I desire to tender my best thanks to my colleagues Professor S. R. Milner, D.Sc., and Mr. W. E. Curtis, A.R.C.Sc., F.R.A.S., also to my friend Dr. Arthur Holmes, A.R.C.Sc., F.G.S., of the Imperial College, for their kindness in reading through the manuscript, for helpful criticism, and for numerous suggestions. I owe an expression of thanks also to Messrs. Methuen for their ready counsel and advice, and for the care they have bestowed on the work during the course of its publication.

ROBERT W. LAWSON

THE PHYSICS LABORATORY
THE UNIVERSITY OF SHEFFIELD
June 12, 1920

CONTENTS

[PART I
THE SPECIAL THEORY OF RELATIVITY]
I. [Physical Meaning of Geometrical Propositions]
II. [The System of Co-ordinates]
III. [Space and Time in Classical Mechanics]
IV. [The Galileian System of Co-ordinates]
V. [The Principle of Relativity (in the Restricted
Sense)]
VI. [The Theorem of the Addition of Velocities employed
in Classical Mechanics]
VII. [The Apparent Incompatibility of the Law of
Propagation of Light with the Principle of
Relativity]
VIII. [On the Idea of Time in Physics]
IX. [The Relativity of Simultaneity]
X. [On the Relativity of the Conception of Distance]
XI. [The Lorentz Transformation]
XII. [The Behaviour of Measuring-Rods and Clocks
in Motion]
XIII. [Theorem of the Addition of Velocities. The
Experiment of Fizeau]
XIV. [The Heuristic Value of the Theory of Relativity]
XV. [General Results of the Theory]
XVI. [Experience and the Special Theory of Relativity]
XVII. [Minkowski's Four-dimensional Space]
[PART II
THE GENERAL THEORY OF RELATIVITY]
XVIII. [Special and General Principle of Relativity]
XIX. [The Gravitational Field]
XX. [The Equality of Inertial and Gravitational Mass
as an Argument for the General Postulate
of Relativity]
XXI. [In what Respects are the Foundations of Classical
Mechanics and of the Special Theory
of Relativity unsatisfactory?]
XXII. [A Few Inferences from the General Principle of
Relativity]
XXIII. [Behaviour of Clocks and Measuring-Rods on a
Rotating Body of Reference]
XXIV. [Euclidean and Non-Euclidean Continuum]
XXV. [Gaussian Co-ordinates]
XXVI. [The Space-time Continuum of the Special
Theory of Relativity considered as a
Euclidean Continuum
] XXVII. [The Space-time Continuum of the General
Theory of Relativity is not a Euclidean
Continuum]
XXVIII. [Exact Formulation of the General Principle of
Relativity]
XXIX. [The Solution of the Problem of Gravitation on
the Basis of the General Principle of
Relativity]
[PART III
CONSIDERATIONS ON THE UNIVERSE
AS A WHOLE]
XXX. [Cosmological Difficulties of Newton's Theory]
XXXI. [The Possibility of a "Finite" and yet "Unbounded"
Universe]
XXXII. [The Structure of Space according to the
General Theory of Relativity]
[APPENDICES]
I. [Simple Derivation of the Lorentz Transformation
[Supplementary to Section XI.]
II. [Minkowski's Four-dimensional Space ("World")
[Supplementary to Section XVII.]
III. [The Experimental Confirmation of the General
Theory of Relativity
(a) Motion of the Perihelion of Mercury
(b) Deflection of Light by a Gravitational Field
(c) Displacement of Spectral Lines towards the
Red]
[BIBLIOGRAPHY]
[INDEX]

RELATIVITY
THE SPECIAL AND THE GENERAL THEORY

PART I
THE SPECIAL THEORY OF RELATIVITY

I
PHYSICAL MEANING OF GEOMETRICAL
PROPOSITIONS

IN your schooldays most of you who read this book made acquaintance with the noble building of Euclid's geometry, and you remember—perhaps with more respect than love—the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers. By reason of your past experience, you would certainly regard everyone with disdain who should pronounce even the most out-of-the-way proposition of this science to be untrue. But perhaps this feeling of proud certainty would leave you immediately if some one were to ask you: "What, then, do you mean by the assertion that these propositions are true?" Let us proceed to give this question a little consideration.

Geometry sets out from certain conceptions such as "plane," "point," and "straight line," with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as "true." Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, i.e. they are proven. A proposition is then correct ("true") when it has been derived in the recognised manner from the axioms. The question of the "truth" of the individual geometrical propositions is thus reduced to one of the "truth" of the axioms. Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning. We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called "straight lines," to each of which is ascribed the property of being uniquely determined by two points situated on it. The concept "true" does not tally with the assertions of pure geometry, because by the word "true" we are eventually in the habit of designating always the correspondence with a "real" object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves.