mathematicians call anequivalence relation known as teleparallelism.

Such a more general notion of parallelity, based on simply transitivegroups, was introduced by the Princeton geometer and later Dean LutherPfahler Eisenhart in a brief paper in 1925 [Eisenhart, 1925]. The concept

was further developed by Elie Cartan together with the Dutch electricalengineer Jan Arnoldus Schouten who had discovered parallel displacementin Riemannian geometry independently of Levi-Civita [Cartan, 1926].

The homogeneity group is characterized by the invariant dierentialforms of (1.B.1) that determine the frames. We get the frames ev-erywhere if we pick a frame in one point and integrate the total dierentialequations (1.B.11) of Ludwig Maurer and Elie Cartan. We call the vectorial2-form

= d = 12C

(1.E.4)

the torsion form. In any case, it is simply determined by the structureconstants, C , of the homogeneity group.

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CHAPTER 2

TORSION AND TELEMOTION

A. Cartans Torsion

Elie Cartan introduced the notion of torsion in a brief note in 1922[Cartan, 1922]. He amplied his sketch in a memoir in the following year.It was reprinted in his collected papers in 1955. This paper Sur les varietesa` connexion ane et la theorie de la relativite generalisee is now accessiblethrough its translation in book form and an introduction that facilitates itsstudy [Cartan, 1923][Cartan, 1924].

It is easiest to introduce torsion by using the notion of the exteriorcovariant derivative of a vector-valued dierential one-form. For the generalformalism see, for example, [EDM2, 1993].

In the tangent vector space attached to each point of a manifold existsa unit operator I. The torsion is described by the vector-valued 2-form

= I . (2.A.1)

Here, indicates the operator of exterior covariant derivation. One writes

I = e , (e) = . (2.A.2)

The connection is then dened by

e = e (2.A.3)

where the are the connection 1-forms. The torsion becomes then

= (e)= e + e d= e ( d

) . (2.A.4)

28

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CHAPTER 2 TORSION AND TELEMOTION 29

Figure 2.A.1: Cartans Torsion

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30 EINSTEINS APPLE

Here, we are interested only in anely connected manifolds for which(2.A.3) holds and where the connection leaves the metric invariant, that is,the scalar product of vectors in the tangent space. We have

( e e ) = e e + e e= e e e e = ( + )= = 0 . (2.A.5)

This means that the connection form with covariant indices must be skewin these indices.

We now want to study the constant torsion in the Minkowski plane.Writing the vector components of the torsion 2-form

=1

2T

, (2.A.6)

we have, from (1.E.4) and (1.C.7)

T 001 = 0 , T101 = . (2.A.7)

This shows that the torsion is constant and thus homogeneous.

We now want to look at the basis vectors e0 and e1 of our distant par-allelism. We have from (1.C.9)

e0 = u , e1 =1

uv . (2.A.8)

We take a point P on the line u = 1 and draw the frame vectors e0(P )and e1(P ). The vector e0(P ) has its tip in the point Q that lies on the line u = 2. According to (2.A.8), the vector e1(Q) is only half as long asthe vector e1(P ). [See Figure 2.A.1.]

We now assume that the connection coecients vanish:

The vector e0(P), issuing from the tip of the vector e1(P ), is obtained

through parallel transfer from e0(P ) and has remained unchanged since uwas constant. The eect of the torsion is that our square PQP Q does notclose. There should be a physical explanation for this phenomenon.

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B. The Gravitational Frequency Shift

For a rst orientation, we draw the two hyperbolic worldlines of twoobservers who experience a homogeneous gravitational eld. [See Fig-ure 2.B.1.] Their 4-velocities V are given by (1.C.14) for negative u

V = V e = V 0 u u 12V 0 u

v . (2.B.1)

Since V 0 is constant, the vectors are parallel in the usual ane sense alonglines u = constant. These are the null lines of rising photons emitted whenthe lower observer has precisely the same velocity as the higher observer atthe reception of the light. There is no frequency shift for a rising photon.It is obvious from the gure that a falling photon is perceived by the lowerobserver to originate from an approaching source and thus be blue-shifted.To calculate the shift, we remember that the ratio of the frequency of theemitter to the frequency of the receiver is given by

=

k Vk V . (2.B.2)

Here, V and V are the 4-velocities of emitter and receiver, respectively,while k is the 4-vector of the photon.

For the falling photon the k null vector has a u-component only, say,

k ( 1, 0 ) . (2.B.3)This is given, with (2.B.1),

=

12V 0 u

:1

2V 0 u= uu

. (2.B.4)Since |u | is smaller than |u |, we obtain a blue-shift. We now also seethat this blue-shift for a falling photon is the physical equivalent of thenon-closure of our square PQP Q. The ratio in length of PP /QQ is theblue-shift /.

As we had seen in (1.C.17), two dierent observers dier only in theirvalue of the parameter v0

u = 12 ( v v0 ) (V 0 )2 . (2.B.5)

This shows that the ratios of their u-values, |u/u |, goes towards 1 forv . The blue-shift vanishes asymptotically.

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32 EINSTEINS APPLE

Figure 2.B.1: The Gravitational Frequency Shift

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C. Fermis Torsion

Almost simultaneously with Elie Cartans 1922 introduction of torsion,this concept and its importance for physics was also discovered by a studentat Italys famous Scuola Normale Superiore in Pisa. Enrico Fermi, thenaged 21, had sent a note to the Rendiconti of the Accademia dei Linceion Sopra i fenomeni che avvengono in vicinanza di una linea oraria (Onthe phenomena that occur in the neighborhood of a time-like worldline)[Fermi, 1922]. The short paper, that appeared in three pieces, introducedwhat became known as Fermi transport of vectors [Walker, 1932].

Let U be a time-like unit vector eld tangent to a bunch of worldlines.We have

U U = 1 . (2.C.1)The acceleration along the worldlines is given by

A = U DUDs

, (2.C.2)

where A is the acceleration vector and s measures proper time along theworldlines. Clearly, because of (2.C.1),

U U = A U = 0 , (2.C.3)

the acceleration vector is orthogonal to the four-velocityU. The two vectorsU andA form a bivectorUA that describes a Lorentz boost for the framethat has the four-velocity vector U.

For a vector V in the frame with four-velocity

U = e0 , (2.C.4)

the time-part V is given by the projection of V on U

V (V U)U = V 0 e0 , (2.C.5)

while the space-part V is dened as

V V j ej = V e V 0 e0 = V (V U)U . (2.C.6)

Fermi transport for a vector V along a worldline with unit tangent U ischaracterized by a boost that compensates the acceleration A. The time-part of V remains unchanged, while the space-part becomes subject toa rotation preserving the length of the space-part of the vector V and,

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34 EINSTEINS APPLE

thus, its four-dimensional length. The Fermi connection is thus a metricalconnection.

As an example of Fermi transport, we return to Born motion in theMinkowski plane, as described in Section 1.A, with the use of polarcoordinates. (They are called Rindler coordinates by some authors.)[See Figure 2.C.1.]

The unit tangent vector, U, along the hyperbolae will remain a unittangent vector along the hyperbolae under Fermi transport. Along theradial geodesics, however, Fermi transport becomes simply Levi-Civitasparallel transport. It is clear then that we have torsion since parallelogramsdo not close. It is easy to see that the torsion becomes the cause of frequencyshifts of light. We have the metric (1.A.3)

ds2 = 2 d2 d2 . (2.C.7)Null lines are given by ds = 0 and thus

d = d. (2.C.8)

A light ray emitted from 1 at time 1 will be received at 2 at time 2:

2+d21+d1

d = 21

d =

21

d

= ln

21

. (2.C.9)

A second light ray emitted at 1+d1, will be received at 2+d2. It followsthat

d1 = d2 . (2.C.10)

Since frequencies go inversely to proper times, ds, we have the gravitationalshift

12

=ds2ds1

=2 d21 d1

=21

. (2.C.11)

Going back to (1.A.3), we can now calculate the torsion. We have

0 = d , 1 = d , (2.C.12)

andds2 =

[0]2 [1]2 . (2.C.13)

The connection form is zero since vectors do not change their componentsunder Fermi transport in the Minkowski plane. We thus have for the torsionforms vector components, ,

0 = d0 = d d = 11 0 , 1 = d1 = 0 . (2.C.14)

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Figure 2.C.1: Fermis Torsion

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36 EINSTEINS APPLE

With

=1

2T

, (2.C.15)

we obtain

T 001 = 1, T 101 = 0 . (2.C.16)

We can express the gravitational shift in terms of torsion coecients.

D. Reference Frames

Enrico Fermi had introduced his vector transport in approaching a phys-ical description of a gravitational eld. It would seem natural for a physi-cist to interpret the hyperbolae of Born motion as the worldlines of rigidat slabs made of incompressible matter. But all attempts to introducerigid bodies into relativity came to naught. This became particularly clearthrough a one-page paper by Pawel Sigmundovich Ehrenfest in Leiden thatwas the death knell for rigidity and led to the introduction of non-Euclideangeometry into relativity theory. The paper posing Ehrenfests paradoxpointed out that a circle of radius R centered on the axis of rotation ina rigid body and orthogonal to it would not show Lorentzs contraction ofits radius while its circumference would. The deviation from 2 in the ratioof circumference to radius of the circle would thus indicate the presence ofnon-Euclidean geometry.

We shall dene our reference system abstractly as an orthonormal framein every point of the spacetime manifold. Nowadays one calls this a sectionof the frame bundle. How this can be physically realized is another question.Mathematically this section is given by

e(x) . (2.D.1)

When Einstein wrote about accelerated reference systems, it was as-sumed that meter sticks and clocks in such systems would be unchanged.Experiments done with muons in synchrocyclotrons have conrmed thisassumption to high accuracy. Since clocks and meter sticks can be repre-sented as vectors in four dimensions, it follows that the frames themselves,their metric, and, thus, their orthonormality is not aected by acceleration.

The acceleration of observers or, as Einstein thought equivalent to it,a gravitational eld are not due to the curvature of the spacetime mani-fold. The falling of apples from Earth-bound trees and the weight we areexperiencing while standing on the surface of this planet is the result of anacceleration of our reference frame that can be described mathematically

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as the torsion eld of Minkowski spacetime. The presence of curvature nearthe surface of the Earth, that is, a non-vanishing Riemann tensor, corre-sponds to second order derivatives of the Newtonian potential and is a smallhigher order eect. The mantra gravitation is curvature is misleading ifone associates gravitation with the falling of apples.

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CHAPTER 3

INERTIAL AND GRAVITATIONAL

FIELDS IN MINKOWSKI SPACETIME

A. Inertial Frames of Reference in Minkowski Spacetime

We begin with a discussion of the Poincare group. We do this to remindthe reader of well known facts and use the occasion to introduce terminologyand notation. More on that can be found in Appendix A and in the booksby Bernard Schutz [Schutz, 1980][Schutz, 2009].

The orthonormal and at hypersurfaces x = constant in Minkowskisspacetime do have a physical meaning. Their intersections dene orthonor-mal inertial reference frames with respect to which kinematical and dy-namical notions like velocity, angular velocity, acceleration, momentum,or energy of a body can be measured. We describe these distinguishedMinkowski coordinates x as a column vector with four entries which weabbreviate by the letter x. Any other set x of Minkowski coordinates isthen related to x by the matrix equation[

x1

]=

[L a0 1

] [x

1

]. (3.A.1)

Here a is another column vector with four entries while L is the 44 matrixof a Lorentz transformation

LT L = (3.A.2)

where is the 4 4 matrix

=

1 0 0 00 1 0 00 0 1 00 0 0 1

(3.A.3)and the superscript T indicates transposition. The set of these transfor-mations form the ten-dimensional Poincare group. The principle of special

38

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CHAPTER 3 INERTIAL AND GRAVITATIONAL FIELDS 39

relativity expresses the covariance of the equations of physics under thePoincare group. By covariance we mean that physical quantities describedby tensors transform as representations of the Lorentz group. The worldlineof a force-free particle remains a straight line in Minkowski spacetime underall transformations of the Poincare group. [If one deals with spinorsaswe shall notone has to go to a covering group of the Poincare group.] Weintroduce the metric through the four dierential forms

= dx (3.A.4)

byds2 =

(3.A.5)

and the frame vectors, also known as four-legs or vierbeine, by

(e) = , e =

x. (3.A.6)

An energy-momentum-stress tensor T(x), to give an example, would bewritten

T(x) = T (x) e e (3.A.7)with components T (x). These components are known as physical compo-nents or leg components. Let us now carry out a Poincare transformationas a coordinate transformation. We have then

= dx = L dx = L

(3.A.8)

and correspondingly,

x= e = L

(

x

)= L e . (3.A.9)

With this transformation of the leg vectors, we have

(e) = L (e) = ( e L ) = L . (3.A.10)

This gives again (e) = . (3.A.11)

For the tensor T we get

T (Lx + a ) = T (Lx + a ) e e . (3.A.12)

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40 EINSTEINS APPLE

This gives the transformation law for the leg components

T (Lx + a ) = L L T

(Lx + a ) . (3.A.13)

We stress that under this coordinate transformation the coecients L ofthe Lorentz transformation do not depend on position in spacetime, nor dothe legs e or e

. We call the inertial frames e , introduced above by the

Minkowski coordinates x, a natural inertial frame. The Poincare grouptransforms natural inertial frames into natural inertial frames.

B. General Leg Transformations

It is often convenient in physics to use accelerated or rotating frames forthe description of phenomena. Such non-inertial frames can be obtainedfrom inertial frames by a general leg transformation given by

= L(x) , e = e L(x) . (3.B.1)

The L(x) are dierentiable functions of x which fulll (3.A.2). They canbe parameterized in terms of six functions of the four variables x. A tensorT(x) under such a general leg transformation gives

T (x) = T (x) e e = T (x) e e (3.B.2)with

T (x) = L(x)L(x)T

(x) . (3.B.3)

If the L are constants, the general leg transformation leads to a new setof inertial frames in the Minkowski spacetime. But it is no longer true that

e =

x. (3.B.4)

The frames then no longer coincide with the natural inertial frames givenby the intersection of the at coordinate hypersurfaces. For the generalleg transformation it is often convenient to work with the dierential legforms which have simpler transformation properties than the leg vectorse. We have introduced a general leg through a transformation from anatural frame. This means that the general leg is described in terms ofthe Minkowski coordinates x. But that is not essential for the notion ofa general leg. We are free to use other coordinates. The reader will not beoended, we hope, if we give a simple example of a general leg. Let

0 = dt , 1 = dx , 2 = dy , 3 = dz (3.B.5)

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CHAPTER 3 INERTIAL AND GRAVITATIONAL FIELDS 4

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