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G. Ferrarese
D. Bini
Introduction to Relativistic
Continuum Mechanics
ABC
Authors
DonatoBini GiorgioFerrarese
IstitutoperleApplicazioni UniversitàRomaLaSapienza
delCalcolo‘MauroPicone’CNR DipartimentodiMatematica
VialedelPoliclinico,137 PiazzaleAldoMoro,2
00161Roma,Italy 00185Roma,Italy
[email protected] [email protected]
G.FerrareseandD.Bini,IntroductiontoRelativisticContinuumMechanics,Lect.Notes
Phys.727(Springer,BerlinHeidelberg2008),DOI10.1007/978-3-540-73168-9
LibraryofCongressControlNumber:2007929739
ISSN0075-8450
ISBN978-3-540-73166-5 SpringerBerlinHeidelbergNewYork
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orpartsthereofispermitted onlyundertheprovisions oftheGermanCopyright LawofSeptember9,
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To Liliana and Daniela
Preface
Instandarduniversitycourses,theteachingofspecialrelativityisoftenlimited
to show the absolute (i.e. four-dimensional) formulation of relativistic kine-
matics,mechanicsandelectromagnetism,whereastheequallyinterestinglong
chapters of geometrical continua and fluid dynamics are left for the general
relativityinvestigationonly.Actually,studentsbecomefamiliarwithrelativis-
tic kinematics andthe relativistic formulationofelectromagnetismevenfrom
their first-year courses,and this is a really important step in their formation.
However, they are given very little information about the so-called relative
observer point of view in relativity and the different ways in which one can
reintroducetheclassicalconceptsofspaceandtime,3-momentumandenergy,
etc. from their space-time counterparts, namely the space-time itself, the 4-
momentum, etc. The formalisms underlying this decomposition process, or
3+1splitting techniques,havebeenwidely developedstartingfromthe 1950s
but can be found only in General Relativity textbooks.
We consider strongly important to make students familiar with Special
Relativity either from the absolute (space-time, four-dimensional) point of
view or from the relative (space plus time, 3+1-dimensional) point of view,
inordertogiveacentralroletothemeasurementproblemandtotheobserver,
even studying Special Relativity.
This book, which apart from standard topics, includes also a geometrical
introduction to continuous media and fluid dynamics, aims at pursuing an
effort in this direction, summarizing the Italian school contribution with the
pioneering works of Carlo Cattaneo since 1950 at the University of Rome,
Italy.
It is therefore a pleasure to acknowledge Prof. Wolfgang Rindler, an “old
friend” of Cattaneo and his coworkers since about 50 years, for many stim-
ulating discussions and useful suggestions in addition to his special effort in
improvingthe contentaswellastheEnglishofthismanuscript.Moreover,we
are also gratefulto Prof.L.Stazi for his comments and the carefulreading of
the hundreds of formulas in the manuscript.
VIII Preface
A final remark concerns the reader: he is supposed to be familiar enough
with ordinary tensor calculus on Riemannian manifolds and endowed with a
good amount of patience!
Rome, Giorgio Ferrarese
February 2007 Donato Bini
Contents
1 Classical Physics: Axiomatic Formulation .................. 1
1.1 Methodology............................................ 1
1.2 General Axioms ......................................... 2
1.3 Axioms of Newtonian Mechanics .......................... 6
1.4 Axioms of Maxwell’s Electromagnetism .................... 8
1.5 Optical Experiments and Classical Physics.................. 12
References .................................................. 20
2 Space-Time Geometry and Relativistic Kinematics ........ 21
2.1 Introduction to Special Relativity ......................... 21
2.2 General Axioms ......................................... 23
2.3 The Minkowski Space-time ............................... 25
2.4 The Minkowski Metric ................................... 29
2.5 Vectors and Their Classification. The Lightcone ............. 31
2.6 Elements of the Geometry of Minkowski Space-time.......... 34
2.7 Proper Time............................................ 37
2.8 Test Particle Kinematics (Absolute and Relative)............ 39
2.9 Lorentz Transformations ................................. 42
2.10 General Lorentz Transformations: I ........................ 45
2.11 Relativity of Lengths and Times........................... 49
2.12 Muon Mean Life and the Time Dilation .................... 51
2.13 Theorem of Relative Motions ............................. 52
2.14 Optical Experiments and Special Relativity................. 56
2.15 Coriolis Theorem........................................ 58
2.16 Vectorial Maps.......................................... 60
2.17 Levi–Civita Indicator and Ricci Tensor..................... 65
2.18 General Lorentz Transformations: II ....................... 67
2.19 General Lorentz Transformations: III ...................... 97
References ..................................................107
X Contents
3 Test Particle Dynamics ....................................109
3.1 Fundamental Laws of Test Particle Dynamics ...............109
3.2 Cauchy Problem ........................................112
3.3 Classification of 4-Forces. Conservative Forces...............114
3.4 Constrained Material Point ...............................116
3.5 The Conservative Case. Hamiltonian Formulation............117
3.6 The Relative Formulation of the Dynamics .................120
3.7 Transformation Laws: Unification Between Mechanics
and Heat Theory ........................................124
3.8 The Cauchy Problem in Relative Dynamics .................127
3.9 The Intrinsically Conservative Case........................129
3.10 Pure Mechanics: Particles with Scalar Structure .............130
3.11 The Conservative Case in a Classical Sense .................131
3.12 Classical Approximation..................................134
3.13 Unified Scheme: Particles and Photons .....................135
3.14 Fundamental Invariants ..................................137
3.15 Particle and Photon Dynamics ............................139
3.16 Unified Relative Dynamics of Particles .....................140
3.17 An Alternative to the Unified Dynamics....................143
References ..................................................145
4 Applications ...............................................147
4.1 Principal Tetrad for Nonnull Curves in M .................147
4
4.2 Frenet–Serret Formalism .................................149
4.3 Curvature and Torsions ..................................150
4.4 Intrinsic Equations ......................................154
4.5 Conservative Lorentz-like Forces...........................156
4.6 Central Forces in a Galilean Frame ........................158
4.7 The Keplerian Case......................................160
4.8 Motion of Charged Particles in a Uniform Magnetic Field.....163
4.9 Extension of Maxwell’s Equations to any Galilean Frame .....165
References ..................................................167
5 RelativisticKinematics for a Three-DimensionalContinuum169
5.1 Continuum Mechanics. Relative Representation of the Motion.169
5.2 Fundamental Kinematical Fields ..........................172
5.3 Shear and Vorticity......................................175
5.4 Christoffel Symbols and Covariant Derivative ...............178
5.5 Local Analysis of the Motion of a Continuum ...............183
5.6 Passing from One Galilean Frame to Another ...............186
5.7 Kinematical Invariants ...................................189
5.8 Dilation Coefficients .....................................193
5.9 Transformation Laws for Angular and Deformation Velocities .196
5.10 Born Rigidity. Thomas Precession .........................201
Contents XI
5.11 Material Continuum. Number and Matter Density ..........203
5.12 Absolute Kinematics. Proper Quantities....................204
References ..................................................206
6 Elements of Classical Dynamics of a Continuum ...........207
6.1 Introduction to Continuum Classical Dynamics..............207
6.2 LagrangianForm of the Fundamental Equations.............211
6.3 Isotropic Systems........................................213
6.4 Symbolic Relation of Continuum Mechanics.................215
6.5 Reversible Transformation Systems. Free Energy ............217
6.6 Perfect Fluids. Characteristic Equation and Specific Heat.....220
6.7 Perfect Gas.............................................222
6.8 General Expression for the Power of the Internal Forces ......224
6.9 Isotropic Systems: Constitutive Equations ..................226
6.10 Dynamical Compatibility of a Continuum ..................229
6.11 Hyperelastic Continua: Intrinsic Dynamics..................234
6.12 Cauchy Problem ........................................235
References ..................................................237
7 Elements of Relativistic Dynamics of a Continuum ........239
7.1 Relativistic Extension....................................239
7.2 Proper Mechanical Stresses and Thermal Energy ............241
7.3 Space-time Splitting Techniques. The Energy Tensor.........244
7.4 Dust Matter and Perfect Fluids ...........................246
7.5 Absolute Dynamics of Ordinary Relativistic Continua ........248
7.6 Relative Dynamics of Ordinary Relativistic Continua ........252
7.7 Transformation Laws of the Fundamental Relative Quantities .254
7.8 Energy Theorem and the First Law of Thermodynamics......258
7.9 Continua Without Material Structure ......................260
References ..................................................262
8 Elements of Relativistic Thermodynamics of a Continuum.263
8.1 Introduction ............................................263
8.2 Nonpolar Continua ......................................264
8.3 Proper Thermodynamics of the Nonpolar Continua ..........267
8.4 Relative Formulation.....................................269
8.5 Transformation Laws of the Fundamental Relative Quantities .272
8.6 Classical Form of the Relativistic Cauchy Equation ..........274
8.7 Transformation Laws of the Dynamical Stresses .............276
8.8 Nonviscous Fluids and Dynamical Pressure .................278
8.9 LagrangianForm of the Relativistic Cauchy Equation........279
8.10 LagrangianForm of the Power of the Internal Forces.........281
8.11 Energy Theorem and First Law of Thermodynamics .........284
8.12 Finite Deformations in Relativity. Isotropic Systems .........287
8.13 Continua Without Material Structure ......................289
