Calssical thery of fields

Giáo trình vật lý lý thuyết về lý thuyết trường, một môn học cơ bản của sinh viên và học viên ngành vật lý lý thuyết. Cuốn sách này viết một cách cơ bản và đầy đủ nhất về những kiến thức cần đạt được. Lý thuyết trường mở ra một tư duy, một ứng dụng của trường toán học vào vật lý. Ở đây là trường lượng tử hay gọi tắt là QFT.

yx = Pn- of equations (113.14) has the form of the equation of one-dimensional motion of a particle in the field of an exponential potential wall, where a plays the role of the coordinate In this analogy, to the initial Kasner regime there corresponds a free motion with first constant velocity aT =p l After reflection from the wall, the particle will again with the opposite sign of the velocity: at az +Px = const, and aT +y t move freely = — p We also note that from equations (113.14), find that flx and y x take the values t = const, hence we Pr = Pm + 2ph V = P» + 2p t l f We emphasize once again that we are considering the evolution of the metric as conditions refer to a later, and not an earlier time / -> 0; thus the "initial" OSCILLATING REGIME OF APPROACH TO A SINGULAR POINT § 113 Now determining a, p, y, e a ~ and then e~ PlX e we using (113.6), t, p ~ e {Pm+2pi)x find ~ y e , 363 e (Pn+2pi)x i.e a ~ t p '\ b ~ t p ' m c , ~ t p '", where Pl If we had p = i^ip-;

0) begins to increase, the rising function a now drops, while the function was increasing, c continues to fall The perturbation itself [~ a in (1 13.7)], which previously now damps out The law of change of metrization the exponents (113.15) is conveniently represented using the para- (113.12): if Pi = Pl(s)> Pm = Pl(s), Pn = PaOO, p„ = p (s-l) then p' l = p (s-l), p'm = p (s-l), (113.16) The larger of the two positive exponents remains positive Thus the action of the perturbation results in the replacement of one Kasner regime by another, with the negative power shifting from the direction to the direction m Further evolution of the metric leads in an analogous way to an increase in the perturbation given ~ b4 in (113.7), another shift of the Kasner regime, etc successive shifts (113.16) with bouncing of the negative exponent/?! between the direc1 are transformed continue so long as s remains greater than Values s tions and according to (113.13); at this moment either p or p m is negative, while />„ is the into s by the terms The < m > t smaller of the two positive numbers (p„ = p )- The following series of shifts will now bounce the negative exponent between the directions n and I or between n and m For an arbitrary (irrational) initial value of s, the process of shifting continues without end In an exact solution of the equations, the powers/?,, p m ,pn will, of course, lose their literal meaning But the regularities in the shifting of exponents allow one to conclude that the course of change of the metric as we approach the singularity will have the following qualitative properties The process of evolution of the metric is made up of successive periods (we shall call them eras), during which distances along two of the axes oscillate, while distances along the third axis decrease On going from one era to the next, the direc- which distances decrease monotonically bounces from one axis to another The order of this bouncing acquires asymptotically the character of a random process The successive eras crowd together as we approach t = But the natural variable for describing the behavior of this time evolution appears to be not the time t, but its logarithm, In t, in terms of which the whole process of approach to the singularity is stretched out tion along to — oo The qualitative analysis presented above must, however, be supplemented with respect to the following point {n) starting from In this analysis there correspond to the «'th era values of the numbers s some largest value sj£> x down to some smallest, s^}n < The length of the era (as measured by the number of oscillations) is the integer s x - s£in- In the infinite sequence of numbers formed in this For the next era, s£i 1) = l/ sminway one will find arbitrarily small (but 364 COSMOLOGICAL PROBLEMS 113 § never zero) values of sg?n and correspondingly arbitrarily large values of s££ 1} ; such values correspond to "long" eras But to large values of the parameter s there correspond exponents (Pi> Pi, Pz) close to the values (0, 0, 1) Two of the exponents which are close to zero are thus close to one another, hence also the laws of the change of two of the functions a, b, c, are close to one another If in the beginning of such a long era these two functions happen to be close also in their absolute magnitude, they shall remain to be such during the larger part of the entire era In such a case it becomes necessary to keep not one term (a4 on ) the right sides of (113.7), but two terms Let c be that one of the functions a, b, c that decreases monotonically in the course of a long era It then rapidly becomes smaller than the other two; let us consider the solution of equations (113.7-8) in just that region of the variable x where we can neglect c compared to a and b Let the upper limit of this region be x = t In this case the first two equations of (113.7) give = arr+ft t att -Ptx while for the third equation we y T (ar We +e (113.18) , +&)= - aJ + \{e *-e 2l >) x write the solution of (113.17) in the where a and £ (113.17) 4li use (113.9), which gives: — 2a a+0 = new =-e 0, 4a (113.19) form (T-T ) + 21na , are positive constants It will be convenient to introduce in place of t a variable f = 2a " ^-(t-t exp £ (113.20) ) Co Then cc + P = In— +2lna (113.21) Co We also transform equations (113.18-19), introducing the notation x 7t * a-/? 1 fe+ = Xt+ = -^+ ~ sinh X = 0, (113.22) £ , g (2/| + cosh 2x-l) (113.23) the decrease of t from oo to there corresponds the drop of x from oo to — oo correspondingly f drops from oo to As we shall see later, a long era is obtained if £ (the value of f corresponding to the instant t = t ) is a very large quantity We shall consider the solution of equations (113.22-23) in the two regions £ > and ^ -4 To For ; large £ the solution of (113.22) in first approximation (in 1/f) X (where A is = a-p = -r~sm(£-£ is: ) a constant); the factor 1/Vc makes x a small quantity, so that we can (113.24) make the OSCILLATING REGIME OF APPROACH TO A SINGULAR POINT § 113 substitution sinh X « 7i 2/ in (113.22) ~ Having determined a and with our approximation, \(xt + X ) From = A2 (113.23) , we now find: = ^ (^-^o) + const from (1 13.21) and (1 13.24) and expanded e* and we e> in accordance finally obtain: ""-Jlk-Tt* ™]- (113 25) ' c The 365 relation of £ to the time = c oe t is -^°-^ gotten by integrating the defining equation (113.6), and is given by the formula 1= The constant c (the value of c We now turn to the region f when < £ = Here e -;i «o-« (113.26) £ ) must satisfy c < a the leading terms in the solution of (113.22) are: z==a -j3 = Kln£ + const, (113.27) the smallness a constant lying in the range - 1< k < + this condition assures 2k K the first two to compared and £ ) of the last term in (113.22) (since sinh 2/ contains 0, = , imposed on the coordinate functions in (114.1) Together with (114.2) there are thus all solution we have in mind a singularity that is t When we speak of a singularity in the cosmological collapse of a finite attainable in all of the space (and not over some restricted part, as in the gravitational body) a each of the general solution is not unambiguous In principle more than one general integral may exist, finite part of it integrals covering not the entire manifold of conceivable initial conditions, but only some Each such integral will contain the whole required set of arbitrary functions, but they may be subject to a singularity specific conditions in the form of inequalities The existence of a general solution possessing does not therefore preclude the existence of other solutions that not have a singularity more detailed presentation, cf I M § This section gives only a general outline of the situation For a Khalatnikov and E M Lifshitz, Adv in Phys 12, 185, 1963; V A Belinskii, I M Khalatmkov, and We emphasize that for a system of nonlinear equations, such as the Einstein equations, the notion oi t E M Lifshitz, Adv in Phys 1970 368 COSMOLOGICAL PROBLEMS § \\4 together four conditions These conditions connect ten different coordinate functions: three components of each of the three vectors 1, and n, and one of the functions that appears as a power of / [any one of the three functions p u pm and pn , which are related by the two equations (113.11)] In determining the number of physically arbitrary functions we must also remember that the synchronous reference system used still allows arbitrary transformations of the three spatial coordinates, not affecting the time Thus the solution (114.1) contains altogether 10-4-3 = physically arbitrary functions, which is one fewer than required for the general solution in empty space m The degree of generality achieved is not reduced when matter is introduced: the matter is "written into" the metric (1 14 1) with its four new coordinate functions needed for assigning the initial distribution of the matter density and the three velocity components (cf the problem in § 113) Of the four conditions that must be imposed on the coordinate functions in (114.1), the three conditions that arise from the equations R° = are "natural"; they follow from the a very structure of the equations of gravitation The imposition of the additional condition (1 14.2) results in the "loss" of one arbitrary function By definition the general solution is completely stable Application of any perturbation is equivalent to changing the initial conditions at some moment of time, but since the general solution admits arbitrary initial conditions, the perturbation cannot change But for the solution (114.1) the presence of the restrictive condition (114.2) its character means, in other words, instability with respect to perturbations that violate this condition The application of such a perturbation should carry the model into a different regime, which ipso facto will be completely general This is precisely the study made in the previous section for the special case of the homogeneous model The structure constants (113.2) mean precisely that for the homogeneous space of type IX all three products -curl I, -curl m, n -curl n are different from zero [cf (112.15)] Thus the condition (114.2) cannot be fulfilled, no matter which direction we assign the negative power of the time The discussion given in § 113 of the equations m (113.3-4) consisted in an explanation of the effects produced on the Kasner regime by the perturbation associated with a nonvanishing X = (1 -curl \)/v Although the investigation of a special case cannot exhibit all the details of the general does give a basis for concluding that the singularity in the general cosmological solution has the oscillating character described in § 113 We emphasize once more that this character is not related to the presence of matter, and is already a feature of empty spacetime itself case, it The regime of approach to the singularity gives a whole new aspect to the conAn infinite set of oscillations are included between any finite moment of world time t and the moment t = In this sense the process has infinite character Instead of the time t, a more natural variable (as already noted in § 113) appears to be oscillating cept of finiteness of time t, in terms of which the process is stretched out to - oo spoken throughout of the direction of approach to the singularity as the direction of decreasing time; but in view of the symmetry of the equations of gravitation under time reversal, we could equally well have talked of an approach to the singularity in the direction of increasing time Actually, however, because of the physical inequivalence of future and past, there is an essential difference between these two cases with respect to the formulation of the problem A singularity in the future can have physical meaning only if it is attainable from arbitrary initial conditions, assigned at any previous instant of time It is the logarithm In We have THE GENERAL COSMOLOGICAL SOLUTION § 114 clear that there is no reason why the distribution of matter and 369 field that is attainable at some instant in the process of evolution of the Universe should correspond to the specific conditions required for the existence of some particular solution of the gravitational equations for the question of the type of singularity in the past, an investigation based solely on the equations of gravitation can hardly give an unambiguous answer It is natural to think that the choice of the solution corresponding to the real universe is connected with some As profound physical requirements, whose establishment solely on the basis of the present theory is impossible and whose clarification will come only from a further synthesis of physical theories In this sense it could, in principle, turn out that this choice corresponds special (for example, isotropic) type of singularity Nevertheless, it appears more natural a priori to suppose that, in view of the general character of the oscillating regime, to some just this regime should describe the early stages of evolution of the universe INDEX Aberration of light 13 Absolute future Absolute past Action function 24, 67, 266 Adiabatic invariant 54 Airy function 149, 183, 200 Angular momentum 40, 79 Antisymmetric tensor 16, 21 Astigmatism 136 Asymptotic series 152 Axial vector 18, 47 Contravariant Babinet's principle 155 Bessel functions 182 Curvature tensor 258, 295 canonical forms of 264 Curved space-time 227 Curvilinear coordinates 229 derivative 239 tensor 16, 229 vector 14 Coriolis force 252, 321 Coulomb field 88 Coulomb law 88 Covariant derivative 236 tensor 16, 229, 313 vector 14 Cross section 34, 215 C-system 31 Current four-vector 69 Bianchi classification 360 Bianchi identity 261 Binding energy 30 Biot-Savart.law 103 Bremsstrahlung 179 magnetic 197 D'Alembert equation D'Alembertian 109 Decay of particles 109 30 Degree of polarization Delta function 29, 70 Caustic 133, 148 Center of inertia 41 Center-of-mass system 31 Centrally symmetric gravitational 141 Characteristic vibrations Charge Depolarization, coefficient Diffraction 145 Dipole field moment 96 282, 287 radiation 75 29 1 6, 254 55, 59 Distribution function density Circular polarization Circulation 67 173 Displacement current 44 69 Christoffel symbols 121 Doppler 238 114 effect Drift velocity Dual tensor 17 Dustlike matter Closed model 336 Coherent scattering 220 Collapse 296 Combinational scattering 220 Comoving reference system 293, 294, 301, 336 Conformal-galilean system 341 Contraction of a field 92 of a tensor 16 Classical mechanics 301 Effective radiation 177 Eikonal 129, 145 angular 134 equation 130 Elastic collision 36 Electric dipole moment Electric field intensity 371 96 47 122 372 INDEX Electromagnetic field tensor 60 Electromagnetic waves 108 Electromotive force 67 Electrostatic energy 89 Electrostatic field 88 Element of spatial distance 233 Elementary particles 26, 43 Elliptical polarization 115 Energy Gravitational Gravitational Gravitational Gravitational Gravitational mass 309 226 284 stability 350 waves 311, 352 Group of motions 356 Guiding center 55, 59 26 density flux 225, 274 centrally symmetric 282, 287 Gravitational field potential radius 75 75 Energy-momentum pseudotensor 304 Energy-momentum tensor 77, 80, 268 for macroscopic bodies 97 Equation of continuity 71 Era 363 Euler constant 185 Events Exact solutions of gravitational equations Fermat's principle 314 Hamiltonian 26 Hamilton-Jacobi equation 349 Hankel function 149 Heaviside system 69 Homocentric bundle 136 Homogeneous space 355 Hubble constant 346 Huygens' principle 146 Hypersurface 19, 73 28,46,94, 112, 130, 132, 252 Field 50 Lorentz transformation of quasiuniform 54 uniform electric 52 uniform magnetic 53 Flat space-time 227 Flux 66 Focus 133 constant Four-acceleration Four-scalar 15 Four- vector 14 Four-velocity 21 Fourier resolution Frame vector 356 Invariants of a field 63 Isotropic coordinates 287 Isotropic space 333 Jacobi identity 358 315, 362 Killing equations 119,124,125 Killing vector Laboratory system 31 Lagrangian 24, 69 density 77 150, 154 333 to fourth order to second order Gauss' theorem 20 Gaussian curvature 262 Gaussian system of units 69 General cosmological solution Generalized momentum 45 Geodesic line 244 Geometrical optics 129 Gravitational collapse 296 Gravitational constant 266 367 236, 367 165 88 Larmor precession 107 Larmor theorem 105 Legendre polynomials 98 Lens 137 Lie group 358 Lienard-Wiechert potentials 160, 171, 179 Light aberration of 13 cone pressure 112 Linearly polarized wave 116 Locally-geodesic system 240, 260 Locally-inertial system 244 Laplace equation Galilean system 228 Galileo transformation Gamma function 185 Gauge invariance 49 269 357 153 152 Friedmann solution 95 Incoherent scattering 220 Incoherent waves 122 Inertial mass 309 Inertial system Kasner solution Fraunhofer diffraction Frequency 114 Fresnel diffraction Impact parameter Interval 22 Four-dimensional geometry Four-force 28 Four-gradient 19 Four-momentum 27 Four-potential 45 integrals 62 1 373 INDEX Polarization Longitudinal waves 125 Lorentz condition 109 Lorentz contraction Lorentz force 48 Lorentz frictional force 204 Lorentz gauge 109 Lorentz transformation 9, 62 L-system 31 circular elliptic 114 114 120 tensor Polar vector 18, 47 Poynting vector 76 138 Principal focus Principal points 138 Principle MacDonald of equivalence 225 of least action 24, 27 of relativity superposition 68 Proper acceleration 52 length 11 time 183 function Macroscopic bodies 85 Magnetic bremsstrahlung 197 Magnetic dipole radiation 188 Magnetic field intensity 47 Magnetic lens 140 Magnetic moment 103 Magnification 142 Mass current vector Mass density 82 volume 83 Mass quadrupole moment tensor Maupertuis' principle 51, 132 Maxwell equations 66, 73, 254 Maxwell stress tensor 82, 112 Metric space-time 227 tensor 16, 230 Mirror 137 Mixed tensor 230 Moment-of-inertia tensor 280 Momentum 25 density 79 four-vector of 27 space 29 Monochromatic wave 114 Multipole moment 97 280 Radiation damping Observed matter 345 Open model 340 Optic axis 136 Optical path length 134 Optical system 134 Oscillator 55 237 Partially polarized light Pascal's law 85 Petrov classification 263 Phase 114 Plane wave 110,129 Poisson equation 88 203,208 of gravitational waves 323 Radius of electron 90 Rays 129 Real image 136 Recession of nebulae 343 shift 249,343 Reference system Renormalization 90, 208 Resolving power 144 Rest energy 26 Rest frame 34 Retarded potentials 158 Ricci tensor 261, 360 Riemann tensor 259 Rotation 253 Rutherford formula 95 Natural light 121 Near zone 190 Newtonian mechanics Newton's law 278 Nicol prism 120 Null vector 1 19 4, 10 Quadrupole moment 98 Quadrupole potential 97 Quadrupole radiation 188 Quantum mechanics 90 Quasicontinuous spectrum 200 Red Parallel translation Pseudo-euclidean geometry Pseudoscalar 17, 68 Pseudotensor 17 Scalar curvature 262 Scalar density 232 Scalar potential 45 Scalar product 15 Scattering 215 Schwarzschild sphere 296 Secular shift of orbit 321 of perihelion 330 Self-energy 90 Signal velocity Signature 228 374 INDEX Space component 15 Spacelike interval Spacelike vector 15 Spatial curvature 286 distance 233 metric tensor 250 Spectral resolution 118, Spherical harmonics 98 Static gravitational field Thomson formula 15 Timelike interval Transverse vector 15 wave 111 Trochoidal motion Ultrarelativistic region 247 Unit four-tensor 16 Unpolarized light 121 247 Vector Telescopic imaging 139 Tensor 15 angular momentum 41 antisymmetric 16, 21 completely antisymmetric unit contraction of 16 contravariant 16, 229 covariant 16, 229, 313 density 232 dual 17 electromagnetic field 60 hermitian 120 irreducible Velocity space 35 theorem 84 Virtual image 136 Virial 17 Wave equation 108 length 114 packet 131 surface 129 vector 116 zone 170 line moment-of-inertia 16, 21 18, density World 98 230 symmetric 47 232 polar 18, 47 potential 45 Poynting 76 axial 236 Synchronous 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"The volume on electrodynamics conveys which is truly astonishing." a sense of mastery of the subject on the part of the authors Nature 08 01 601 Pergamon ... direction of propagation; whereas mechanics the velocity of light should be smaller in the direction of the the principle of relativity Measurements first complete lack of dependence of the velocity of. .. of the components of see p 77) (for the definition of its components, see p 78) CHAPTER THE PRINCIPLE OF RELATIVITY § Velocity of propagation of interaction For the description of processes taking... velocity of light that the assumption that the does not materially affect the accuracy of the results The combination of the principle of relativity with the finiteness of the velocity of propagation