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A tension within Kant’s Transcendental Analytic, regarding the combination of the “active” faculty of understanding with the “passive” faculty of sensibility, underlies the distinct appraisals in 1920 by Hans Reichenbach and Ernst Cassirer of constitutive but “relativized” a priori principles in the GTR. Reichenbach’s “principles of coordination” presuppose Schlick’s conception of cognition as a coordination of formal concepts to objects of perceptual experience, and are shown to be consonant only with the commitments of scientific realism. Cassirer’s rejection of the “active”/“passive” dichotomy promoted his conception of general covariance as a high level principle of objectivity, much in accord with Einstein’s own later views, as recently articulated in the literature on the “Hole Argument.” In particular, the principle of general covariance is shown to place significant constraints on field theories, a point noted by David Hilbert and implicit in the work of Emmy Noether.

This chapter discusses the idea that the treatment of time in present-day physical theories, general relativity and quantum theory, might be an approximation to a very different treatment in the as yet unknown quantum theory of gravity. It considers the general idea that one theory could be emergent from another, emergence being a relation analogous to, but weaker than, intertheoretic reduction. It also gives a broad description of the search for a quantum theory of gravity and some of its interpretative problems. Thereafter, the discussion focuses on the emergence of time in two specific quantum gravity programmes: quantum geometrodynamics and the Euclidean programme. It also addresses the so-called ‘problem of time’. It is really a cluster of problems; technical and conceptual, arising from how time is treated very differently in general relativity and quantum theory.

It is shown how the epistemological thesis that physics can provide knowledge only of the structure of the physical world emerged in Arthur Eddington’s semi-popular, philosophical and technical writings on the general theory of relativity. The implicitly Kantian character of Eddington’s conception of “world building” in a geometrized physics is developed through examination of Eddington’s two principal works on general relativity. Eddington’s structuralism is contrasted with that associated with Bertrand Russell, and his conception of the mind’s role in “world building” is linked to earlier views of the mathematician William Kingdom Clifford.

This chapter charts the complicated legacy of Mach's critique of absolute space and time. In 1902, Poincaré achieved a clear formulation of what a truly Machian mechanics should accomplish: it should permit a unique prediction of future motion on the basis of just the relative separations of bodies, and these separations' rates of change. However, this work made no impact on Einstein, despite his admiration for Mach. The discussion explains how several independent ideas that dominated Einstein's thinking about space, time and matter led him to a quite different interpretation (or misinterpretation) of Mach. This chapter also argues that, despite the misinterpretation, general relativity is perfectly Machian (in a sense that is the analogue for field theories of Poincaré's criterion), and that this shows general relativity to be ‘timeless’ in a certain sense, which is suggestive of quantum gravity.

This chapter presents and defends an interpretation of classical gauge theories, including electromagnetism. It argues that Yang-Mills gauge theories (but not general relativity) are best understood as postulating non-localized properties of loops in space (or more generally space-time). It raises both semantic and epistemological objections to an alternative interpretation in terms of localized gauge properties, and argues that the empirical success of a classical Yang-Mills gauge theory warrants an inference to the existence of a certain kind of non-localized property. It responds to both epistemological and metaphysical objections to this conclusion, and draws more general morals for both metaphysics and scientific epistemology.

This chapter reviews various interpretations of structural realism and then adopts a definition that allows both relations between things that are already individuated (‘relations between things’) and relations that individuate previously un-individuated entities (‘things between relations’). Since both spacetime points in general relativity and elementary particles in quantum theory fall into the latter category, the chapter proposes a principle of maximal permutability as a criterion for the fundamental entities of any future theory of ‘quantum gravity’; i.e., a theory yielding both general relativity and quantum field theory in appropriate limits. It reviews a number of current candidates for such a theory. The chapter ends by suggesting a new approach to the question of which spacetime structures should be quantized.

Albert Einstein is the only scientist who's genius was comparable to that of Newton's. Their personalities and lifestyles were completely different, but they were both consumed by the desire to know. In 1905, the miracle year, Einstein gave quantum mechanics its true beginning by working out the theory of photoelectricity, created a new statistical mechanics by studying Brownian motion, gave a fully formed theory of special relativity, and derived his famous mass-energy equation. This monumental accomplishment was matched only by Newton's work during the plague years. A decade later, Einstein single handedly developed general relativity, the deepest and most beautiful of all scientific theories.

The validity suggested by beauty must be verified by experiment. Three major predictions of general relativity are: gravitational fields bend light, slow down time beyond the effect of special relativity, and slightly alter the orbits of the planets. The first effect was spectacularly verified by the observation of light passing near the Sun during a solar eclipse. Eddington's study of the bending of light, more than anything else, made Einstein a world celebrity. Precise time measurements showed that gravity indeed does slow down time. Also, previously mysterious discrepancies in the orbits of Mercury were resolved by general relativity. The predictions of general relativity have been shown to be true.

Mortiz Schlick’s article of this title was highly influential in convincing several generations of philosophers that GTR outrightly falsified any variety of Kantian epistemology. In fact, the empiricism Schlick countered to transcendental idealism had not yet appeared in his previous writings but was quickly cobbled together from disparate elements: Henri Poincaré’s geometric conventionalism and selective readings of Einstein’s “Geometry and Experience” and earlier texts of Hermann von Helmholtz. The result of Schlick’s improvisation is that the empiricist interpretation of the spacetime metric rests on conventions regarding the behavior of rigid rods and clocks.

Newtonian mechanics is revisited in the light of Einsteinian relativity, and the repercussions shake physics to its core. Special relativity, by placing all inertial observers on the same footing, leads to the equivalence of mass and energy: E = mc² . General relativity, by granting the same rights to observers even in accelerated reference frames, leads to a revolutionary new theory of gravity: a force-free warping of space-time in the presence of mass.

This chapter shows that interpretative issues belonging to classical General Relativity (GR) might be preliminary to a deeper understanding of conceptual problems stemming from ongoing attempts at constructing a quantum theory of gravity. It focuses on the meaning of general covariance and the related question of the identity of points, by basing the investigation on the Hamiltonian formulation of GR as applied to a particular class of spacetimes. In particular, it argues that the adoption of a specific gauge-fixing within the canonical reduction of Arnowitt–Deser–Misner metric gravity provides a new solution to the debate between substantivalists and relationists, by suggesting a tertium quid between these two age-old positions. Such a third position enables the evaluation of the controversial relationship between entity realism and structural realism in a well-defined case study. After indicating the possible developments of this approach in Quantum Gravity, the chapter discusses the structuralist and holistic features of the class of spacetime models that are used in the above-mentioned canonical reduction.

General relativity is the extension of the ideas of relativity to accelerated motion. It starts with the premise that the laws of physics should be the same for all, whatever the state of motion. It then builds on the observation that an elevator falling in a gravitational field experiences no forces, and that a gravitational field is fully equivalent to an elevator being accelerated upwards. From these common-place foundations, Einstein found that mass causes a curvature in space-time and that gravity is the result of bodies moving along this curvature. Gravitational forces are reduced to the geometry of space-time.

Part I. Distinctive features of Newton’s method: Successively more accurate approximations and increasing empirical support from measurements. Part II. The Mercury perihelion problem: A proposal to alter the inverse-square law ruled out by a more precise measurement. Einstein’s theory accounts for the extra precession and recovers the successful measurements of Newton’s theory. An alternative to general relativity that would answer a new challenge from Mercury is ruled out by a more precise measurement. Part III. Newton does not require or endorse scientific progress as progress toward Laplace’s ideal limit of a final theory. Part IV. Newton’s conception of scientific progress through successively more accurate approximations is not undermined by the classic argument against convergent realism. Part V: Agreeing measurements from diverse phenomena play a decisive role of in transforming dark energy from a dubious hypothesis into part of the accepted background framework guiding empirical research in cosmology today.

Relativity means that physically it is impossible to detect absolute motion. This can be stated as a “symmetry in physics”. Special relativity (SR) is the symmetry with respect to coordinate transformations among inertial frames, general relativity (GR) among more general frames, including the accelerating coordinate systems. GR is also the relativistic theory of gravitation, and SR is valid only in the absence of gravity. Einstein's motivations to develop GR are reviewed, and his basic idea of curved spacetime as the gravitation field is outlined. Relativity represents a new understanding of space and time. GR provides the natural conceptual framework for cosmology. Experimental foundation of GR will be emphasized in our presentation. The necessary mathematics is introduced as it is needed.

This section presents annotations of the manuscript of Albert Einstein's canonical 1916 paper on the general theory of relativity. It begins with a discussion of the foundation of the general theory of relativity, taking into account Einstein's fundamental considerations on the postulate of relativity, and more specifically why he went beyond the special theory of relativity. It then considers the spacetime continuum, explaining the role of coordinates in the new theory of gravitation. It also describes tensors of the second and higher ranks, multiplication of tensors, the equation of the geodetic line, the formation of tensors by differentiation, equations of motion of a material point in the gravitational field, the general form of the field equations of gravitation, and the laws of conservation in the general case. Finally, the behavior of rods and clocks in the static gravitational field is examined.

The general theory of relativity (GTR) brought a revolutionary transformation in philosophical as well as physical outlook. The philosopher Mortiz Schick, student of Max Planck, played a pivotal role in fashioning the received view that GTR implied the untenability of any type of Kantian philosophy. Schlick’s assessment ignored the philosophically motivated contributions to GTR by Hermann Weyl and Arthur Eddington. Paul Dirac in 1931 recognized the significance of a new method of a priori mathematical speculation in theoretical physics, tying it to Eddington (and to Weyl).