The General Theory of Relativity

The General Theory

of Relativity

A Mathematical Exposition

Simon Fraser University Burnaby, BC

Canada

Simon Fraser University Burnaby, BC

Canada

ISBN 978-1-4614-3657-7 ISBN 978-1-4614-3658-4 (eBook) DOI 10.1007/978-1-4614-3658-4

Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2012938036 © Springer Science+Business Media New York 2012

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

Printed on acid-free paper

General relativity is to date the most successful theory of gravity. In this theory, the gravitational field is not a conventional force but instead is due to the geometric properties of a manifold commonly known as space–time. These properties give rise to a rich physical theory incorporating many areas of mathematics. In this vein, this book is well suited for the advanced mathematics or physics student, as well as researchers, and it is hoped that the balance of rigorous mathematics and physical insights and applications will benefit the intended audience. The main text and exercises have been designed both to gently introduce topics and to develop the framework to the point necessary for the practitioner in the field. This text tries to cover all of the important subjects in the field of classical general relativity in a mathematically precise way.

This is a subject which is often counterintuitive when first encountered. We have therefore provided extensive discussions and proofs to many statements, which may seem surprising at first glance. There are also many elegant results from theorems which are applicable to relativity theory which, if someone is aware of them, can save the individual practitioner much calculation (and time). We have tried to include many of them. We have tried to steer the middle ground between brute force and mathematical elegance in this text, as both approaches have their merits in certain situations. In doing this, we hope that the final result is “reader friendly.” There are some sections that are considered advanced and can safely be skipped by those who are learning the subject for the first time. This is indicated in the introduction of those sections.

The mathematics of the theory of general relativity is mostly derived from tensor algebra and tensor analysis, and some background in these subjects, along with special relativity (relativity in the absence of gravity), is required. Therefore, in Chapter1, we briefly provide the tensor analysis in Riemannian and pseudo-Riemannian differentiable manifolds. These topics are discussed in an arbitrary dimension and have many possible applications.

In Chapter2, we review the special theory of relativity in the arena of the four-dimensional flat space–time manifold. Then, we introduce curved space–time and Einstein’s field equations which govern gravitational phenomena.

In Chapter3, we explore spherically symmetric solutions of Einstein’s equations, which are useful, for example, in the study of nonrotating stars. Foremost among these solutions is the Schwarzschild metric, which describes the gravitational field outside such stars. This solution is the general relativistic analog of Newton’s inverse-square force law of universal gravitation. The Schwarzschild metric, and perturbations of this solution, has been utilized for many experimental verifications of general relativity within the solar system. General solutions to the field equations under spherical symmetry are also derived, which have application in the study of both static and nonstatic stellar structure.

In Chapter4, we deal with static and stationary solutions of the field equations, both in general and under the assumption of certain important symmetries. An important case which is examined at great length is the Kerr metric, which may describe the gravitational field outside of certain rotating bodies.

In Chapter 5, the fascinating topic of black holes is investigated. The two most important solutions, the Schwarzschild black hole and the axially symmetric Kerr black hole, are explored in great detail. The formation of black holes from gravitational collapse is also discussed.

In Chapter6, physically significant cosmological models are pursued. (In this arena of the physical sciences, the impact of Einstein’s theory is very deep and revolutionary indeed!) An introduction to higher dimensional gravity is also included in this chapter.

In Chapter7, the mathematical topics regarding Petrov’s algebraic classification of the Riemann and the conformal tensor are studied. Moreover, the Newman– Penrose versions of Einstein’s field equations, incorporating Petrov’s classification, are explored. This is done in great detail, as it is a difficult topic and we feel that detailed derivations of some of the equations are useful.

In Chapter8, we introduce the coupled Einstein–Maxwell–Klein–Gordon field equations. This complicated system of equations classically describes the self-gravitation of charged scalar wave fields. In the special arena of spherically symmetric, static space–time, these field equations, with suitable boundary condi-tions, yield a nonlinear eigenvalue problem for the allowed theoretical charges of gravitationally bound wave-mechanical condensates.

Eight appendices are also provided that deal with special topics in classical general relativity as well as some necessary background mathematics.

C2 and the conventions for the definitions of the Riemann, Ricci, and conformal tensors follow the classic book of Eisenhart.

We would like to thank many people for various reasons. As there are so many who we are indebted to, we can only explicitly thank a few here, in the hope that it is understood that there are many others who have indirectly contributed to this book in many, sometimes subtle, ways.

I (A. Das) learned much of general relativity from the late Professors J. L. Synge and C. Lanczos during my stay at the Dublin Institute for Advanced Studies. Before that period, I had as mentors in relativity theory Professors S. N. Bose (of Bose– Einstein statistics), S. D. Majumdar, and A. K. Raychaudhuri in Kolkata. During my stay in Pittsburgh, I regularly participated in, and benefited from, seminars organized by Professor E. Newman. In Canada, I had informal discussions with Professors F. Cooperstock, J. Gegenberg, W. Israel, and E. Pechlaner and Drs. P Agrawal, S. Kloster, M. M. Som, M. Suvegas, and N. Tariq. Moreover, in many international conferences on general relativity and gravitation, I had informal discussions with many adept participants through the years.

I taught the theory of relativity at University College of Dublin, Jadavpur University (Kolkata), Carnegie-Mellon University, and mostly at Simon Fraser University (Canada). Stimulations received from the inquiring minds of students, both graduate and undergraduate, certainly consolidated my understanding of this subject.

Finally, I thank my wife, Mrs. Purabi Das. I am very grateful for her constant encouragement and patience.

I (A. DeBenedictis) would like to thank all of the professors, colleagues, and students who have taught and influenced me. As mentioned previously, there are far too many to name them all individually. I would like to thank Professor E. N. Glass of the University of Michigan-Ann Arbor and the University of Windsor, who gave me my first proper introduction to this fascinating field of physics and mathematics. I would like to thank Professor K. S. Viswanathan of Simon Fraser University, from whom I learned, among the many things he taught me, that this field has consequences in theoretical physics far beyond what I originally had thought.

I would also like to thank my colleagues whom I have met over the years at various institutions and conferences. All of them have helped me, even if they do not know it. Discussions with them, and their hospitality during my visits, are worthy of great thanks. During the production of this work, I was especially indebted to my colleagues in quantum gravity. They have given me the appreciation of how difficult it is to turn the subject matter of this book into a quantum theory, and opened up a fascinating new area of research to me. The quantization of the gravitational field is likely to be one of the deepest, difficult, and most interesting puzzles in theoretical physics for some time. I hope that this text will provide a solid background for half of that puzzle to those who choose to tread down this path.

Not least, I extend my deepest thanks and appreciation to my wife Jennifer for her encouragement throughout this project. I do not know how she did it.

We both extend great thanks to Mrs. Sabine Lebhart for her excellent and timely typesetting of a very difficult manuscript.

Finally, we wish the best to all students, researchers, and curious minds who will each in their own way advance the field of gravitation and convey this beautiful subject to future generations. We hope that this book will prove useful to them.

Vancouver, Canada Anadijiban Das

1 Tensor Analysis on Differentiable Manifolds. . . 1

1.1 Differentiable Manifolds.. . . 1

1.2 Tensor Fields Over Differentiable Manifolds. . . 16

1.3 Riemannian and Pseudo-Riemannian Manifolds. . . 40

1.4 Extrinsic Curvature. . . 88

2 The Pseudo-Riemannian Space–Time ManifoldM4. . . 105

2.1 Review of the Special Theory of Relativity. . . 105

2.2 Curved Space–Time and Gravitation.. . . 136

2.3 General Properties ofTij . . . 174

2.4 Solution Strategies, Classification, and Initial-Value Problems.. . . 195

2.5 Fluids, Deformable Solids, and Electromagnetic Fields. . . 210

3 Spherically Symmetric Space–Time Domains. . . 229

3.1 Schwarzschild Solution. . . 229

3.2 Spherically Symmetric Static Interior Solutions . . . 246

3.3 Nonstatic, Spherically Symmetric Solutions . . . 258

4 Static and Stationary Space–Time Domains. . . 277

4.1 Static Axially Symmetric Space–Time Domains. . . 277

4.2 The General Static Field Equations. . . 290

4.3 Axially Symmetric Stationary Space–Time Domains.. . . 317

4.4 The General Stationary Field Equations. . . 331

5 Black Holes. . . 351

5.1 Spherically Symmetric Black Holes. . . 351

5.2 Kerr Black Holes. . . 384

5.3 Exotic Black Holes. . . 403

6 Cosmology. . . 419

6.1 Big Bang Models. . . 419

6.2 Scalar Fields in Cosmology. . . 440

6.3 Five-Dimensional Cosmological Models. . . 456

7 Algebraic Classification of Field Equations. . . 465

7.1 The Petrov Classification of the Curvature Tensor. . . 465

7.2 Newman–Penrose Equations. . . 503

8 The Coupled Einstein–Maxwell–Klein–Gordon Equations. . . 537

8.1 The General E–M–K–G Field Equations. . . 537

8.2 Static Space–Time Domains and the E–M–K–G Equations. . . 542

8.3 Spherical Symmetry and a Nonlinear Eigenvalue Problem for a Theoretical Fine-Structure Constant. . . 551

Appendix 1 Variational Derivation of Differential Equations. . . 569

Appendix 2 Partial Differential Equations. . . 585

Appendix 3 Canonical Forms of Matrices. . . 605

Appendix 4 Conformally Flat Space–Times and “the Fifth Force”. . . 617

Appendix 5 Linearized Theory and Gravitational Waves. . . 625

Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines. . . 633

Appendix 7 Gravitational Instantons. . . 647

Appendix 8 Computational Symbolic Algebra Calculations. . . 653

References. . . 661

Fig. 1.1 A chart.; U /and projection mappings. . . 2

Fig. 1.2 Two charts inM and a coordinate transformation. . . 3

Fig. 1.3 The polar coordinate chart. . . 4

Fig. 1.4 Spherical polar coordinates . . . 5

Fig. 1.5 Tangent vector inE3andR3. . . 6

Fig. 1.6 A parametrized curveintoM. . . 11

Fig. 1.7 Reparametrization of a curve. . . 13

Fig. 1.8 The Jacobian mapping of tangent vectors. . . 22

Fig. 1.9 A vector fieldUE.x/along an integral curve.; x/. . . 36

Fig. 1.10 A classification chart for manifolds endowed with metric. . . 65

Fig. 1.11 Parallel propagation of a vector along a curve . . . 74

Fig. 1.12 Parallel transport along a closed curve. . . 76

Fig. 1.13 Parallel transport along closed curves on several manifolds. Although all manifolds here are intrinsically flat, except for the apex of (c), the cone yields nontrivial parallel transport of the vector when it is transported around the curve shown, which encompasses the apex. The domain enclosed by a curve encircling the apex is non-star-shaped, and therefore, nontrivial parallel transport may be obtained even though the entire curve is located in regions where the manifold is flat. . . 76

Fig. 1.14 Two-dimensional surface generated by geodesics. . . 79

Fig. 1.15 Geodesic deviation between two neighboring longitudes. . . 81

Fig. 1.16 A circular helix inR3_{. . . . 83}

Fig. 1.17 A two-dimensional surface†2embedded inR3. . . 89

Fig. 1.18 A smooth surface of revolution. . . 92

Fig. 1.19 The image†N1of a parametrized hypersurface . . . 94

Fig. 1.20 Coordinate transformation and reparametrization of hypersurface . . . 95

Fig. 1.21 Change of normal vector due to the extrinsic curvature. . . 98

Fig. 2.1 A tangent vectorEvp0inM4and its imageEvx0inR 4 _{. . . 106}

Fig. 2.2 Null coneNx₀ with vertex atx0(circles represent suppressed spheres). . . 108

Fig. 2.3 A Lorentz transformation inducing a mapping between two coordinate planes. . . 109

Fig. 2.4 ImagesS; T, andNof a spacelike, timelike, and a null curve.. . . 112

Fig. 2.5 The three-dimensional hyperhyperboloid representing the 4-velocity constraint.. . . 114

Fig. 2.6 A world tube and a curve representing a fluid streamline. . . 118

Fig. 2.7 A doubly sliced world tube of an extended body. . . 120

Fig. 2.8 Mapping of a rectangular coordinate grid into a curvilinear grid in the space–time manifold.. . . 128

Fig. 2.9 A coordinate transformation mapping half linesLO_C andLOinto half linesLOO_CandLOO. . . 129

Fig. 2.10 Three massive particles falling freely in space under Earth’s gravity. . . 138

Fig. 2.11 (a) Space and time trajectories of two geodesic particles freely falling towards the Earth. (b) A similar figure but adapted to the geodesic motion of the two freely falling observers in curved space–timeM4 . . . 139

Fig. 2.12 Qualitative representation of a swarm of particles moving under the influence of a gravitational field. . . 140

Fig. 2.13 (a) shows the parallel transport along a nongeodesic curve. (b) depicts the F–W transport along the same curve. . . 147

Fig. 2.14 Measurement of a spacelike separation along the image . . . 149

Fig. 2.15 A material world tube in the domainD.b/. . . 165

Fig. 2.16 Analytic extension of solutions from the original domainD.e/intoDQO. . . 168

Fig. 2.17 Five two-dimensional surfaces with some peculiarities. . . 169

Fig. 2.18 (a) shows a material world tube. (b) shows the continuousU field overPE ˙ . . . 184

Fig. 2.19 A doubly sliced world tube of an isolated, extended material body. . . 188

Fig. 2.20 DomainDWDD.0/.0; t1/R4for the initial-value problem.. 203

Fig. 3.1 Two-dimensional submanifoldM2 of the Schwarzschild space–time. The surface representing M2here is known as Flamm’s paraboloid [102]. . . 233

Fig. 3.2 Rosette motion of a planet and the perihelion shift. . . 238

Fig. 3.3 The deflection of light around the Sun. . . 240

Fig. 3.5 Qualitative representation of a spherical body inside a

concentric shell. . . 248

Fig. 3.6 A convex domainDin a two-dimensional coordinate plane.. . . 260

Fig. 4.1 The two-dimensional and the corresponding axially symmetric three-dimensional domain. . . 279

Fig. 4.2 Two axially symmetric bodies in “Euclidean coordinate spaces”. . . 281

Fig. 4.3 A massive, charged particle at x.1/and a point x in the extended body. . . 308

Fig. 5.1 Qualitative picture depicting two mappings from the Lemaˆıtre chart. . . 353

Fig. 5.2 The graph of the semicubical parabola.r /O 3D. /O 2 . . . 355

Fig. 5.3 The mappingXand its restrictionsX_j::. . . 360

Fig. 5.4 The graph of the LambertW-function.. . . 361

Fig. 5.5 Four domains covered by the doubly null, uv coordinate chart.. . . 364

Fig. 5.6 The maximal extension of the Schwarzschild chart.. . . 365

Fig. 5.7 Intersection of two surfaces of revolution in the maximally extended Schwarzschild universe. . . 369

Fig. 5.8 Eddington–Finkelstein coordinates.uO;vO/describing the black hole. The vertical linesrO D2mandrOD0 indicate the event horizon and the singularity, respectively. . . 370

Fig. 5.9 Qualitative graph ofM.r/. . . 372

Fig. 5.10 Collapse of a dust ball into a black hole in a Tolman-Bondi-Lemaˆıtre chart. . . 373

Fig. 5.11 Collapse of a dust ball into a black hole in Kruskal–Szekeres coordinates. . . 375

Fig. 5.12 Qualitative representation of a collapsing spherically symmetric star in three instants. . . 376

Fig. 5.13 Boundary of the collapsing surface and the (absolute) event horizon.. . . 377

Fig. 5.14 Various profile curves representing horizons in the submanifold' D=2; t Dconst in the Kerr space–time.. . . 387

Fig. 5.15 Locations of horizons, ergosphere, ring singularity, etc., in the Kerr-submanifoldx4_{Dconst. . . 389}

Fig. 5.16 The region of validity for the metric in (5.100iii) and (5.99).. . . 396

Fig. 5.17 The region of validity for the metric in (5.101). . . 396

Fig. 5.18 The submanifoldM₂and its two coordinate charts. . . 397

Fig. 5.19 The maximally extended Kerr submanifoldMQ₂. . . 398

Fig. 5.20 Qualitative representation of an exotic black hole in theT-domain and the Kruskal–Szekeres chart. . . 409

Fig. 5.22 Qualitative graphs ofy D Œ.s/ 1_{and the straight}

lineyD˝.s/WDy0C.1=3/.ss0/. . . 418

Fig. 6.1 Qualitative graphs for the “radius of the universe” as a

function of time in three Friedmann (or standard) models. . . 426 Fig. 6.2 Qualitative representation of a submanifoldM2of the

spatially closed space–timeM4. . . 426

Fig. 6.3 Qualitative graphs ofy D Œ.s/ 1_{and the straight}

lineyDy0C 1₃.ss0/. . . 435

Fig. 6.4 Comparison of the square of the cosmological scale factor,a2_{.t /. The dotted lines represent the}

numerically evolved Cauchy data utilizing the scheme outlined in Sect. 2.4 to various orders int t0

(quadratic, cubic, quartic). The solid line represents

the analytic result.e2t_{/. . . 452}

Fig. 6.5 Qualitative graphs of two functions.ˇ/and z.ˇ/

corresponding to the particular functionh.ˇ/WD"ˇ1_{. . . 460}

Fig. 6.6 The qualitative graph of the function.ˇ/for0 < ˇ < 1. . . 461 Fig. 6.7 Qualitative graphs of a typical functionh.ˇ/and the

curve comprising of minima for the one-parameter

family of such functions.. . . 462 Fig. 6.8 Graphs of evolutions of functions depicting the scale

factorsA.t;O w/andˇ.t;O w/. (Note that at late times the compact dimension expands at a slower rate than

the noncompact dimensions).. . . 464 Fig. 7.1 A tetrad field containing two spacelike and two null

vector fields. . . 468 Fig. 8.1 A plot of the function in (8.39) with the following

parameters:c0D1,eD1,0D1andx0D0. . . 550 Fig. 8.2 A plot of the functionW .x1/ D x1 V .x1/

subject to the boundary conditionsW .0/ D 0,

@1W .x1/_jx1_D0D0:5; 1;and 5 representing increasing

frequency respectively. The constant.0/is set to unity.. . . 552 Fig. 8.3 The graph of the functionr Dcothx1 > 0. . . 556 Fig. 8.4 Graphs of the eigenfunctionsU.0/.x/,U.1/.x/andU.2/.x/. . . 560 Fig. 8.5 (a) Qualitative graph of eigenfunction u.0/.r/: (b)

Qualitative plot of the radial distanceR.r/versusr:

(c) Qualitative plot of the ratio of circumference divided by radial distance. (d) Qualitative

two-dimensional projection of the three-dimensional,

spherically symmetric geometry.. . . 564 Fig. 8.6 Qualitative plots of three null cones representing

Fig. 8.7 (a) Qualitative graph of the eigenfunction u.2/.r/: (b) Qualitative plot of the radial distance R.r/WDR₀r

Cˇˇu.2/.w/ˇˇdw: (c) Qualitative plot of the ratio of circumference divided by radial distance. (d) Qualitative, two-dimensional projection of the

three-dimensional, spherically symmetric geometry.. . . 566 Fig. A1.1 Two twice-differentiable parametrized curves intoRN_{. . . 570}

Fig. A1.2 The mappings corresponding to a tensor field

y.rCs/_D.rCs/_{.x/. . . 573}

Fig. A1.3 Two representative spacelike hypersurfaces in an

ADM decomposition of space–time. . . 583 Fig. A2.1 Classification diagram of p.d.e.s. . . 588 Fig. A2.2 Graphs of nonunique solutions. . . 603 Fig. A5.1 An illustration of the quantities in (A5.13) in

the three-dimensional spatial submanifold. The coordinatesxs, known as the source points, span the entire source (shaded region).O represents an

arbitrary origin of the coordinate system. . . 628 Fig. A5.2 TheC(top) and (bottom) polarizations of

gravitational waves. A loop of string is deformed as shown over time as a gravitational wave passes out of the page. Inset: a superposition of the two most extreme deformations of the string for theCand

polarizations. . . 630 Fig. A5.3 The sensitivity of the LISA and LIGO detectors. The

dark regions indicate the likely amplitudes (vertical axis, denoting change in length divided by mean length of detector) and frequencies (horizontal axis, in cycles per second) of astrophysical sources of gravitational waves. The approximately “U”-shaped lines indicate the extreme sensitivity levels of the LISA (left) and LIGO (right) detectors. BHDblack hole, NSDneutron star, SNDsupernova (Figure

courtesy of NASA). . . 631 Fig. A6.1 A possible picture for the space–time foam.

Space-time that seems smooth on large scales (left) may actually be endowed with a sea of nontrivial topologies (represented by handles on the right) due to quantum gravity effects. (Note that, as discussed in the main text, this topology is not necessarily changing.)

Fig. A6.2 A qualitative representation of an interuniverse wormhole (top) and an intra-universe wormhole (bottom). In the second scenario, the wormhole could

provide a shortcut to otherwise distant parts of the universe. . . 634 Fig. A6.3 Left: A cross-section of the wormhole profile curve

near the throat region. Right: The wormhole is

generated by rotating the profile curve about thex3_{-axis. . . 635}

Fig. A6.4 A “top-hat” function for the warp-drive space–time with one direction (x3_{) suppressed. The center}

of the ship is located at the center of the top hat,

corresponding tosrD0 . . . 639

Fig. A6.5 The expansion of spatial volume elements, (A6.12), for the warp-drive space–time with thex3_coordinate

suppressed. Note that, in this model, there is contraction of volume elements in front of the ship and an expansion of volume elements behind the ship. The ship, however, is located in a region with no

expansion nor compression. . . 640 Fig. A6.6 A plot of the distribution of negative energy density in

a plane (x3 _{D0) containing the ship for a warp-drive}

space–time. . . 641 Fig. A6.7 Two examples of closed timelike curves. In

(a) the closure of the timelike curve is introduced by topological identification. In (b) the time

coordinate is periodic . . . 642 Fig. A6.8 The embedding of anti-de Sitter space–time in a

five-dimensional flat “space–time” which possesses two timelike coordinates,U andV (two dimensions

suppressed).. . . 643 Fig. A6.9 The light-cone structure about an axis%D 0in the

G¨odel space–time. On the left, the light cones tip forward, and on the right, they tip backward. Note that at%Dln.1Cp2/the light cones are sufficiently tipped over that the'direction is null. At greater%, the'direction is timelike, indicating the presence of

Table 1.1 The number of independent components of the

Riemann–Christoffel tensor for various dimensionsN . . . 60 Table 2.1 Correspondence between relativistic and

non-relativistic physical quantities. . . 127 Table 4.1 Comparison between Newtonian gravity and Einstein

static gravity outside matter.. . . 292 Table 7.1 Complex Segre characteristics and principal null

directions for various Petrov types. . . 497 Table 8.1 Physical quantities associated with the first five

eigenfunctions ofU.j /.x/.. . . 567

Symbols

d’Alembertian operator, completion of an example

Q.E.D., completion of proof

Central dot denotes multiplication (used to make

crowded equations more readable)

Identity

WD,DW Definition, which is an identity involving new notation

Hodge star operation, tortoise coordinate designation

j:: Constrained to a curve or surface

2 Belongs to

r

s0, O .rCs/th order zero tensor. (In the latter the number of dots indicaterands.)

E

O Zero vector

[j k; i] Christoffel symbol of the 1st kind

i j k

Christoffel symbol of the 2nd kind

Œc$d ,f$g, etc. Represents the previous term in brackets of an expression but with the given indices interchanged

a Angular momentum parameter, expansion factor in

F–L–R–W metric

k EAk Norm or length of a vector

A[B Union of two sets

A\B Intersection of two sets

AB Cartesian product of two sets

1_{For common tensors, only the coordinate component form is shown in this list.}

2_{Occasionally the symbols listed here will also have other definitions in the text. We tabulate the}

most common definitions here as it should be clear in the text where the meanings differ from those in this list.

AB Ais a subset ofB

.AE EB/gg:: .AE;BE/ Inner product between two vectors

pA^ qB Wedge product between ap-form and aq-form

A Electric potential, (also a function used in

five-dimensional cosmologies)

Ai Components of the electromagnetic 4-potential

˛ Affine parameter for a null geodesic, Newman–Penrose

spin-coefficient

B Magnetic potential, bivector set

ˇ Expansion coefficient for 5th dimension, Newman–

Penrose spin-coefficient

c Speed of light (usually set to 1)

C Conformal group, causality violating region

Cr Differentiability classr

Ci _{Coordinate conditions}

u

vCŒrsT Contraction operation of a tensor fieldrsT

Ci_{j kl} Components of Weyl’s conformal tensor

C The set of all complex numbers

.; U / Coordinate chart for a differentiable manifold

.p/Dx Local coordinates of a pointpin a manifold.

.x1; x2; : : : ; xN/ In some placesx2R.

ij Extended extrinsic curvature components

D A domain inRN (open and connected)

Di Gauge covariant derivative

@D .N 1/-dimensional boundary ofD

ri Covariant derivatives

D

@t Covariant derivative along a curve

Laplacian in a manifold with metric, determinant ofij

r2 _{Laplacian in a Euclidean space}

ıi

j Components of Kronecker delta (or identity matrix)

p pı; ı

i1;:::;ip

j1;:::;jp Generalized Kronecker delta

df, dŒpW Exterior derivative off orpW

dij Components of flat space metric

@.bx1_{; : : : ;bx}N_/

@.x1_{; : : : ; x}N_/ Jacobian of a coordinate transformation

e Electric charge, exponential

Eij,EQij,El_{ij k} Components of Einstein equations (in various forms)

E

E; E˛ Electric field and its components

EN N-dimensional Euclidean space

fEe.a/gN1 A basis set for a vector space

WD fEe.1/; : : : ;Ee.N /g

fEE.a/g41 A complex null basis set for Newman–Penrose formalism

WDnmE;mE;El;kE

o

" A small number, Newman–Penrose spin coefficient, co-efficient of a perturbation

"i1i2;:::;iN Totally antisymmetric permutation symbol (Levi-Civita)

i _{Components of the geodesic deviation vector}

.a/.b/ _{Components of metric tensor relative to a complex null} tetrad

i1;i2;:::;iN Totally antisymmetric pseudo (or oriented) tensor (Levi-Civita)

f˛ _{Newtonian force}

fij.x;u/ Finsler metric components

Fi _{4-force components}

Fij Electromagnetic tensor field tensor components

g;jgj Metric tensor determinant and its absolute value

G Gravitational constant (usually set to 1)

gij Metric tensor components

Gi

j Einstein tensor components

A parametrized curve into a manifold, Newman–Penrose

spin coefficient

X WDı A parametrized curve intoRN

The image of a parametrized curve intoRN,

characteris-tic matrix

ij Characteristic matrix components

L

.a/.b/.c/ Complex Ricci rotation coefficients .a/.b/.c/ Ricci rotation coefficients

k

ij Independent connection components in Hilbert-Palatini

variational approach

„ Reduced Planck’s constant (usually set to 1)

hi_,hij_,hk

ij Variations of vector, second-rank tensor, Christoffel con-nection respectively

E

H; H˛ Magnetic field and its components

H Relativistic Hamiltonian

I Identity tensor

J Action functional or action integral

Ji _{4-current components}

Ji k _{Total angular momentum components}

E

k Real null tetrad vector

ki _{Wave vector (or number) components}

k0 Curvature of spatial sections of F–L–R–W metric

K.u/ Gaussian curvature

E

K; Ki A Killing vector and corresponding components

Einstein equation constant (D8G=c4_{in common units)} .A/,.0/ Athcurvature, Newman–Penrose spin coefficient

E_{l Real null tetrad vector}

li

j,Lij Components of a generalized Lorentz transformation

ŒL T Transposed matrix

L A Lagrangian function

L_VE Lie derivative

L Lagrangian density

L.I/ Lagrangian function from super-Hamiltonian

Eigenvalue, Lagrange multiplier, electromagnetic gauge

function, Newman–Penrose spin coefficient

.i / ith eigenvalue

E

.A/.s/ Ath normal vector to a curve i_.a/,.a/_i Components of orthonormal basis

Cosmological constant

m,M.s/ Mass, mass function

E

m;mE Complex null tetrad vectors

M,MN A differentiable manifold,N dimensional differentiable manifold

M “Total mass” of the universe

Mi_,Mi _{Maxwell vector (and dual) components}

Mass density, Newman–Penrose spin coefficient

N Dimension of tangent vector space, lapse function in

A.D.M. formalism

ni _{Unit normal vector components}

N˛ _{Shift vector in A.D.M. formalism}

Frequency, Newman–Penrose spin coefficient

O.p; nIR/ Generalized Lorentz group

IO.p; nIR/ Generalized Poincar´e group

p Point in a manifold, polynomial equation, pressure

p# _{Polynomial equation for invariant eigenvalues}

p_k,p_? Parallel pressure and transverse pressure respectively

pi_;Pi _{4-momentum components}

.0/ Newman–Penrose spin coefficient

k _{Projection mapping}

Pi

j Projection tensor field components

Characteristic surface function of a p.d.e., scalar field

˚ Born-Infeld (or tachyonic) scalar field, (also a function

used in five-dimensional cosmologies) ˛

(ext) External force density

'ij _{Complex electromagnetic field tensor components}

˚.A/.B/ Complex Ricci components (A; B 2 f0; 1; 2g) ˚i1;:::;ir

Complex Klein-Gordon field

.J / ComplexJth Weyl components (J 2 f0; : : : ; 4g) Q.a/.d / Complex Weyl tensor with second and third index

pro-jected in a timelike direction

R Ricci curvature scalar (or invariant)

Rij Components of Ricci tensor

Ri

j k Components of Cotton–Schouten–York tensor

Ri

j kl Components of Riemann-Christoffel tensor

R The set of real numbers, complex Ricci scalar

RN _{Cartesian product ofN copies of the setR}

WDR_„R_ƒ‚ R_…

N

Mass density, proper energy density, Newman–Penrose

spin coefficient

sij_{; S}ij _{Components of relativistic stress tensor (special and} general respectively)

Sij kl Components of symmetrized curvature tensor

s Arc separation parameter

S2 _{Two-dimensional spherical surface}

Electrical charge density, Newman–Penrose spin

coeffi-cient, separation of a vector field, Klein-Gordon equation ˛ˇ_,ij _{Stress density, shear tensor components}

P

; ˙ Arc separation function, function in Kerr metric, summa-tion

E_{tx Tangent vector of the image at the pointx}

Tx Tangent vector space of a manifold

Q

Tx Cotangent (or dual) vector space of a manifold

Tij _{Components of energy–momentum–stress tensor}

T::; Ti

j k Torsion tensor and the corresponding components

r

sT Tensor field of order.rCs/

Ti1;:::;ir

j1;:::;js Coordinate components of the (same) tensor field T.a1/;:::;.ar/

.b1/;:::;.bs/ Orthonormal components of the (same) tensor field

T.a1/;:::;.ar/

.b1/;:::;.bs/ Complex tensor field components r

sT˝ p

qS Tensor (or outer product) of two tensor fields

Ti _{Conservation law components}

Affine parameter along geodesic (usu. proper time),

Newman–Penrose spin coefficient r

sT.Tx.RN// Tensor bundle

ij Expansion tensor components, T-domain energy–

momentum–stress tensor r

s Components of a relative tensor field

U an open subset of a manifold

ui_{; U}i_;Ui _{4-velocity components}

V˛_{.t /,V}˛ _{Newtonian or Galilean velocity}

W, w Effective Newtonian potential

W Lambert’s W-function, (symbol also used for other

func-tions in axi-symmetric metrics)

W Work function

pW; Wi1;:::;ip p-form and its antisymmetric components

!ij Vorticity tensor components

˝ Synge’s world function

xDX.t /; A parametrized curve inRN xi _DXi_{.t /}

xD.u/; A parametrized submanifold

xi _Di_.u1_{; : : : ;u}D_/

Y Coefficient of spherical line element in Tolman-Bondi

coordinates

z;z A complex variable and its conjugate

Tensor Analysis on Differentiable Manifolds

1.1

Differentiable Manifolds

We will begin by briefly defining an N-dimensional differentiable manifoldM. (See [23,38,56,130].) There are a few assumptions in this definition. A set with a topology is one in which open subsets are known. Furthermore, if for every two distinct elements (or points)pandqthere exist open and disjoint subsets containing

pandq, respectively, then the topology is called Hausdorff. A connected Hausdorff manifold is paracompact if and only if it has a countable basis of open sets. (See [1,

130,132].)

1. The first assumption we make about an applicable differentiable manifoldM is that it is endowed with a paracompact topology.

(Remark: This assumption is necessary for the purpose of integration in any domain.)

We also consider only a connected setM for physical reasons. Moreover, we

mostly deal with situations whereM is an open set.

Now we shall introduce local coordinates forM. A chart .; U / or a local

coordinate system is a pair consisting of an open subsetU M together with

a continuous, one-to-one mapping (homeomorphism) from U into (codomain)

D RN_{. Here,Dis an open subset of R}N _{with the usual Euclidean topology.}1

For a pointp 2M, we havex .x1_{; x}2_{; : : : ; x}N_{/D.p/2 D. The coordinates}

.x1_{; x}2_{; : : : ; x}N_{/are the coordinates of the pointpin the chart.; U /.}

Each of theN coordinates is obtained by the projection mappingsk _{W D}

RN _{! R,k 2 f1; : : : ; Ng. These are defined by}k_.x/ k_.x1_{; : : : ; x}N_{/ DW}

xk_{2R. (See Fig.1.1.)}

1_{We can visualize the coordinate spaceR}N_{as anN-dimensional Euclidean space.}

A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical

Exposition, DOI 10.1007/978-1-4614-3658-4 1,

© Springer Science+Business Media New York 2012

Fig. 1.1 A chart.; U /and projection mappings

Consider two coordinate systems or charts.; U /and.b;U /b such that the point pis in the nonempty intersection ofU andbU. From Fig.1.2, we conclude that

bxD.bı1/.x/;

xD.ıb1/.x/;O (1.1)

wherex2DsDandbx2DbsDb.

From the preceding considerations, the mappingsbı 1 _{and ıb}1 _are continuous and one-to-one. By projection of these points, we get

bxkDŒkıbı1 .x/DWXbk.x/Xbk.x1; : : : ; xN/;

xkDŒkııb1 .bx/DWXk.bx/Xk.bx1; : : : ;bxN/: (1.2)

Fig. 1.2 Two charts inMand a coordinate transformation

By the first assumption of paracompactness, we can conclude [38,130] that open subsets Uh exist2 _{such thatM D} S

hUh. The second assumption aboutM is the following:

2. There exist countable charts .h; Uh/ for M. Moreover, wherever there is a nonempty intersection between charts, coordinate transformations as in (1.2) of classCr _{can be found.}

Such a basis of charts forM is called a Cr_{-Atlas. A maximal collection ofC}r -related atlases is called a maximalCr_{-Atlas. (It is also called the complete atlas.)} Finally, we are in a position to define a differentiable manifold.

3. AnN-dimensionalCr_{-differentiable manifold is a setM with a maximalC}r -atlas.

(Remark: Forr D0, the setM is called a topological manifold.)

A differentiable manifold is said to be orientable if there exists an atlas.h; Uh/ such that the Jacobian det

h

@bXk_.x/ @xj

i

is nonzero and of one sign everywhere.

Example 1.1.1. Consider the two-dimensional Euclidean manifoldE2. One global chart.; E2/is furnished byx D .x1; x2/ D .p/; p 2 E2; D D R2. Let .; E2/be one of infinitely many Cartesian coordinate systems.

Another chart,.;O U /O , is given byxO .xO1; xO2/ .r; '/ D O.p/I OD WD

˚

.xO1; xO2/2R2W Ox1 > 0; <xO2< .

χ χ^−1

χ^

χ χ^ −1

x x

χ

^ p

x x2

x1 x1

E2

2

π

−_π 2

2

Fig. 1.3 The polar coordinate chart

The transformation to the polar coordinate chart is characterized by

x1 DX1.x/O D Ox1cosxO2; x2 DX2.x/O D Ox1sinxO2;

and the inverse transformation by

O

x1D OX1.x/D Cp.x1_/2_C.x2_/2_DWr;

O

x2D OX2.x/Darc.x1; x2/'WD

8 ˆ ˆ ˆ < ˆ ˆ ˆ :

Arctan.x2_=x1_{/ for x}1_{> 0;}

2sgn.x

2_{/ for x}1_{D0 and x}2 _¤0;

Arctan.x2_=x1_/Csgn.x2_{/ for x}1_{< 0} and x2_¤0:

Note that=2 < Arctan.x2_=x1_{/ < =2, so that < arc.x}1_{; x}2_{/ < . (See}

Fig.1.3.)

Fig. 1.4 Spherical polar coordinates

two-dimensional differentiable manifold in its own right. This coordinate chart is characterized by

x D.; '/D.p/; p2U S2I

DW D˚.; '/2R2W0 < < ; < ' < R2: Another distinct spherical polar chart is furnished by

b D.; '/b WDArc cos.sinsin'/;

b' Db˚ .; '/WDarc.sincos';cos /;

b

DW D

n

.b ;b'/2R2_{W0 <}b _{< ; <b' <} o_R2_:

Neither of the two charts is global. However, the unionU [bU D S2. Thus, the collection of two charts.; U /and.b;U /b constitutes an atlas forS2. (The minimum number of charts for an atlas ofS2 is two.) See Fig.1.4for the illustration. (For

future use, a vectorEvpatpis shown.)

Fig. 1.5 Tangent vector inE3andR3

definition of a tangent vectorEvpor its imageEvxinR3, we shall visualize an “arrow” inR3 with the starting pointx .x1; x2; x3/inR3and the directed displacement E

v.v1;v2;v3/necessary to reach the point.x1Cv1; x2Cv2; x3Cv3/inR3. (See Fig.1.5.)

Example 1.1.3. Let us choose the following as standard (Cartesian) basis vectors at

xinR3: E_i

x Ee1x WD.1; 0; 0/x; Ejx Ee2xWD.0; 1; 0/x; Ekx Ee3x WD.0; 0; 1/x:

Let a three-dimensional vector be given byvE WD 3Ee1C2Ee2 C Ee3 D .3; 2; 1/. Let us choose a pointx .x1_{; x}2_{; x}3_{/D.1; 2; 3/. Therefore, the tangent vectorvE}

x D .3; 2; 1/.1;2;3/starts from the point.1; 2; 3/and terminates at.4; 4; 4/.

However, such a simple definition runs into problems in a curved manifold. An example of a simple curved manifold is the spherical surfaceS2_{we have previously} discussed and as shown in Fig.1.4. We have drawn an intuitive picture of a tangent vectorEvpon S2. The starting pointp ofvEp is onS2. However, the end point of

E

vpis not onS2. The problem is how to define a tangent vector intrinsically onS2, without going out of the spherical surface. One logical possibility is to introduce

directional derivatives of a smooth functionF defined atpin a subset ofS2_{. Such} a definition involves only the pointpand its neighboring points all onS2_{. Thus, we} shall represent tangent vectors by the directional derivatives. This concept appears to be very abstract at the beginning. (See [56,121,197].)

ukvkWD N

X

kD1

ukvk N

X

jD1

ujvj Dujvj ;

gijaibj WD N

X

iD1 N

X

jD1

gijaibj N

X

kD1 N

X

lD1

gklakbl Dgklakbl:

The summation indices are also called dummy indices, since they can be replaced by another set of repeated indices over the same range, as demonstrated in the above equations.

Now, let us define generalized directional derivatives. We shall use a coordinate chart.; U / instead of the abstract manifoldM. Let x.x1; : : : ; xN/D.p/. Let anN-tuple of vector components be given by.v1; : : : ;vN/. Then, the tangent

vectorEvxinDRN is defined by the generalized directional derivative:

E

vxW Dvi @ @xi;

E

vxŒf W Dvi@f .x/

@xi : (1.3)

Here,f belongs toC1_.DRN_{IR/. (Note that in the notation of the usual calculus,}

E

vxŒf D Evgradf.)

The set of all tangent vectorsEvxconstitutes the N-dimensional tangent vector

space Tx.RN/ in D RN. It is an isomorphic image of the tangent vector space Tp.M /. The coordinate basis set fEe1x; : : : ;eEN xgforTx.RN/is defined by the differential operators

E

e1xW D @

@x1; : : : ;EeN xWD @ @xN I

E

ekxŒf W D @f .x/

@xk : (1.4)

(See [23,197].)

Now, we shall define a tangent vector field Ev.p/ in U M or equivalently the tangent vector field vE.x/ in D RN. It involves N real-valued functions

v1.x/; : : : ;vN.x/. The tangent vector field is defined by the operator

E

v.x/W Dvj.x/ @ @xj;

E

v.x/Œf W Dvj.x/@f .x/

@xj : (1.5)

Example 1.1.4. LetD WD f.x1_{; x}2_{/ 2 R}2 _{W x}1 _{2 R; 1 < x}2_{g. Moreover, let} f .x/WDx1_Œ.x2_/x2

and

E

v.x1; x2/WD e x2

2 @

@x1 .2coshx 1_/ @

@x2:

Therefore, by (1.5), we get

E

v.x1; x2/Œf DŒ.x2/x2

"

ex2 2 2.x

1_coshx1_/.1Clnx2_/

#

;

lim x1_!1x2lim_!1C

˚ E

v.x1_{; x}2_{/Œf D} e 2Ce

1_:

The Kronecker delta is defined by the (real) numbers:

ıi_j WD

(

1 for iDj;

0 for i¤j: (1.6)

TheN N matrix with entriesıi

j is the unit matrixŒI DŒıij .

Example 1.1.5. Consider a vector ˛E D .˛1_{; ˛}2_{; : : : ; ˛}N_{/. Using the summation} convention,

ı1_j˛j Dı1₁˛1Cı1₂˛2C Cı1_N˛N D˛1; ıi

j˛j D˛i;

ıi_jıj_kDıi_k; ıi_jıj_kık_lıl_i DN:

Example 1.1.6. The coordinate basis vectors from (1.4) can be expressed as

Eej.x/D @ @xj:

Therefore, for the functionf .x/WDxi,

E

ej.x/Œxi D @.xi_/

@xj Dı i

j;

E

v.x/Œxi Dvj.x/ @ @xj.x

i_/Dvj_.x/ıi

j Dvi.x/ : (1.7)

The main properties of tangent vector fields can be summarized in the following theorem:

.i /

Œf .x/Ev.x/Cg.x/wE.x/ Œh Df .x/.vE.x/Œh /Cg.x/.wE.x/Œh /; (1.8)

.ii/

E

v.x/Œcf Ckg Dc.Ev.x/Œf /Ck.vE.x/Œg /; (1.9)

for all constantscandk .iii/

E

v.x/Œfg D.vE.x/Œf /g.x/Cf .x/.vE.x/Œg / : (1.10)

The proof is left as an exercise.

Now we shall define a cotangent (or covariant) vector field. Consider a function Q

f.x/which maps the tangent vector spaceTx.RN/intoRsuch that

Q

f.x/g.x/Ev.x/Ch.x/wE.x/Dg.x/

h Q

f.x/ vE.x/iCh.x/

h Q

f.x/ wE.x/i (1.11)

for all functionsg.x/; h.x/and all tangent vector fieldsEv.x/;wE.x/inTx.RN_{/. Such} a function is called a cotangent (or covariant) vector field.

Example 1.1.8. The (unique) zero covariant vector field0Q.x/is defined by

Q

0.x/ Ev.x/WD0

for allEv.x/inTx.RN_/.

(Remark: In Newtonian physics, the gradient of the gravitational potential is a covariant vector field.)

We define the linear combinations of covariant vector fields as

.x/Qf.x/C˝.x/gQ.x/ Ev.x/WD.x/fQ.x/.Ev.x//C˝.x/gQ.x/.Ev.x// (1.12)

for all functions.x/,˝.x/ and all tangent vectorsEv.x/inTx.RN/. Under such a rule, the set of all covariant vector fields constitutes anN-dimensional cotangent (or dual) vector spaceTx.Q RN/.

Now we shall introduce the notion of a 1-form which will be identified with a covariant vector field. We need to define a (totally) differentiable functionf over DRN. In casef satisfies the following criterion,

lim .h1_;:::;hN_/!.0;:::;0/

8 < : h

f .x1Ch1; : : : ; xNChN/f .x1; : : : ; xN/hj @f .x/_@xj

i

p

.h1_/2_{C C.h}N_/2

9 = ;D0;

for arbitrary.h1_{; : : : ; h}N_{/, we call the functionf a totally differentiable function at} x. The usual condensed form for denoting (1.13) is to write

df .x/D @f .x/ @xj dx

j_{: (1.14)}

Each side of (1.14) is called a 1-form. It is customary to identify a 1-form, df .x/, with a covariant vector field,Qf.x/, with the following rule of operation:

df .x/ŒEv.x/ Qf.x/Ev.x/WDvj.x/@f .x/

@xj : (1.15)

Here,vE.x/is an arbitrary vector field.

Example 1.1.9. Letf .x/ f .x1_{; x}2_{/ WD .1=3/Œ.x}1_/3_3ex2

,.x1_{; x}2_{/ 2 R}2_. Therefore, by (1.14) and (1.15),

df .x/D.x1/2dx1ex2dx2;

df .x/ŒEv.x/ D.x1/2v1.x/ex2v2.x/;

df .0; 0/ŒvE.0; 0/ D v2.0; 0/:

Example 1.1.10. Consider the functionf .x/WDxk_{. Then}

df .x/Ddxk;

@ @xi Dı

j i

@ @xj:

Furthermore, by (1.15),

dxk

@ @xi

Dıj_i@ x k

@xj Dı j

iıkj Dıki:

Therefore, we identify the coordinate covariant basis field forTx.Q RN/as fQe1.x/; : : : ;eQN.x/g DW

˚

dx1; : : : ;dxN; (1.16)

Q

ek.x/Eei.x/

Dık_i: (1.17)

Thus, every covariant vector (or 1-form) admits the linear combination

Q

W.x/DWj.x/dxj (1.18)

in terms of the basis covariant vectors dxj_{’s. The (unique) functions,W}

Fig. 1.6 A parametrized curveintoM

Example 1.1.11. Consider the two-dimensional Euclidean manifold and a Cartesian

coordinate chart. Let another chart be given by

O

x1D OX1.x/WD.x1/3;

O

x2D OX2.x/WD.x2/3; .x1; x2/2R2 f.0; 0/g I .xO1; xO2/2R2 f.0; 0/g: Then, by direct computations, we deduce that

@ @xO1 D

1 3.x1_/2

@ @x1;

@ @xO2 D

1 3.x2_/2

@ @x2;

dxO1D3.x1/2dx1; dxO2 D3.x2/2dx2:

It can now be verified that

dxOi

@ @xOj

Ddxi

@ @xj

Dıi_j:

Now we shall discuss a different topic, namely, a parametrized curve. Consider an intervalŒa; b R.

(Remark: Open or semiopen intervals are also allowed. Moreover, unbounded intervals are permitted too.)

(Note that the function is called the parametrized curve.) The image of the composite functionX WD ı inD RN _{is denoted by the symbol. The} coordinates on are furnished by

xDŒı .t /DX.t /;

xj DŒj ıı .t /Dj ıX.t /DWXj.t /: (1.19)

Here,t 2 Œa; b R. The functionsj _{are usually assumed to be differentiable} and piecewise twice-differentiable. The condition of the nondegeneracy (or

regu-larity) is

N

X

jD1

dXj_{.t /}

dt

2

> 0: (1.20)

In Newtonian physics,t is taken to be the time variable and M D E3, the physical space. Moreover, is a particle trajectory relative to a Cartesian coordinate system. The nondegeneracy condition (1.20) implies that the speed of the motion is strictly positive.

Example 1.1.12. Let the image inR3be given by

xDX.t /WD.2cos2t; sin2t; 2sint /I 0 < t < =2:

The curve is nondegenerate and the coordinate functions are real-analytic. Consider a circular cylinder inR3such that it intersects thex1_x2 _{plane on the unit circle} with the center at.1; 0; 0/. Now, consider a spherical surface in R3 given by the equation.x1_/2_C.x2_/2_C.x3_/2 _{D 4. The image lies in the intersection of the}

cylinder and the sphere.

Let us consider the tangent vectorEtx of the image at the point x in RN (see Fig.1.7). In calculus, the components of the tangent vectorEtX.t / are taken to be

dX1_{.t /}

dt ; : : : ; dXN_{.t /}

dt

. Therefore, the tangent vectorEt_X.t / EX

0

.t /along (according to (1.3)) must be defined as the generalized directional derivative:

E_{tX.t / EX}0_{.t /WD} dXj.t / dt

@ @xj

ˇˇ_X.t /

: (1.21)

The above tangent vector field belongs to the tangent vector space T_{X.t /}.R3_/ along.

Example 1.1.13. Let a real-analytic, nondegenerate curveX intoR4be defined by x DX.t /WD..t /3; t; et;sinht /; 1< t <1I

Fig. 1.7 Reparametrization of a curve

The corresponding tangent vector field along is furnished by the generalized directional derivative:

E

X0.t /D.3t2/

@ @x1

ˇˇ_X.t /

C

@ @x2

ˇˇ_X.t /

C.et/

@ @x3

ˇˇ_X.t /

C.cosht /

@ @x4

ˇˇ_X.t /

;

E X0.0/D

@ @x2

ˇˇ_X.0/C

@ @x3

ˇˇ_X.0/C

@ @x4

ˇˇ_X.0/:

The tangent vectorXE0.t /can act on a differentiable functionf (restricted to) by (1.22) to follow. On this topic, we state and prove the following theorem.

Theorem 1.1.14. Let a parametrized curveX WŒa; b R!RN _{be differentiable}

and nondegenerate. Letf W D RN _{! Rbe a (totally) differentiable function.}

Then

E

X0.t /Œf .X.t // D d

dt Œf .X.t // : (1.22)

Proof. By (1.15) and the chain rule of differentiation, the left-hand side of the above equation yields

dXj_{.t /} dt

@ @xj

ˇˇ_X.t /

Œf .X.t // dX j_{.t /}

dt

@f .x/ @xj

ˇˇ_X.t /

D d

dtŒf .X.t // :

Now, we shall discuss the reparametrization of a curve. Lethbe a differentiable and one-to-one mapping fromŒc; d RintoR. (See Fig.1.7.).

The image inRN _{is given by the points} t Dh.s/;

Here,

X#_{WDXıh (1.23)}

is the reparametrized curve intoRN.

Theorem 1.1.15. IfX# _{is the reparametrization of the differentiable curveX by}

the functionh, then the tangent vector

E

X#0.s/D dh.s/ ds

E

X0.h.s//: (1.24)

Proof. By (1.23), we get

X#j_.s/DXj_.h.s//:

By the assumption of differentiabilities and the chain rule, we obtain

dX#j_.s/

ds D

dXj_{.t /} dt ˇˇtDh.s/

dh.s/ ds :

Therefore, the tangent vector

E

X#0.s/D dX #j_.s/

ds

@ @xj

ˇˇ_X#_.s/

D dh.s/ ds

dXj_{.t /}

dt

ˇˇtDh.s/

@ @xj

ˇˇ_X.h.s//

dh.s/ ds

E X0.h.s//:

Example 1.1.16. Consider the Euclidean plane E2 and a parametrized (real-analytic) curveXWDıgiven by

X.t /WD.cost;sint /; 2t 2:

The image is a unit circle inR2 with self-intersections. The winding number of this circle is exactly two.

Let a reparametrization be defined by

t Dh.s/WD2s; sI

dh.s/

ds 2 > 0:

The reparametrized curveX#_{is furnished by}

The new tangent vector is given by

E

X#0.s/D2

.sin2s/ @

@x1 C.cos2s/ @ @x2

ˇˇ_X#_.s/

D2XE0.2s/:

Exercises 1.1

1. Consider the three-dimensional Euclidean manifold and a global Cartesian chart .;E3/given by

xD.x1; x2; x3/D.p/; x inDDR3: The spherical polar chart.;O U /O is given by

O

x D .xO1;xO2;xO3/.r; ; '/D O.p/;

O

DWD˚.xO1_;xO2_;xO3_/2R3 _{W Ox}1_{> 0; 0 <xO}2_{< ; <xO}3_{< :} The coordinate transformation is characterized by

x1 DX1.x/O WD Ox1sinxO2cosxO3;

x2 DX2.x/O WD Ox1sinxO2sinxO3;

x3 DX3.x/O WD Ox1cosxO2:

Obtain the inverse functionsXO1_{.x/; X}O2_{.x/; X}O3_{.x/explicitly.} 2. Prove (1.10) in the text.

3. Consider a function of two variables given by

f .x1; x2/WD

8 ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ :

.x1_/2_sin.1=x1_/C.x2_/2_sin.1=x2_{/ for x}1_x2_¤0;

.x1_/2_sin.1=x1_{/ for x}1_{¤0 and x}2_D0;

.x2_/2_sin.1=x2_{/ for x}1_{D0 and x}2 _¤0;

0 for x1_Dx2_D0:

(i) Prove thatf is totally differentiable at the origin.0; 0/. (ii) Prove that@f .x_@x1; x1 2/ and

@f .x1; x2/

4. Evaluate the 1-formwQ.x/ WD ıijxi_dxj _{on the tangent vector field V}E_{.x/ WD}

PN

jD1.xj/5 @@xj. Prove thatwQ.x/ŒVE.x/ 0. 5. Consider the semicubical parabola inR2_{given by}

X.t /WD.t2; t3/; t2R:

The curve is degenerate atX.0/ D .0; 0/. Prove that reparametrization of this curve cannot remove the degeneracy.

(Remark: The degenerate point is called a cusp.)

Answers and Hints to Selected Exercises

1.

O

x1D OX1.x/Dp.x1_/2_C.x2_/2_C.x3_/2_;

O

x2D OX2.x/DArccos h

x3=p.x1_/2C_.x2_/2C_.x3_/2i_;

O

x3D OX3.x/Darc.x1; x2/:

3.

@f .x1_{; x}2_/ @x1 D

8 < :

2x1sin

1 x1

cos

1 x1

for x1 ¤0;

0 for x1_D0I

@f .x1; x2/ @x2 D

8 < :

2x2sin

1 x2

cos

1 x2

for x2 ¤0;

0 for x2_D0:

(See [112].)

1.2

Tensor Fields Over Differentiable Manifolds

Consider a pointx0 .x1

0; : : : ; x0N/ D .p0/ 2 RN. The corresponding tangent vector space and the cotangent (or covariant) vector space are denoted byTx₀RN and TQx0R

a function3 _{from the Cartesian product T}Q

x0 QTx0

„ ƒ‚ …

r

T_„x0 _ƒ‚ Tx_…0

s

into R. Moreover, it has to satisfy the following multilinearity conditions:

r sTx0 uQ

1

x0; : : : ; uQ k x0CvQ

k x0; : : : ;uQ

r

x0I Ea1 x0; : : : ;aEs x0

Dr_sTx0 uQ 1 x0; : : : ;uQ

k x0; : : : ;uQ

r

x0I Ea1 x0; : : : ;aEs x0

Cr_sTx0 uQ 1 x0; : : : ;vQ

k x0; : : : ; uQ

r

x0I Ea1 x0; : : : ; Eas x0

; (1.25i)

r sTx0

Q

u1_x0; : : : ; uQr_x0I Ea1 x0; : : : ; Eaj x0CbEj x0; : : : ; aEs x0

DTx0 uQ 1

x0; : : : ; uQ r

x0I Ea1 x0; : : : ;Eaj x0; : : : ; aEs x0

C

h

r sTx0

Q

u1_x

0; : : : ; uQ r

x0I Ea1 x0; : : : ;bEj x0; : : : ; aEs x0

i

; (1.25ii)

for all , in R, all uQ1_x0; : : : ;uQr_x0I Qvk_x0 in TQx0 R

N_{, all aE}

1 x0; : : : ; aEs x0I Eb j x0 in Tx₀ RN_{, allkinf1; : : : ; rg, and alljinf1; : : : ; sg. Here, the nonnegative integer} r represents the contravariant order and the nonnegative integers represents the

covariant order. Note that the functionr

sTx0 is linear in each of therCsslots in the argument. (See [23,56,195,197].)

Example 1.2.1. The (unique).rCs/th order zero tensor is defined by

r s0x0 uQ

1 x0; : : : ;uQ

r

x0I Ea1 x0; : : : ; Eas x0

WD0 (1.26)

for alluQ1_x0; : : : ; uQr_x0inTxQ 0 R

N_{and allEa}

1 x0; : : : ; Eas x0 inTx0 R

N_.

Example 1.2.2. LetbQkx0 WDˇ k

1eQ1x0C Cˇ k

NeQNx0andEaj x0D˛ 1

jEe1 x0C C˛ N

jEeN x0. (Here,˚Ee1 x0; : : : ;EeN x0

and˚eQ1_x0; : : : ;eQN_x0are basis sets forTx0.R

N_/andTxQ 0.R

N_/,

respectively.) Let a function be defined by

r sFx0 bQ

1 x0; : : : ; bQ

r

x0I Ea1 x0; : : : ; Eas x0

WD ˇ1₁ˇ2₂ ˇr_r ˛1₁˛2₂ ˛s_s:

The functionr_sFx0can be proved to be a mixed tensor of order.rCs/.

3_{The rank of a tensor is different from the rank of a matrix. The rank of a matrix is defined as the}

The linear combination of two mixed tensors are defined by:

r_sTx0C r

sWx0

Q

u1_x

0; : : : ;uQ r

x0I Ea1 x0; : : : ; Eas x0

WDr_sTx0 uQ 1 x0; : : : ;uQ

r

x0I Ea1 x0; : : : ;aEs x0

Cr_sWx0 uQ 1 x0; : : : ;uQ

r

x0I Ea1 x0; : : : ;aEs x0

(1.27)

for all, inR, alluQ1_x0; : : : ;uQr_x0 inTQx0.R

N_{/and allEa}

1 x0; : : : ; Eas x0 in Tx0.R N_/.

(Note that addition between two tensors of different order is not permitted.) It can be proved via (1.27) that the set of all mixed tensors of order.r Cs/ constitutes a (real) vector space of dimensionNrCs.

The tensor product (or outer product) between two tensors r_sTx0 and p

qWx0 is defined by the function r

sTx0˝ p

qWx0fromTxQ 0.R

N_{/ QTx} 0.R

N_/

„ ƒ‚ …

rCp

Tx0.R

N_{/ Tx} 0.RN/

„ ƒ‚ …

sCq

intoRsuch that _r

sTx0˝ p

qWx0

Q

u1_x0;: : : ;uQr_x0I Qv1_x0;: : : ;vQxp0I Ea1 x0;: : : ;Eas x0I Eb1 x0; : : : ;bEq x0

DWr

sTx0 uQ1x0; : : : ;uQ r

x0I Ea1 x0; : : : ;Eas x0

p

qWx0 vQ1x0; : : : ;vQ p

x0I Eb1 x0;: : : ;bEq x0

(1.28) for allaE1 x0; : : : ; Eas x0I Eb1 x0; : : : ;bEq x0 inTx0.RN/and alluQ1x0; : : : ;uQ

r x0I Qv

1 x0; : : : ;

Q

vpx0inTxQ0.R

N_{/. Note that}r sTx0˝

p

qWx0 is a tensor of order.rCp/C.sCq/.

Example 1.2.3. LetN D 2and˚Ee1 x0; Ee2 x0

be a basis set forTx0.R

N_{/. Let two}

covariant tensors,0₁Tx0and 0

2Wx0, be defined by

E

ax0D˛ 1_Ee

1 x0C˛ 2_Ee

2 x0; Eb1 x0 Dˇ 1

1Ee1 x0Cˇ 2

1Ee2 x0 ; bE2 x0Dˇ 1

2Ee1 x0Cˇ 2

2Ee2 x0;

E

cx0 D1Ee1 x0C2Ee2 x0I

0

1Tx0.aEx0/WD˛1; 0

2Wx0.bE1 x0; bE2 x0/WDˇ 1

1ˇ22:

Then, by the tensor product rule (1.28), we derive that ₀

1Tx0˝ 0

2Wx0 Eax0I Eb1 x0; bE2 x0