The Theory of Relativity: and Other Essays Read online

  The Theory of Relativity and Other Essays

  Albert Einstein

  Contents

  1. The Theory of Relativity (1949)

  2. E = MC2 (1946)

  3. Physics and Reality (1936)

  General Consideration Concerning the Method of Science

  Mechanics and the Attempt to Base all Physics Upon It

  The Field Concept

  The Theory of Relativity

  Quantum Theory and the Fundamentals of Physics

  Relative Theory and Corpuscles

  Summary

  4. The Fundamentals of Theoretical Physics (1940)

  5. The Common Language of Science (1941)

  6. The Laws of Science and the Laws of Ethics (1950)

  7. An Elementary Derivation of the Equivalence of Mass and Energy (1946)

  A Biography of Albert Einstein

  Acknowledgments

  1. The Theory of Relativity

  MATHEMATICS DEALS EXCLUSIVELY with the relations of concepts to each other without consideration of their relation to experience. Physics too deals with mathematical concepts; however, these concepts attain physical content only by the clear determination of their relation to the objects of experience. This in particular is the case for the concepts of motion, space, time.

  The theory of relativity is that physical theory which is based on a consistent physical interpretation of these three concepts. The name “theory of relativity” is connected with the fact that motion from the point of view of possible experience always appears as the relative motion of one object with respect to another (e.g., of a car with respect to the ground, or the earth with respect to the sun and the fixed stars). Motion is never observable as “motion with respect to space” or, as it has been expressed, as “absolute motion.” The “principle of relativity” in its widest sense is contained in the statement: The totality of physical phenomena is of such a character that it gives no basis for the introduction of the concept of “absolute motion”; or shorter but less precise: There is no absolute motion.

  It might seem that our insight would gain little from such a negative statement. In reality, however, it is a strong restriction for the (conceivable) laws of nature. In this sense there exists an analogy between the theory of relativity and thermodynamics. The latter too is based on a negative statement: “There exists no perpetuum mobile.”

  The development of the theory of relativity proceeded in two steps, “special theory of relativity” and “general theory of relativity.” The latter presumes the validity of the former as a limiting case and is its consistent continuation.

  A. Special theory of relativity.

  Physical interpretation of space and time in classical mechanics.

  Geometry, from a physical standpoint, is the totality of laws according to which rigid bodies mutually at rest can be placed with respect to each other (e.g., a triangle consists of three rods whose ends touch permanently). It is assumed that with such an interpretation the Euclidean laws are valid. “Space” in this interpretation is in principle an infinite rigid body (or skeleton) to which the position of all other bodies is related (body of reference). Analytic geometry (Descartes) uses as the body of reference, which represents space, three mutually perpendicular rigid rods on which the “coordinates” (x, y, z) of space points are measured in the known manner as perpendicular projections (with the aid of a rigid unit-measure).

  Physics deals with “events” in space and time. To each event belongs, besides its place coordinates x, y, z, a time value t. The latter was considered measurable by a clock (ideal periodic process) of negligible spatial extent. This clock C is to be considered at rest at one point of the coordinate system, e.g., at the coordinate origin (x = y = z = O). The time of an event taking place at a point P (x, y, z) is then defined as the time shown on the clock C simultaneously with the event. Here the concept “simultaneous” was assumed as physically meaningful without special definition. This is a lack of exactness which seems harmless only since with the help of light (whose velocity is practically infinite from the point of view of daily experience) the simultaneity of spatially distant events can apparently be decided immediately.

  The special theory of relativity removes this lack of precision by defining simultaneity physically with the use of light signals. The time t of the event in P is the reading of the clock C at the time of arrival of a light signal emitted from the event, corrected with respect to the time needed for the light signal to travel the distance. This correction presumes (postulates) that the velocity of light is constant.

  This definition reduces the concept of simultaneity of spatially distant events to that of the simultaneity of events happening at the same place (coincidence), namely the arrival of the light signal at C and the reading of C.

  Classical mechanics is based on Galileo’s principle: A body is in rectilinear and uniform motion as long as other bodies do not act on it. This statement cannot be valid for arbitrary moving systems of coordinates. It can claim validity only for so-called “inertial systems.” Inertial systems are in rectilinear and uniform motion with respect to each other. In classical physics laws claim validity only with respect to all inertial systems (special principle of relativity).

  It is now easy to understand the dilemma which has led to the special theory of relativity. Experience and theory have gradually led to the conviction that light in empty space always travels with the same velocity c independent of its color and the state of motion of the source of light (principle of the constancy of the velocity of light—in the following referred to as “L-principle”). Now elementary intuitive considerations seem to show that the same light ray cannot move with respect to all inertial systems with the same velocity c. The L-principle seems to contradict the special principle of relativity.

  It turns out, however, that this contradiction is only an apparent one which is based essentially on the prejudice about the absolute character of time or rather of the simultaneity of distant events. We just saw that x, y, z and t of an event can, for the moment, be defined only with respect to a certain chosen system of coordinates (inertial system). The transformation of the x, y, z, t of events which has to be carried out with the passage from one inertial system to another (coordinate transformation), is a problem which cannot be solved without special physical assumptions. However, the following postulate is exactly sufficient for a solution: The L-principle holds for all inertial systems (application of the special principle of relativity to the L-principle). The transformations thus defined, which are linear in x, y, z, t, are called Lorentz transformations. Lorentz transformations are formally characterized by the demand that the expression

  dx2 + dy2 + dz2 − c2dt2,

  which is formed from the coordinate differences dx, dy, dz, dt of two infinitely close events, be invariant (i.e., that through the transformation it goes over into the same expression formed from the coordinate differences in the new system).

  With the help of the Lorentz transformations the special principle of relativity can be expressed thus: The laws of nature are invariant with respect to Lorentz-transformations (i.e., a law of nature does not change its form if one introduces into it a new inertial system with the help of a Lorentz-transformation on x, y, z, t).

  The special theory of relativity has led to a clear understanding of the physical concepts of space and time and in connection with this to a recognition of the behavior of moving measuring rods and clocks. It has in principle removed the concept of absolute simultaneity and thereby also that of instantaneous action at a distance in the sense of Newton. It has shown how the law of motion must be modified in dealing with mo
tions that are not negligibly small as compared with the velocity of light. It has led to a formal clarification of Maxwell’s equations of the electromagnetic field; in particular it has led to an understanding of the essential oneness of the electric and the magnetic field. It has unified the laws of conservation of momentum and of energy into one single law and has demonstrated the equivalence of mass and energy. From a formal point of view one may characterize the achievement of the special theory of relativity thus: it has shown generally the role which the universal constant c (velocity of light) plays in the laws of nature and has demonstrated that there exists a close connection between the form in which time on the one hand and the spatial coordinates on the other hand enter into the laws of nature.

  B. General theory of relativity.

  The special theory of relativity retained the basis of classical mechanics in one fundamental point, namely the statement: The laws of nature are valid only with respect to inertial systems. The “permissible” transformations for the coordinates (i.e., those which leave the form of the laws unchanged) are exclusively the (linear) Lorentz-transformations. Is this restriction really founded in physical facts? The following argument convincingly denies it.

  Principle of equivalence. A body has an inertial mass (resistance to acceleration) and a heavy mass (which determines the weight of the body in a given gravitational field, e.g., that at the surface of the earth). These two quantities, so different according to their definition, are according to experience measured by one and the same number. There must be a deeper reason for this. The fact can also be described thus: In a gravitational field different masses receive the same acceleration. Finally, it can also be expressed thus: Bodies in a gravitational field behave as in the absence of a gravitational field if, in the latter case, the system of reference used is a uniformly accelerated coordinate system (instead of an inertial system).

  There seems, therefore, to be no reason to ban the following interpretation of the latter case. One considers the system as being “at rest” and considers the “apparent” gravitational field which exists with respect to it as a “real” one. This gravitational field “generated” by the acceleration of the coordinate system would of course be of unlimited extent in such a way that it could not be caused by gravitational masses in a finite region; however, if we are looking for a field-like theory, this fact need not deter us. With this interpretation the inertial system loses its meaning and one has an “explanation” for the equality of heavy and inertial mass (the same property of matter appears as weight or as inertia depending on the mode of description).

  Considered formally, the admission of a coordinate system which is accelerated with respect to the original “inertial” coordinates means the admission of non-linear coordinate transformations, hence a mighty enlargement of the idea of invariance, i.e., the principle of relativity.

  First, a penetrating discussion, using the results of the special theory of relativity, shows that with such a generalization the coordinates can no longer be interpreted directly as the results of measurements. Only the coordinate difference together with the field quantities which describe the gravitational field determine measurable distances between events. After one has found oneself forced to admit non-linear coordinate transformations as transformations between equivalent coordinate systems, the simplest demand appears to admit all continuous coordinate transformations (which form a group), i.e., to admit arbitrary curvilinear coordinate systems in which the fields are described by regular functions (general principle of relativity).

  Now it is not difficult to understand why the general principle of relativity (on the basis of the equivalence principle) has led to a theory of gravitation. There is a special kind of space whose physical structure (field) we can presume as precisely known on the basis of the special theory of relativity. This is empty space without electromagnetic field and without matter. It is completely determined by its “metric” property: Let dx0, dy0, dz0, dt0 be the coordinate differences of two infinitesimally near points (events); then

  (1) ds2 = dx02 + dy02 + dz02 − c2dt02

  is a measurable quality which is independent of the special choice of the inertial system. If one introduces in this space the new coordinates x1, x2, x3 x4 through a general transformation of coordinates, then the quantity ds2 for the same pair of points has an expression of the form

  (2) ds2 = Σgikdxidxk (summed for i and k from 1 to 4)

  where gik = gki The gik which form a “symmetric tensor” and are continuous functions of x1, … x4 then describe according to the “principle of equivalence” a gravitational field of a special kind (namely one which can be retransformed to the form [1]). From Riemann's investigations on metric spaces the mathematical properties of this gik field can be given exactly (“Riemann-condition”). However, what we are looking for are the equations satisfied by “general” gravitational fields. It is natural to assume that they too can be described as tensor-fields of the type gik, which in general do not admit a transformation to the form (1), i.e., which do not satisfy the “Riemann condition,” but weaker conditions, which, just as the Riemann condition, are independent of the choice of coordinates (i.e., are generally invariant). A simple formal consideration leads to weaker conditions which are closely connected with the condition. These conditions are the very equations of the pure gravitational field (on the outside of matter and at the absence of an electromagnetic field).

  These equations yield Newton’s equations of gravitational mechanics as an approximate law and in addition certain small effects which have been confirmed by observation (deflection of light by the gravitational field of a star, influence of the gravitational potential on the frequency of emitted light, slow rotation of the elliptic circuits of planets—perihelion motion of the planet Mercury). They further yield an explanation for the expanding motion of galactic systems, which is manifested by the red-shift of the light omitted from these systems.

  The general theory of relativity is as yet incomplete insofar as it has been able to apply the general principle of relativity satisfactorily only to gravitational fields, but not to the total field. We do not yet know with certainty, by what mathematical mechanism the total field in space is to be described and what the general invariant laws are to which this total field is subject. One thing, however, seems certain: namely, that the general principle of relativity will prove a necessary and effective tool for the solution of the problem of the total field.

  2. E = MC2 (1946)

  IN ORDER TO UNDERSTAND the law of the equivalence of mass and energy, we must go back to two conservation or “balance” principles which, independent of each other, held a high place in pre-relativity physics. These were the principle of the conservation of energy and the principle of the conservation of mass. The first of these, advanced by Leibnitz as long ago as the seventeenth century, was developed in the nineteenth century essentially as a corollary of a principle of mechanics.

  Drawing from Dr. Einstein’s manuscript.

  Consider, for example, a pendulum whose mass swings back and forth between the points A and B. At these points the mass m is higher by the amount h than it is at C, the lowest point of the path (see drawing). At C, on the other hand, the lifting height has disappeared and instead of it the mass has a velocity v. It is as though the lifting height could be converted entirely into velocity, and vice versa. The exact relation would be expressed as with g representing the acceleration of gravity. What is interesting here is that this relation is independent of both the length of the pendulum and the form of the path through which the mass moves.

  The significance is that something remains constant throughout the process, and that something is energy. At A and at B it is an energy of position, or “potential” energy; at C it is an energy of motion, or “kinetic” energy. If this concept is correct, then the sum must have the same value for any position of the pendulum, if h is understood to represent the height above C, and v the velocity at that point in the pe
ndulum's path. And such is found to be actually the case. The generalization of this principle gives us the law of the conservation of mechanical energy. But what happens when friction stops the pendulum?

  The answer to that was found in the study of heat phenomena. This study, based on the assumption that heat is an indestructible substance which flows from a warmer to a colder object, seemed to give us a principle of the “conservation of heat.” On the other hand, from time immemorial it has been known that heat could be produced by friction, as in the fire-making drills of the Indians. The physicists were for long unable to account for this kind of heat “production.” Their difficulties were overcome only when it was successfully established that, for any given amount of heat produced by friction, an exactly proportional amount of energy had to be expended. Thus did we arrive at a principle of the “equivalence of work and heat.” With our pendulum, for example, mechanical energy is gradually converted by friction into heat.

  In such fashion the principles of the conservation of mechanical and thermal energies were merged into one. The physicists were thereupon persuaded that the conservation principle could be further extended to take in chemical and electromagnetic processes—in short, could be applied to all fields. It appeared that in our physical system there was a sum total of energies that remained constant through all changes that might occur.

  Now for the principle of the conservation of mass. Mass is defined by the resistance that a body opposes to its acceleration (inert mass). It is also measured by the weight of the body (heavy mass). That these two radically different definitions lead to the same value for the mass of a body is, in itself, an astonishing fact. According to the principle—namely, that masses remain unchanged under any physical or chemical changes—the mass appeared to be the essential (because unvarying) quality of matter. Heating, melting, vaporization, or combining into chemical compounds would not change the total mass.