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`

` RELATIVITY: THE SPECIAL AND GENERAL THEORY `

` `

` BY ALBERT EINSTEIN `

` `

` `

` `

` `

` CONTENTS `

` `

` Preface `

` `

` Part I: The Special Theory of Relativity `

` `

` 01. Physical Meaning of Geometrical Propositions `

` 02. The System of Co-ordinates `

` 03. Space and Time in Classical Mechanics `

` 04. The Galileian System of Co-ordinates `

` 05. The Principle of Relativity (in the Restricted Sense) `

` 06. The Theorem of the Addition of Velocities employed in `

` Classical Mechanics `

` 07. The Apparent Incompatability of the Law of Propagation of `

` Light with the Principle of Relativity `

` 08. On the Idea of Time in Physics `

` 09. The Relativity of Simultaneity `

` 10. On the Relativity of the Conception of Distance `

` 11. The Lorentz Transformation `

` 12. The Behaviour of Measuring-Rods and Clocks in Motion `

` 13. Theorem of the Addition of Velocities. The Experiment of Fizeau `

` 14. The Hueristic Value of the Theory of Relativity `

` 15. General Results of the Theory `

` 16. Expereince and the Special Theory of Relativity `

` 17. Minkowski's Four-dimensial Space `

` `

` `

` Part II: The General Theory of Relativity `

` `

` 18. Special and General Principle of Relativity `

` 19. The Gravitational Field `

` 20. The Equality of Inertial and Gravitational Mass as an Argument `

` for the General Postulate of Relativity `

` 21. In What Respects are the Foundations of Classical Mechanics `

` and of the Special Theory of Relativity Unsatisfactory? `

` 22. A Few Inferences from the General Principle of Relativity `

` 23. Behaviour of Clocks and Measuring-Rods on a Rotating Body of `

` Reference `

` 24. Euclidean and non-Euclidean Continuum `

` 25. Gaussian Co-ordinates `

` 26. The Space-Time Continuum of the Speical Theory of Relativity `

` Considered as a Euclidean Continuum `

` 27. The Space-Time Continuum of the General Theory of Relativity `

` is Not a Eculidean Continuum `

` 28. Exact Formulation of the General Principle of Relativity `

` 29. 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 `

` `

` 30. Cosmological Difficulties of Netwon's Theory `

` 31. The Possibility of a "Finite" and yet "Unbounded" Universe `

` 32. The Structure of Space According to the General Theory of `

` Relativity `

` `

` `

` Appendices: `

` `

` 01. Simple Derivation of the Lorentz Transformation (sup. ch. 11) `

` 02. Minkowski's Four-Dimensional Space ("World") (sup. ch 17) `

` 03. The Experimental Confirmation of the General Theory of Relativity `

` 04. The Structure of Space According to the General Theory of `

` Relativity (sup. ch 32) `

` 05. Relativity and the Problem of Space `

` `

` Note: The fifth Appendix was added by Einstein at the time of the `

` fifteenth re-printing of this book; and as a result is still under `

` copyright restrictions so cannot be added without the permission of `

` the publisher. `

` `

` `

` `

` PREFACE `

` `

` (December, 1916) `

` `

` The present book is intended, as far as possible, to give an exact `

` insight into the theory of Relativity to those readers who, from a `

` general scientific and philosophical point of view, are interested in `

` the theory, but who are not conversant with the mathematical apparatus `

` of theoretical physics. The work presumes a standard of education `

` corresponding to that of a university matriculation examination, and, `

` despite the shortness of the book, a fair amount of patience and force `

` of will on the part of the reader. The author has spared himself no `

` pains in his endeavour to present the main ideas in the simplest and `

` most intelligible form, and on the whole, in the sequence and `

` connection in which they actually originated. In the interest of `

` clearness, it appeared to me inevitable that I should repeat myself `

` frequently, without paying the slightest attention to the elegance of `

` the presentation. I adhered scrupulously to the precept of that `

` brilliant theoretical physicist L. Boltzmann, according to whom `

` matters of elegance ought to be left to the tailor and to the cobbler. `

` I make no pretence of having withheld from the reader difficulties `

` which are inherent to the subject. On the other hand, I have purposely `

` treated the empirical physical foundations of the theory in a `

` "step-motherly" fashion, so that readers unfamiliar with physics may `

` not feel like the wanderer who was unable to see the forest for the `

` trees. May the book bring some one a few happy hours of suggestive `

` thought! `

` `

` December, 1916 `

` A. EINSTEIN `

` `

` `

` `

` PART I `

` `

` THE SPECIAL THEORY OF RELATIVITY `

` `

` 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 our 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 form 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 "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. `

` `

` It is not difficult to understand why, in spite of this, we feel `

` constrained to call the propositions of geometry "true." Geometrical `

` ideas correspond to more or less exact objects in nature, and these `

` last are undoubtedly the exclusive cause of the genesis of those `

` ideas. Geometry ought to refrain from such a course, in order to give `

` to its structure the largest possible logical unity. The practice, for `

` example, of seeing in a "distance" two marked positions on a `

` practically rigid body is something which is lodged deeply in our `

` habit of thought. We are accustomed further to regard three points as `

` being situated on a straight line, if their apparent positions can be `

` made to coincide for observation with one eye, under suitable choice `

` of our place of observation. `

` `

` If, in pursuance of our habit of thought, we now supplement the `

` propositions of Euclidean geometry by the single proposition that two `

` points on a practically rigid body always correspond to the same `

` distance (line-interval), independently of any changes in position to `

` which we may subject the body, the propositions of Euclidean geometry `

` then resolve themselves into propositions on the possible relative `

` position of practically rigid bodies.* Geometry which has been `

` supplemented in this way is then to be treated as a branch of physics. `

` We can now legitimately ask as to the "truth" of geometrical `

` propositions interpreted in this way, since we are justified in asking `

` whether these propositions are satisfied for those real things we have `

` associated with the geometrical ideas. In less exact terms we can `

` express this by saying that by the "truth" of a geometrical `

` proposition in this sense we understand its validity for a `

` construction with rule and compasses. `

` `

` Of course the conviction of the "truth" of geometrical propositions in `

` this sense is founded exclusively on rather incomplete experience. For `

` the present we shall assume the "truth" of the geometrical `

` propositions, then at a later stage (in the general theory of `

` relativity) we shall see that this "truth" is limited, and we shall `

` consider the extent of its limitation. `

` `

` `

` Notes `

` `

` *) It follows that a natural object is associated also with a `

` straight line. Three points A, B and C on a rigid body thus lie in a `

` straight line when the points A and C being given, B is chosen such `

` that the sum of the distances AB and BC is as short as possible. This `

` incomplete suggestion will suffice for the present purpose. `

` `

`

` RELATIVITY: THE SPECIAL AND GENERAL THEORY `

` `

` BY ALBERT EINSTEIN `

` `

` `

` `

` `

` CONTENTS `

` `

` Preface `

` `

` Part I: The Special Theory of Relativity `

` `

` 01. Physical Meaning of Geometrical Propositions `

` 02. The System of Co-ordinates `

` 03. Space and Time in Classical Mechanics `

` 04. The Galileian System of Co-ordinates `

` 05. The Principle of Relativity (in the Restricted Sense) `

` 06. The Theorem of the Addition of Velocities employed in `

` Classical Mechanics `

` 07. The Apparent Incompatability of the Law of Propagation of `

` Light with the Principle of Relativity `

` 08. On the Idea of Time in Physics `

` 09. The Relativity of Simultaneity `

` 10. On the Relativity of the Conception of Distance `

` 11. The Lorentz Transformation `

` 12. The Behaviour of Measuring-Rods and Clocks in Motion `

` 13. Theorem of the Addition of Velocities. The Experiment of Fizeau `

` 14. The Hueristic Value of the Theory of Relativity `

` 15. General Results of the Theory `

` 16. Expereince and the Special Theory of Relativity `

` 17. Minkowski's Four-dimensial Space `

` `

` `

` Part II: The General Theory of Relativity `

` `

` 18. Special and General Principle of Relativity `

` 19. The Gravitational Field `

` 20. The Equality of Inertial and Gravitational Mass as an Argument `

` for the General Postulate of Relativity `

` 21. In What Respects are the Foundations of Classical Mechanics `

` and of the Special Theory of Relativity Unsatisfactory? `

` 22. A Few Inferences from the General Principle of Relativity `

` 23. Behaviour of Clocks and Measuring-Rods on a Rotating Body of `

` Reference `

` 24. Euclidean and non-Euclidean Continuum `

` 25. Gaussian Co-ordinates `

` 26. The Space-Time Continuum of the Speical Theory of Relativity `

` Considered as a Euclidean Continuum `

` 27. The Space-Time Continuum of the General Theory of Relativity `

` is Not a Eculidean Continuum `

` 28. Exact Formulation of the General Principle of Relativity `

` 29. 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 `

` `

` 30. Cosmological Difficulties of Netwon's Theory `

` 31. The Possibility of a "Finite" and yet "Unbounded" Universe `

` 32. The Structure of Space According to the General Theory of `

` Relativity `

` `

` `

` Appendices: `

` `

` 01. Simple Derivation of the Lorentz Transformation (sup. ch. 11) `

` 02. Minkowski's Four-Dimensional Space ("World") (sup. ch 17) `

` 03. The Experimental Confirmation of the General Theory of Relativity `

` 04. The Structure of Space According to the General Theory of `

` Relativity (sup. ch 32) `

` 05. Relativity and the Problem of Space `

` `

` Note: The fifth Appendix was added by Einstein at the time of the `

` fifteenth re-printing of this book; and as a result is still under `

` copyright restrictions so cannot be added without the permission of `

` the publisher. `

` `

` `

` `

` PREFACE `

` `

` (December, 1916) `

` `

` The present book is intended, as far as possible, to give an exact `

` insight into the theory of Relativity to those readers who, from a `

` general scientific and philosophical point of view, are interested in `

` the theory, but who are not conversant with the mathematical apparatus `

` of theoretical physics. The work presumes a standard of education `

` corresponding to that of a university matriculation examination, and, `

` despite the shortness of the book, a fair amount of patience and force `

` of will on the part of the reader. The author has spared himself no `

` pains in his endeavour to present the main ideas in the simplest and `

` most intelligible form, and on the whole, in the sequence and `

` connection in which they actually originated. In the interest of `

` clearness, it appeared to me inevitable that I should repeat myself `

` frequently, without paying the slightest attention to the elegance of `

` the presentation. I adhered scrupulously to the precept of that `

` brilliant theoretical physicist L. Boltzmann, according to whom `

` matters of elegance ought to be left to the tailor and to the cobbler. `

` I make no pretence of having withheld from the reader difficulties `

` which are inherent to the subject. On the other hand, I have purposely `

` treated the empirical physical foundations of the theory in a `

` "step-motherly" fashion, so that readers unfamiliar with physics may `

` not feel like the wanderer who was unable to see the forest for the `

` trees. May the book bring some one a few happy hours of suggestive `

` thought! `

` `

` December, 1916 `

` A. EINSTEIN `

` `

` `

` `

` PART I `

` `

` THE SPECIAL THEORY OF RELATIVITY `

` `

` 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 our 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 form 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 "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. `

` `

` It is not difficult to understand why, in spite of this, we feel `

` constrained to call the propositions of geometry "true." Geometrical `

` ideas correspond to more or less exact objects in nature, and these `

` last are undoubtedly the exclusive cause of the genesis of those `

` ideas. Geometry ought to refrain from such a course, in order to give `

` to its structure the largest possible logical unity. The practice, for `

` example, of seeing in a "distance" two marked positions on a `

` practically rigid body is something which is lodged deeply in our `

` habit of thought. We are accustomed further to regard three points as `

` being situated on a straight line, if their apparent positions can be `

` made to coincide for observation with one eye, under suitable choice `

` of our place of observation. `

` `

` If, in pursuance of our habit of thought, we now supplement the `

` propositions of Euclidean geometry by the single proposition that two `

` points on a practically rigid body always correspond to the same `

` distance (line-interval), independently of any changes in position to `

` which we may subject the body, the propositions of Euclidean geometry `

` then resolve themselves into propositions on the possible relative `

` position of practically rigid bodies.* Geometry which has been `

` supplemented in this way is then to be treated as a branch of physics. `

` We can now legitimately ask as to the "truth" of geometrical `

` propositions interpreted in this way, since we are justified in asking `

` whether these propositions are satisfied for those real things we have `

` associated with the geometrical ideas. In less exact terms we can `

` express this by saying that by the "truth" of a geometrical `

` proposition in this sense we understand its validity for a `

` construction with rule and compasses. `

` `

` Of course the conviction of the "truth" of geometrical propositions in `

` this sense is founded exclusively on rather incomplete experience. For `

` the present we shall assume the "truth" of the geometrical `

` propositions, then at a later stage (in the general theory of `

` relativity) we shall see that this "truth" is limited, and we shall `

` consider the extent of its limitation. `

` `

` `

` Notes `

` `

` *) It follows that a natural object is associated also with a `

` straight line. Three points A, B and C on a rigid body thus lie in a `

` straight line when the points A and C being given, B is chosen such `

` that the sum of the distances AB and BC is as short as possible. This `

` incomplete suggestion will suffice for the present purpose. `

` `

`