Search Results

In his 1923 book The Mathematical Theory of Relativity, Arthur Eddington distinguished between two chains of reasoning in general relativity. The first familiar one starts with the existence of the four-dimensional space-time interval ds, whose meaning is the usual one associated with the readings of physical rods and clocks and possibly light rays. The other less familiar chain of reasoning ‘binds the physical manifestations of the energy tensor and the interval; it passes from matter as now defined by the energy-tensor to the interval regarded as the result of measurements made with this matter.’ This chapter takes up the challenge of outlining this second chain of reasoning. In developing this reasoning, it argues that the dynamical underpinning of relativistic kinematics that has been defended in this book is consistent with the structure and logic of GR.

After a review of the Newtonian theory of gravitation in terms of its potential function, we start the study of general relativity (GR) with the introduction of the equivalence principle (EP). The Weak EP (the equality of the gravitational and inertial masses) is extended by Einstein to the Strong EP, the equivalence between inertia and gravitation for all interactions. This implies the existence of “local inertial frames” at every spacetime point. In a sufficiently small region, the “local inertial observer” will not sense any gravity effect. The equivalence of acceleration and gravity means that GR (physics laws valid in all coordinate systems, including accelerating frames) must necessarily be a theory of gravitation. The strong EP is used to deduce the results of gravitational redshift and time dilation, as well as gravitational bending of a light ray.

This chapter examines how the notion of indeterminacy challenges the equivalence principle (“Saying something is true is equivalent to just saying it”). Indeterminacy involves cases where we have a question “Are things thus?” to which neither the affirmative nor the negative answer seems appropriate. If the question “Are things thus?” cannot be answered “Yes” or “No,” then it seems the corresponding declarative “Things are thus” cannot be called “true” or “false,” and so it seems we have a counterexample to the bivalence principle, according to which every proposition is either true or false. The chapter first considers two classes of examples, involving phenomena of presupposition and vagueness, before discussing various conceivable lines of response to the problems they raise. In particular, it looks at denial strategy, disqualification strategy, deviance strategy, doublespeak strategy, dependency strategy, and defeatism. Finally, it analyzes a third type of case, involving a purported special kind of relativity.

This is a brief introduction to dynamic programming and the method of using linear quadratic (LQ) approximations to the return function; the method is an approximation because it computes the solution to a quadratic expansion of the utility function about the steady state or the stable growth path of model economies. The main purpose of the chapter is to review the theoretical basis for the LQ approximation and to illustrate its use with a detailed example (social planning). The author demonstrates that, using the LQ approximation approach and the certainty equivalence principle, solving for the value function is a relatively easy task. The different sections of the chapter describe the standard neoclassical growth model, present a social planner problem that can be used to solve the model, give a recursive formulation of the social planner's problem, and describe an LQ approximation to this problem. Exercises are included throughout, and an appendix presents a MATLAB program to illustrate the LQ method.

This chapter presents a unified theory of tunneling phenomenon and covalent bond force, as a result of the similarity between the Bardeen theory of tunneling and the Herring-Landau theory of the covalent bond. Three general theoretical treatments are presented, which show that tunneling conductance is proportional to the square of the covalent bond interaction energy, or equivalently, the square of covalent bond force. The constant of proportionality is related to the electronic properties of the materials. For the case of a metal tip and a metal sample, an explicit equation contains only measurable physical quantities is derived. Several experimental verifications are presented. The equivalence of covalent bond energy and tunneling conductance provides a theoretical explanation of the threshold resistance observed in atom-manipulation experiments, and points to a method of predicting the threshold resistance for atom manipulation.

This chapter starts by emphasizing that we can construct a picture of the universe as a whole only by extrapolating the laws of physics as we know them on earth to conditions almost unimaginably different from those prevailing here. It then reviews the information we can get from optical and other observations, and the picture of the current universe and its contents which emerges from it, with a discussion inter alia of the cosmic distance scale and the cosmological red shift. The ideas of special and general relativity are sketched, and some consequences such as gravitational radiation and black holes are discussed; possible futures of the universe are considered. Finally, it is indicated how extrapolation of the equations of general relativity into the past leads inexorably to the idea of a “hot big bang”.

This chapter examines the more purely philosophical aspect of the question of the paradoxes of truth. It first considers a particular paradoxical derivation that uses the equivalence principle not in the form of the T-biconditionals (which immediately raise questions about what kind of conditional is involved), but of rules of T-introduction and T-elimination. It then explains the concept of revenge as well as logical and contextualist “solutions” and the so-called “paraconsistency.” It concludes with a discussion of inconsistency theories such as defeatism and deflationism. The background assumptions about meaning behind an integrated deflationist/defeatist theory might run as follows. Meaning can be given by rules, but rules can be inconsistent. There is even a result in mathematical logic (Church's theorem) to the effect that there is no mechanical test for inconsistency of rules, making it unlikely we have any filter preventing us from ever internalizing inconsistencies.

This section discusses the development of Albert Einstein's ideas and attitudes as he struggled for eight years to come up with a general theory of relativity that would meet the physical and mathematical requirements laid down at the outset. It first considers Einstein's work on gravitation in Prague before analyzing three documents that played a significant role in his search for a theory of general relativity: the Zurich Notebook, the Einstein–Grossmann Entwurf paper, and the Einstein–Besso manuscript. It then looks at Einstein's completion of his general theory of relativity in Berlin in November 1915, along with his development of a new theory of gravitation within the framework of the special theory of relativity. It also examines the formulation of the basic idea that Einstein termed the “equivalence principle,” his Entwurf theory vs. David Hilbert's theory, and the 1916 manuscript of Einstein's work on the general theory of relativity.

General relativity describes gravity as a byproduct of the geometry of spacetime. This chapter explains why this notion makes sense, introducing the principle of equivalence and the universality of gravitational interactions.

After a review of the Newtonian theory of gravitation in terms of its potential function, this chapter starts the study of general relativity (GR) with the introduction of the equivalence principle (EP). The weak EP (equality of gravitational and inertial masses) was extended by Einstein to the strong EP (equivalence between inertia and gravitation for all interactions). This implies the existence of “local inertial frames” at every spacetime point. In a sufficiently small region, a “local inertial observer” will not sense any gravitational effect. The equivalence of acceleration and gravity means that GR (with physics laws valid in all coordinate systems, including accelerating frames) must necessarily be a theory of gravitation. The strong EP is used to deduce results for gravitational redshift and time dilation, as well as gravitational bending of a light ray. The operation of GPS is shown to depend crucially on relativistic time dilation effects.

In general relativity (GR) physics laws are not changed under general coordinate transformations. Among Einstein’s motivation for GR was his wish to have a deeper understanding of the empirically observed equality between gravitational and inertial masses. Gravity naturally enters into the theory when accelerated frames are considered. The generalization of this equivalence principle to electromagnetism implies gravitational redshift, gravitational time dilation, and gravitational bending of a light ray. Most importantly, such considerations led Einstein to the idea that the gravitational effect on a body can be attributed directly to some underlying spacetime feature and the gravitational field is simply curved spacetime. We present some elements of Riemannian geometry (the mathematics of curved space): Gaussian coordinates metric tensor, the geodesic equation, and curvature.

The equivalence principle is an important thinking tool to bootstrap our thinking from the inertial coordinate systems of special relativity to the more complex coordinate systems that must be used in the presence of gravity (general relativity). The equivalence principle posits that at a given event gravity accelerates everything equally, so gravity is equivalent to an accelerating coordinate system.This conjecture is well supported by precise experiments, so we explore the consequences in depth: gravity curves the trajectory of light as it does other projectiles; the effects of gravity disappear in a freely falling laboratory; and gravitymakes time runmore slowly in the basement than in the attic—a gravitational form of time dilation. We show how this is observable via gravitational redshift. Subsequent chapters will build on this to show how the spacetime metric varies with location.

This chapter attempts to formulate a consistent extension of the theory of general relativity. The starting point of the general theory of relativity is the recognition of the unity of gravitation and inertia (principle of equivalence). From this principle, it follows that the properties of “empty space” were to be represented by a symmetrical tensor expressed in the theory. The principle of equivalence, however, does not give any clue as to what may be the more comprehensive mathematical structure on which to base the treatment of the total field comprising the entire physical reality. As such, this chapter considers the problem of how to find a field structure which is a natural generalization of the symmetrical tensor as well as a system of field equations for this structure which represent a natural generalization of certain equations of pure gravitation.

General relativity. The equivalence principle and the derivation of the Einstein–Hilbert equations. The geometrical notions of curvature and affine connection are introduced. Geodesics and the bending of light by a gravitational field. General relativity as a gauge invariant classical field theory.

This chapter describes how gravity provided the backdrop for one of the most important paradigm shifts in the history of physics. Prior to Albert Einstein’s general theory of relativity, trajectories were paths described by geometry. After the theory of general relativity, trajectories are paths caused by geometry. This chapter explains how Einstein arrived at his theory of gravity, relying on the space-time geometry of Hermann Minkowski, whose work he had originally harshly criticized. The confirmation of Einstein’s theory was one of the dramatic high points in twentieth-century history of physics when Arthur Eddington journeyed to an island off the coast of Africa to observe stellar deflections during a solar eclipse. If Galileo was the first rock star of physics, then Einstein was the first worldwide rock star of science.

This introductory chapter is mainly about the locality postulate of the standard relativity theory. The fundamental laws of microphysics have been formulated with respect to inertial observers. However, inertial observers do not in fact exist, since actual observers are accelerated. What do accelerated observers measure? Lorentz invariance is extended to accelerated observers by assuming that they are pointwise inertial. That is, an accelerated observer at each instant is equivalent to an otherwise identical momentarily comoving inertial observer. This hypothesis of locality, which underlies the special and general theories of relativity, is described in detail. The locality postulate fits perfectly together with Einstein’s local principle of equivalence to ensure that every observer in a gravitational field is pointwise inertial. When coupled with the hypothesis of locality, Einstein’s principle of equivalence provides a physical basis for a field theory of gravitation that is consistent with local Lorentz invariance.

This chapter describes two of the three most influential findings macroeconomics has available to it. One is known as the First Fundamental Theorem of Welfare Economics, or the “invisible-hand” theorem. It tells us that under some conditions, competitive market outcomes deliver stable non-wasteful outcomes. This result holds even if consumers care only about the goods and services they consume, and know nothing about the others, or the world, beyond the prices they face. Next, the chapter describes work showing that in general, there will be prices that, all by themselves, can guide self-interested buyers and sellers to such orderly outcomes. But who cares that this can happen? In practice, will it? To answer this, we discuss some reasons to believe in the relevance of Walrasian Equilibrium for the real-world.

Is gravity nonlocal? Einstein interpreted the principle of equivalence of inertial and gravitational masses to mean that there exists a profound relationship between inertia and gravitation. Based on Einstein’s fundamental insight, it would seem natural to extend history dependence to the gravitational domain. However, it is not clear how to develop a nonlocal extension of Einstein’s local principle of equivalence. To go forward, we therefore choose an indirect approach based on a certain analogy with electromagnetism. In a material medium, the electromagnetic constitutive relations are nonlocal and this fact leads to the nonlocal electrodynamics of media. It turns out that general relativity can be formulated in a form that resembles the electrodynamics of media. Making the corresponding gravitational constitutive relations nonlocal would then lead to nonlocal GR. This indirect approach is adopted in the rest of this book.