SPACE, TIME AND ELEMENTARYINTERACTIONS IN RELATIVITYTo express in a unified formalism all the interactions of matter,ranging from the elementary particles to astronomical bodies, we shouldbegin with a simple study of the concepts of Einstein's space-time.

MENDEL SACHS

A REASONABLE COURSE of investigationin fundamental physics should fullyexploit Einstein's approach to themeaning of space-time; it would pro-ceed by exploring, in a unified way,the predictions of generalized equa-tions of general relativity that wouldencompass in one formalism all of thedomains of interaction—from fermis tolight years.

SEARCH FOR SIMPLICITY

When the theory of relativity was firstproposed and had attained some suc-cess in its initial predictions, a rumorstarted to circulate that throughout theworld only a very small number ofmen actually understood this theory.The general impression was therebycreated that this was so profound anapproach to the laws of nature thatmost ordinary scientists should noteven attempt to understand the foun-dations of this theory. But this im-pression in the scientific communitywas incompatible with one of Ein-stein's prime motivations in his inves-tigation. For he was very stronglyguided in this study, and indeedthroughout his scientific career, by theheuristic value of simplicity.

Around the same time that relativitytheory started to develop, the otherrevolution in 20th-century physics—the quantum theory—was discovered toreveal a formalism that explained

atomic phenomena. This theory even-tually developed into its present formof a nondeterministic theory of mea-surement. It is my contention that thequantum theory is much more difficultto understand from the conceptualpoint of view than is the theory of rela-tivity. It is partly for this reason thatsome of the original founders of thequantum theory (in its old form),such as Max Planck and Einstein, andlater Erwin Schrodinger, could not ac-cept the notions that were proposedby the Copenhagen school to underliethe fundamental description of matter.On the other hand, the mathematicalexpression of the new quantum theory—its equations and the rules for relat-ing their solutions to observables—isfar more simple than is the generalmathematical expression of the theoryof relativity.

We see here that a distinction mustbe made between conceptual simplic-ity and mathematical simplicity.When Einstein expressed the beliefthat "God may be subtle but He is notmalicious," I believe that he was re-ferring to his faith in the simplicityof the conceptual content of the lawsof nature. On the other hand, if manhas thus far been unable to formulateconceptually simple, though subtle,natural laws in equally simple lan-guage, this is not God's fault! It ap-pears to me that this inability is ratherdue to the relatively primitive stage of

man's intelligence. I should like, then,to concentrate here on the conceptualcontent of relativity theory, as I see itand without writing down a singleequation, to see what this approachimplies with regard to the propertiesof nature.

A REVOLUTIONARY CONCEPT

It is commonly admitted that thetheory of relativity brought a revolu-tion in ideas to the 20th century.Nevertheless, there seems to be somecontroversy among physicists on pre-cisely what it is that is revolutionary

Mendel Sachs, who took his bachelor's,master's and doctor's degrees in phys-ics at UCLA, is now a theoreticalphysicist at the State University of NewYork at Buffalo. A member of the edi-torial board of the International Journalof Theoretical Physics, he has workedrecently on the physical implications inthe elementary-particle domain of gen-eralizations of general-relativity theory.

PHYSICS TODAY FEBRUARY 1969 • 51

here. My contention is that Einstein'soriginal intention in the change thathe instigated had to do with an en-tirely new view of space and time.That is, except for the argumentationof some of the preceding philosophers,Einstein gave a view of space-timethat was different from all the previousviews in physical theories. Thischange, in turn, implied an alterationof the form of the laws of nature to amore generalized expression that couldbe tested by experimentation.

Pre-Einsteinian relativity

What, then, is the actual difference inthe view of space and time betweenEinstein's approach and that of hispredecessors (and conservative con-temporaries)? Consider first the ear-lier theories. In all the previous in-terpretations of the physical universe,it was assumed that space and timeare just there, once and for all, inde-pendent of any matter content. Mat-ter is then supposed to be in the pre-existing space and time, as a childmight be on a jungle gym in the play-ground. One then describes how mat-ter interacts with matter in terms of itslocation in the fixed space-time. Inphilosophical terms, this descriptionmay be restated as the view thatspace-time has objective significance,that it is a thing in itself.

It is important at this point to notethat the relativity of coordinates withinan underlying fixed space-time in thedescription of the natural laws is nota new idea and that the idea of rela-tivity in this sense is really not such arevolutionary concept. Consider, forexample, the physical laws before thetime of Galileo. It would have beenclaimed by most in that period that"up" is "up" to anyone-independent

SPEED OF LIGHT

In my opinion it is not necessary toassert the universality of the speedof light as a separate postulate, asis conventional in special-relativitytheory. The assertion appearsrather to follow as a logical conclu-sion of the single underlying postu-late of this theory—the principle otrelativity (page 53)—as soon as thecoordinates of relatively moving sys-tems are defined in a way that as-signs the temporal as well as spatialparameters to the language of oneobserver relative to the others whoare comparing their respective de-ductions about the form of the lawsof nature.

of his position or state of motion rela-tive to anyone else. Similarly it wouldhave been claimed that "before,""present" and "after" have absolutemeaning in the sense of being inde-pendent of the condition of motion orthe relative position of any observer.On the other hand, it would have beengenerally agreed that in the descrip-tion of a physical law the location of apoint in a two-dimensional plane isonly significant in relation to someother points in the plane-that is, phys-ical laws depend on distances be-tween things. For example, everyoneexperienced the diminishing heat ef-fect of a flame as he correspondinglyincreased his distance from the loca-tion of the fire.

Then a time came, with the appear-ance of Copernicus and Galileo, whenthe natural philosopher was forced toincorporate the "up" and the "down"with the surface coordinates to yield athree-dimensional space where, forexample, "up" to one observer mightbe "down," "sideways" or "oblique"to another observer of the same phe-nomena. Finally, more than 300 yearsafter Galileo, Einstein recognized thatthe time coordinate must also neces-sarily be incorporated with the threespatial cobdinates in one relativisticspace-time to describe the laws of na-ture correctly. This discovery wasmade when Einstein recognized thatthe laws of electrodynamics, in theform that they were originally discov-ered by Michael Faraday and JamesClerk Maxwell, have the same form toall observers, independent of theirstates of motion relative to each other,only if it can be assumed that the timecoordinate is included with the spatialcoordinates in the relativistic sensethat to describe an observation, all fourcoordinates must be specified relativeonly to the space-time point of a givenobserver. At this stage, the theory ofspecial relativity was born.

Unification of space-time

Incorporation of the time coordinatewith the spatial coordinates impliedsome interesting consequences, notpredicted previously. First, to be ableto convert the time coordinate of oneobserver to a combination of time andspace coordinates of another (who isin motion relative to the first) onemust express time and length in thesame units—in all frames of reference.Thus the time coordinate had to bemultiplied by a universal constant withthe dimension of length divided by

GALILEO

time—a constant speed. Indeed, theidentification of the laws of electrody-namics with this number revealed it tobe the speed of propagation of theelectromagnetic interaction betweenmatter and matter in a vacuum-nu-merically determined from the speedof light in a vacuum. This determina-tion of the universal speed c then fixedit for all other applications (see boxon this page). A further implicationhere was that the speed of light in avacuum is independent of the speedof its source. That is, according tothis theory, if light is emitted from amoving train or from a fixed positionnext to the tracks, a fixed observerwill measure the same speed in eachcase.

Another new consequence of rela-tivity theory was that no longer couldwe speak of the absolute simultaneityof events. If two events are simul-taneous to one observer, they are notgenerally so to a different observer inmotion relative to the first one. Athird interesting consequence of spe-cial relativity came from the modifica-tion that results when we pass fromclassical mechanics of point particles(where time is absolute) to relativis-tic mechanics of point particles (wherethe time, along with the spatial coor-dinates, is relative to the point of ob-servation). It turns out in this gen-eralization that although the inertialmass of an interacting body is con-stant when the body is at rest, it in-creases in a definite way as the bodymoves relative to a fixed observer.

52 • FEBRUARY 1969 • PHYSICS TODAY

Further, when at rest, the body has anintrinsic energy that depends on theproduct of its rest mass and the squareof the speed of light—an explosivelylarge amount of energy that was notsuspected until relativity theory ap-peared! These and other implicationsof special-relativity theory in regard towave motion, for example, that werenot predicted by the preceding the-ories were impressively substantiatedby experimental facts.

Thus we see that the rules of spe-cial-relativity theory refuted the previ-ous laws of mechanics by requiringthat the time coordinate must be in-corporated with the three spatial co-ordinates as a fourth parameter, to bespecified only relative to the observer.But this change was not too much of arevolution! It was rather a natural ex-tension to a more general way of ex-pressing the laws of nature with aspace-time eooordinate system. Inthis case, then, what was the real revo-lution that came with Einstein's the-ory? It was the abandonment of theidea that the space-time coordinatesystem has objective significance as aseparate physical entity. Instead ofthis idea, relatively theory implies thatthe space and time coordinates areonly the elements of a language that isused by an observer to describe his en-vironment. However, the convention-alist view is not fully adopted sinceit is further asserted with this theorythat the relation between the pointsof the space-time language of anyobserver is in fact a representation ofthe intrinsic interaction within thematter distribution that comprises thephysical system. It then follows thatif there should be no matter in the uni-verse, there would be no space-timeto talk about! It implies that if thematter distribution should be variable,then the relation between the points ofthe space-time (that is, the geometry)that is used, to describe the environ-ment of any point would be corre-spondingly variable. This view takesspace-time as a passive entity that isused to describe nature—perhaps forwant of a better language!

THE PRINCIPLE OF RELATIVITY

The fundamental starting point of rela-tivity theory—the principle of relativity-asserts that, if the laws of nature thatare substantiated by one observer'sdata are indeed bona fide natural laws,their form cannot change when theyare deduced by any other observer

". . . the theory of relativity introduced

a revolutionary concept into physics by relegating

space and time to the subjective role of the

elements of a language that one observer or another

may use to describe the natural laws."

who may be at any other space-timepoint and in arbitrary relative motionwith reference to the first observer.Let us examine this assertion a littlecloser. It means that after the naturallaws have been deduced and substan-tiated in one space-time frame of ref-erence, with the language (x,y,z,t) ofone observer (who need not be a hu-man or his equipment; it may also bean electron!) then in any other frameof reference, with the language(x'y',z',t') of the different observerwho is in arbitrary relative motioncompared with the first observer, thederived natural laws must be identi-cal in form. (See box on page 52).

Of course, the observer who usesthe language (x,y,z,t) and the otherobservers who use the languages(x",y",z",t"), (x",y",z",t"), . . . wouldnever be able to check that theirrespective laws of nature for the

same sorts of phenomena are actuallythe same until they learn to translatefrom one language to the other. Thecollection of translations of all words(space-time points) from one coordi-nate frame to the others, which may bein arbitrary relative motion, is a con-tinuous transformation group; it iscalled the "Einstein group." The rep-resentations of the group provide apowerful mathematical device for ab-stracting many of the implications ofrelativity theory that are not at allobvious at first glance. Some of theseimplications in regard to general rela-tivity will be discussed later.

We see, then, that the theory of rela-tivity introduced a revolutionary con-cept into physics by relegating spaceand time to the subjective role of theelements of a language that one ob-server or another may use to describethe natural laws. Use of the theory is

EINSTEIN

PHYSICS TODAY • FEBRUARY 1969 • 53

subject to the restriction that the formof these laws must be unchanged un-der the coordinate transformations be-tween any space-time frames of refer-ence that are in arbitrary relative mo-tion.

ELEMENTARITY OF INTERACTION

If space and time are indeed to beconsidered in the subjective role thatwas discussed above, and if at thesame time we adopt the realist viewthat there are fundamental objectiveelements underlying the laws of na-ture, then what does relativity theoryimply these to be? That is to say, inthe earlier theories, and even in thecontemporary elementary-particle the-ory, the world is mad,e out of things—indestructible bits of matter thatmove along space-time trajectories.The quantum-mechanical approach isof course not quite as deterministic asclassical mechanics, because the clas-sical mechanics makes precise predic-tions about the trajectories of inter-acting things, and the quantum theorymust resort only to probability state-ments about the states of motion ofthese things. Nevertheless, it is thebit of matter, with its own space-timetrajectory, that is, in both of thesetheories, the objective reality out ofwhich the world is constructed. Eachof these mechanical theories is indeeda different version of an atomistic ap-proach to the natural laws.

On the other hand, when one fol-lows the axiomatic basis of relativitytheory to its logical extreme, it ap-pears to me that it is not the free bitof matter that is the fundamentalbuilding block with which to constructa theory of matter. It is rather a re-lation that forms the basic entity here.With relativity theory, one must takethe "observer-observed" relation as afundamental starting point. That isto say, the "observer" is here an entitywith meaning only in relation to the"observed," and vice versa. Also, incontrast with the quantum theory,"observer" has no anthropomorphicdenotation. It refers only to an intrin-sic component of a fundamental inter-action. It might refer to a componentof a physical system that could beidentified with a star, a man or a pro-ton! On the other hand, the "ob-server" and the "observed" in classicaland quantum mechanics do indeedhave meaning as things in themselves.Again, in contrast to the quantum the-ory, the relativistic theory requires a

"Relativistic theory requires a theoretical structurein which it makes no difference to thepredictions of the theory which part of the physicalsystem is identified with the 'observer'and which part is called 'observed.' "

theoretical structure in which it makesno difference to the predictions of thetheory which part of the physical sys-tem is identified with "observer" andwhich part is called "observed,"These names are merely chosen forreasons of convenience to describe thecomponents of a single interaction mone space-time—without actual parts!

To compare the predictions of thistheory with certain (but not all) ex-periments, one must be able to ex-amine this one relation in the asymp-totic region, where there is suffi-ciently weak intrinsic coupling to re-veal an apparent uncoupling into sepa-rated parts. It is important, however,that with the assumption about theelementarity of this relation (whichmay be called "elementary interac-tion") there can never in principle beany actual separation, no matter howclosely one may approach this ap-parent situation in the theoretical de-scription. Such a conclusion about theintrinsic inseparability of the elemen-tary interaction does not refute anyexperimental evidence because it is, inprinciple, impossible to observe a com-pletely uncoupled component of aphysical system. The very act of mea-surement automatically couples an ap-paratus (not necessarily macroscopic!)to the "observed." But, independentof approximation, it is still the one re-lation—"elementary interaction"—thatis the fundamental entity in a fully ex-ploited theory of relativity. Such anentity is of course entirely objective asits fundamental description is inde-pendent of the interchange of thenames for the subject (the "observercomponent") and the object (the "ob-served component").

A metaphor: a dog and a flea

This idea can be illustrated with thefollowing simple allegory. Considera world that consists of a dog, a flea,the sky and the ground. Let us saythat the dog's life span is of the orderof 15 years and that of the flea is about5 hours. Suppose now that the flea

lives on the dog's back, and let us in-quire about the flea's and the dog'srespective accounts of the world ac-cording to the atomistic and rela-tivistic approaches. According to theatomistic view, the flea will think thathe lives in a forest of tall soft trees, onwarm, undulating ground. He wouldoften recall the painful experience thatoccurs every few years of his life, whena major earthquake throws him upinto the sky—only to land again in astrange land, but still surrounded bytall soft trees on warm moving ground.The flea would also think about him-self—that he is strong, handsome,clever, etc. . . (this is called "self-energy" in modern physics). Thedog's idea of the world, on the otherhand, would be quite different. Hewould be aware of the cool stillground. But he would also worry andbe annoyed by a terrible itch on hisbody that recurs every few minutes,even though he has thoroughly shakenhis body. The dog would also thinkabout himself—that he is a tall hand-some dog with a beautiful long tail,and that he is clever, modest, etc. . . .

After considering both stories, itwould be impossible for us to decidewhich is the true description of theworld. The reason, of course, is thatneither is the complete story! Each ofthese accounts presents a subjectiveview; one belongs to the flea and theother to the dog.

Suppose now that we consider therelativistic view in which it is the re-lation that is the elementary entity. Inthis case, the flea would proceed as be-fore to think about the warm undulat-ing ground with the tall soft trees andthe periodic earthquakes every fewyears. However, instead of going intoself-attributes, the flea would make acareful study of his environment (thedog). If he is a sufficiently percep-tive investigator, the flea may be ableto deduce precisely how the dog isreacting to his presence. Similarly,the dog would, proceed to think abouthis environment as he did before. But

54 • FEBRUARY 1969 • PHYSICS TODAY

he would not try to relate his self-at-tributes to an objective description ofthe world; instead he would learn howhis environment (the flea) is reactingto him. We see, then, that with thisapproach the flea and the dog wouldhave identical descriptions of theworld—even though the languageswith which they express these viewsare entirely different. With this rela-tivistic approach, the world would bedescribed with a single relation, dog-flea, without parts! The description isnow entirely objective because it is in-deed independent of which compo-nent of the coupled system is express-ing the laws of nature.

THE UNIFIED FIELD CONCEPT

The basic elements of the language ofa fully exploited theory of relativityare continuous field variables. Theseare continuously mapped functions ofthe space and time coordinates. Thisfeature is a consequence of the as-sumption that the valid laws of natureare invariant in form under the trans-formations of the Einstein group (thegroup of continuous transformationsamong the space-time frames of refer-ence that are in arbitrary motion rela-tive to each other). Einstein showed,for example, that even with the ideaof a finite propagation time for the in-teraction between distant bits of mat-ter, it is not possible to consider thismatter in discrete quantities withinrelativity theory. It would not be pos-sible in this case, for example, to definethe conserved quantities of the system,such as energy and momentum. Theapproach of relativity theory then ne-cessitated an expression in terms offield equations, where the field vari-ables relate to densities. In the finalanalysis, to compare the predictions ofthe theory with the experimental facts,it becomes necessary to integrate cer-tain prescribed functions of these fieldvariables over all of the coordinates inwhich they are mapped. Thus we seethat the basic elements of the languagein relativity theory are not the spaceand time coordinates themselves butrather a certain set of functions thatare continuously mapped in the space-time coordinate system.

Faraday s view of gravitation

The field concept was originally intro-duced to physics by Faraday about 50years before the discovery of relativ-ity theory. Before Faraday there waslittle (recorded) doubt in any physi-

cist's mind that the fundamental de-scription of the material world mustnecessarily be in terms of bits of mat-ter that act on each other at a distance,for example in the way that Newtonenvisioned the force of the sun onEarth and the other planets. On theother hand, Faraday recognized thatNewton's gravitational force need notbe viewed in terms of this model. Herather saw the effect of the sun on theplanets in terms of a continuous fieldof force—the continuous function 1/R-of the distance R from the center ofthe sun to any exterior point of obser-vation. Thus, instead of assigningspecial meaning to the separate spatialpoints that locate the sun and theplanets, the field approach of Faradayconsiders one space, and a special con-tinuous function of its coordinates pre-dicts how a test particle would moveshould it be placed at any spatial lo-cation. That is to say, Faraday's ap-proach took the potential field of force(a continuous entity) to replace thebit of matter as a fundamental con-struct. The conflict of these two ap-proaches is indeed as old as the studyof physics itself; it is the conflict be-tween discreteness and continuity inthe fundamental description of matter.

An interpretation of the field

Faraday also believed that all the dif-ferent manifestations of the influencethat matter exerts on matter are de-rivable from a single unified-field de-scription. Of course he was unable tomove planets around in order to test

this hypothesis in regard to the motionof the heavenly bodies. He did initiatehis investigation of this idea in hisstudies of electricity and magnetism,physical phenomena that were previ-ously thought to be unrelated. Afterstudying the topological features of theelectric and magnetic fields, by respec-tively observing the electrostatic po-tential in an electrolyte and the pat-tern of iron filings in the vicinity of amagnet, Faraday proceeded to investi-gate whether or not these two fieldswere unified, in a single electromag-netic force field. He saw that this wasindeed the case by observing that,when an electrical conducting wire ismoved across a magnetic field of force,an electric potential is generatedthereby causing an electric current \oflow. Similarly when a current iscaused to flow7 through a wire he ob-served that a compass needle willpolarize in a plane perpendicular cothe direction of the electric-currentflow.

An important feature of Faraday'sview of the field as a continuouslymapped potential of force is that thisentity would lose all meaning as suchshould there not be any test chargeas a probe. That is to say, becausethe electromagnetic field was inter-preted by Faraday solely as a pre-existing cflw.se for an observable effect—the motion of a charged test particle—and because cause and effect arelogically inseparable, the electromag-netic field of force in Faraday's viewwas meaningful only when a test

FARADAY MAXWELL

PHYSICS TODAY • FEBRUARY 1969 • 55

". . . if space and time are only to servethe passive function of providing a languagefor the observer, why shouldthe inertial frame of reference in particularbe singled out?"

charge was taken to exist at the sametime. However, the structure of Fara-day's electromagnetic field of forcewas taken to be uninfluenced by thepresence of a test charge in the physi-cal system. This field was taken,rather, as a fundamental representa-tion of charged matter outside of thetest charge. Indeed the electromag-netic field of force was assumed byFaraday to be the thing in itself thatreplaces the bit of matter of "particle"theories as the basic entity from whichthe elementary description must bebuilt.

Thus, in contrast with the "particle"theories (for example, Newton's theoryof gravitation) the fundamental thingin itself in Faraday's field approachplayed the role of the cause in acause-effect relation—a relation thatis logically inseparable! Atomistictheories, on the other hand, are basedon the "free particle" (say, the sun)as the fundamental reality—an entitythat has meaning with or without otherparticles in the system. Later we willsee that the assumption of the test

RIEMANN

56 • FEBRUARY 1969 • PHYSICS TODAY

charge, as uncoupled from the forcefield, must be removed in the exactform of a relativistic theory, simplybecause of the elementarity of theinteraction that follows from a logicalanalysis of this theory.

In some of his later experimentalstudies, Faraday was unfortunatelyunsuccessful in extending his earlierresults to an electromagnetic-gravita-tional force field. However, in addi-tion to the lack of experimental verifi-cation, there were still some basictheoretical questions that had to beanswered about such a unification,even at that time. For example, ifelectromagnetic and gravitationalphenomena are in fact manifestationsof the same force field, why is it thatgravitational forces are only attractivebut electromagnetic forces can beeither attractive or repulsive? LaterI w7ill show how the answer may liein the necessary incorporation of theelectromagnetic and gravitational forcefields with fields that relate to theinertia of matter—fields that were notsuspected to exist in 19th-centuryphysics!

Maxwell's field equations

It was the genius of Maxwell thatcreated the exact mathematical de-scription for Faraday's electromagneticforce field. After formulating "Max-well's equations" he discovered thatnot only did some of their solutionsrelate to the predictions of electromag-netic phenomena that were alreadyknown, but that there were othersolutions of these equations, describ-ing the faraway effect of oscillatingcharged matter, that precisely describethe properties of light. Thus Max-well found that all optical phenomenacould be described with the "radia-tion" solutions of the electromagneticequations, thereby unifying the opti-cal with the electromagnetic manifes-tations of interacting matter in a singlefield theory. This discovery settled avery old dispute (at least temporarily!)

about the corpuscular or continuousnature of light. Of course, other mani-festations of electromagnetic radia-tion were discovered not long afterthis—x radiation, radio waves, gammaradiation, etc.

The salient feature of Maxwell'sequations that Einstein discovered isthat they are form invariant under aspecial class of continuous transforma-tions among the space-time coordi-nates (x,y,z,t) of one frame of refer-ence and the coordinates (x\\j',z\t')of other frames of references that arein relative motion. These are thegeometrical transformations that leavethe interval between any two suchsets of coordinates, [c~At2 — (A.V2 +Ay- + A~2)]1/2, unchanged. Herec is the constant speed that appearsin Maxwell's equations. The particu-lar form of this interval in the four-dimensional space is a characteristicfeature of special-relativity theory. Ifone should choose to keep the timecoordinate fixed, as a special case, (bytaking At = 0) these transformationswould reduce to the set of coordinatechanges in a three-dimensional spacethat characterizes the symmetryGalileo discovered to underlie the lawsof classical mechanics. But, accord-ing to the structure of Maxwell's equa-tions, one may not generally fix thetime coordinate, as was done in clas-sical mechanics, if the particular lawsof nature that are embodied in Max-well's equations are to be the same loany set of observers who are in rela-tive motion. It then follows, from thisanalysis of the laws of electromagneticfields, that the laws of mechanics thatobey this more general transformation-group invariance (the theory of spe-cial relativity) are necessarily differentfrom the laws of classical mechanics.However, it turns out that if the rela-tive speed between interacting bits ofmatter is small compared with theuniversal speed c (the speed of lightin a vacuum) the two laws of me-chanics are practically the same. Thedifferences in the predictions of therespective formalisms are too small lohave been detected by the type ofexperimentation that preceded the20th century.

UNIVERSAL INTERACTION

The Maxwell formulation was the firstset of field equations satisfying theinvariance requirement of the theoryof special relativity to be discovered.Einstein also noticed that special rela-

tivity, although it incorporates thetime coordinate with the spatial co-ordinates, relates only to inert ialframes of reference.

These are space-time frames thatare moving with the special featureof constant rectilinear speed (or areat rest) relative to each other. Thetheory of special relativity was not ac-ceptable as the most general descrip-tion of matter because the inertialframe refers only to an extrapolationfrom observations that actually entailnoninertial frames of reference. Thereason is that the inertial frame refersonly to a description in which thereare no forces involved and thereforewhere there is no energy and momen-tum transfer between interactingmatter. On the other hand, anymeasurement (whether it involves thecoupling of only microscopic matter,macroscopic matter, or a combinationof these) necessarily entails a transferof energy and momentum, if some-thing is to be recorded about the"observed." Secondly, Einsteinargued that if space and time are onlyto serve the passive function of pro-viding a language for the observer,why should the inertial frame of refer-ence in particular be singled out? Hethen concluded that to exploit thistheory fully, the principle of rela-tivity must necessarily imply that thelaws of nature are invariant in formunder the transformations betweenspace-time frames of reference thatare in arbitrary relative motion.

Extension of his theory to noninertialframes of reference then led to thetheory of general relativity, with spe-cial relativity serving only as a par-ticular limiting case. I should sayhere that in view of the elementarityof the interaction that follows fromthe conceptual starting point of rela-tivity theory, the actual limit of spe-cial relativity cannot in principle bereached, even though it can be ap-proached arbitrarily closely. Withthis view, then, the success of themathematical formulations in modernphysics that are consistent with specialrelativity is not more than an indica-tion, in specific applications, of howgood special relativity is as a mathe-matical approximation for field equa-tions that are normally expressed ingeneral relativity.

Had Einstein stopped with specialrelativity, although it was eminentlysuccessful in explaining and predict-ing experimental facts not previouslyunderstood, not too much of a revolu-

tion would have occurred. For, as inthe previous "classical" theories, therelation between points in space-timein special relativity is the same every-where, and it is independent of thematter content of the physical systemthat is described. Fully to exploit theidea that the space and time coordi-nates serve only as a language used torepresent the matter content of aphysical system, it becomes necessaryto extend from the flat (Euclidean)space-time geometry of special rela-tivity to a curved (non-Euclidean)space-time of general relativity. Thevariable curvature of the geometricalsystem in the general description is in-deed a function of the variation of themutual interaction within the physi-cal system. According to this view, ifthe matter content of a system wasdepleted the curvature of space-timewould correspondingly diminish. Inthe limit, where the system is emptiedof matter (corresponding to no mutualinteraction of matter), the curvaturewould be zero, thereby yielding aEuclidean geometrical system andspecial relativity. The particular non-Euclidiean geometry that has thisproperty of asymptotically approach-ing a Euclidean geometry is that dis-covered by Georg Riemann. ThusEinstein looked for a relationship be-tween the field of mutual interactionof the matter content of a physicalsystem and the field properties of aRiemannian space-time.

The Einstein field, equations are aspecial representation of a relationshipbetween the metrical field (the func-tion that prescribes the relation be-tween points in a Riemannian four-space) and the mutual interaction ofmatter fields. In particular, the left-hand side of Einstein's equations arecertain nonlinear differential forms inthe metric tensor, while the right-handside depends on the energy-momen-tum tensor for the matter content ofthe physical system. Thus, if one isgiven the energy-momentum tensor forthe system, the solutions of these

equations thereby yield the cor-responding metric tensor, which inturn prescribes the variable relationbetween the points of space-time. Ifone now takes seriously the contentionthat the metrical field is a representa-tion of the matter that is being de-scribed, according to the restraint ofEinstein's equations, then it appearsto me to follow that these field equa-tions must be considered as "if-and-only-if" relations, rather than "if-then"relations. With this conclusion, thesolutions of these equations that cor-respond to empty space must be re-jected as physically unacceptable.Similarly the inversion of these equa-tions and the insertion of the metrictensor for a flat space must yieldmatter-field solutions that are physi-cally unacceptable. It follows thenthat space-time can only be describedas curved, in any realistic situation,even though it can come arbitrarilyclose to flatness. This conclusion iscompatible with the one that wasreached earlier indicating that thelimit of special relativity may be ap-proached arbitrarily closely (indeedthe empirical facts require this to beso) but that the limit cannot actuallybe reached within the framework ofthis theory. Some of the physical im-plications of this result will be dis-cussed below in connection with theconcept of inertial mass.

Equations of motion

One of the very important features ofEinstein's field equations is their con-tainment of the equations of motion ofinteracting masses. In this respectthey contrast with Maxwell's equa-tions for electromagnetism. Indeed,to predict the equations of motion ofinteracting charged bodies, Lorentzhad to adjoin further equations tothe Maxwell formalism; these equa-tions then define the electric and mag-netic forces (in terms of the couplingof matter source variables to theelectromagnetic field-intensity solu-tions of Maxwell's equations). In the

"In principle, the same set of field equationsthat describe the domain of elementary-particle physics must also describe the domainof laboratory dimensions and the domainof astronomical dimensions."

PHYSICS TODAY • FEBRUARY 1969 • 57

MACH

gravitational problem, however, whenone recognizes the experimental factthat the inertial mass of a body innonuniform motion (for example, cen-trifugal motion) is equal to the massthat is acted on by a gravitational field,then one sees that the effects on bodiesthat result from the gravitationalforces exerted by other bodies are al-ready incorporated in Einstein's metri-cal field equations. The relation be-tween the points in space-time, whenthere is nonzero curvature, predictsthe gravitational force between mas-sive bodies—without the need to addfurther equations of motion. Einsteinconcluded from this feature of generalrelativity that perhaps not only thegravitational forces but all other forcesbetween matter and matter may bedifferent manifestations of an evenmore generalized geometry for space-time. This conclusion, of course, sup-ported Faraday's intuitive feelingsabout a unified field theory. Anequivalent statement about Einstein'sconclusion regarding the role of ge-ometry in physics follows from an in-terpretation of his equations as "if-and-only-if" relations. For it would thenfollow that a generalized geometry ofspace-time could be viewed as a man-ifestation of all possible forces be-tween matter and matter—a manifesta-tion of a universal interaction.

The first test of Einstein's equationswas to see if they would predict all ofthe effects that Newton had alreadysuccessfully predicted about gravita-tional forces. Einstein found that,under the special approximating cir-

cumstances that are accurate whereNewton's equations had worked, histensor-field equations reduce preciselyto Newton's equation. Thus, withoutevoking the action-at-a-distance idea,Einstein's field formalism made theidentical successful predictions in theway that Faraday had originally an-ticipated. However, because Ein-stein's equations were more generalthan Newton's equation, it was im-plied that the tensor equations shouldbe able to make additional predictionsabout gravity that are not at all im-plied by the Newtonian theory. Thiswas indeed the case. The three pre-dictions of his equations that were notpreviously made were: a frequencyshift of monochromatic light as itpropagates through a gravitationalfield; the bending of the trajectory oflight as it passes through a gravita-tional field, and a precession of theaxes of a planetary orbit. Each ofthese effects was observed to be invery good quantitative agreementwith the predictions of Einstein's fieldequations.

Towards a unified field theory

But Einstein did not consider hisequations as another theory of gravity.Rather, according to what we saidearlier, he considered them to be notmore than one step towards the con-struction of a unified field theory inwhich electromagnetism, nuclearforces, weak interactions (and anyother phenomena not yet discovered)might be incorporated with gravita-tion in terms of one geometrical fieldthat expresses most generally the re-lation between points of the space-time language to describe the laws ofnature. He also anticipated that ifsuch a unification could eventually beachieved, then perhaps the peculiar-looking consequences of the equationsof quantum mechanics (for examplethe appearance of a wave-particleduality and nondeterminism) mayemerge from an approximation for de-terministic unified field equations.

Although both Faraday and Ein-stein anticipated a unified field theoryto underlie the fundamental descrip-tion of the material universe, Einsteinextended Faraday's conceptual ap-proach when he incorporated the prin-ciple of relativity. The implication inthe studies of both Faraday and Ein-stein was that although it is convenientto utilize a space-time coordinate sys-tem to facilitate a mapping of forcefields, space and time do not by them-

selves have objective connotation.However, a logical implication of theincorporation of the principle of rela-tivity with Faraday's field concept isthe appearance of physical significancein the relation between points ofspace-time as having to do with themutual interaction of the matter con-tent of the system. Thus a logicalimplication of the Einstein generaliza-tion of Faraday's approach is the nec-essary incorporation of the "testcharge" with the field of force, to yieldas the elementary entity one field ofmutual interaction. A mathematicalconsequence of this generalization isthat of passing from a formalism interms of linear differential equations(such as Maxwell's equations) to aformalism in terms of nonlinear dif-ferential equations. Here we have astriking example in the history of sci-ence of passage towards increased con-ceptual simplicity (from one forcefield plus an arbitrarily defined testcharge to a single field of mutual in-teraction) accompanied by decreasedmathematical simplicity (from linearto nonlinear differential equations).

A GENERALIZED MACH PRINCIPLE

One of the important manifestationsof interacting matter that I have notyet mentioned is inertia. Indeed, Ein-stein acknowledged that Ernst Mach'sconclusions about the source of inertiahad a major influence on his own the-oretical studies of relativity theory.Let us then examine the conceptualnotion that Mach introduced.

MacKs view of inertia

Mach's argumentation led to the con-clusion that the inertial mass of anyamount of matter (whether it be ofmicroscopic, macroscopic or astronom-ical proportions) is not an intrinsicproperty of a thing. Rather it wasargued that inertia is rooted in thedynamical coupling between the quan-tity of matter being studied and all ofthe other matter contained in theclosed physical system that is de-scribed. Thus it is implied here thatthe most minute amount of matter,say an electron, has an inertial massthat is in fact a measure of its couplingto its entire environment, including theapparatus that measures the propertiesof the electron, the laboratory, theearth, the solar system, the galaxy,and so on, until all of the content ofthe universe is exhausted! Thus Machrejected all previous views that as-

58 • FEBRUARY 1969 • PHYSICS TODAY

sume that the mass of any bit of mat-ter is one of its intrinsic properties.

When the principle that asserts thisfeature of inertia—the Mach principle-is incorporated with the theory ofgeneral relativity, it follows that in-ertial mass must be derivable fromthe field properties of space-time.Thus a full exploitation of the Machprinciple implies that inertia must beincorporated with the other manifesta-tions of interacting matter in the uni-fied-field description. Einstein ac-knowledged the validity of this asser-tion but he did not reach the stage inhis own studies where he would at-tempt to construct such a unification.

Derivation of inertia! ?nass

How then can we set about discover-ing this relation between the fieldproperties of space-time and the fea-ture of inertia of matter? I have beentaking the following approach in myown investigations of this problem.1

Let us start out by defining the in-ertial mass most primitively in termsof its appearance in the most funda-mental equations that describe matterat the microscopic level. These weretaken to be the field equations thathave the Dirac form; they are first-order differential equations in spinorvariables, although they are not in-terpreted within this field theory asthey are in the quantum theory, norare they linear equations (see box onthis page). Nevertheless, these mat-ter field equations are constructed toapproach asymptotically the lineareigenvalue form of the quantum-me-chanical equations in the limit ofsufficiently small energy-momentumtransfer within the physical system.Normally one starts with this equationin special relativity and inserts a massparameter. After the solutions of theequation have been found, the energyof the matter that is described can becomputed. The resulting expressiondepends on the inserted mass param-eter. One then adjusts the magnitude(and sign) of this parameter to fit thedata.

On the other hand, my plan is notto introduce any mass parameter, butinstead to express the matter-fieldequation in a curved space-time.When this is done, it turns out thatwith the most general type of spinorequation (in terms of the two-com-ponent spinor) a derived field appearsin the place where the mass param-eter was formerly inserted. The pre-dictions of the theory then compel

one to identify this field with the in-ertial mass of the described matter.The derived mass field depends on thecurvature of space-time. The lattergeometrical property is, in turn, amanifestation of the mutual couplingof all of the matter within the closedsystem. Thus, if the rest of the uni-verse should be depleted of all matter,the mass of the remaining electron,say, should correspondingly go tozero. The derived field relationshipis then a quantitative expression ofthe Mach principle because here theinertial mass of any amount of matter

is indeed a well defined function of itsdynamical coupling with all of theother matter within the entire closedsystem.

Why is gravity only attractive?

To check this result further, it wasfound that the inertial mass field soderived is positive-definite. That is,with all possible changes in the metri-cal field, the particular combination offield variables that relate to the in-ertial mass is always positive. It thenfollows that even in the local limit,where the equations in special rela-

SPINORS

Dirac discovered that to expressSchrodinger's wave mechanics in arelativistically invariant form, it wasnecessary to generalize the (com-plex) scalar field to a (complex) two-component field. This was called a"spinor variable" because of its re-lation to the previously discoveredspin degrees of freedom that wereempirically necessitated by the Zee-man spectrum of atoms. The spinorvariable itself was actually discov-ered before Dirac's work by themathematician E. Cartan. (Althoughthe conventional expression of Di-rac's equation is in terms of thefour-component "bispinor" variable,these, in turn, are a union of twotwo-component spinors. The latterare the most primitive variables fromwhich the Dirac formulation is built.)

After Dirac's discovery, Einsteinand Mayer studied the followingquestion: "Are the spin degrees offreedom a consequence of the postu-lates of the quantum theory or a con-sequence of the postulates of relativ-ity theory?" To answer the ques-tion, they investigated the structureof the most primitive irreducible rep-resentations of the underlying groupof relativity theory. They discoveredthat indeed the four-dimensional(real) representations (which de-scribe the transformations of a four-vector between relatively moving ob-servers) reduce to the direct prod-uct of two two-dimensional (com-plex) representations. The two-component (complex) functions thatare the basis of these representa-tions (analogous to the four-vectorbasis of the four-dimensional repre-sentations) were found to be thespinor variables that Dirac discov-ered to describe the electron!

Thus, Einstein and Mayer discov-ered the very important fact that thespinor variable is the most primitiveexpression of a relativistically invari-ant theory and that the spin degreesof freedom that appeared in Dirac'sequation were actually a conse-

quence of relativity theory and notof the quantum theory. (Of course,once a relativistic theory is con-structed in this, the most primitiveform, it can then be "quantized" ornot, depending on the particulartheoretical study that is being pur-sued.) For this reason my study ofinertia necessitated a formalism interms of spinor variables in the mostelementary description.

To mention one of the most im-portant explicit features that definethe spinor field variable, let us com-pare this two-component object witha two-component vector field in con-figuration space. If A and B are anytwo-dimensional (real) vector fields,then an invariant is the scalar prod-uct A • B = A,Bi + A,B,. If

are any two (complex) spinor fieldvariables, the invariant combinationis (s,t) = Sit2 — s-jti. The compo-nents, S],s^,1:2, are all continuouscomplex functions of the space-timecoordinates. Note that this spinorinvariant is then complex so that itactually corresponds to two invari-ants—the real and imaginary partsof (s,t). Finally it follows from thetransformations between the space-time coordinates of relatively movingreference frames in relativity theorythat the spinor field variables alsotransform in a definite way from oneframe to the others. Because of thisresult, and the fact that there is amaximal number of invariants in thistheory (compared with all other rela-tivistic formulations) this type offormulation predicts the maximalnumber of consequences. That is,a spinor theory will contain the samepredictions as a vector theory, butit will make additional predictionsthat have no counterpart in thehigher dimensional form. The elec-tron spin coupling to an externalmagnetic field is an important ex-ample.

PHYSICS TODAY • FEBRUARY 1969 • 59

tivity have empirical validity, the in-ertial mass can have only one possiblepolarization. This result, in turn, im-plies that a gravitational force canhave only one sign as this force de-pends on the product of the massesof the interacting matter. The latterresult, of course, is in agreement withthe experimental facts. To refute thisresult, it would be necessary to dem-onstrate the existence of negativemass, which is to say that we wouldhave to observe the gravitational re-pulsion of one massive body by an-other, thereby implying the existenceof masses with opposite polarities.

One other feature of the spinormatter-field equations that led to thepreceding result is the necessary ap-pearance of a coupling term corre-sponding exactly to the form of theelectromagnetic coupling that even-tually leads to the Lorentz force.That force, in turn, also dependswithin this theorv on the metrical field,but in such a way that its form is non-positive-definite; this term could bepositive or negative, depending on thesigns of certain derived functions ofthe metrical field. If one should de-fine electromagnetic coupling mostprimitively in terms of its appearancein the matter-field equation that de-scribes microscopic matter, then theconclusion must be drawn from thisanalysis that electromagnetic forcescan be either attractive or repulsive—a result that is also in agreement withthe experimental facts. Thus we seethat with the expression of the inertialfeatures of matter in terms of the

DIRAC

primitive matter fields that solve rela-tivistic spinor equations in a curvedspace-time (equations that asymp-totically approach the equations ofquantum mechanics) two importantfeatures of interacting matter, in alldomains of interaction, are predictedfrom first principles to be in agree-ment with the experimental facts.These are the known properties thatelectromagnetic forces can be attrac-tive or repulsive while gravitationalforces are only attractive.

Finally, when one extends this anal-ysis further it is found that the elec-tromagnetic field intensity that corre-sponds to charged matter in motion isalso dependent on the curvature ofspace-time and therefore on the mat-ter content of the entire closed phys-ical system.2 As in the case of inertialmass, the limit of a matterless uni-verse corresponds here to an identicalvanishing of the electromagnetic fieldintensity of any remaining bit ofcharged matter. This feature of thefield equations of general relativity,which includes the inertial propertiesof matter, then relates to a generalizedversion of the Mach principle whereall manifestations of interacting mat-ter are in fact a feature of their dy-namical coupling with the rest of theentire closed system.

Implications

The application of the generalizedMach principle necessarily implies thatthe universe is "closed", which meansthat there is no matter in the universethat, in an exact sense, is uncoupled(that is "free"). In principle, then,the same set of field equations thatdescribe the domain of elementary-particle physics must also describe thedomain of laboratory dimensions andthe domain of astronomical dimen-sions. The conclusion that follows isvery difficult to believe at first glance;it is the possibility of discovering fea-tures about the structure of the uni-verse in cosmological terms (the do-main of light years) by studying thedomain of elementary-particle inter-actions (the domain of fermis). For,according to the field theory that wehave been discussing, there are in-deed global features of the entire uni-verse that appear in the quantitativedescription of elementary-particle in-teractions. One of these is the de-pendence of the inertial mass of aninteracting elementary particle on thecurvature of space-time. It is true, ofcourse, that in such studies we are only

looking at an extremely small portionof space-time, corresponding to veryhigh momentum transfer between mat-ter and matter in the microscopic do-main. Yet we are indeed looking at aportion of a continuous field that ex-tends into the astronomical domain.If we should have at our disposal thebasic equations that determine theseglobal fields, a knowledge of their be-havior in the very small domain couldwell act as a boundary condition toextend our knowledge of the mappingof the metrical field to any domain ofinteraction. This ability would beanalogous to predicting a large dis-tortion on a radio wave front, beingcaused by a small bump on the emit-ting antenna, many miles away fromthe point of observation.

Another test of the implications ofthe generalized Mach principle wouldbe to study the dependence of theelectromagnetic field amplitudes onthe curvature of space-time. Theimplication above was that as the ef-fective curvature of space-time dimin-ishes, the amplitudes of the electro-magnetic force field correspondinglydiminish. This effect can be testedby probing sufficiently small domains,which in turn correspond to sufficientlylarge quantities of energy-momentumtransfer in the microscopic domain.The prediction here is that as the mo-mentum transfer between matter in-the microscopic domain approachesinfinity, the electromagnetic coupling!correspondingly approaches zero. Itis as though the charge of interacting,matter effectively reduces to zero atthe origin. Experimentation on theseeffects is indeed within the domainof experimental high-energy physicsstudies that are presently in the plan-ning stage in high-energy electronscattering programs. Current experi-mental studies on high energy elec*tromagnetic scattering already hint ajthe validity of this theoretical conchl

sion.-1

This article is adapted from a lecturt,given to the physics club at the-State Unfyversify of New York College at Cortland

References

1. M. Sachs, Nuovo Cimento 53B, 398(1968).

2. M. Sachs, Nuovo Cimento 55B, 199(1968).

3. K. W. Chen, A. Cone, J. Dunning, NF. Ramsey, J. K. Walker, R. Wilson.in Nucleon Structure, eds. R. Hofstad-ter, L. I. Schiff, Stanford UniversityPress (1964), p. 55. D

60 • FEBRUARY 1969 • PHYSICS TODAY