This chapter addresses generalizations of the Schrödinger equation. It tries to convey that the Schrödinger equation is not the whole story when it comes to quantum physics. This is illustrated by expanding the framework in two rather orthogonal directions: relativistic quantum physics and open quantum systems. The former is introduced by taking the Klein–Gordon equation as the starting point, before shifting attention to the Dirac equation. Its time-independent version is solved numerically for a one-dimensional example, and its relation to the Schrödinger equation is derived. Also here, the Pauli matrices play crucial roles. The notion of open quantum systems is motivated by the fact that it is hard to keep a quantum system completely isolated from its surroundings – and that this necessitates a different approach than the one provided by wave functions. To this end, reduced density matrices and the notion of master equations are introduced. It is explained why master equations of the form of the generic Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) equation are desirable. Two particular phenomena following this equation are studied quantitatively: amplitude damping for a single quantum bit system and particle capture in a confining potential. Again, these examples draw directly on previous ones.

Newtonian gravity is reviewed and an attempt is made to combine it with special relativity, first by expanding the sources from mass to more general mass-energy, and then by considering relativistic force predictions. The gravito-electro-magnetic field equations are developed by analogy with Maxwell’s equations, and using dynamical source configurations familiar from the study of E&M. In addition to the fields, there are predicted particle interactions, like the bending of light, that go beyong Newtonain gravitational forces. Finally, it is clear that this attempt to combine gravity and special relativity lacks the necessary self-coupling of the gravitational field, which carries energy and therefore acts as its own source.

Chapter 1 contains the problem statements of the 150 problems in special relativity theory. The chapter is divided into nine sections with problems organized by different topics defined by the keywords in the section headings.

Chapter 3 contains the complete and elaborated solutions to all 300 problems stated and described in Chapters 1 and 2, respectively. The idea to present the solutions in a separate chapter is to help the reader to avoid the temptation of peeking at the solutions too soon.

Many have difficulties understanding what Kuhn meant when he spoke of “world change” due to revolutions. I reconstruct the historical path in which the idea emerged that reality is not something purely object-sided. The path starts with Copernicus’ new planetary system. The motions of the Sun and the planets, previously seen as purely object-sided, were now seen as containing genetically subject-sided contributions. A similar process, also at the center of the constitution of modern science, was the introduction of secondary qualities in the seventeenth century. In these historical processes, the reality status of something, whose reality seemed beyond doubt, changed dramatically. Philosophical reflection of such processes culminates in Kant’s critical philosophy. Ever since, this kind of “post-Copernican thinking” has been an indispensable part of the Western intellectual tradition, and it surfaced in the development of special relativity and quantum mechanics. I argue that Kuhn is continuing this tradition. Understanding this genealogy may make Kuhn’s metaphysics accessible to those realists who maintain that talk of genetically subject-sided contributions to reality is utterly inconsistent.

Einstein’s 1905 special theory of relativity requires a profound revision of the Newtonian ideas of space and time that were reviewed in the previous chapter. In special relativity, the Newtonian ideas of Euclidean space and a separate absolute time are subsumed into a single four-dimensional union of space and time, called spacetime. This chapter reviews the basic principles of special relativity, starting from the non-Euclidean geometry of its spacetime. Einstein’s 1905 successful modification of Newtonian mechanics, which we call special relativity, assumed that the velocity of light had the same value, c, in all inertial frames, which requires a reexamination, and ultimately the abandonment, of the Newtonian idea of absolute time. Instead, he found a new connection between inertial frames that is consistent with the same value of the velocity of light in all of them. The defining assumption of special relativity is a geometry for four-dimensional spacetime.

This introductory chapter gives a brief survey of some of the phenomena for which classical general relativity is important, primarily at the largest scales, in astrophysics and cosmology. The origins of general relativity can be traced to the conceptual revolution that followed Einstein’s introduction of special relativity in 1905. Newton’s centuries-old gravitational force law is inconsistent with special relativity. Einstein’s quest for a relativistic theory of gravity resulted not in a new force law or a new theory of a relativistic gravitational field, but in a profound conceptual revolution in our views of space and time. Four facts explain a great deal about the role gravity plays in physical phenomena. Gravity is a universal interaction, in Newtonian theory, between all mass, and, in relativistic gravity, between all forms of energy. Gravity is always attractive. Gravity is a long-range interaction, with no scale length. Gravity is the weakest of the four fundamental interactions acting between individual elementary particles at accessible energy scales.

This chapter covers the Special Theory of Relativity, introduced by Einstein in a pair of papers in 1905, the same year in which he postulated the quantization of radiation energy and showed how to use observations of diffusion to measure constants of microscopic physics. Special relativity revolutionized our ideas of space, time, and mass, and it gave the physicists of the twentieth century a paradigm for the incorporation of conditions of invariance into the fundamental principles of physics.

In this chapter we extend our review of mechanics to include Einstein’s special theory of relativity. We will see that our previous Newtonian framework is a useful description of the mechanical world only when speeds are much less than that of light. We also use this chapter to introduce index notation and general technical tools that will help us throughout the rest of the book. Then, in the following chapter, we will show how relativity provides insights for an entirely different formulation of mechanics -- the so-called variational principle.

The basic principles of general relativity are reviewed, in particular the different forms of the equivalence principle: the weak, Einstein, and strong equivalence principles. The concept of a metric is introduced within special relativity. The Einstein equations are derived in an heurisitic manner including the Christoffel symbols, the Ricci tensor, and the Ricci scalar. The Schwarzschild as the solution of Einstein‘s equation in vacuum are explicitly derived. The notion of the energy–momentum tensor, as the source term of the Einstein equations, is discussed in terms of the four-momentum of particles. For bulk matter, the definition of an ideal fluid is given. The conservation of the energy–momentum tensor in curved space-time is discussed. The Einstein equations are solved for a sphere of an ideal fluid to arrive at the Tolman–Oppenheimer–Volkoff equations, the central equations for the investigation of compact stars. Finally the analytically known solution for a sphere of an incompressible fluid, the Schwarzschild solution, is derived and used to set the Buchdahl limit on the compactness of a compact star.