The foundation of the generalized theory of relativity E-Book

Einstein Albert

• Herausgeber: Balungi Francis
• Kategorie: Lebensstil
• Sprache: Englisch
• Veröffentlichungsjahr: 2020

Beschreibung

The theory which is sketched in the following pages forms the most wide-going generalization conceivable of what is at present known as "the theory of Relativity;" this latter theory I differentiate from the former "Special Relativity theory," and suppose it to be known. The generalization of the Relativity theory has been made much easier through the form given to the special Relativity theory by Minkowski, which mathematician was the first to recognize clearly the formal equivalence of the space like and time-like co-ordinates, and who made use of it in the building up of the theory. The mathematical apparatus useful for the general relativity theory, lay already complete in the "Absolute Differential Calculus", which were based on the researches of Gauss, Riemann and Christoffel on the non-Euclidean manifold, and which have been shaped into a system by Ricci and Levi-Civita, and already applied to the problems of theoretical physics. I have in part B of this communication developed in the simplest and clearest manner, all the supposed mathematical auxiliaries, not known to Physicists, which will be useful for our purpose, so that, a study of the mathematical literature is not necessary for an understanding of this paper. Finally in this place I thank my friend Grossmann, by whose help I was not only spared the study of the mathematical literature pertinent to this subject, but who also aided me in the researches on the field equations of gravitation.

Leseprobe

The Foundation of the Generalized Theory of Relativity

By ALBERT EINSTEIN

Contents

Preface

A. Principal considerations about the Postulate of Relativity.

§ 1. Remarks on the Special Relativity Theory.

§ 2. About the reasons which explain the extension of the relativity-postulate.

§ 3. The time-space continuum. Requirements of the general Co-variance for the equations expressing the laws of Nature in general.

§ 4. Relation of four co-ordinates to spatial and temporal measurements. Analytical expression for the Gravitation-field.

B. Mathematical Auxiliaries for Establishing the General Covariant Equations.

§ 5. Contravariant and covariant Four-vector.

§ 6. Tensors of the second and higher ranks.

§ 7. Multiplication of Tensors.

§ 8. A few words about the Fundamental Tensor .

§ 9. Equation of the geodetic line (or of point-motion).

§ 10. Formation of Tensors through Differentiation.

§ 11. Some special cases of Particular Importance.

C. The Theory of the Gravitation-Field

§ 13. Equation of motion of a material point in a gravitation-field. Expression for the field-components of gravitation.

§ 14. The Field-equation of Gravitation in the absence of matter.

§ 15. Hamiltonian Function for the Gravitation-field. Laws of Impulse and Energy.

§ 16. General formulation of the field-equation of Gravitation.

§ 17. The laws of conservation in the general case.

§ 18. The Impulse-energy law for matter as a consequence of the field-equations.

D. The "Material" Phenomena.

§ 19. Euler's equations for frictionless adiabatic liquid.

§ 20. Maxwell's Electro-Magnetic field-equations for the vacuum.

E. § 21. Newton's Theory as a First Approximation.

§ 22. Behaviour of measuring rods and clocks in a statical gravitation-field. Curvature of light-rays. Perihelion-motion of the paths of the Planets.

Preface

The theory which is sketched in the following pages forms the most wide-going generalization conceivable of what is at present known as "the theory of Relativity;" this latter theory I differentiate from the former "Special Relativity theory," and suppose it to be known. The generalization of the Relativity theory has been made much easier through the form given to the special Relativity theory by Minkowski, which mathematician was the first to recognize clearly the formal equivalence of the space like and time-like co-ordinates, and who made use of it in the building up of the theory. The mathematical apparatus useful for the general relativity theory, lay already complete in the "Absolute Differential Calculus", which were based on the researches of Gauss, Riemann and Christoffel on the non-Euclidean manifold, and which have been shaped into a system by Ricci and Levi-Civita, and already applied to the problems of theoretical physics. I have in part B of this communication developed in the simplest and clearest manner, all the supposed mathematical auxiliaries, not known to Physicists, which will be useful for our purpose, so that, a study of the mathematical literature is not necessary for an understanding of this paper. Finally in this place I thank my friend Grossmann, by whose help I was not only spared the study of the mathematical literature pertinent to this subject, but who also aided me in the researches on the field equations of gravitation. [ 770 ]

A. Principal considerations about the Postulate of Relativity.

§ 1. Remarks on the Special Relativity Theory.

The special relativity theory rests on the following postulate which also holds valid for the Galileo-Newtonian mechanics.

If a co-ordinate system K be so chosen that when referred to it the physical laws hold in their simplest forms, these laws would be also valid when referred to another system of co-ordinates K′ which is subjected to an uniform translational motion relative to K. We call this postulate "The Special Relativity Principle". By the word special, it is signified that the principle is limited to the case, when K′ has uniform translatory motion with reference to K, but the equivalence of K and K′ does not extend to the case of non-uniform motion of K′ relative to K.

The Special Relativity Theory does not differ from the classical mechanics through the assumption of this postulate, but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the special relativity postulate, gives in a well-known way, the relativity of synchronism as well as the Lorentz transformation, with all the relations between moving rigid bodies and clocks.

The modification which the theory of space and time has undergone through the special relativity theory, is indeed a profound one, but a weightier point remains untouched. According to the special relativity theory, the theorems of geometry are to be looked upon as the laws about any possible relative positions of solid bodies at rest, and more generally the theorems of kinematics, as theorems which describe the relation between measurable bodies and clocks. Consider two material points of a solid body at rest; then according to these conceptions there corresponds to these points a wholly definite extent of length, independent of kind, position, orientation and time of the body.

Similarly let us consider two positions of the pointers of a clock which is at rest with reference to a co-ordinate system; then to these positions, there always corresponds a time-interval of a definite length, independent of time and place. It would be soon shown that the general relativity theory cannot hold fast to this simple physical significance of space and time. [ 771 ]

§ 2. About the reasons which explain the extension of the relativity-postulate.

To the classical mechanics (no less than) to the special relativity theory, is attached an epistemological defect, which was perhaps first clearly pointed out by E. Mach. We shall illustrate it by the following example; Let two fluid bodies of equal kind and magnitude swim freely in space at such a great distance from one another (and from all other masses) that only that sort of gravitational forces are to be taken into account which the parts of any of these bodies exert upon each other. The distance of the bodies from one another is invariable. The relative motion of the different parts of each body is not to occur. But each mass is seen to rotate by an observer at rest relative to the other mass round the connecting line of the masses with a constant angular velocity (definite relative motion for both the masses). Now let us think that the surfaces of both the bodies ( and ) are measured with the help of measuring rods (relatively at rest); it is then found that the surface of is a sphere and the surface of the other is an ellipsoid of rotation. We now ask, why is this difference between the two bodies? An answer to this question can only then be regarded as satisfactory[1] from the epistemological standpoint when the thing adduced as the cause is an observable fact of experience. The law of causality has the sense of a definite statement about the world of experience only when observable facts alone appear as causes and effects.

The Newtonian mechanics does not give to this question any satisfactory answer. For example, it says:— The laws of mechanics hold true for a space relative to which the body is at rest, not however for a space relative to which is at rest.

The Galiliean space, which is here introduced is however only a purely imaginary cause, not an observable thing. It is thus clear that the Newtonian mechanics [ 772 ] does not, in the case treated here, actually fulfil the requirements of causality, but produces on the mind a fictitious complacency, in that it makes responsible a wholly imaginary cause for the different behaviours of the bodies and which are actually observable.

A satisfactory explanation to the question put forward above can only be thus given:— that the physical system composed of and shows for itself alone no conceivable cause to which the different behaviour of and can be attributed. The cause must thus lie outside the system. We are therefore led to the conception that the general laws of motion which determine specially the forms of and must be of such a kind, that the mechanical behaviour of and must be essentially conditioned by the distant masses, which we had not brought into the system considered. These distant masses, (and their relative motion as regards the bodies under consideration) are then to be looked upon as the seat of the principal observable causes for the different behaviours of the bodies under consideration. They take the place of the imaginary cause . Among all the conceivable spaces and etc. moving in any manner relative to one another, there is a priori, no one set which can be regarded as affording greater advantages, against which the objection which was already raised from the standpoint of the theory of knowledge cannot be again revived. The laws of physics must be so constituted that they should remain valid for any system of co-ordinates moving in any manner. We thus arrive at an extension of the relativity postulate.

Besides this momentous epistemological argument, there is also a well-known physical fact which speaks in favour of an extension of the relativity theory. Let there be a Galiliean co-ordinate system K relative to which (at least in the four-dimensional region considered) a mass at a sufficient distance from other masses move uniformly in a line. Let be a second co-ordinate system which has a uniformly accelerated motion relative to K. Relative to any mass at a sufficiently great distance experiences an accelerated motion such that its acceleration and its direction of acceleration is independent of its material composition and its physical conditions.

Can any observer, at rest relative to , [ 773 ] then conclude that he is in an actually accelerated reference-system? This is to be answered in the negative; the above-named behaviour of the freely moving masses relative to can be explained in as good a manner in the following way. The reference-system has no acceleration. In the space-time region considered there is a gravitation-field which generates the accelerated motion relative to .