SPACETIME: Opened, Dec. 21, 1998; last update, August 10, 2001

This seminar is finished; but many materials will be used again in my new lecture on the philosophy of space and time, Fall 2001; anyone who wishes to attend this lecture should read these materials.

For Supplementary materials by Uchii, refer to the links! And for the assignments (2000), see the bottom of this page! See also Space and Time Index.

Lawrence Sklar, Space, Time, and Spacetime, University of California Press, 1974, 1976.



Euclidean differential geometry


Parallel transportation

Equivalence Principle

Einstein Equation

Cosmological Term Revised, Apr. 3

Notes on Poincare

Gruenbaum on the Duhemian Thesis and Einstein's Philosophy of Geometry New

Reichenbach on Helmholtz Revised

Gruenbaum on Reichenbach's "Universal Forces" New

Gruenbaum on Eddington's Criticism of Poincare Revised

Riemann and Helmholtz New

Gruenbaum's reply to Malament and Friedman appeared→ PhilSci Archive, "David Malament and the Conventionality of Simultaneity: A reply" New

Malament's Theorem and Allen Janis's Construction New, Mar. 14


Mach's Principle Revised

Absoluteness and Relationism

Newton's Scholium

Spacetime 2000

Neo-Newtonian spacetime

Maxwell Equations

A Comparison, Newtonian, Neo-Newtonian, and Minkowskian space-and-time Revised, Mar. 23

General Principle of Relativity Revised, March 23

Galilean Relativity and Galilean Transformation

Galilean Relativity [Japanese] Revised, Mar.13

The Aristotelian Universe and Space Revised, Mar. 15

General Relativity and Absolute Space

Sklar's Maneuver

Coriolis' Force

Perihelion of Mercury

Eddington on 1919 Expeditions Revised

The Parable of the Apple New


Causal Theory of Time

Michelson-Morley Experiment Revised

Lorentz Transformation

Affine Structure

Einstein on Fizeau's Experiment


The Genesis of General Relativity (1)

The Genesis of General Relativity (2)

The Genesis of General Relativity (3)

The Genesis of General Relativity (4)

Interlude: Einstein in Japan (1922); Einstein with European Physicists

The Genesis of General Relativity (5) Revised (4 Sept)

The Genesis of General Relativity (6)

The Genesis of General Relativity (7)

The Genesis of General Relativity (8) the series completed

For Conventionality of Simulataneity, see Allen Janis' Paper (Norton's discussion is also good: Norton, John D., "Philosophy of Space and Time", in Introduction to the Philosophy of Science (Salmon et al.), Prentice-Hall, 1992.)

Wormholes in General Relativity

Conventionality of Topology

Comments on the 2nd Assignment

Sklar on the Causal Theory of Time New


A "Flow" of Time?

McTaggart on the Unreality of Time



The whole structure of Sklar's book is described by the author briefly as follows:

Chapter II is concerned with the spistemology of geometry. Given that mathematics provides us with more than one consistent description of possible spaces oor spacetimes, and given that some physical theories utilize these possibilities to characterize worlds with quite different geometric structures, to what extent can we empirically determine just what geometric structure of the space or spacetime of the world is? Is this an "empirical " matter? Can we, instead, fix upon a structure for the world independently of any observation or experiment? Or is the situation more appropriately described as one in which the geometric structure of the world we posit is one we can choose to please ourselves, "as a matter of convention"?

Chapter III is concerned with a closely related issue. When we deal with the spatiotemporal structure of the world, must we view ourselves as attempting to discover the nature of an entity of the world, space or spacetime itself, or should we instead view ourselves as attempting merely to determine certain general truths about the structure of a set of relations holding among concrete material happenings?

Chapter IV is devoted to discussing the relation between the temporal order of the world and its causal order. The first part of the chapter discusses the revolution in our views about time brought about by the adoption of the special theory of relativity and deals with such questions as: the empirical basis fot this conceptual change, the "philosophical" ingredients in the adoption of special relativity, the problem of whether the special-relativistic theory of time is a matter of "convention", etc. The later parts of the chapter examine the claim that temporal relations among events can be "reduced" to causal relations among events in some plausible way----the so-called causal theory of time.

In Chapter V[,] I discuss the philosophical issue sometimes called the question of the direction of time. The primary focus of the chapter is on the question of whether the intuitive notion of the direction of time received any clarification by the study of various physical processes that we might intuitively describe as behaving asymmetrically in time. (pp.2-3)


+ New Material: For the history of the relativity theories, an up-to-date and readable exposition by John Stachel is recommended:

Twentieth Century Physics, vol. l, ed. by Laurie M. Brown, Abraham Pais, and Brian Pippard, Institute of Physics, 1995; Japanese translation 『20世紀の物理学』vol. 1、第4章「相対性理論の歴史」、丸善、1999年

[Sklar's photo by Robert Kalmbach, fron The Research News, Vol. 31, Nos. 1-2,

The University of Michigan, 1980]



一般相対性理論ではなぜ曲がった時空が必要になるのか、簡潔に説明せよ。 2000字以内。締め切り6月25日(金)


空間の実体説と関係説の間の争点(ライプニッツとニュ−トンまでの段階に限定してよい)を簡潔に説明せよ。 2000字以内。締め切り12月3日(金)

2000年度 (In order to update your information, you may also consult Michael Friedman's book: Foundations of Space-Time Theories, Princeton Univ. Press; another, advanced book is John Earman's World enough and Space-Time, MIT, 1989, but since Earman is not kind enough to untrained readers, you need a considerable preparation.)

Spacetime 2000

This year, we will begin from chapter 3, section D "The evolution of physics and the problem of substantival space", page 194. For the newcomers, I will briefly summarize the most important points of Sklar's discussions so far.

Chapter 2 contains basic materials for discussing the epistemology of geometry.

(1) History of geometry

The structure of space has been regarded as clarified by geometry, and that's why Sklar's discussion begins with the epistemology of geometry. You've got to know the Euclidean geometry and its axiomatic structure; you have to know the status of the fifth postulate (the parallel postulate). Non-Euclidean geometry appeared in the 19th century, and roughly speaking, it replaces the fifth postulate in one way or another, producing Lobachevski geometry (many parallel postulate) and Riemann geometry (no parallel).

(2) Some mathematics

Also, you have to know at least basic results of Minkowski spacetime which is indispensable for special relativity, and the Gaussian and Riemannian analytic geometry, or differential geometry, which is indispensable for general relativity.

(3) Some physics

The contemporary spacetime philosophy naturally centers on relativity theories. So that you've got to know at least elementary portion of special relativity and general relativity. The essential difference is that the graviation is treated in the latter. Spacetime in special relativity is flat, spacetime in general relativity is curved, because gravity cannot be treated without curvature. Local spacetime can be approximated by special relativity, but global structure cannot be. Grasp at least the intuitive significance of the Equivalence Principle and the Einstein Equation in general relativity.

(4) Poincare's conventionalism

Poincare's conventionalism (as regards geometry) is also an indispensable topic. He argued that geometrical axioms are neither a priori truths nor empirical truths, and claimed that they are conventions. What did he mean by this? And how did some empiricists argued against this? These questions lead us to the Duhemian problem: how can we check the geometry of the actual world by means of empirical tests conducted in the midst of a whole network of physics, geometry, and many other experimental hypotheses?

Chapter 3 treats a long-standing problems of motion, space and time.

(5) Absolute vs. Relative, Substantivalist vs. Relationist

What is space, what is time? What is the nature of each? Some argue that space is nothing but a bunch of relations among physical objects in the world, and there is no space apart from such relations (relationist). Others argue that there is space independent from physical objects within it, and space is a kind of substance, different in nature from physical objects (substantivalist).

Their controversy center around the notions of absolute motion, relative motion, absolute space, absolute acceleration, etc. Major opponents are Leibniz and Newton. Leibniz argued that only relative motions can be observed, so that any claim unsupported by such observations are meaningless; Newton introduced a new epoch by his argument for absolute space, based on empirical grounds, i.e. forces produced by accelerated motions. Read carefully Newton's Scholium.






Comments on the 2nd Assignment New

Last modified, April 16, 2006. (c) Soshichi Uchii