Book Review: The Classical Theory of Fields

Time for another book review. You can purchase this book at my book store below.

This time I am going for volume 2 of the Course of Theoretical Physics by Landau and Lifshitz. This volume is titled, "The Classical Theory of Fields," and covers, in one volume, special and general relativity, electrodynamics in a vacuum, and optics in a vacuum. The authors assume you are familiar with vector analysis and the electromagnetic phenomena equivalent to that covered in a general physics course (charges, electric and magnetic fields, and induction).

The first chapter is on special relativity and it begins by establishing the principle of relativity, the fact that there must be a maximum velocity for the transfer of information, and the requirement that the principle of relativity apply in all inertial reference frames. Next the authors define the concept of the event, and then show that the spacetime interval for an event is invariant of the inertial reference frame. This idea is then extended to multiple events and in this way the idea of spacetime is built-up. Next the idea of proper time is developed. The authors then derive the Lorentz transformations, the idea of proper length and length contraction, proper volume, and time dilation. This is followed by a discussion of the transformation of velocities, and the relativistic aberration of light as a consequence. Then there is a lengthy section on the mathematics of four-vectors and four-tensors. The chapter ends with a presentation of four-velocity.

The second chapter can be thought of as a quick survey of how you might apply special relativity to volume one of the Course. This begins with deriving the relativistic version of the least action principle. Then the ideas of relativistic momentum and energy, including the rest energy (you know, E=mc^2), the momentum four-vector, and the relativistic Hamilton-Jacobi equation, are presented. Then there is a very interesting section on the distribution in phase space of particles having given components of momentum. An application of special relativity is to the decay of particles, the next section covers that in detail. This is followed by a section on the invariant cross-section, and that leads into a section on elastic collisions. The chapter ends with a section on relativistic angular momentum.

The third chapter is where electrodynamics begins with a discussion of fields, the impossibility of rigid bodies in relativity, and the conclusion that this leads to the notion that particles must be treated as points. The book then establishes the four-potential with spatial components corresponding to the components of the vector potential and the time component being the scalar potential, this leads to the Hamiltonian. Then the equations of motion of a charged particle in a field are presented, along with the definitions of the electric and magnetic fields, and the Lorentz force. The next section is on gauge invariance, where the equations are invariant under transformations of the potentials. Standard electrostatics and magnetostatics are presented as aspects of a constant electromagnetic field. Then motion in a constant uniform electric field, a constant uniform magnetic field, and then in constant uniform electric and magnetic fields are presented. This is followed by a discussion of the electromagnetic field tensor based on the least action principle. Then the authors derive the Lorentz transformations of the field. Then they wrap the chapter up with the invariants of the field.

The fourth chapter covers Maxwell's equations. These are presented in their tensor form and in their more traditional vector forms. There is a nice section that establishes the action functional of the electroamgnetic field. Other nicely handled topics are the four-current vector, the electromagnetic equation of continuity, energy density and energy flux (including the Poynting vector), the stress-energy tensor (called the energy-momentum tensor) and its application to the electromagnetic field and to systems of particles. The chapter ends with the relativistic virial theorem.

The fifth chapter picks up one of the themes of chapter three, that of constant fields. This begins with Coulomb's law, the Poisson equation, and the Laplace equation. This is then extended to the idea of electrostatic energy. This is then applied to the field of a uniformly moving charge and then to motion in a Coulomb field. Another important situation covered is the dipole moment, and then the multipole moment (specifically the quadrupole moment where Legendre polynomials and spherical harmonics are introduced). This is then extended to a system of charges in an external electric field. Then the focus shifts to the constant magnetic field, then to magnetic moments, ending in a discussion of Larmor's theorem.

The sixth chapter introduces waves. The chapter begins with the wave equation and the Lorentz gauge. Then there is a section on transverse plane waves, energy flux, and momentum flux. The authors then define monochromatic waves (waves whose field is a periodic function of time), the wave vector, and explore the idea of polarization. There is also a section on Fourier analysis. Then there is a discussion of partially polarized waves (waves whose polarization changes with time), and the polarization tensor and the Stokes parameters make their appearance. The next section describes how the field produced by charges can be expanded in plane waves by Fourier transforms, and introduces longitudinal waves. The idea of Fourier series is then extended to the problem of the energy and the momentum of a field.

The seventh chapter is about optics. This begins with geometrical optics, the eikonal equation, and Fermat's principle of least time. This is extended to caustics and the intensity of light. The next three sections cover the ray theory of light. This is followed by a section on resolving power. The emphasis then shifts to wave optics with diffraction, Frensel diffraction, and Fraunhofer diffraction. Teh treatment is very abstract since it deals only with light propagation in the absence of matter.

The eighth chapter describes the field of moving charges. This begins with the so-called retarder potentials. This is then extended to the Lienard-Weichert potentials. Then Fourier analysis is applied to expand the field of moving charges into monochromatic waves. The final section deals with the issue of a finite interaction velocity and its ramifications for the Lagrangian and Hamiltonian.

The ninth chapter is long and wraps up electrodynamics in a vacuum. This begins by exploring the field of a system of charges at long distances by applying Fourier analysis. This is applied to the dipole radiator. This result is then extended into the dipole radiation due to colliding charges. This in turn is extended to the idea of bremsstrahlung. Bessel functions are discussed in the context of Coulomb interactions. Then dipole, quadrupole, and magentic dipole radiation are discussed. Then the focus shifts to fields close to a radiator. Then the focus shifts again to the fields produced by relativistic motion and synchrotron radiation. The authors then discuss radiation damping, the relativistic, and ultra-relativistic cases (where the particle energy is large when compared to the rest energy). The chapter ends with three sections on scattering.

Beginning with the tenth chapter the book switches over the general relativity. This chapter begins with a description of gravitational fields and the Lagrangian of the gravitational potential. Then the book discusses the metric and the idea of general covariance. The authors then present tensor analysis. This is followed by the motion of a particle in a gravitational field, the constant gravitational field, and a uniformly rotating frame of reference. The final section of this chapter deals with the electromagnetic field in a gravitational field.

The eleventh chapter presents Einstein's relativistic field equations. This begins with the Riemann curvature tensor and its properties, this is essentially a primer on differential geometry. Then they introduce the action functional of the gravitational field and its stress-energy tensor. This leads to the Einstein equations of general relativity. Then some consequences of the stress-energy tensor are introduced. After some more differential geometry the authors present the famous tetrad formulation, which ends the chapter.

The twelfth chapter explores the theory of gravitational fields beginning with a Newtonian treatment in tensorial language. Moving into the full relativistic treatment is the section on the centrally symmetric field including the famous Schwarzschild solution, leading to the prediction of black holes. Then they discuss motion in the Schwarzschild geometry. Leading into black hole theory is the section on the gravitational collapse of a spherical body, it is here that the idea of the event horizon is presented. These results are then extended to studying the internal condition of a collapsing body. The idea of a black hole is extended to a body in rotation described by the Kerr geometry. The authors then discuss gravitational fields at long distances from their sources. The chapter ends with the post-Newtonian approximation.

The thirteenth chapter is extremely relevent in the days of LIGO and LISA; gravitational waves. These are oscillations in the structure of spacetime itself. Beginning with weak and then strong gravitational waves, the authors discuss gravitational waves in curved space and the radiation of gravitational waves from sources.

The book wraps up with a chapter on cosmology. This begins with a discussion of cosmology of an isotropic space (the Friedmann model), then a closed isotropic space, and then an open isotropic space. This is followed by a discussion of the red shift, including the Hubble constant. Then the issue of the stability of an isotropic universe is addressed. Then the authors discuss homogeneous spaces. Then there is a section on the flat anisotropic model. This is followed by two sections on the initial singularity.

Bear in mind that this is not the normal treatment of relativity found in Misner, Thorne, and Wheeler or in Wald. This is full-blown relativity in a Lagrangian and Hamiltonian mechanical world, not for the faint of heart; but well worth the effort to work through.