The Theory of Relativity and Other Essays ? Albert Einstein ?
? Contents ? 1. The Theory of Relativity (1949) ?
2. E = MC 2 (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 withoutconsideration of their relation to experience. Physics too deals with mathematical concepts;however, these concepts attain physical content only by the clear determination of theirrelation to the objects of experience. This in particular is the case for the concepts ofmotion, space, time.
The theory of relativity is that physical theory which is based on a consistent physicalinterpretation of these three concepts. The name “theory of relativity” is connected with the
relativefact that motion from the point of view of possible experience always appears as the motion of one object with respect to another (e.g., of a car with respect to the ground, or theearth with respect to the sun and the fixed stars). Motion is never observable as “motion withrespect to space” or, as it has been expressed, as “absolute motion.” The “principle ofrelativity” in its widest sense is contained in the statement: The totality of physicalphenomena 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 thereexists an analogy between the theory of relativity and thermodynamics. The latter too is basedon a negative statement: “There exists no perpetuum mobile.”
The development of the theory of relativity proceeded in two steps, “special theory ofrelativity” and “general theory of relativity.” The latter presumes the validity of theformer 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 bodiesmutually at rest can be placed with respect to each other (e.g., a triangle consists of threerods whose ends touch permanently). It is assumed that with such an interpretation theEuclidean laws are valid. “Space” in this interpretation is in principle an infinite rigidbody (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, threemutually perpendicular rigid rods on which the “coordinates” (x, y, z) of space points aremeasured 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 placecoordinates x, y, z, a time value t. The latter was considered measurable by a clock (idealperiodic process) of negligible spatial extent. This clock C is to be considered at rest at onepoint of the coordinate system, e.g., at the coordinate origin (x = y = z = O). The time of anevent taking place at a point P (x, y, z) is then defined as the time shown on the clock Csimultaneously with the event. Here the concept “simultaneous” was assumed as physicallymeaningful without special definition. This is a lack of exactness which seems harmless onlysince with the help of light (whose velocity is practically infinite from the point of view ofdaily experience) the simultaneity of spatially distant events can apparently be decidedimmediately.
The special theory of relativity removes this lack of precision by defining simultaneityphysically with the use of light signals. The time t of the event in P is the reading of theclock C at the time of arrival of a light signal emitted from the event, corrected with respectto 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 thesimultaneity of events happening at the same place (coincidence), namely the arrival of thelight signal at C and the reading of C.
Classical mechanics is based on Galileo’s principle: A body is in rectilinear and uniformmotion as long as other bodies do not act on it. This statement cannot be valid for arbitrarymoving 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 classicalphysics laws claim validity only with respect to all inertial systems (special principle ofrelativity).
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 alwaystravels with the same velocity c independent of its color and the state of motion of the sourceof 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 raycannot move with respect to all inertial systems with the same velocity c. The L-principleseems to contradict the special principle of relativity.
It turns out, however, that this contradiction is only an apparent one which is basedessentially on the prejudice about the absolute character of time or rather of the simultaneityof distant events. We just saw that x, y, z and t of an event can, for the moment, be definedonly with respect to a certain chosen system of coordinates (inertial system). Thetransformation of the x, y, z, t of events which has to be carried out with the passage fromone inertial system to another (coordinate transformation), is a problem which cannot be solvedwithout special physical assumptions. However, the following postulate is exactly sufficientfor 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 linearin x, y, z, t, are called Lorentz transformations. Lorentz transformations are formallycharacterized by the demand that the expression
dx 2 + dy 2 + dz 2 c 2 dt 2 ,
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 beexpressed 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 withthe 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 ofspace and time and in connection with this to a recognition of the behavior of moving measuringrods and clocks. It has in principle removed the concept of absolute simultaneity and therebyalso that of instantaneous action at a distance in the sense of Newton. It has shown how thelaw of motion must be modified in dealing with motions that are not negligibly small ascompared with the velocity of light. It has led to a formal clarification of Maxwell’sequations of the electromagnetic field; in particular it has led to an understanding of theessential oneness of the electric and the magnetic field. It has unified the laws ofconservation of momentum and of energy into one single law and has demonstrated the equivalenceof mass and energy. From a formal point of view one may characterize the achievement of thespecial theory of relativity thus: it has shown generally the role which the universal constantc (velocity of light) plays in the laws of nature and has demonstrated that there exists aclose connection between the form in which time on the one hand and the spatial coordinates onthe other hand enter into the laws of nature.
General theory of relativity.B.
The special theory of relativity retained the basis of classical mechanics in one fundamentalpoint, namely the statement: The laws of nature are valid only with respect to inertialsystems. The “permissible” transformations for the coordinates (i.e., those which leave theform 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 heavymass (which determines the weight of the body in a given gravitational field, e.g., that at thesurface of the earth). These two quantities, so different according to their definition, areaccording to experience measured by one and the same number. There must be a deeper reason forthis. The fact can also be described thus: In a gravitational field different masses receivethe same acceleration. Finally, it can also be expressed thus: Bodies in a gravitational fieldbehave as in the absence of a gravitational field if, in the latter case, the system ofreference 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” gravitationalfield 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 away that it could not be caused by gravitational masses in a finite region; however, if we arelooking for a field-like theory, this fact need not deter us. With this interpretation theinertial system loses its meaning and one has an “explanation” for the equality of heavy andinertial mass (the same property of matter appears as weight or as inertia depending on themode of description).
Considered formally, the admission of a coordinate system which is accelerated with respect tothe original “inertial” coordinates means the admission of non-linear coordinatetransformations, hence a mighty enlargement of the idea of invariance, i.e., the principle ofrelativity.
First, a penetrating discussion, using the results of the special theory of relativity, showsthat with such a generalization the coordinates can no longer be interpreted directly as theresults of measurements. Only the coordinate difference together with the field quantitieswhich describe the gravitational field determine measurable distances between events. After onehas found oneself forced to admit non-linear coordinate transformations as transformationsbetween equivalent coordinate systems, the simplest demand appears to admit all continuouscoordinate transformations (which form a group), i.e., to admit arbitrary curvilinearcoordinate systems in which the fields are described by regular functions (general principle ofrelativity).
Now it is not difficult to understand why the general principle of relativity (on the basis of
) has led to a theory of gravitation. There is a special kind ofthe equivalence principle
space whose physical structure (field) we can presume as precisely known on the basis of thespecial theory of relativity. This is empty space without electromagnetic field and withoutmatter. It is completely determined by its “metric” property: Let dx 0 , dy 0 , dz 0 , dt 0 be the coordinate differences of two infinitesimally near points (events); then
(1)??????????????????????ds 2 = dx 0 2 + dy 0 2 + dz 0 2 c 2 dt 0 2
is a measurable quality which is independent of the special choice of the inertial system. Ifone introduces in this space the new coordinates x1, x2, x3 x4 through a general transformationof coordinates, then the quantity ds2 for the same pair of points has an expression of the form
(2)??????????????????????ds 2 = Σg ik dx i dx k (summed for i and k from 1 to 4)
where g ik = g ki The g ik which form a “symmetric tensor” and are continuous functions ofx 1 , … x 4 then describe according to the “principle of equivalence” a gravitational fieldof a special kind (namely one which can be retransformed to the form ). From Riemann'sinvestigations on metric spaces the mathematical properties of this g ik field can be givenexactly (“Riemann-condition”). However, what we are looking for are the equations satisfiedby “general” gravitational fields. It is natural to assume that they too can be described as
admit a transformation to the form (1),nottensor-fields of the type g ik, which in general do
i.e., which do not satisfy the “Riemann condition,” but weaker conditions, which, just as theRiemann condition, are independent of the choice of coordinates (i.e., are generallyinvariant). A simple formal consideration leads to weaker conditions which are closelyconnected with the condition. These conditions are the very equations of the pure gravitationalfield (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 andin addition certain small effects which have been confirmed by observation (deflection of lightby the gravitational field of a star, influence of the gravitational potential on the frequencyof emitted light, slow rotation of the elliptic circuits of planets—perihelion motion of theplanet Mercury). They further yield an explanation for the expanding motion of galacticsystems, 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 thegeneral principle of relativity satisfactorily only to gravitational fields, but not to thetotal field. We do not yet know with certainty, by what mathematical mechanism the total fieldin space is to be described and what the general invariant laws are to which this total fieldis subject. One thing, however, seems certain: namely, that the general principle of relativitywill prove a necessary and effective tool for the solution of the problem of the total field.
2. E = MC 2 (1946)
IN ORDER TO UNDERSTAND the law of the equivalence of mass and energy, we must go back to twoconservation or “balance” principles which, independent of each other, held a high place inpre-relativity physics. These were the principle of the conservation of energy and theprinciple of the conservation of mass. The first of these, advanced by Leibnitz as long ago asthe seventeenth century, was developed in the nineteenth century essentially as a corollary ofa 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 thepath (see drawing). At C, on the other hand, the lifting height has disappeared and instead ofit the mass has a velocity v. It is as though the lifting height could be converted entirelyinto velocity, and vice versa. The exact relation would be expressed as with g representingthe acceleration of gravity. What is interesting here is that this relation is independent ofboth 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 somethingis energy. At A and at B it is an energy of position, or “potential” energy; at C it is anenergy of motion, or “kinetic” energy. If this concept is correct, then the sum must havethe same value for any position of the pendulum, if h is understood to represent the heightabove C, and v the velocity at that point in the pendulum's path. And such is found to beactually the case. The generalization of this principle gives us the law of the conservation ofmechanical 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 theassumption that heat is an indestructible substance which flows from a warmer to a colderobject, seemed to give us a principle of the “conservation of heat.” On the other hand, fromtime 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 ofheat “production.” Their difficulties were overcome only when it was successfully establishedthat, for any given amount of heat produced by friction, an exactly proportional amount ofenergy had to be expended. Thus did we arrive at a principle of the “equivalence of work andheat.” With our pendulum, for example, mechanical energy is gradually converted by frictioninto heat.
In such fashion the principles of the conservation of mechanical and thermal energies weremerged into one. The physicists were thereupon persuaded that the conservation principle couldbe further extended to take in chemical and electromagnetic processes—in short, could beapplied 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 abody 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 themass of a body is, in itself, an astonishing fact. According to the principle—namely, thatmasses remain unchanged under any physical or chemical changes—the mass appeared to be theessential (because unvarying) quality of matter. Heating, melting, vaporization, or combininginto chemical compounds would not change the total mass.
Physicists accepted this principle up to a few decades ago. But it proved inadequate in theface of the special theory of relativity. It was therefore merged with the energyprinciple—just as, about 60 years before, the principle of the conservation of mechanicalenergy had been combined with the principle of the conservation of heat. We might say that theprinciple of the conservation of energy, having previously swallowed up that of theconservation of heat, now proceeded to swallow that of the conservation of mass—and holds thefield alone.
It is customary to express the equivalence of mass and energy (though somewhat inexactly) bythe formula E = mc 2 , in which c represents the velocity of light, about 186,000 miles persecond. E is the energy that is contained in a stationary body; m is its mass. The energy thatbelongs to the mass m is equal to this mass, multiplied by the square of the enormous speed oflight—which is to say, a vast amount of energy for every unit of mass.
But if every gram of material contains this tremendous energy, why did it go so long unnoticed?The answer is simple enough: so long as none of the energy is given off externally, it cannotbe observed. It is as though a man who is fabulously rich should never spend or give away acent; no one could tell how rich he was.
Now we can reverse the relation and say that an increase of E in the amount of energy must beaccompanied by an increase of in the mass. I can easily supply energy to the mass—forinstance, if I heat it by 10 degrees. So why not measure the mass increase, or weight increase,connected with this change? The trouble here is that in the mass increase the enormous factor c
2 occurs in the denominator of the fraction. In such a case the increase is too small to bemeasured directly; even with the most sensitive balance.
For a mass increase to be measurable, the change of energy per mass unit must be enormouslylarge. We know of only one sphere in which such amounts of energy per mass unit are released:namely, radioactive disintegration. Schematically, the process goes like this: An atom of themass M splits into two atoms of the mass M′ and M″, which separate with tremendous kineticenergy. If we imagine these two masses as brought to rest—that is, if we take this energy ofmotion from them—then, considered together, they are essentially poorer in energy than was theoriginal atom. According to the equivalence principle, the mass sum M′ + M″ of thedisintegration products must also be somewhat smaller than the original mass M of thedisintegrating atom—in contradiction to the old principle of the conservation of mass. Therelative difference of the two is on the order of of one percent.
Now, we cannot actually weigh the atoms individually. However, there are indirect methods formeasuring their weights exactly. We can likewise determine the kinetic energies that aretransferred to the disintegration products M′ and M″. Thus it has become possible to test andconfirm the equivalence formula. Also, the law permits us to calculate in advance, fromprecisely determined atom weights, just how much energy will be released with any atomdisintegration we have in mind. The law says nothing, of course, as to whether—or how—thedisintegration reaction can be brought about.
What takes place can be illustrated with the help of our rich man. The atom M is a rich miser
energy). But in his will he bequeaths his fortune towho, during his life, gives away no money (
his sons M′ and M″, on condition that they give to the community a small amount, less thanone thousandth of the whole estate (energy or mass). The sons together have somewhat less than
the father had (the mass sum M′ + M″ is somewhat smaller than the mass M of the radioactive
). But the part given to the community, though relatively small, is still so enormouslyatom
large (considered as kinetic energy) that it brings with it a great threat of evil. Avertingthat threat has become the most urgent problem of our time.
3. Physics and Reality
? 1. General Consideration Concerning the Method of Science
IT HAS OFTEN been said, and certainly not without justification, that the man of science is apoor philosopher. Why then should it not be the right thing for the physicist to let thephilosopher do the philosophizing? Such might indeed be the right thing at a time when thephysicist believes he has at his disposal a rigid system of fundamental concepts andfundamental laws which are so well established that waves of doubt can not reach them; but, itcan not be right at a time when the very foundations of physics itself have become problematicas they are now. At a time like the present, when experience forces us to seek a newer and moresolid foundation, the physicist cannot simply surrender to the philosopher the criticalcontemplation of the theoretical foundations; for, he himself knows best, and feels more surelywhere the shoe pinches. In looking for a new foundation, he must try to make clear in his ownmind just how far the concepts which he uses are justified, and are necessities.
The whole of science is nothing more than a refinement of everyday thinking. It is for thisreason that the critical thinking of the physicist cannot possibly be restricted to theexamination of the concepts of his own specific field. He cannot proceed without consideringcritically a much more difficult problem, the problem of analyzing the nature of everydaythinking.
On the stage of our subconscious mind appear in colorful succession sense experiences, memorypictures of them, representations and feelings. In contrast to psychology, physics treatsdirectly only of sense experiences and of the “understanding” of their connection. But eventhe concept of the “real external world” of everyday thinking rests exclusively on senseimpressions.
Now we must first remark that the differentiation between sense impressions and representationsis not possible; or, at least it is not possible with absolute certainty. With the discussionof this problem, which affects also the notion of reality, we will not concern ourselves but weshall take the existence of sense experiences as given, that is to say as psychic experiencesof special kind.
I believe that the first step in the setting of a “real external world” is the formation ofthe concept of bodily objects and of bodily objects of various kinds. Out of the multitude ofour sense experiences we take, mentally and arbitrarily, certain repeatedly occurring complexesof sense impression (partly in conjunction with sense impressions which are interpreted assigns for sense experiences of others), and we attribute to them a meaning—the meaning of thebodily object. Considered logically this concept is not identical with the totality of senseimpressions referred to; but it is an arbitrary creation of the human (or animal) mind. On theother hand, the concept owes its meaning and its justification exclusively to the totality ofthe sense impressions which we associate with it.
The second step is to be found in the fact that, in our thinking (which determines ourexpectation), we attribute to this concept of the bodily object a significance, which is to ahigh degree independent of the sense impression which originally gives rise to it. This is whatwe mean when we attribute to the bodily object “a real existence.” The justification of sucha setting rests exclusively on the fact that, by means of such concepts and mental relationsbetween them, we are able to orient ourselves in the labyrinth of sense impressions. Thesenotions and relations, although free statements of our thoughts, appear to us as stronger andmore unalterable than the individual sense experience itself, the character of which asanything other than the result of an illusion or hallucination is never completely guaranteed.On the other hand, these concepts and relations, and indeed the setting of real objects and,generally speaking, the existence of “the real world,” have justification only in so far asthey are connected with sense impressions between which they form a mental connection.