Course Materials


Introduction to the Philosophy of Space and Time

This site is renewed as an independent site, for students of space and time. The original site (at Kyoto Univ.) will not be updated anymore.


(1) Tracing Einstein's Ideas, Minerva, 2004   (2) The Riddle of Space and Time, Chuko Shinsho, 2006

The first book provides basic materials for the philosophy of space and time, including the essentials of general relativity. Its sequel appeared as 『空間の謎・時間の謎』(中公新書, 2006), and this provides historical background such as the Leibniz-Clarke Correspondence, as well as newer materials of cosmology and unified theories. And more importantly, it gives a more detailed exposition of the Machian frameless dynamics (due to Julian Barbour, in conformity with Leibniz's and Mach's ideas) based only on the relational concepts. In the following, additional and supplementary materials will be provided, most of which are in English.

Let us examine what Einstein thought in the course of his struggle for constructing his two theories of relativity, the special and the general. For any students of the philosophy of space and time, it is a quite rewarding experience to examine Einstein's writings. I have finished, more than a year ago, a book on Einstein and the philosophy of space and time. For some reasons, its publication has been delayed---a publisher accepted the manuscript for publication, but left it more than a year doing nothing; thus I withdrew it. So I decided to publish its PDF version---now containing only 129 figures without text ---on the web, for our students. My lecture will be based on this material, so the students planning to attend this course are required to read this material carefully, and also to consult, if necessary, other references listed on the attached REFERENCES.

For seeing this material, you need a viewer such as Acrobat Reader, which can be easily downloaded.

(In the meantime, another publisher accepted the manuscript for publication; the book appeared in November from Minerva, Kyoto; but all figures will be changed into grey-scale color, and the reader is referred to the original color pictures presented here. Since the book is forthcoming, I have deleted all text from chapter 1 to chapter 6, and only figures are available in the following PDF version.)


Tracing Einstein's Ideas---Introduction to the Philosophy of Space and Time (ENGLISH ABSTRACT)


図18のにように 図18のように
65ページ下から5 行目
AL(v)の間で LL(v)の間で
記述に誤りがあるので段落全体を書き換え ここをクリック
記述に誤りがあるので段落と図を訂正 ここをクリック
図とタイトルが重なっている 図を1行下げ
  56. 重力場での空間の曲がり方
  100. 重力場での空間の曲がり方


Poincare ...  l'Hypothese


たことは たことは
Grunbaum ウムラウトを加える


For Further Studies

We have concentrated on the controversy of absolutism-relationalism in the last chapter. However, the philosophy of space and time have a much wider scope, and we have to take into consideration recent developments in cosmology and various attempts of unified theories, including "superstring theories"; as well as deepening your understanding of general relativity. Here is a brief list of recommended readings:

Alan Guth, The Inflationary Universe, Perseus, 1998. [A very careful and readable presentation of the inflation theories and related matters; Japanese translation available, but Prof. K. Sato should write his own version, comparable to this, I would say!]

Brian Greene, The Elegant Universe, Vintage Books, 2000. [If you wish to know what string theories are all about, this is the book you should consult; Japanese translation available. You need skepticism, though!]

Brian Greene, The Fabric of the Cosmos: space, time and the texture of reality, Allen Lane, 2004. [This time, the author does not begin with string but with a more familiar example: Newton's Bucket. How can we interpret this experiment in terms of classical mechanics, special relativity, and general relativity? Then he goes on to the nature of time, cosmology, and the fabric of the universe according to the superstring theory. The readability and quality is as good as the first book.]

Michio Kaku, Einstein's Cosmos, Atlas Books, Norton, 2004. [Although this book contains no figures, it is quite readable, combining Einstein's biography and physics; part iii is quite useful for seeing the significance of Einstein's "unified field theory" and its connection with superstring theories.]

Wolfgang Rindler, Relativity: special, general and cosmological, Oxford Univ. Press, 2001. [A compact but balanced treatment of relativity and related fields. By the way, Taylor & Wheeler 2000 is now available in Japanese tralslation, Dec. 2004.]

Lee Smolin, Three Roads to Quantum Gravity, Basic Books, 2001. [Smolin is a leading figure of the loop quantum gravity, an alternative to the superstring theory. This is quite interesting in that he emphasizes the relational point of view in constructing a theory of quantum gravity.]

(Why English books rather than much cheaper and accessible Japanese books by famous or popular authors? BECAUSE such books are not useful for us in that they do not contain enough information, such as references for clarifying many "why's" which many sensible readers should feel! I will prepare a site for cosmology soon, although I have no more occasion to give a lecture on it. See Leibniz-Clarke. See also my review of Greene and Smolin.)

(Beware, most chapters are heavy!)

まえがき、もくじほか (Preface, Table of Contents, etc.) [246K]

第1章 古典力学と電磁気学から特殊相対性理論へ(1905) (From Classical Mechanics to Special Relativity)

参考 Galilean Relativity / Michelson-Morley Experiment / Galilean Relativity and Galilean Transformation / Einstein on Galileian System / Inertia / A Commentary on Newton's Scholium

第2章 等価原理の着想(1907-1911) (Led by the Equivalence Principle) [pdf file corrected Oct. 21]

参考 The Genesis of General Relativity: (1) Einstein's 1907 Paper / The Genesis of General Relativity: (2) Einstein's 1911 Paper /

第3章  曲がった時空( 1912-1915) (Time Warps, Space Warps)

参考 Gauss' Theory of Curved Surface / Curvature and Riemann Tensor / Eddington on Kinds of Space and Field Equations / The Genesis of General Relativity: (3) Einstein and Grossmann's 1913 Paper

第4章  穴の謎( 1913-1915) (The Riddle of the Hole)

参考 The Genesis of General Relativity: (4) Einstein's "Hole Argument" in his 1914 Paper / The Genesis of General Relativity: (5) Perihelion of Mercury / The Genesis of General Relativity: (6) How did Einstein overcome the "Hole Argument"? / Appendix: Howard & Norton (1993) / The Parable of the Apple

第5章  シュヴァルツシルト 幾何学とブラックホール (Schwarzschild Geometry and the Black Hole) [pdf file corrected Jan. 11, 2005]

参考 Schwarzschild Geometry / Mass in terms of Length / The Fate of Massive Stars / Approaching the Horizon / Inside the Horizon / Orbits and Trajectories (how to derive planetary motions, paths of light, etc.)

第6章  時空の哲学入門 (Ready for the Philosophy of Space and Time)

参考 Newton's Scholium / A Commentary on Newton's Scholium / Leibniz-Clarke Correspondence, Part 1 / Leibniz-Clarke Correspondence, Part 2 / Mach's Principle / The Machian Problem / The Invisible Framework and the Ultimate Arena / The Deep Structure of General Relativity


1st assignment, due Oct. 26 (Tue)

2nd assignment, due Nov. 19 (Fri)

For additional materials, see the following sites:


The Riddle of Space and Time ---The cosmos speculated by physics and philosophy (English Abstract)




第1章 空間とは?時間とは? (What's Space, What's Time?)


第2章 ライプニッツとニュートンは何を争ったか (Newton and Leibniz: What did they fight for?)  Chapter 2 figures pdf

1.論争の文脈 ── 論争の始まり/科学のヴィジョンに関わる論争/重力の遠隔作用/絶対空間と絶対時間/「絶対時空」の背景/デカルト運動論に対する反発/デカルトの事情

2.科学の二つのヴィジョン ── 空間は神の感覚器官か/神は腕の悪い時計職人か/クラークのニュートン弁護「感覚器官は比喩にすぎない」/ニュートンの見解「自然哲学は第一原因を指し示す」/「神の、いわば、感覚器官としての空間」から何が明らかになるか/科学を動かす二つのヴィジョン/神の力と神の知恵/充足理由律の二つの解釈/ライプニッツの充足理由律は無い物ねだりか/ニュートンの宇宙論、ベントリーへの手紙(1692-3年)/絶対時空のイドラ

3.時空の関係説 ── 時空の関係説の要点/ライプニッツ自身による解説/関係説は(物理)幾何学の基礎づけも提供しうる/関係説はなぜ絶対説の難点を回避できるか/クラークの反論(1)──慣性力、創造の時間、時空の量/天下りで絶対空間に引き入れられたユークリッド幾何学/クラークの反論(2)──回転運動による遠心力/アインシュタインの先駆け

4.真空および原子と不可識別者同一の原理 ── ライプニッツは真空と原子も否定する/「小説よりも奇なり」/引力はワケのわからない性質か/見分けられないものは同一か/統計力学からの例/モナドロジー/情報の流れによる宇宙のダイナミクス

第3章 ニュートンのバケツから相対性理論まで (From Newton's Bucket to the Theories of Relativity) Chapter 3 figures pdf

1.ニュートンからマッハまで ── ニュートンのバケツ/慎重なニュートン/オイラーのひねり/水の中の物体はどんな場合に「動く」のか/「同じ方向への運動」にどう意味を与えるか/空間と時間の「等しい量」/オイラーの貢献/カントはスキップ/マッハ『力学の発展史』/マッハのバケツ?/マッハと慣性の法則

2.二つの相対性理論で論争を裁く ── 特殊相対性理論/同時性の相対性/時間の本質?/特殊相対論では空間も時間も相対的/特殊相対論において加速度は絶対的/特殊相対論で成り立つ幾何学は?/時空は絶対的/特殊相対論とニュートンのバケツ/等価原理/一般相対論とニュートンのバケツ/一般相対論とマッハのバケツ/一般相対性理論から学んだこと

第4章 マッハ流力学の行方 (Whereabouts of the Machian Mechanics) Chapter 4 figures pdf

1.関係説力学、第一の路線 ── マッハ流力学のシナリオ/ポアンカレの洞察/関係説における位置エネルギーと運動エネルギー/シュレーディンガーの1925年論文

2.関係説力学、第二の路線 ── バーバーとベルトッティの研究/相対的配置空間またはプラトニア/惑星系の運動/最小作用の原理/内在的な差/プラトニアの測地線/ニュートンとライプニッツの調停/時間の等しさはどうなるか

3.関係説力学、相対性理論への拡張 ── 特殊相対性理論はどうなるか/一般相対性理論と関係説/一般相対性のためのプラトニア/プラトニアでの軌跡の多様性/一般相対性はマッハ流力学の条件を満たす/量子力学はどうなるか

第5章 宇宙と量子 (Cosmos and Quantum) Chapter 5 figures pdf

1.静的宇宙から動的宇宙へ ── 重力は引力だけとは限らない/ニュートンゆかりの問題/閉じた有限の空間/宇宙定数/アインシュタインとド・ジッターの論争/膨張宇宙論/フリードマン宇宙/膨張宇宙における空間と時間

2.ビッグバンと素粒子 ── ビッグバン宇宙論/宇宙論と素粒子論/素粒子と相互作用/クォークモデル/ヒッグス場と統一理論/ゲージ理論

3.インフレーションと宇宙の創成 ── 宇宙背景放射と地平問題/平坦性問題/インフレーション宇宙論/平坦性問題の解決/地平問題の解決/インフレーションはどのように起きるか/インフレーション宇宙論とエネルギー収支/宇宙と時空の創造/プランク長さとプランク時間/エピローグ

文献リスト (References) References pdf


Additional Materials

Gravitation and Cosmology (重力理論と宇宙論, Japanese)

From the History of Cosmology

Goedel's Universe

Quantum Gravity and Philosophy (in Japanese) New

Brief review: Lee Smolin, The Trouble with Physics New

Last modified Apr. 20, 2008. (c) Soshichi Uchii