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CIRI/03-smw04

Einsteinian Field Theory as a Program in Fundamental Physics

Sanjay M. WaghCentral India Research Institute,

Post Box 606, Laxminagar,

Nagpur 440 022, India

E-mail:cirinag ngp@sancharnet.in

(Dated: April 18, 2004)

I summarize here the logic that leads us to a program for the Theory of the Total Field in Einstein’ssense. The purpose is to show that this theory is a logical culmination of the developments of(fundamental) physical concepts and, hence, to initiate a discussion of these issues.

PACS numbers: 01.55+b; 01.65+g; 01.70+w

I. INTRODUCTION

Our purpose is to analyze in one place Einstein’sreasoning that led him to the program of a field-theory that, in his own words, is: Continuous func-tions in the four-dimensional [continuum] as basicconcepts of the theory [1].Einstein had summarized these reasons in [1].

However, we reconsider these from a different per-spective, particularly, of [2]. Then, we also modifyEinstein’s program suitably.Furthermore, von Laue summarized in [3] the

developments related to conservation postulates inPhysics. In the present article, we shall often usethis excellent, historically as well as scientificallyimportant [13], article to illustrate and substanti-ate our physical arguments.

II. INERTIA, ENERGY ANDCONSERVATION LAWS

To begin with, and following von Laue, considerGalileo’s famous analysis leading to the concept ofan inertia of a physical body.Galileo observed that a body falling on the sur-

face of the earth from a certain altitude must ob-tain precisely that velocity which it requires to re-turn to its former level. Any deviation from thislaw would but be able to furnish us a method formaking the body ascend by means of its own grav-ity, a conceivable perpetuum mobile. We considerthis to be impossible.Now, following Galileo, let a body ascend, after

it has fallen downwards a certain distance, on aninclined plane. The lower the inclination of theinclined plane toward the horizontal, the longer thepath the body requires to obtain its former levelon the plane. Galileo verified this experimentally.Then, if the inclined plane were made horizontal,we may conclude that the body will keep movingon it to infinity with undiminished velocity.

This last conclusion is not any experimental re-sult but an inference to be drawn by imagining aninfinite horizontal plane tangential to the surfaceof the earth, an obvious impossibility.

Now, continuing with the earlier analysis, thesimplest form of the law of inertia was obtained byGalileo, that the velocity of each (force-free) bodyis maintained in direction and magnitude, both.Here, the word inertia refers to the tendency of aphysical body to oppose a change in its state ofmotion. An inertial mass is then a measure of theinertia of a physical body.

Further, in those times, collisions were consid-ered as the simplest form of interaction of physicalbodies. For m as the inertial mass and v as the ve-locity of a body relative to an observer, Descartesformed the quantity of motion, mv, the linear mo-mentum, and asserted that it is conserved in a colli-sion. He, however, considered velocity and, hence,also the linear momentum as scalar quantities, forus today, an erroneous notion.

Huygens, on the other hand, realized correctlythat the sum (formed with correct signs) of mvhas the same value before and after a perfectlyelastic collision, and that the sum of mv2 is alsoconserved in such a collision.

Here entered Newton. He adopted the geometricapproach in defining appropriate physical quan-tities for physical bodies. Newton therefore con-ceptualized velocity and, hence, linear momentumalso, as vector quantities. From this, Newton de-veloped (difference) calculus needed to deal withthe notion of velocity of a body as a tangent toits path. On this basis, Newton then proposed hisfamous three laws of motion.

The concept of a mass point was introduced byNewton as it was needed in his geometric approach.(If not mass point, then what would move withthe tangential velocity along the one-dimensionalpath?) We note, however, that, for Galileo, thenotion of an inertial mass was that of the measureof the inertia of a physical body.

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In Newton’s Principia, two pronouncements ap-pear: first, the rate of change of vector of (linear)momentum of a mass point per unit time equalsvector of the force acting on it, and the second,the forces between two mass points are equal andopposite. Consequently, the interaction of an ar-bitrary number of mass points never changes theirtotal momentum, it remains constant for any sys-tem not subject to external forces.Then was formulated the law of conservation of

angular momentum which is one of the importantconsequences of the law of conservation of linearmomentum. In Newton’s formulation, it is alsobased on the concept of a mass point because the

angular momentum is a vector quantity: ~ℓ = ~r× ~pwhere ~r is the radius vector of the mass point rela-tive to a given fixed point, ~p = m~v is the vector ofits linear momentum and we use the cross productof vectors ~r and ~p to obtain the angular momentum

vector ~ℓ of the mass point.Simultaneously, Newton also formulated his fa-

mous (inverse square) law of gravitation, furnish-ing us the gravitational force of attraction betweentwo mass points. Here, Newton introduced a newnotion of the source properties of a particle. Pre-cisely, if M is the source or gravitational mass ofone particle and m is the inertial mass of anotherparticle located at distance d from the first parti-cle, then the gravitational force of attraction (pro-duced by M and acting on m) is given by the fa-mous expression:

~Fg = −GmM

d2d

where G is Newton’s constant of gravitation and dis a unit vector along the line joining the two par-ticles with origin of the coordinates at the locationof the gravitational mass M .It is essential to distinguish between the inertial

mass and the gravitational mass of the newtonianparticle. These two are conceptions of very differ-ent physical origins.However, as first shown by Galileo’s experiments

at the leaning tower of Pisa, the inertial and thegravitational mass of a physical body are equal toa high degree of accuracy. That is an experimentalresult. But, the fact that these two quantities areequal is to be recognized as an assumption of thenewtonian theory. (This recognition played a cru-cial role in the formulation of the General Theoryof Relativity.)Newton’s three laws of motion and his law of

gravitation then provided us a complete solutionto the problem of motions (mechanics) of physi-cal bodies as mechanical systems, as collections ofnewtonian particles.

Here, the newtonian third law of motion, the lawof equality of action-reaction pair, and his law ofgravitation show very distinctly that his mechanicsassumes action at a distance. Of course, there isnothing objectionable in this and it is beside thepoint as to whether Newton himself considered hislaw of gravitation as some sort of approximationto be replaced eventually by a law incorporatingfinite speeds of propagation.

On the basis of the mathematical formulationdeveloped by him, Newton could then calculatethe planetary motions. Newtonian mechanics ob-tained its major confirmation from Kepler’s as-tronomical observations. Moreover, all the othergreat achievements of Newton’s work, the theoryof tides, the equilibrium configurations of rotatingbodies, the calculation of the speed of sound etc.lent credence to various laws of conservation, of lin-ear momentum, of angular momentum and, hence,also to this monumental newtonian formulation ofmechanics of physical bodies.

There however existed one problem with thenewtonian framework, that of optics - the theoryof light. Newton’s corpuscular theory for light didnot explain every phenomenon of light, and New-ton recognized this. In particular, the existence ofumbra and penumbra indicated that the light cor-puscles penetrated “forbidden” region in the shad-ows behind objects illuminated by light. Forbiddenregion exists as Newton’s laws show that a particleof light should not move there.

These problems of optical phenomena got ne-glected and did not hinder in any way with thescientific developments of mathematical character,of those times, in newtonian mechanics. Some, inparticular, Huygens, however considered these anddeveloped the wave theory of light which could ex-plain the optical phenomena.

Primarily, one of the simplest forms of a gen-eral law of Nature is to assert the conservation ofsome particular physical quantity. That a givenphysical quantity is really subject to a conservationprinciple is, however, to be decided only by exper-imentations. Different experimentations then con-firmed the conservation principles obtainable fromthe newtonian mechanics.

That is why the impact of the newtonian formu-lation of mechanics completely overshadowed thedevelopments in Physics for the next few centuries.That the mathematical structure of the newtonianmechanics, erected by many others after Newton,required no new experiments or observations is tes-timony enough to say that the physical foundationlaid by Newton was completely sufficient to sup-port it. This led the physicists of those times to(erroneous) conviction that all of physics could bereduced to newtonian mechanics.

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Of course, the newtonian concept of a mass pointhad been the basis of these developments becauseNewton’s laws of motion presuppose it. As we havealready noted, this mass point is a notion derivedfrom the (point-wise) geometric approach to (one-dimensional) paths of physical bodies. Then, wehave also noted earlier that the galilean notion ofinertial mass is simply that of the measure of theinertia of a physical body.The concept of kinetic energy, 1

2mv2, was then

proposed and developed by Bernoulli (who gaveus the word “energy”) and Euler. That a changein this quantity in a closed mechanical system didnot at all result in a reduction in its “capacity ofaction” was emphasized by them.The stimulus for a generalization of the concept

of energy to include other forms of “energy” existedfrom the experiences with the thermal processes.That the kinetic energy or mechanical work couldbe lost while the temperature of the bodies underconsideration increased was known. Thus, the factthat kinetic energy could be transformed to heatwas known and that led to thermodynamical con-siderations of conservation postulates.J R Mayer provided the reasonably accurate the-

oretical value for the mechanical equivalent of heatand L A Colding obtained almost the same valuein his experiments involving friction.Independently of Mayer, Helmholtz developed

the principle of conservation of energy and its im-plications. Helmholtz derived his expressions forenergy directly from the impossibility of the per-petuum mobile. He, then, reached the concept ofpotential energy for mechanical systems, of the po-tential energy of a body experiencing gravitationalforce, of a charged body experiencing electric force,of a magnetized body experiencing magnetic forceetc. His energy considerations applied to the pro-duction of currents in galvanic cells, thermocouple,electromagnetic induction etc.Historically, Helmholtz’s considerations were not

generally accepted in the beginning. However, GJ Jacobi emphasized that in them we essentiallyobtain the logical continuation of the earlier ideasbehind the science of newtonian mechanics. Theinitial opposition to Helmholtz’s ideas graduallydisappeared and “energy considerations” becameimportant in Physics.In the words of William Thomson (Lord Kelvin

of Largs), we finally had: We denote as energy ofa material system in a certain state the contribu-tion of all effects (measured in mechanical units ofwork) produced outside the system when it passesin an arbitrary manner from its state to a referencestate which has been defined ad hoc. The words “inan arbitrary manner” embody the physical law ofthe conservation of energy.

Thus, the law of conservation of energy was,gradually, found to hold beyond the sphere of new-tonian mechanics. The same could also be calledthe situation with the law of conservation of linearmomentum, to begin with. It took a gradual whilefor this law to emerge out of the sphere of newto-nian mechanics but that needed new conceptionsin the form of locality of actions.

Although Newton’s laws involved action at a dis-tance, the collisions of a specific mass point withthe others in its immediate vicinity in a mechanicalsystem (of closely packed mass points) are think-able as local phenomena. A local disturbance couldthen propagate in such a mechanical system toother regions from the region of its origin. Theseconsiderations lead us to the calculation of thespeed of sound in such fluid configurations. Thisis exactly how finite speeds of propagation obtainin Newton’s theory, although all its basic laws arebased on the action at a distance.

Concept of locality of action and related ideasdeveloped from those of fluid properties of matter.Considerations of closely packed newtonian par-ticles led to development of concepts of density,pressure, fluxes, stresses etc.

We define density of newtonian particles (iner-tial masses) as a function of space coordinates andintegrate it over the volume under considerationto obtain the inertial mass contained within thatvolume. Motions of newtonian particles lead usto concepts of momentum transfer, pressure, etc.Here, the mathematical procedure for switchingfrom the particle picture to the fluid picture is awell-defined one, we note.

From the viewpoint of present mathematics, dif-ferent physical quantities are measurable functionsdefined over the underlying [continuum] space.Measure Theory tells us as to how to perform thenthe integration (averaging) procedures involved inabove mentioned considerations.

The principle of local action (as opposed to ac-tion at a distance) and that of finite velocity ofpropagation of disturbances of physical quantities,even in a vacuum, meaning a region with no (new-tonian) particles present in it, first gained promi-nence in electromagnetism. (It is this connotationof the word vacuum that is generally taken to beimplied by it and will be used here.)

Considerations of static electricity promptedCoulomb to propose the (inverse square) law ofelectrostatic force between two charged bodies.Similar to the occurrence of inertial/gravitationalmasses in Newton’s law of gravitation, Coulombintroduced the electric charge as a measure/sourceof the quantity of electricity in his law of force be-tween electrically charged bodies. Similar consid-erations were also used in magnetism.

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The motion of a charged physical body along a(point-wise) geometric path was again the pivotalconcept behind these laws. Therefore, of necessity,the electric charge became the intrinsic propertyof a newtonian point-particle.At this point, we therefore note that the new-

tonian particle is then endowed with two distinctsource attributes or properties, gravitational massand electric charge. The gravitational mass actsas a source of its gravitational force (defined asper Newton’s law of gravitation) while the electriccharge acts as a source of its electric force (definedas per Coulomb’s law).It then gradually emerged that the motion of a

charged body results into the (simultaneous) exis-tence of its magnetic force along with its electricforce. Ampere’s experiments and his laws of mag-netism associated with current (of charges) werethe reasons behind this realization.J C Maxwell then connected all these empiri-

cal laws to provide a sound mathematical founda-tion to the theory of electromagnetism. Maxwell’stheory then provided us the prediction of elec-tromagnetic radiation, an electromagnetic distur-bance propagating from the region of the source toother regions at the speed of light.Predictions of Maxwell’s theory of electromag-

netism were confirmed in numerous experiments.In particular, contributions of Faraday were note-worthy. Also, Hertz’s spectacular confirmation ofthe existence of electromagnetic radiation lent duecredence to Maxwell’s theory.For electromagnetic processes, Helmholtz’s en-

ergy methods yield merely a formula for the totalenergy. This is as long as one believed in action ata distance without a transmitting medium. But,the question of localization of action or disturbancethen lacked meaning.It is against this background that Michael Fara-

day developed the concept of field as the mediumtransmitting such localized action or disturbance.The field was considered as a change in the physi-cal state of a system which was essentially locatedin the dielectric. It is equally necessary to invokethe same conceptions for even the empty space be-tween the carriers of electric charge, electric cur-rents and magnets.Note that Maxwell’s theory does provide the

energy density (of the field) which is composedadditively of an electric and a magnetic term:1

8π (E2+B2), where E is the electric field strengthand B is the magnetic field strength. It is a nec-essary supplement of the field concept, field hasenergy associated with, and inseparable from, it.The question therefore arose of the nature of phys-ical processes involving this energy of the field ofFaraday’s conception.

J H Poynting then introduced the notion of a fluxof electromagnetic energy entirely on the basis ofthe mathematical formalism of Maxwell’s theory.He showed that there is a flux of electromagneticenergy whenever an electric and a magnetic fieldare present at the same time.

With this recognition, it is now possible to de-termine the route by which the chemical energy,which in the galvanic cell is transformed into elec-tromagnetic energy, gets to wire that completes thecircuit, where that energy gets converted into heat.Similarly, we can also trace the energy flow in a cir-cuit involving an electric motor that transforms itto mechanical work.

It was then recognized that the electromagneticenergy must also exist in the space intervening itsemitter and its absorber. Emitter looses energywhile the absorber gains energy only on the ab-sorbtion of radiation. Then, during the transit ofthe electromagnetic radiation from the time of itsemission to the time of its absorbtion, the sum to-tal of all energies can be constant only if we takeinto account the energy of radiation.

Similar considerations also apply for the law ofmomentum. The emitter of electromagnetic radia-tion experiences the opposite force of the absorberof radiation. But, during the transit of radiation,the electromagnetic radiation must carry the mo-mentum with it and “deposit” that momentum atthe absorber. In fact, Maxwell showed that a bodywhich absorbs a light ray experiences a force in thedirection of that ray. This must also hold for allelectromagnetic fields.

Henri Poincare showed that, if S denotes themagnitude of the flux of electromagnetic energy,then the field must contain momentum of the mag-nitude S/c2 per unit volume where c is the speed oflight. Electromagnetic momentum was then shownto be observable not only in phenomena with lightand heat but also with static fields.

As a matter of historical interest, this approachwas the cause of considerable difficulties and tooka long time to gain acceptance. The chief reasonsbehind the difficulties were the required general-izations of the laws of conservation of energy andof momentum.

Of certain importance is the conclusion of theinertia of electromagnetic energy that follows fromPoincare’s expression S/c2. If we displace a car-rier of electric charge, then the motion of the cor-responding electric field gives rise to a magneticfield, and their coexistence leads both to a currentof energy and momentum. This additional mo-mentum represents an additional inertial mass inthe system under consideration. For an electron,this electromagnetic mass is of the same order ofmagnitude as the observed mass.

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The newtonian picture of a mechanical systemof closely packed mass points had always been atthe background of these electromagnetic consider-ations. Then, the question arises: what consti-tutes field. It must be newtonian particles, eachone of some electromagnetic inertia, making upthe medium or the field. This medium was theether. Interactions of these ethereal particles wouldthen provide us the mechanical interpretation ofMaxwell’s equations.Also, the ether was required to be incompress-

ible since the electromagnetic radiation was onlyof transverse type. But, if a body moved throughether there must be observable effects of the pres-ence of ether on its motion. Such observationaleffects were fruitlessly sought.Ultimately, one got used to the concept of the

“field” existing independently of newtonian parti-cles. Thus emerged the field-particle dualism. Amaterial particle in Newton’s sense and the (elec-tromagnetic) field as a continuum existed side byside with the material particle acting as a source ofits field. The newtonian (source) particle appearedhere as a singularity of the (electromagnetic) fieldit generated around it.We owe this clear formulation of the field and the

particle dualism to H A Lorentz. In this formula-tion of Lorentz, the newtonian action at a distancegets replaced by that of the field which also repre-sented radiation.Disturbing here are two facts: firstly, kinetic en-

ergy (of a newtonian particle) and the energy ofthe field appear as physically unrelated entities,and secondly, the field energy carried inertia butnot that of a newtonian particle.An obvious question is then of the nature of the

inertia of the field energy. But, it is thinkable thatthe inertia of field energy is the same as the inertiaof a newtonian particle. Then, the concept of anewtonian particle would be simply that of a re-gion of special density of field energy. In that caseone could hope to deduce the concept of a particleand its equations of motion from that of only theequations of the field.Einstein then comments [1] about these ideas

as: H A Lorentz knew this very well. However,Maxwell’s equations did not permit the derivationsof the equilibrium of the electricity which consti-tutes a particle. Only other, non-linear field equa-tions could possibly accomplish such a thing. Butno method existed by which this kind of field equa-tions could be discovered without deteriorating intoadventurous arbitrariness.In other words, Lorentz was aware of the above

mentioned disturbing facts and clearly recognizedthat the total inertia of a newtonian particle couldpossess origin in the field conception.

However, the problem Lorentz faced was thatof the linearity of Maxwell’s equations. Solutionsof (linear) Maxwell’s equations obey superpositionprinciple. Then, one could always superpose re-quired number of solutions to obtain the solutionof any assumed field configuration. The newtonianparticle would still continue to be the singularity ofthe final field configuration. Therefore, there wereno means here of removing all together the newto-nian particle that had the nature of the singularityof the field it generated.

Some non-linear field equations could conceiv-ably possess singularity-free solutions for the field.Solutions of such (non-linear) field equations wouldalso not obey the superposition principle. Then,one could hope that these (intrinsically non-linear)equations for the (total) field would permit someappropriate treatment of newtonian particles assingularity-free regions of concentrated field en-ergy. But, an obviously vexing question was nowthat of the appropriate (non-linear) field equationsof this type. There did not exist (with Lorentz) anyphysical guidelines (principles) for getting to these(non-linear) field equations.

This however does not mean that such an ap-proach is an impossibility. In fact, it is a logicalcontinuation of the newtonian framework in a def-inite sense. This is what led Einstein to expressthe following optimism.

Einstein remarks [1] that: In any case one couldbelieve that it would be possible by and by to find anew and secure foundation for all of physics uponthe path which had been so successfully begun byFaraday and Maxwell. —

Then, we recall here that the galilean notion of(inertial) mass of a physical body is that of themeasure of the tendency of a physical body to op-pose a change in its state of motion. It is the(point-wise) geometric approach of Newton thatactually gave us the concept of a mass point or anewtonian particle in Physics.

By giving up the point-wise geometric pictureof Newton as basic approach (but retaining it insome appropriate form), one may still hope to treatinertia in the galilean sense of it being a measureof the tendency of a physical body, an extendedregion of the field energy, to oppose a change inits state of motion. This is then the meaning ofEinstein’s above mentioned optimism.

The question is, of course, of reaching the (non-linear) field equations of appropriate nature with-out venturing into meaningless arbitrariness. Wethen need some definite physical principles to reachto the non-linear field equations of the desiredtype. This is what we turn to. However, we notethat the route to appropriate field equations hadbeen long and difficult.

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Now, as separate remarks, we firstly recall thatthe equality of the inertial mass and the gravita-tional mass is certainly an assumption of the new-tonian mechanics. Newton’s theory then does notoffer any explanation of this experimental result.This is apart from the fact that Newton’s theorydoes not provide satisfactory explanations of theoptical phenomena. Huygens’s wave theory of lightthen has existence separate, meaning independent,from that of Newton’s theory.

Secondly, from a geometric point of view, all co-ordinate systems are among themselves logicallyequivalent. But, Newton himself had realized (thebucket experiment) that the validity of his lawsof motion (for example, the law of inertia) is re-stricted to only certain types of such reference sys-tems, the inertial frames of reference. Why thisspecial status to inertial frames of reference amongall the possible others? This fact therefore neededan explanation.

Newton’s analysis (of the bucket experiment)then showed, quite to his own dislike, that thisexplanation required the introduction of absolutespace as an omnipresent active participant in allmechanical events, but the one which is not af-fected by the masses and their motions. Else, hislaws, in particular, the law of inertia, could nothave any physical content.

Newton himself could not resolve this impasseand nor could any one else in his own times. Lateron, it was also thought for some while that theether provided this absolute space of Newton’s the-ory. But, the ether, as a mechanical system madeup of newtonian mass points, must get affected bymotions of masses.

It was then soon realized that, since the massesand their motions did not affect the absolute space,there could not be any way of establishing the ex-istence of the absolute space. The absolute spacethen came to possess the ghostly existence in thenewtonian framework.

Furthermore, in Newton’s theory, there are es-sentially two distinct concepts: firstly, that of thelaw of motion and, secondly, that of the law offorce. The law of motion is then empty of contentwithout the law of force. One may then be temptedto ask if any (arbitrary) law of force could work.But, Newton’s law of gravitation and Coulomb’slaw of electrostatic attraction possess, both, only aspecific form: that of the inverse-square type. Thenewtonian theoretical framework but provided noexplanation of this fact.

In essence, we therefore gradually came to rec-ognize that various of the basic concepts of thenewtonian theoretical framework needed to be re-placed by suitable others.

III. SPECIAL RELATIVITY ANDSPACETIME

It is then History that Einstein realized: manyof the simultaneous problems of electromagnetismand galilean principle of relativity of newtonianmechanics could be resolved by invoking a prin-ciple of the constancy of the speed of light in allthe (inertial) frames of reference.

Maxwell’s equations do not permit a situationof spatially oscillatory electromagnetic wave, the“standing” electromagnetic wave. An electromag-netic wave always travels and, in vacuum, withthe speed of light. Moreover, various experimentsdid not indicate the existence of any such stand-ing electromagnetic wave. Then, Maxwell’s equa-tions had experimental verification of certain im-portance. Therefore, the galilean principle of rel-ativity (the galilean coordinate transformations)needed to be modified suitably so that no stand-ing electromagnetic wave could be observed by anyinertial observer.

That is to say, what are needed here are some“new” transformation laws (of coordinates) whichkeep Maxwell’s equations invariant. Then, nostanding electromagnetic wave would be obtainedin all the (inertial) frames of reference if it can-not be obtained in one frame of reference, that ofMaxwell’s standard equations.

The coordinate transformations needed for thispurpose were already available to Einstein in theform of the Lorentz transformations. This situa-tion then led him to understand in a clear mannerwhat the spatial coordinates and the temporal du-ration of events meant in the (new) formulationwith the Lorentz transformations.

Einstein’s analysis led him to abandon the no-tion of the absolute simultaneity of events and tothe demand that the Laws of Physics remain in-variant under the Lorentz transformations. Thisis similar in spirit to the demand of the newtoniantheory that the Laws of Physics be invariant underthe galilean transformations.

Minkowski’s important contributions then madeit possible to write the Laws of Physics in a math-ematical form that ensured their invariance underthe Lorentz transformations.

Now, to the criticism of the basis of the SpecialTheory of Relativity.

Firstly, this theory provides us a (point-wise)geometric description of physical bodies as masspoints. This is evident from the laws of motion inSpecial Relativity.

Secondly, the law of the force is also to be sep-arately stated and, hence, the relevant criticism ofthe newtonian mechanics also applies.

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Thirdly, as Einstein himself recognized [1], thespecial theory of relativity treats, essentially inde-pendently, two kinds of physical things, (1) mea-suring rods and clocks and (2) all other things, likethe electro-magnetic field, the material point etc.This is unsatisfactory.

Surely, measuring rods and even clocks must bemade up of material points. Then, measuring ap-paratuses would also have to be represented as ob-jects consisting of moving material configurations.Their treatment in Special Relativity is then not aconsistent one.

Next, the special status of the inertial frames ofreference among all the possible others remains un-explained in Special Relativity. Therefore, the cor-responding criticism of the newtonian theory (ab-solute space) also applies to the theory of specialrelativity as well.

Next, this theory does not explain the (observed)equality of the inertial mass and the gravitationalmass, although optical phenomena are explainedby way of Maxwell’s equations in a consistent man-ner in this theory.

Clearly, then, the theory of special relativity isnot an entirely satisfactory one since it leaves somuch completely unexplained.

This, however, does not mean that it is not astep forward in the direction of some satisfactorytheory. There are certain noteworthy and impor-tant achievements of Special Relativity.

In this theory, the inertial mass m of a closedsystem is identical with its energy E, ie, E = mc2

where c is the speed of light in vacuum. Further,the inertial mass m is dependent not only on therest mass of the particle but also on its velocity:

m = mo/√

1− v2/c2

where mo is the rest mass of the particle. Thisvariation ofm with velocity is a direct consequenceof the Lorentz transformations.

Conceptually, if a physical body were to “move”in a “field” surrounding it, that body should “dis-play” opposition to a change in its state of motionand this opposition may be expected to depend onits “state of motion” - the velocity that body pos-sesses. Recall that the inertial mass of a body is ameasure of its tendency to oppose a change in itsstate of motion.

As a consequence, the principles of conservationof linear momentum and the conservation of energyare here fused into one thing.

It also clearly recognizes that the coordinateshave no absolute physical meaning. This recogni-tion is a very significant and important step awayfrom that of Newton’s theory.

Anyway, it was very clear to Einstein [1] thatthe Theory of Special Relativity was only a tran-sition from the newtonian framework to incorpo-rate electromagnetism in a consistent way. Thatthe special relativistic laws are linear laws obey-ing superposition principle for their solutions wasindication enough for him that it is only a tran-sitory phase in the formulation of an appropriate(non-linear) theory of the field.

However, what emerged in the special theoryof relativity is the Minkowskian four-dimensionalcontinuum indicating clearly that the coordinateshave no absolute physical meaning. But, differ-ences of coordinates had some definite physicalmeaning in this theory, that of physical length andof duration of physical time.

It then gradually became clear to Einstein [1]that even the differences of coordinates need notpossess any absolute physical meaning. It was notany “easy to come by” realization. This is whatwe turn to next.

IV. GENERAL RELATIVITY ANDCURVED SPACETIME

There were many problems with the theory ofspecial relativity and one could easily have chosento rectify them individually starting from differentpossible perspectives. However, as is well known,Einstein chose to concentrate only on one of them:the problem of the equality of the inertial and thegravitational mass.

This fact is not just mere luck. Einstein’s lineof argument has been clearly stated [1] by him asthe following one. Firstly, one can see very clearlythat, in the special theory of relativity, if the in-ertial mass depended on the velocity of a materialpoint, then its gravitational mass would also de-pend on its velocity (kinetic energy) in exactly thesame manner if, of course, the equality of the iner-tial and the gravitational mass holds. This equalityis but known experimentally to be true to a highdegree of accuracy.

Then, the weight of a physical body would de-pend on its total energy in a precise manner sincethe inertial mass now includes also the kinetic en-ergy of the body. In essence, the acceleration ofa material system falling freely in a given gravita-tional field is then independent of the nature (totalenergy) of the falling system.

It then occurred to Einstein [1] that: In a gravi-tational field (of small spatial extent) things behaveas they do in a space free of gravitation, if one in-troduces in it, in place of an “inertial system,” areference system which is accelerated relative to aninertial system.

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The above is a statement of the equivalence prin-ciple, that represents the equality of the inertialand the gravitational mass. Then, in a real gravi-tational field, it is therefore possible to regard anappropriate reference frame, the frame of referenceof a falling body, as an “inertial” frame of refer-ence. The concept of the inertial frame becomescompletely vacuous, then.The equivalence principle then implies that the

demand of the Lorentz invariance of physical lawsis too narrow. Therefore, it is necessary to demandthat the physical laws be invariant relative to non-linear transformations of coordinates in the four-dimensional continuum. This is then the principleof general covariance.Locally, meaning in a small region of the four-

dimensional continuum, the laws of special relativ-ity must, however, hold in an approximation. Inthe context of modern terminology, a general four-dimensional spacetime manifold is then required tobe locally Lorentzian. The Lorentz group is there-fore a subgroup of the group of general coordinatetransformations.Einstein then raised [1] the following two ques-

tions: Of which mathematical type are the variables(functions of the coordinates) which permit theexpression of the physical properties of the space(“structure”)? Only after that: Which equationsare satisfied by those variables?Interestingly, he writes below these questions in

[1] that: The answer to these questions is todayby no means certain. (That was in 1949, somedecades after the (standard) field equations of gen-eral relativity were given by him!)We now trace here the path [1] that Einstein

chose to reach the standard (Einstein) field equa-tions of general relativity.Although we do not, to begin with, know which

mathematical type are the variables of the theory,we do know with certainty one special case, thecase of “no gravitational field” - the spacetime ofthe special theory of relativity.The line element of the Minkowski space is

ds2 = − dx2

1+ dx2

2+ dx2

3+ dx2

4

for a properly chosen coordinate system. It is ameasurable distance between events.Now, referring to an arbitrary coordinate sys-

tem, the same (Minkowski) line element (metric)is expressible as

ds2 = gijdxidxj

where gijare the (real) metric coefficients forming

a symmetric tensor of rank two and i, j take valuesfrom 1 to 4. (Here, we have also used the (Einstein)summation convention.)

If the first derivatives of gij

do not vanish afterthe coordinate transformation, then there exists aspecial gravitational field in the (new) coordinatesystem in the sense that it is “accelerated” withrespect to the earlier coordinate system.

Detailed mathematical investigations of metricspaces were carried out by Riemann and variousrelevant conceptions existed before Einstein beganhis search for the variables of the physical proper-ties of the space structure.

From Riemann’s investigations of metric spaces,the Minkowski spacetime can be uniquely charac-terized as the one for which the Riemann curvaturetensor vanishes. That is, the Minkowski spacetimeis a flat manifold. Then, it can also be seen easilythat the path of a mass point (not acted upon byany force) in the Minkowski spacetime is a straightline and also a geodesic of the Minkowski space-time. A geodesic (straight line) is then already acharacterization of the Law of Motion in the (flat)Minkowski spacetime.

In Einstein’s words [1]: ”The universal law ofphysical space must now be a generalization of thelaw just characterized.”

He then assumes that there are two steps of gen-eralization, namely,

(a) pure gravitational field

(b) general field (in which quantities correspond-ing somehow to the electromagnetic field oc-cur, too).

The situation of (a), that of the pure gravita-tional field, then is characterized by a symmet-ric (Riemannian) metric (tensor of rank two) forwhich the Riemann curvature tensor does not van-ish. Now, if the universal field law is required tobe of the second order of differentiation and alsolinear in the second derivatives of the metric co-efficients g

ij, then only the vanishing of the Ricci

tensor comes under consideration as an equationfor the (vacuum) field.

It may also be noted that the geodesics of themetric can then be taken to represent the law ofmotion of the material point. Therefore, the lawof motion (of a mass point) is already incorporatedin that for the field.

Einstein [1] then goes on elaborating further: Itseemed hopeless to me at that time to venture theattempt of representing the total field (b) and toascertain field-laws for it. I preferred, therefore,to set up a preliminary formal frame for the rep-resentation of the entire physical reality; this wasnecessary in order to be able to investigate, at leastpreliminarily, the usefulness of the basic ideas ofgeneral relativity.

9

Now, let us therefore consider, following Einstein[1], this preliminary (formal) formulation of the(non-linear) field theory based on the principle ofgeneral covariance.Then, in Newton’s theory, the gravitational field

obeys the law (Poisson’s equation):

∇2Φ = 4πGρ

where Φ denotes the gravitational potential and ρdenotes the density of the sources generating thatgravitational potential.In general relativity, it is the Ricci tensor which

takes the place of ∇2Φ. Therefore, Einstein pro-posed that the preliminary formulation of generalrelativity be based on the equations

Rij−

1

2Rg

ij= − κT

ij

where Rij

denotes the Ricci tensor, R denotesthe Ricci scalar, κ denotes a proportionality con-stant and, importantly, T

ijdenotes the energy-

momentum tensor of matter. In so far as theseequations are concerned, the energy-momentumtensor does not contain the energy (or inertia) ofthe pure gravitational field.The form of the left hand side of these equa-

tions (geometric part) is chosen to be such thatits divergence in the sense of absolute differentialcalculus vanishes since a similar divergence of theright hand side (matter part) must vanish fromconservation principles.In this connection, Einstein expresses [1] his

judgement and concerns about these preliminaryequations as: The right side is a formal condensa-tion of all things whose comprehension in the senseof a field theory is still problematic. Not for a mo-ment, of course, did I doubt that this formulationwas merely a makeshift in order to give the generalprinciple of relativity a preliminary closed expres-sion. For it was essentially not anything more thana theory of the gravitational field, which was some-what artificially isolated from a total field of as yetunknown structure.Then, keeping in mind these important remarks,

let us consider these two preliminary formulationsof the above field equations.It is then History that within a few months

of Einstein’s publication of these field equations,K Schwarzschild obtained (under heroic circum-stances) the first solution of these highly non-lineardifferential equations in a spherically symmetriccase of pure gravitational field. The Schwarzschildsolution, a spherically symmetric spacetime geom-etry, represents the exterior or the vacuum gravi-tational field of a point mass.Further solutions of these field equations (and

history of some) can be found in [4].

However, noteworthy for us here are also theKerr-Newman spacetimes which represent a masspoint with electric charge and spin. Of course,when the electric charge and the spin are vanish-ing, the Kerr-Newman spacetimes reduce to theSchwarzschild spacetime.

We shall refer to this “preliminary formulation”as the “standard general relativity” since, over thedecades, it has indeed become one of the standardformulations of general relativity.

Since our purpose here is not to consider thehistory of these developments but rather that ofconsidering, partially, the history of the develop-ment of associated ideas, we shall now turn to thecriticism of this “preliminary formulation” or thestandard general relativity.

V. CRITICISM OF STANDARDGENERAL RELATIVITY

We recall that Einstein himself was never at easewith this standard general relativity. In [1], hesays: If anything in the theory as sketched - apartfrom the demand of the invariance of the equationsunder the group of the continuous coordinate trans-formations - can possibly make the claim to finalsignificance, then it is the theory of the limitingcase of the pure gravitational field and its relationto the metric structure of space.

However, “matter” is not any part of the theoryof the “vacuum” or pure gravitational field. Ein-stein, of course, recognized this but hinted onlythat it is the relation of the metric structure withthe field which holds in this case can possibly makethe claim to final significance.

Since physical objects must be made up of ma-terial particles, there simply cannot be physicalobjects, hence, measuring rods and clocks, in thiscase of the pure gravitational field. Therefore, thecriticism that Einstein himself levelled against thespecial relativity applies here.

It also does not therefore make “real sense” tosay that the geodesics of the field provide the law ofmotion for the material points since these are notthe part of the equations of the pure gravitationalfield. However, such a relation may be expected inthe “final theory” of the general field, the case (b)referred to earlier.

Because of this problem, we also cannot claimthat the equivalence principle has been incorpo-rated in the theory of the pure gravitational field.That is, the equality of the inertial and the gravi-tational mass is not explainable on the basis of thetheory of the pure gravitational field. This factthen invites the same criticism as that of the spe-cial theory of relativity.

10

Furthermore, because of the same problem, italso cannot be claimed that the formalism incor-porates the law of force, which must be separatelystated. Then, again, this fact invites the same crit-icism as that of the special theory of relativity andof the newtonian theory.Next, the inertia associated with the field energy

is also not the inertia of the material particle in thisframework. This is then very similar to the case ofMaxwell’s electromagnetism. Therefore, the rele-vant criticism of the field-particle dualism in thattheory also applies to this case of the pure gravi-tational field.But, what is more offending here is that, in this

case of the pure gravitational field, there definitelyare the laws for the field, but there simply cannotbe laws for the material particles generating thatgravitational field.To see this, notice that a mass point in this case

is, necessarily, a curvature singularity of the space-time manifold.A point-particle has existence only at one spatial

location and everywhere else, except at this speciallocation, it is the non-singular gravitational field ofthe particle that “exists”. In its spherically sym-metric spacetime geometry, Schwarzschild or theReissner-Nordstrom geometry, all the points, ex-cept for the location of the mass-point, are then“equivalent” to each other in the sense of matter.That is, there does not exist a material particle atany of these points.This gravitational field only diverges at the loca-

tion of the mass-point unless, perhaps, we includethe self-field in some way, which, of course, is notthe case with the equations of the pure gravita-tional field. Hence, this geometry should not “in-clude” the location of the mass-point.Then, any mass-point is a curvature singularity

of the manifold describing its “exterior” gravita-tional field which, like in the newtonian case, di-verges at the mass point.Addition of other source attributes of a physical

body, like charge, spin etc., does not change thissituation. The Kerr-Newman family is obtainable(in the case of the pure gravitational field) andthe point-mass with charge and spin is a spacetimesingularity in it.The issue of the inadequacy of such a descrip-

tion of a physical body would have arisen if theKerr-Newman family of spacetimes were not ob-tainable for the case (a) of the pure gravitationalfield. We could then have said that the pure mass-point is an inadequate description of a “physicalbody” and the addition of other attributes “takesus away from the case (a)” to some new situation,the case (b) of the general field, with some newdescription of a physical body.

Restricting to the case (a) of the pure gravita-tional field, we then note that, strictly speaking,a path of a mass point is not a geodesic since themathematical structure defining a geodesic breaksdown at every point along the path of the singu-larity. We may try to circumvent this problem inthe following ways.

We may surround the singular trajectory of apoint-particle by an appropriately “small” world-tube, remove the singular trajectory and call thistube the “geodesic” of the particle. But, the newspacetime is not the original spacetime and, by thephysical basis of general relativity, it correspondsto different gravitational source. Therefore, thisprocedure is not at all satisfactory.

We may, along a geodesic, change the spatial co-ordinate label(s), by which we “identify the singu-larity”, with respect to the time label of the space-time geometry and may term this change the “mo-tion” of a particle along a geodesic. However, thisis only some “special” simultaneous relabelling ofthe coordinates. But, a singularity in the space-time does not “move” at all.

In the case (a) of the pure gravitational field,the motion of a particle must be a “singular tra-jectory” in the spacetime if a particle, the singular-ity, “moves” at all. It is also equally clear that anysingular trajectory is not a geodesic in the space-time since such a trajectory is not the part of thesmooth spacetime geometry.

Then, strictly speaking, geodesics of a space-time do not provide the law of motion of particles.Thus, strictly speaking, we have to specify sepa-rately the law of motion for the particles. It is alsoclear that the equations of pure gravitational fielddo not (strictly speaking, cannot) specify this lawof motion for the particles.

But, mathematically speaking, there simply can-not be any laws for the motion of a geometric sin-gularity! That is why, in the case (a) of the puregravitational field, we have only the (non-linear)equations for the field but no equations of motionfor the sources of that field.

In Newton’s theory, the laws of motion for theparticles were Newton’s laws of motion, but therewere none for the field. The field laws had to bepostulated separately in the newtonian frameworkand it was possible to formulate such laws for thefield independently. This is what Maxwell’s elec-tromagnetism achieved.

Then, the case (a) of the pure gravitational fieldleads us to a situation which is the other extremeof that of Newton’s theory. We have the laws forthe field in this case and there are none for theparticles. But, it is more offending that, with it, wecannot even formulate separate laws for the motionof the mass points or particles.

11

To further clarify this situation, let us considera mass point executing a simple harmonic motionalong a straight line, the X-axis, say, in relation totwo other mass points located some distance awayfrom the first one along the X-axis. At t = 0, letthe first mass point be located at, say, x = 0 andthe second one at, say, x = x2 and the third oneat, say, x = x3.To be able to describe the simple harmonic mo-

tion of the first mass point in relation to othermass points under consideration, we need a space-time having two singular curves at x = x2 and atx = x3 which run parallel to the time axis (in theMinkowski-type diagram) and having another sin-gular curve, of sinusoidal character, running alongthe time axis.It is then thinkable that such a solution of the

field equations of the pure gravitational field existsand is obtainable explicitly. This is however notthe issue under consideration here.The issue is of the “cause” behind the harmonic

motion of a curvature singularity, and there cannotbe any laws providing us this cause of the motion.The solution to the field equations of pure gravi-tational field is then “ghostly.”That is why, if we continue to consider, with

complete disregard to this fundamental difficulty,the case (a) of the pure gravitational field, we arebound to face inconsistencies. That this is indeedthe case can be seen as under.Further, if there existed other particles, then the

spacetime geometry would be different from thatof a single point-particle, now having singularitiesat every location of particles in it. Then, a physi-cal body is, for our considerations, a collection ofcurvature singularities.Now, we may want to replace a collection of par-

ticles by that of their smooth (fluid) distribution.To achieve this, let us attach a “weight function”to each mass-point in our collection. Each of theseweight-functions must “diverge at a suitable rate”at its mass-point to balance the field-singularity atthat location. Then, the resulting “weighted-sum”should provide a smooth “volume-measure” as wellas a smooth source density.Mathematically, dµ = ρ(x) dµo, where µ is the

volume-measure of the smooth geometry, µo thatof the geometry with singularities and ρ(x) is therequired source density.But, any smooth ρ(x) is impossible if the geom-

etry of µo-measure has curvature singularities. Tokeep µ smooth, ρ(x) must diverge at the singu-larities. Thus, associated difficulties arise for con-structing a smooth spacetime here.Now, ignore also these difficulties and consider a

spherical star, using the fluid approximation, witha smooth spherical spacetime.

Consider two copies of such a star separated bya very large distance today. Now, the spacetimeof two stars taken together is not globally spheri-cally symmetric even though the spacetimes of sin-gle stars are globally spherically symmetric. How-ever, the evolution of each star would be “weakly”affected by the other distant star, an expectationjustifiable on general considerations.

Let stars collapse. Then, if we had some definiteoutcome in the collapse of the star, exactly thesame outcome is observable at locations of eachstar in the above (symmetric) situation.

We may add further copies of the same star, sep-arated by the same large distance from originalstars and from each other. If stars collapse, thesame end-result will be seen for each of the starsin this, arbitrary, situation too.

Now, if gravitational collapse leads to a nakedsingularity for the spherical spacetime, two totallyinequivalent descriptions obtain for a particle, thatit is sometimes naked and is sometimes covered bya horizon, an inconsistency.

There are then enough indications for us herethat the case of the pure gravitational field is aninternally inconsistent one to consider.

We then note that Einstein, in recognition ofsome of these problems, wrote in ([1], p. 675) that:Maxwell’s theory of the electric field remained atorso, because it was unable to set up laws for thebehavior of electric density, without which therecan, of course, be no such thing as an electromag-netic field. Analogously the general theory of rel-ativity furnished then a field theory of gravitation,but no theory of the field-creating masses. (Theseremarks presuppose it as self-evident that a field-theory may not contain any singularities, i.e., anypositions or parts in space in which the field-lawsare not valid.)

However, any considerations of the pure gravi-tational field in general relativity provide us theequations for the field but, in complete contrast toMaxwell’s theory, there clearly can never be any,not even faintest, possibility of our formulatingany dynamical laws for the sources of that field,may those be even independent of these field laws.These considerations provide us, similar to that ofNewton’s absolute space, not just a mere torso buta ghost [14] for us to deal with.

Recall that, since the masses and their mo-tions did not affect Newton’s absolute space, nomeans are possible of establishing the existence ofthe absolute space. Then, the absolute space hasthe ghostly existence in the newtonian framework.Similarly, the field equations of the pure gravita-tional field also posses a ghostly existence since thelaws of motions of the sources of the pure gravita-tional field are an impossibility here.

12

Without the laws for the motion of sources (gen-erating the field under consideration), there is nomeaning to the equations of the pure gravitationalfield because these equations for the field do notpermit any understanding of the physical situa-tions, as the above example shows.Now, we may be tempted to imagine that the

solutions of the equations of the pure gravitationalfield will be able to approximate the “true” situa-tion in some useful way.Unfortunately, this optimism ignores some very

important aspects of the general theory of rela-tivity, those related to the fact that its solutionsdo not follow any superposition principle. It maythen be noticed that the geometry with (smooth)matter fields cannot be approximated by the col-lection of those representing the pure gravitationalfield since the latter would possess the curvaturesingularities at the locations of sources in it whilethe former geometry would not.Some (apparently) physically reasonable results

could also be obtained using some solutions of theequations of the pure gravitational field. For ex-ample, the bending of light, the perihelion preces-sion, the prediction of gravitational radiation etc.The observations may also “agree” with such re-sults. However, such cases are the “pathologicalsituations” of definite kind.Just as we cannot conclude the correctness of the

newtonian corpuscular theory of light on the ba-sis of the explanation it provides for the existenceof penumbra on the assumption of “some suitablehypothetical force” acting on the light corpusclesmaking them enter the shadow of an object illumi-nated by light, we also cannot conclude that thesolutions of the equations of the pure gravitationalfield provide us physically reasonable explanationsof the observed phenomena.The meaning of the phrase “the equations of the

pure gravitational field are ghostly” is then thisabove. Clearly, any such physical explanations ofobserved phenomena must, therefore, be based onthe case (b) mentioned earlier.Of course, it may happen in some situations that

the mathematical expression for some phenomenaobtained on the basis of case (b) is “identical” withthat of the case of some solution of the equationsof the pure gravitational field. But, that is for thecase (b) to tell us.However, this does not mean that such solu-

tions of the pure gravitational field are some “goodapproximation” to the corresponding situation ofcase (b). Just as the explanation of the formationof penumbra on the basis of some suitable force act-ing on light corpuscles is “ghostly,” the solutionsof the equations of the pure gravitational field alsoremain “ghostly.”

In essence, one must then wait for the case (b) toprovide us such physically acceptable explanationsof various observed phenomena.

Such considerations of fundamental difficultiesthen prompt us to abandon the case (a) of the puregravitational field or of the field-particle dualism inGeneral Relativity, ie, the notion of a (curvature)singularity as a mass point or particle. This is asfar as the case (a) of the pure gravitational field isconcerned, then.

(We must therefore, unassumingly, seek the par-don of all those whose sincere as well as herculeanefforts gave us the many solutions of these highlynon-linear equations of the pure gravitational field.Unfortunately, being ghostly, these equations can-not, however, further our understanding of the Na-ture for the above reasons.)

At this place, we may also note that any (open orconcealed) increasing of the number of dimensionsfrom four does not change this situation. Clearly,all of the above fundamental problems associatedwith a (curvature) singularity will arise in higherdimensions in exactly the same manner as holdsfor the case of the pure gravitational field whichwe have considered above.

Therefore, unless there are some other specificreasons, it does not appear compelling to considerhigher dimensional situations.

Restricting to four dimensions, we then considernext the situation of Einstein’s “preliminary equa-tions” containing matter in the form of the energy-momentum tensor.

We then recall here the concern expressed byEinstein regarding these equations. The point ofEinstein’s concern [1] is that of the comprehensionof the concept of an energy-momentum tensor inthe sense of a field theory.

In this connection, we then note that the energy-momentum tensor is (usually) obtained on the ba-sis of only the particle considerations. Recall thatto obtain the related basic physical quantities weconsider a collection of particles. By defining var-ious physical quantities (such as, for example, theflux of particles across a surface), we average thesequantities over this collection of particles. Theserelated (averaged) quantities then help us definethe energy-momentum tensor.

(For a mass point as a singularity of the space-time, the notion of its motion is not mathemati-cally available. Therefore, it is clear that we wouldnot be able to define, for example, the flux ofsingularities across a surface. Hence, the energy-momentum tensor is not obtainable by consideringparticles as spacetime singularities.)

Since there cannot be any point-particle, it is notclear what we mean by a particle in General Rela-tivity. It is only after we have specified this concept

13

that the question of defining physical quantitiesand averaging them over a collection of particlescan be tackled.

Clearly, difficulties arise with the comprehen-sion of the energy-momentum tensor because it isnot clearly specified in the “field theoretical frame-work” of general relativity what exactly we meanby a “particle” (which, of course, cannot be asingular-particle now). Moreover, it is also notclear as to how one can unambiguously define theconcept of a particle in this “field-theoretic frame-work” of geometric character, except perhaps asan extended region of energy.

Then, the energy-momentum tensor of the Ein-stein field equations is not a well-defined conceptto begin with. Therefore, it is unclear whether theEinstein field equations can be formulated withoutfirst specifying what we mean by a particle. It isthen equally unclear as to what the solutions ofthese equations mean in the absence of a clear for-mulation of the concept of a particle (which, now,cannot be a singular-particle).

However, it should be clear by now that onlysmooth (singularity-free) spacetime geometries inGeneral Relativity need to be considered sincethe energy density should be non-vanishing every-where in a spacetime. Many such solutions of Ein-stein’s makeshift equations, spacetime geometries,are available in the literature [4].

The issue therefore arises of extracting physicalresults out of these spacetimes, in particular, inthe absence of the availability of the notion of par-ticle. Then, which of these smooth geometries areto be considered relevant to this Physics withoutthe field-particle dualism?

It may, however, be noted at this stage thatthere simply cannot be any considerations of“singular-particles” and “appropriate fluid de-scription approximating some collection of parti-cles” in all these spacetimes because of the reasonsalready considered by us earlier.

It may be stressed once again that such consider-ations are not justifiable in any manner. To stressthe same thing again, we note that the concept ofenergy-momentum tensor is itself not yet definedby us because we do not have any well-defined no-tion of what a particle is in this framework. Thisis the primary reason behind the current situationof the above kind.

However, the makeshift equations, the Einsteinfield equations, may provide us a useful startingpoint. This is true provided we use other physicalprinciples to some advantage. We therefore needto explore this issue further.

We will, for the time being, postpone these con-siderations of smooth spacetimes.

VI. QUANTUM AND ITS THEORY

Historically, the theory of the quantum owes itsorigin to the dualism of a particle and a wave. Itcan be traced to Newton’s times.

As seen earlier, Newton’s geometric considera-tions led him to postulate a mass point and hislaws (of mechanics and gravitation) are primarilybased on this conception of physical bodies. If ev-ery physical body were to follow these laws, thenit must be possible to treat it using the concept ofa mass point.

Therefore, Newton proposed the corpusculartheory for light. That a ray of light propagatesin a straight line, that it is reflected from a surface(of a mirror) etc. are then explainable on the basisof the corpuscular hypothesis.

However, in Newton’s own experiments in op-tics, it emerged that the corpuscular hypothesisdoes not explain, in a natural manner, all the phe-nomena displayed by light.

As an example, in the situation of an object illu-minated by light, umbra and penumbra form. Asa corpuscle, light is not expected to penetrate theregion forming the shadow of the object. Then,the existence of penumbra needed some corpuscu-lar explanation.

In another experiment, Newton observed “New-ton’s rings”. This experiment shows that corpus-cles of light gather in some regions forming the(bright) rings while they do not at all gather inother (dark) regions. This fact also needed somecorpuscular explanation.

On the basis of Newton’s laws, it then followsthat some force (acting on light) is causing this be-havior of the light corpuscles. However, this forcethen acts on only “some” corpuscles to pull themto the bright regions but not on some other lightcorpuscles (which escape in a straight path behindthe object, for example).

Furthermore, in the case of Newton’s rings, thereare more than one bright/dark such regions. Then,the force acting on the light must be (compara-tively) “larger” for some corpuscles and “smaller”for some other corpuscles.

Such an explanation is definitely thinkable. But,an important question is now that of the simplic-ity of this explanation. The simplicity of a theoryhas always been the driving impetus behind thescientific investigations.

It was clear to Newton that this explanationis not appealing. It requires us to postulate the“switching on and off” of the force whose strengthis also different for different corpuscles. Whatcauses such a behavior of this force? Is there someuniversal rational explanation for this?

14

There did not exist in the newtonian frameworkany conceivable explanations of such behavior ofthe force under consideration here.Huygens, on the other hand, considered these

phenomena from an entirely different perspective.His wave theory of light postulated undulatory be-havior for light on the basis of the continuum pos-tulate. Similar to waves on the surface of ocean,he imagined light as a wave phenomenon in some(unknown) medium.Huygens’s wave theory of light then provided

satisfactory explanations of the optical phenomenaon the basis of constructive and destructive inter-ference of the waves. The wave theory of light thengained wide acceptance.This situation persisted till almost the end of

the nineteenth century. Newton’s corpuscular the-ory for light was then forgotten or, at least, notconsidered seriously.Incorporation of various optical phenomena in

Maxwell’s electromagnetism was a climactic stagefor the wave theory. It showed that light consistsof oscillations of electric and magnetic fields trans-verse to the direction of a propagating light ray.The polarization of light then received an explana-tion here. (We also note that Newton’s corpuscu-lar theory had no explanation whatsoever for thispeculiar behavior of light.)It then came as a real surprise that the corpuscu-

lar behavior of light surfaced in some experimentsagain, of special significance is the photo-electriceffect discovered by Hertz in 1887.In Hertz’s experiments, a metal is subjected to

an incident radiation. Such a metal ejects elec-trons if the frequency of incident radiation is abovea certain threshold which depends on the metalproperties. The kinetic energy of ejected electronsdoes not, however, depend upon the intensity ofincident radiation but only on the difference of thefrequency of incident radiation and the thresholdfrequency of the metal.In the beginning, it was not at all clear as to

how one could explain this Hertzian photo-electriceffect in any manner. G Kirchhoff’s investigationsinto radiation had also provided us various lawsregarding the behavior of heat radiation interact-ing with matter. Efforts then began of providingexplanations of these empirical laws on the basisof Maxwell’s theory.No one realized that a stage was slowly getting

prepared for a fundamental crisis with Maxwell’stheory of electromagnetism in this era of hecticexperimental activities.In 1900, Max Planck’s remarkable intuition

in dealing with deep physical problems suddenlybrought into focus the seriousness of this crisis withclassical theories.

These investigations are all the more remarkablein that they were based primarily on only the the-ories regarding the behavior of heat radiation in-teracting with matter.

In what follows, we (partly) adopt Einstein’s(method of) exposition [1] of these theoreticaldevelopments, because his exposition remarkablyclearly describes the (theoretical) nature of thisfundamental crisis.

Then, let us first note that Kirchhoff had con-cluded, on thermodynamical grounds, that the en-ergy density and the spectral composition of radi-ation in a cavity (Hohlraum) with impenetrablewalls of absolute temperature T is independentof the material of the walls. That is to say, themonochromatic density, , of radiation is some uni-versal function of the frequency ν of radiation andof the absolute temperature T .

Thus arose the problem of determining this uni-versal function (ν, T ).

As per Maxwell’s theory, radiation exerts pres-sure on the walls of the cavity and this pressure isdetermined by the total energy density. From this,Boltzmann then concluded that the entire energydensity of radiation,

∫

dν, is proportional to T 4

and thereby provided a theoretical explanation ofStefan’s empirical law on the basis of Maxwell’stheory. W Wien then used ingenious thermody-namical arguments, also using Maxwell’s theory,to deduce that

∼ ν3 f( ν

T

)

Clearly, f is a universal function of only one vari-able ν/T and its theoretical determination is ofdefinite importance.

Basing his faith on the empirical form of thefunction f , Planck firstly succeeded in reaching thefollowing form for

=8π h ν3

c31

ehν/kT − 1

whereby he had two universal constants k and hin the expression for the monochromatic energydensity of the radiation.

If this formula were correct, it permitted the cal-culation of the average energy E of an oscillatorinteracting with the radiation in the cavity:

E =h ν

ehν/kT − 1

For fixed ν but high temperature, this gives E =kT - an expression obtainable from the kinetictheory of gases. From this theory, we have E =(R/N)T where R is the gas constant and N is thefamous Avogadro’s number.

15

Then, N = R/k and its numerical value agreedreasonably well with that of the kinetic theory ofgases. Therefore, Planck’s investigations were inagreement with the size of the atom since Avo-gadro’s number tells us about it.Using Boltzmann’s (and Gibbs’s) entropy meth-

ods, Planck then partitioned the total energy intoa large but finite number of identical bins of sizeǫ and asked in how many different ways can ǫ bedivided among the oscillators. This number wouldthen furnish the entropy and, hence, the tempera-ture of the cavity-radiation system.As is well known, Planck obtained the aforemen-

tioned radiation formula if ǫ = h ν. His expres-sion then yields correctly the Rayleigh-Jeans, theStefan-Boltzmann, the Wien laws.However, Planck’s reasoning camouflaged the

fact that his derivation demands that energy canbe emitted and absorbed by an oscillator only inquanta of energy ǫ = h ν.Planck’s formula implies that the energy of any

arbitrary mechanical system capable of oscilla-tions can be transferred only in “packets” or thesequanta. The same is also the situation with the en-ergy of radiation. This contradicts fundamentallythe laws of Newton’s mechanics as well as those ofMaxwell’s electrodynamics.Then, we note that Planck’s formula is compat-

ible with Maxwell’s electrodynamics, although itis not a necessary consequence of its equations.Therefore, the contradiction with Newton’s me-chanics is (more) fundamental here than that withthe electrodynamics.This was clear to Einstein soon after the ap-

pearance of Planck’s fundamental work. Althoughhe had no ideas on what framework(s) shouldsubstitute Newton’s and Maxwell’s theories, Ein-stein’s intuition nonetheless permitted him to ap-ply Planck’s formula to explain the photo-electriceffect in a remarkably simple fashion.If the implication of Planck’s reasoning were cor-

rect, then an electron can absorb the energy of ra-diation only in quanta. An electron (bound to anatom) could then be set free (ejected) on absorp-tion of energy of the incident radiation only if thefrequency of incident radiation were larger thancertain minimum corresponding to the binding en-ergy of the electron. There is a threshold then forthe frequency of incident radiation below whichelectron ejection does not occur in this hertzianphoto-electric effect.N Bohr’s remarkable insights developed the

quantum theoretical explanations for the empiricallaws of atomic spectra. Sommerfeld then also in-cluded special relativistic considerations in Bohr’stheory of the atom. These inputs were of radicaltheoretical nature indeed.

Einstein followed with interest these works and,in his own turn, revealed the deep connection be-tween Planck’s formula and Bohr’s law of frequen-cies, thereby introducing the probabilities of quan-tum transitions of atomic systems. This is wherethe Einstein coefficients for induced and sponta-neous transitions made their inroad.

This is the return, in a sense, of Newton’s cor-puscle - a particle of light and, hence, of the wave-particle dualism (for light). We will however notgo into details of other developments here. Instead,we turn to the next important step that was takenfor the concept of a quantum.

Next, L de Broglie proposed that if light, pri-marily an electromagnetic wave in Maxwell’s the-ory, displays particle-like phenomena, then parti-cles (of Newton’s type) should display wave-likephenomena in order that the basic symmetry ofwave versus particle is maintained.

This radical proposition, of course, needed ex-perimental confirmation and it was soon obtainedin diffraction experiments.

Einstein then generalized S N Bose’s statisticalmethods (for light particles) to particles indistin-guishable from each other. This goes in the nameof the Bose-Einstein statistics and the particlesobeying these statistical methods are now calledas the Bosons.

The fundamental difference between the statis-tical properties of like and unlike particles is inti-mately connected with the circumstance that, dueto Heisenberg’s indeterminacy relations, the possi-bility of distinguishing between like particles, withthe help of the continuity of their motion in spaceand time, is getting lost. Pauli’s analytical mindgrasped this fundamental issue immediately andthat is what led him to (Pauli’s) exclusion princi-ple for electrons.

Fermi and Dirac then showed that electrons,in particular, follow a different statistical methodsince they obey Pauli’s exclusion principle. Thisgoes in the name of the Fermi-Dirac statistics andthe particles obeying these statistical methods arenow called as the Fermions.

It is still a deep mystery as to whether the parti-cles of Nature are only of these two types, Bosonsand Fermions. If yes, why.

The need for replacement(s) of Newton’s and ofMaxwell’s theories became urgent.

E Schrodinger then formulated the Wave Me-chanics and, almost simultaneously, W Heisenbergformulated the Matrix Mechanics for the quantum.These works provided the bridge between the par-ticle and wave conceptions. M Born then providedthe “probability interpretation” of Schrodinger’sWave Mechanics and Bohr supported it with hiscomplementarity arguments.

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The phenomenon of barrier penetration (tun-nelling) was also discovered in such considerations.Special relativistic considerations were taken up byP A M Dirac and others. These provided us theconcept of particle and anti-particle pair. The im-plications of such considerations were “confirmed”in numerous experiments. Today, these considera-tions are the basis of numerous technological equip-ments and also of their theories.The method of second or field quantization was

subsequently developed. The primary thesis of thismethod is that we should be able to count the num-ber of quanta, if these are really the packets of en-ergy. It is this (second quantization) method thathas come to be recognized as the genuine theoryof the quantum.The theory of the quantum that developed as

a result was mathematically satisfactory but ex-tremely counter-intuitive. We then note here thatmany vigorous discussions and efforts were neededto reconcile the results, in particular, those con-cerning Heisenberg’s indeterminacy relations, withthe physical intuition. Rather than entering thesehistorical details, we then refer to various excellentessays, in particular, Bohr’s article and Einstein’sReply to Criticisms in [1].

VII. CRITICISM OF THE THEORY OFTHE QUANTUM

Let us first recollect that Einstein, in spite of hisinitial resistance as reported in Bohr’s essay in [1],finally agreed [5] to the correctness of Heisenberg’sindeterminacy relations. However, he differed frommost other of his contemporary physicists on theissue of probability being the only basis of under-standing the entire physical world.In Einstein’s own words [5] (my curly brackets):

... On the strength of the successes of this theorythey {Born, Pauli, Heitler, Bohr, and Margenau}consider it proved that a theoretically completedescription of a system can, in essence, involveonly statistical assertions concerning the measur-able quantities of this system. They are apparentlyall of the opinion that Heisenberg’s indeterminacyrelation (the correctness of which is, from my ownpoint of view, rightfully regarded as finally demon-strated) is essentially prejudicial in favor of thecharacter of all thinkable reasonable physical theo-ries in the mentioned sense. ...He further hastened to add to the above that:

... I am, in fact, firmly convinced that the essen-tially statistical character of contemporary quan-tum theory is solely to be ascribed to the fact thatthis [theory] operates with an incomplete descrip-tion of physical systems.

Still, it is undoubtable that the self-consistentformalism of the Quantum Theory provides us thetheoretical explanations of variety of experimentalresults. In fact, the amount of experimental data itsupports is so enormous, so unparalleled in the his-tory of Physics, that there must be some elementof the finality in its formalism.

Einstein did not want to leave behind any doubtsthat he did not recognize the importance of thecontributions from the Quantum Theory. So, headded further: ... This theory is until now the onlyone which unites the corpuscular and undulatorydual character of matter in a logically satisfactoryfashion; and the (testable) relations, which are con-tained in it, are, within the natural limits fixed bythe indeterminacy relation, complete. The formalrelations which are given in this theory - i.e., itsentire mathematical formalism - will probably haveto be contained, in the form of logical inference, inevery useful future theory.

Then, Einstein went on to tell us what exactlyit is that does not satisfy him with this Theory ofthe Quantum: What does not satisfy me in thattheory, from the standpoint of principle, is its at-titude towards that which appears to me to be theprogrammatic aim of all of physics: the completedescription of any (individual) real situation (as itsupposedly exists irrespective of any act of obser-vation or substantiation).

In what follow we recall (although not in a ver-batim manner) Einstein’s arguments in support ofhis above statement [1].

Let us then consider a radioactive atom as aphysical system. For practical purposes, we canconsider that it is located exactly at a point ofthe coordinate system. We may also neglect themotion of the residual atom after its radioactivedisintegration process in which a (comparativelylight) particle is emitted by the atom. Then, fol-lowing Gamow’s theory, we may replace the restof the atom by a potential barrier which surroundsthe particle to be emitted. The radioactive disin-tegration process is then the “tunnelling” of theparticle out of this potential barrier.

We then solve Schrodinger’s equation and obtainSchrodinger’s Ψ-function which is initially nonzeroonly inside the potential barrier, but which, withtime, becomes non-vanishing outside the barrier.Essentially, the Ψ-function yields the probability offinding the initially “trapped” particle to be out-side of the trapping barrier, at some later time, insome specific portion of the space outside of thatpotential barrier or the atom.

However, this does not imply any assertion of thetime-instant of the disintegration of the radioactiveatom. That is, no observable exists in the quantumtheory for this time-instant.

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Einstein then raised the question: Can this the-oretical description be taken as the complete de-scription of the disintegration of a single individualatom? The immediately plausible answer is: No,he wrote. For we are inclined to assume the exis-tence of an instant of the disintegration and sucha definite value of the time-instant is not impliedby the Ψ-function.Einstein then answered for a quantum theorist:

This alleged difficulty arises from the fact thatone postulates something not observable as “real.”That is to say, this difficulty arises because one isassuming the (reality of) time-instant of the dis-integration of an individual atom that is not anobservable of its (quantum) theory.Einstein then phrased the question as: Is it,

within the framework of our theoretical total con-struction, reasonable to assume the existence of adefinite time-instant of the disintegration of an in-dividual atom?Then, if one takes the viewpoint that the de-

scription in terms of a Ψ-function refers only to anideal systematic totality (ensemble) but not to anindividual system, then one may assume the ex-istence of the time-instant of disintegration of anindividual atom.But, if one represents the assumption that the

description in terms of the Ψ-function is a com-plete description of an individual system, then onemust reject the existence of the time-instant of anindividual atom and can justifiably point to thefact that a determination of the exact time-instantof disintegration is not possible for an individualatom. Any such attempt would mean disturbances(of the atom) of such nature which would then de-stroy the very phenomenon whose time-instant weare trying to determine.Now, following Schrodinger’s reasoning, one may

construct a contraption which kills a cat (macro-scopic object) sitting close to the radioactive atomonly if the decay particle is emitted by the atom.One may then ask: Is the cat alive or dead at somelater instant of time? Here, one would expect toget an answer with certainty since a beam of torch-light falling on the cat cannot be expected to dis-turb (kill it, if alive) its state. But, the theory ofthe quantum can only tell us the probability of thecat being alive or dead.Then, to begin with we have the Ψ-function for

an alive cat. With time, the Ψ-function is a super-position of two components, one for the alive catand one for the dead cat. It is only when an ob-server makes an observation (of the cat) that theΨ-function reduces or collapses to one of (thesetwo of) its components. This collapse of the Ψ-function represents the interference of an observerwith the system under observation.

Surely, the quantum theory as represented bySchrodinger’s Ψ-function is self-consistent. Now,if this probabilistic theory of the quantum is ofuniversal character, that is to say, if the basic lawsof nature are intrinsically probabilistic in charac-ter, then this formalism applies to microscopic aswell as macroscopic systems.

Some of the macroscopic systems constitute ex-perimental equipments and, hence, their behavioris also probabilistic in character then. By associ-ating Schrodinger’s Ψ-function with every physi-cal system, the theory of the quantum treats thenmeasuring apparatuses and all other things on anequal footing in its formalism.

But, a question can then be asked as to when,within this formalism of the quantum theory, canwe say that the “measuring apparatus” has madeits observation.

It is evident that this question is related to thecollapse of the Ψ-function because there is no an-other way of answering it within the realm of thequantum theory. Therefore, we have to say thatthe system makes an observation only when thecollapse of the Ψ-function takes place to one of itsvarious components, to that which uniquely corre-sponds to the value of the observable measured bythe macroscopic apparatus.

Then, we can consider an apparatus which hasbeen “suitably prepared” to measure a specificphysical observable of a certain physical system.But, let us not interfere with this apparatus in anymanner whatsoever and let us even not attempt to‘look” at the reading of the apparatus. But, now,let us ask a question as to whether the apparatushas “performed” the measurement of that quantityit were prepared to measure by “prearrangement ofsome special kind.”

For example, we prepare the apparatus to locatethe position of an electron in Heisenberg’s micro-scope and place a charge-coupled device (CCD) atthe eyepiece so that a photon, reflected after thecollision with an electron, makes a mark on it. Thisaction in Heisenberg’s microscope takes place, but,we choose not to look at that mark made by thatphoton on the CCD plate.

Then, the question is: whether this apparatushas performed the measurement of the position ofan electron, particularly when we have not lookedat the CCD plate.

Evidently, if the formalism of the quantum the-ory were to tell us that the above apparatus has“never” performed any observation, then the resultof the observation is not known to any observer,that is to say, the collapse of the Ψ-function hasnot occurred for any observer. This is indicativeof the importance of an observer in the frameworkof the quantum theory.

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On the other hand, if we say that an observationhas been performed in the above prearrangementwith Heisenberg’s microscope, then the Ψ-functionwill always be in some collapsed state because thephenomenon occurring in Heisenberg’s microscope(for that matter, any other apparatus) occurs ev-erywhere in the space. This then results in an ob-vious absurdity with the entire formalism of thequantum theory.Therefore, we are forced to assume here that no

observation is made in the concerned prearrange-ment with Heisenberg’s microscope or with anyother prearranged apparatus. An observer is there-fore of definite importance in this interpretation ofthe quantum theory.Now, an observer is also made up of the same

thing (matter) that an apparatus is made up of.Hence, the above situation also applies to an ob-server (as a measuring apparatus, may be of specialkind). It is therefore not specified by the quantumtheory as to what it means by an observer and,hence, what precisely constitutes an act of obser-vation within its formalism.Surely, although complicated constructs, as is

an apparatus or an observer, are not explicitly de-scribable, one can ascribe a corresponding total Ψ-function to them, thereby rendering the phenom-ena of the macroscopic world also probabilistic incharacter. Surely, the measuring apparatuses arethen at par with everything the quantum theorytreats. However, some questions of serious scien-tific concern then arise.Clearly, an observer is then given an exceptional

importance in the theory of the quantum but thistheory does not tell us what constitutes an ob-server and how does an observer act to collapsethe Ψ-function on having made an observation ofsome (quantum) system.Then, the issue arises as to where, in some

appropriate sense, does the collapse (of the Ψ-function) occur? How is it to be described in someunderstandable language? Many such questionscan be raised.In response to such questions, we may, for exam-

ple, then propose that the collapse occurs in themind of a conscious observer. Such an approachbut transgresses the obvious limits of the scientificinquiry and, hence, invites the corresponding crit-icism. This is then the problem of the collapse ofthe Ψ-function.To this date, there do not appear to exist any

genuinely satisfactory resolutions [15] of such para-doxes of the quantum theory.Schrodinger’s Cat Paradox and other paradoxes

therefore highlight such problematic issues of thequantum theory if this theory were assumed to beuniversal of character.

This is surely then a problematic issue for thequantum theory (description using Schrodinger’sΨ-function) if it is to be of universal character,that is, if its laws are to be universally applicableto all the physical systems.

Hence, from the point of view of a complete the-ory forming the basis of the totality of physicalphenomena, in a sense similar to that of Newton’stheory (supposedly) forming the basis for the to-tality of physical phenomena, we therefore note thefollowing important lacunae with the theory of thequantum described above.

Firstly, the theory of the quantum, as it standseven today, does not incorporate the explanationsneeded for the equality of the inertial and the grav-itational mass of a physical body, which is to betaken as an experimental result.

Secondly, this theory separates artificially “anobserver” (as definable in the theory of the quan-tum) and all other things that it treats since thereis no consistent demarkation line between whatconstitutes “an act of observation” in this theo-retical framework. The considered example of the“prearranged” Heisenberg’s microscope highlightsprecisely this issue.

In this connection, it cannot be forgotten thatthe measuring apparatuses are also made up of thesame things that the formalism of the theory issupposed to treat. Therefore, their treatment in atheory must be at par with everything else that thetheory intends to treat, this is if the theory claimsuniversality of its formalism. Then, we would notexpect to find any serious paradoxical situationsraised in the theory.

Then, we note that, in relation to the standardviewpoint regarding the probabilistic interpreta-tion of the laws of the quantum theory, seriousparadoxical situations have been constructed andthe resolutions of these paradoxical situations havenot been achieved. Clearly, the emphasis of theseparadoxes is on whether the quantum theory is ofuniversal character, whether the basic laws of na-ture are probabilistic of character.

If we assume that the quantum theory is of uni-versal character, then we end up with the paradox-ical situations of serious character, with the quan-tum theory not offering us any satisfactory resolu-tions of these paradoxes.

On the other hand, if we assume that it is notof universal character, then we must search for analternative formulation which evidently cannot bebased on the (probabilistic) formalism of the quan-tum theory. That is to say, the laws as are appli-cable to all the physical systems cannot then beprobabilistic of character. Question is then of suchan alternative formulation for the description ofthe totality of physical phenomena.

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It may therefore be noted here that various para-doxes of the quantum theory then indicate that [1]:The attempt to conceive the quantum-theoreticaldescription as the complete description of the indi-vidual systems leads to unnatural theoretical inter-pretations, which become immediately unnecessaryif one accepts the interpretation that the descrip-tion refers to ensembles of systems and not to in-dividual systems.Einstein [1] then continued: There exists, how-

ever, a simple psychological reason for the factthat this most nearly obvious interpretation is beingshunned. For if the statistical quantum theory doesnot pretend to describe the individual system (andits development in time) completely, it appears un-avoidable to look elsewhere for a complete descrip-tion of the individual system; in doing so it wouldbe clear from the very beginning that the elementsof such a description are not contained within theconceptual scheme of the statistical quantum the-ory. With this one would admit that, in principle,this scheme could not serve as the basis of the-oretical physics. Assuming the success of effortsto accomplish a complete physical description, thestatistical quantum theory would, within the frame-work of future physics, take an approximately anal-ogous position to the statistical mechanics withinthe framework of classical mechanics. ...Perhaps, there is an element of truth in these

(last quoted) Einstein’s opinions. Next, we shallsee that a general relativistic theory of the totalfield does offer such a possibility.

VIII. A THEORY OF THE TOTAL FIELD

Let us then return to the considerations of thesmooth spacetime geometries in general relativ-ity that are obtainable on the basis of Einstein’smakeshift field equations.As noted before, in this case, to extract physical

results in the absence of any conception of a par-ticle, which cannot be a point-particle, we need toemploy other physical principles to some advan-tage. That this is indeed possible is what is thesubject of the present section.It is of course not very clear to begin with as to

which physical principles to use to extract mean-ingful results in this situation. However, we mustlook for some hints here.To begin with, as we have seen before, consider-

ations of the “pure or vacuum gravitational field”lead us to an internal inconsistency in the formal-ism. Then, we note that the energy density of“matter” must be non-vanishing at every spatiallocation in any such (smooth) spacetime.

Also, the volume-form, in Cartan’s sense, is well-defined at every location in this (smooth) space-time. Consequently, just as it was permissible toattribute the concept of inertia to a point of spacein the newtonian situation, we should also be ableto define the concept of inertia for (every) spatiallocation of the (smooth) spacetime, of course, inonly some non-singular sense.

Then, a well-behaved (Cartan’s) volume-formcan be expected to allow us appropriate defini-tions of non-vanishing “gravitational,” “inertial”and also “total” mass for (every) point of the spacein such a spacetime. (See later.)

Next, the case (b) under consideration is, clearly,that of the general field in which quantities corre-sponding somehow to an electric field occur as well.Therefore, a point of the space in such a spacetimecan also be prescribed an electric charge as a sourceattribute of a physical body in a manner similar tothe case of the total mass attribute.

This must also be the case with all the permis-sible source attributes of a physical body since thesame procedure can be expected to work for eachof such source attributes.

Thus, it must be possible to incorporate all thesource attributes of a physical body in it. Then,in this spacetime, physical bodies are concentratedtotal energy [1] and would everywhere in space bedescribable as singularity-free. We may also lookat a physical body in the newtonian, now non-singular, sense of a point particle possessing thesource attributes as outlined above.

Next, any local motion of a physical body can,clearly, be a change in the local energy distributionin this spacetime.

However, the global properties of the spacetimemust not change with any “local” motions of aphysical body in it. Otherwise, infinite speeds ofcommunication exist in the spacetime. This is,physically speaking, undesirable.

A spacetime for which all the spatial propertiesare arbitrary has these desired properties. (Seebelow.) The required spacetime is given by [6]:

ds2 = − P 2Q2R2dt2 + P ′2Q2R2B2dx2

+ P 2Q2R2C2dy2 + P 2Q2R2D2 dz2 (1)

where P ≡ P (x), Q ≡ Q(y), R ≡ R(z), B ≡ B(t),C ≡ C(t), D ≡ D(t). We also use P ′ = dP/dx,

Q = dQ/dy and R = dR/dz.Now, consider an energy-momentum tensor

(with heat flux) for the fluid in the spacetime andEinstein’s (makeshift) field equations using it. Theenergy-density in the spacetime of (1) then variesas ρ ∝ 1/P 2Q2R2 and is seen [6] to be arbitrarybecause the field equations do not determine thespatial functions P , Q, R.

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But, we must remember that we do not knowwhether the energy-momentum tensor is abinitiodefinable one for this spacetime under considera-tion. (However, see later.) Thus, it is not truly jus-tifiable to use it to obtain the Einstein makeshiftfield equations for (1).However, what we realize here is that it is also

the 3-space, any constant-time section of (1), thatis endowed with various properties such as curva-ture, pseudo-metric nature etc.Let us recall here that the characteristic of New-

tonian Physics is that it ascribes independent andreal existence to space and time as well as to mat-ter. In this newtonian formulation, space and timeplay a dual role.Firstly, they play the role of a background for

things happening physically. Secondly, they alsoprovide us the inertial systems which happen to beadvantageous to describe the law of inertia. There-fore, if matter were to be somehow removed com-pletely, the space and time of the newtonian frame-work would “remain” behind.Let us also recall Descartes’s opposition to con-

sider space as independent of material objects [7]:space is identical with extension, but extension isconnected with physical bodies; thus there shouldbe no space without physical bodies and hence noempty space [16].We then notice that this expectation is indeed

true of the spatial sections of (1). Physical bodiesare “extended regions of space” in it.The issue, however, remains of incorporating

time in the framework of (1), in particular, inthe absence of a well-defined concept of a parti-cle. Here, Einstein’s makeshift equations provideus the required clue to this issue.The makeshift equations essentially provide the

laws for the temporal evolution of physical bod-ies in (1). Then, the makeshift equations can, inturn, be determined on the basis of the temporalevolution of physical bodies in (1).It may now be stressed that the makeshift equa-

tions are based on the as-yet undefined concept ofthe energy-momentum tensor for matter fields, un-defined because it is not clear as to how to definethe concept of a physical particle in this theoreticalframework as yet.It is therefore logically compelling to investigate

whether the second alternative, that of determin-ing the makeshift equations on the basis of the tem-poral evolution of physical bodies, extended spacein (1), is the realizable one.In fact, this last approach is really the (logically)

appropriate one since the temporal evolution of“physical bodies” in (1) is a mathematically well-definable concept (dynamical systems) while themakeshift equations are not.

We therefore abandon [2] the four-dimensionalspacetime in favor of a three-dimensional pseudo-Riemannian manifold admitting a pseudo-metric,called the Einstein pseudo-metric, given as:

dℓ2 = P ′2Q2R2 dx2

+ P 2Q2R2 dy2

+ P 2Q2R2 dz2 (2)

where, as before, P ≡ P (x), Q ≡ Q(y), R ≡ R(z)

and P ′ = dP/dx, Q = dQ/dy, R = dR/dz. Wedenote the space of (2) by the symbol B. Thethree spatial functions P , Q, R are initial data forthe space B. The vanishing of any of these spatialfunctions is a curvature singularity, and constancy(over a range) is a degeneracy of (2).

A particular choice of functions, say, Po, Qo,Ro is a specific spatial distribution of energy inthe space of (2). As some “concentrated” energy“moves” in the space, we have the original set offunctions changing to the “new” set of correspond-ing functions, say, P1, Q1, R1.

Then, “motion” as described above is, basically,a change of one set of initial data to another setof initial data with “time”.

Clearly, we are considering the isometries of (2)while considering “motion” of this kind. Then, wewill remain within the group of the isometries of (2)by restricting to the triplets of nowhere-vanishingfunctions P , Q, R. We also do not consider anydegenerate situations for (2).

If we denote by d, a metric function canonically[8] obtainable [17] from the pseudo-metric (2), thenthe space (B, d) is an uncountable, separable, com-plete metric space. If Γ denotes the metric topol-ogy induced by d on B, then (B,Γ) is a Polish topo-logical space. Further, we also obtain a StandardBorel Space (B,B) where B denotes the Borel σ-algebra of the subsets of B, the smallest one con-taining all the open subsets of (B,Γ) [9].

But, the Einstein pseudo-metric (2) is a metricfunction on certain “open” sets, to be called the P-sets, of its Polish topology Γ. A P-set of (B, d) istherefore never a singleton subset, {{x} : x ∈ B},of the space B. Note also that every open set of(B,Γ) is not a P-set of (B, d).

Now, the differential of the volume-measure onB, defined by (2), is

dµ = P 2Q2R2

(

dP

dx

dQ

dy

dR

dz

)

dx dy dz (3)

This differential of the volume-measure vanisheswhen any of the derivatives, of P , Q, R with re-spect to their arguments, vanishes. (Functions P ,Q, R are non-vanishing over B.)

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A P-set of the space B is then also thinkable asthe interior of a region of B for which the differ-ential of the volume-measure, (3), is vanishing onits boundary while it being non-vanishing at anyof its interior points.Any two P-sets, Pi and Pj , i, j ∈ N, i 6= j,

are, consequently, pairwise disjoint sets of B. Also,each P-set is, in its own right, an uncountable,complete, separable, metric space.A P-set is the mathematically simplest form of

“localized” total energy in the space B. This sug-gests that we should use set-theoretic concepts forit. One such concept is of a suitable (Lebesgue)measure definable on sets.We then recall that the Galilean concept of the

(inertial) mass of a physical body is that of themeasure of its inertia. Therefore, some appropriatemeasure definable for a P-set is the property ofinertia of a physical body, a P-set in question. Soalso should be the case with the gravitational massof a physical body. Such should also be the casewith other relevant properties of physical bodies,for example, its electric charge.Thus, we associate with every attribute of a

physical body, a suitable class of (Lebesgue) mea-sures on such P-sets. Therefore, a P-set is a physi-cal particle, always an extended body, since a P-setcannot be a singleton set of (B, d).Now, in the absence of the field-particle dualism,

the field and the source-particle are indistinguish-able. The source-properties are then also the field-properties. Thus, a P-set (the total field) and themeasures on P-sets (sources) are, then, are amalga-mated into one thing here, ie, are indistinguishablefrom each other, in a sense.Therefore, various source-properties (measures)

change only when the field (P-set) changes. Thisunion of the field and the source-properties is thenclearly perceptible here.Moreover, a given measure can be (Haar) inte-

grated over the underlying P-set in question. Theintegration procedure is always a well-defined onehere as a P-set, being an “open” subset of a con-tinuum, is a non-empty locally compact Hausdorffgroup in an obvious sense.The (Haar) integral provides then an “averaged

quantity characteristic of a P-set” under question.Of course, this is a property of the entire P-setunder consideration.For example, let us define an almost-everywhere

finite-valued positive-definite measurable function,ρ, on (B,B) and call it the energy density. Inte-grating it over the volume of a P-set, the resultantquantity can be called a total mass, m

T, of that

P-set under consideration. The total mass, mT, is

a property of that entire P-set and, hence, of everypoint of that P-set.

(Clearly, to define the notions of “gravitationalmass” and of “inertial mass” of a P-set, we need toconsider the “motion” of a P-set and also an appro-priate notion of the “force” acting on that P-set.Since we are yet to define any of these associatednotions, we call the integrated energy density as,simply, the total mass of the P-set.)

We note that every point of the P-set is thenthinkable as having these averaged properties ofthe P-set and, in this precise mathematically non-singular sense, is thinkable as a point-particle pos-sessing those averaged properties. It is in this non-singular sense that we can recover the notion of apoint particle in the present framework. That thisis indeed permissible in a mathematically precisesense is then an indication of the internal consis-tency of the present approach.

Clearly, the “location” of the mass mT

will beintrinsically indeterminate over the size of that P-set because the averaged property is also the prop-erty of every point of the set under consideration.We may then associate a Dirac δ-distribution withthe mass m

Tover that P-set.

Thus, “averaging a given measure” over any P-set and associating a Dirac δ-distribution with thataveraged measure, an intrinsic indeterminacy oflocation over the size of that P-set is obtained forthat averaged measure.

Now, any two P-sets of the same cardinality, be-longing either to the same metric-space (B, d) orto two different metric-spaces (B, d1) and (B, d2),are Borel-isomorphic [10]. Then, copies (P-sets) ofa physical particle are indistinguishable from eachother except for their spatial locations.

Let us reserve the word particle for a P-set sinceit is the simplest form of “localized” energy in thepresent framework.

Then, in a precise mathematical sense [8], setscan be touching and that describes our intuitivenotion of touching physical bodies. Of course, thecorresponding point particles are then “touching”within the limits of the sizes of the correspondingP-sets. (The size of a P-set then acts in the mannerof the de Broglie wavelength for a point particle, apoint of the P-set.)

Further, if a P-set splits into two or more P-sets, we have the process of creation of particlessince the measures are now definable individuallyover the split parts, two or more P-sets. On theother hand, if two or more P-sets unite to becomea single P-set, we have the process of annihilationof particles since the measures are now definableover a single P-set.

Clearly, the laws of creation and annihilation ofparticles will require of us to specify the corre-sponding transformations causing the splitting andthe merger of the P-sets.

22

Now, we call as an object a region of B boundedby the vanishing of (3) but containing interiorpoints for which it vanishes (so such a region isnot a P-set). Such a region of B is then a collec-tion of P-sets. But, a P-set is a particle. Therefore,an object is a collection of particles.Objects may also unite to become a single object

or an object may also split into two or more thantwo objects under transformations of P-sets. Wemay then also think of the corresponding laws forthese processes involving objects.Obviously, various concepts such as the density

of particles, a flux of particles across some surfaceetc. are then well definable in terms of the transfor-mations of P-sets and the effects of these transfor-mations on the measures definable over the P-setsunder consideration.Then, such “averaging procedures” are well-

defined over any collection of P-sets and, also, ofobjects. Thus, we may, in a mathematically mean-ingful way, define a suitable “energy-momentumtensor” [18] and some relation between the aver-aged quantities, an “equation of state” definingappropriately the “state of the fluid matter” un-der consideration.(Such conceptions require however the notion of

transformations of P-sets and objects. Moreover,this averaging is a “sum total” of the effects ofvarious such transformations of P-sets and objectsand, hence, will require corresponding mathemat-ical machinery. This is, then, the premise of theergodic theory. Recall that (B,B) is a StandardBorel Space.)Einstein’s makeshift field equations are then de-

finable in the sense (only) of these averages. There-fore, Einstein’s makeshift equations are “obtain-able” on the basis of the temporal evolution ofpoints of the space B, physical particles as elementsof the 3-space of B. This is also the sense in whichDescartes’s conceptions are then realizable in thepresent formalism. However, we will not pursuethis obvious issue of details here.Now, we can “count” P-sets and, also, objects.

This precise mathematical notion of countability ofP-sets and objects here then agrees well with ourvery general experience that arbitrary physical ob-jects (chairs, stones, persons etc.) are “countable”in Nature in an obvious sense.Moreover, the metric of (B, d) allows us the pre-

cise definition of the sizes of P-sets and objects.Then, given an object of specific size, we may useit as a measuring rod to measure “distance” be-tween two other objects.A measurable, one-one map of B onto itself is

a Borel automorphism. Now, the Borel automor-phisms of (B,B), forming a group, are natural forus to consider here.

Let a Borel automorphism φ of B act to take apoint x1 7→ y1 and point x2 7→ y2 where x1 ∈ P1

and x2 ∈ P2; P1, P2 being P-sets. Let y1 ∈ P ′

1

and y2 ∈ P ′

2 with P ′

1 and P ′

2 being the images ofP1 and P2 under φ. Then, the canonical distance“d” between P1 and P2 can evidently change underthe action of φ continuously (with respect to thecorresponding Polish topologies).

A Borel automorphism of (B,B) then induces anassociated transformation of (B, d), say, to (B, d′),and that “moves” P-sets about in B, since (suitablydefined) distance between the P-sets can changeunder that Borel automorphism.

We call this the “physical” distance separatingP-sets (as extended bodies). We also (naturally)define distance separating objects.

Now, the Borel automorphisms of B can be clas-sified as follows:

(1) those which preserve measures defined on aspecific P-set and

(2) those which do not preserve measures definedon a specific P-set

Note that we are restricting our attention to onlya specific P-set/Object and not every P-set/Objectis under consideration here.

Measure-preserving Borel automorphisms then“transform” a P-set maintaining its characteristicclasses of (Lebesgue) measures on a P-set, its phys-ical properties.

Non-measure-preserving Borel automorphismschange the characteristic classes of Lebesgue mea-sures (physical properties) of a P-set while “trans-forming” it. Evidently, such considerations alsoapply to objects.

It is therefore permissible that a particular pe-riodic Borel automorphism leads to an oscillatorymotion of a P-set or an object while preserving itsclass of characteristic measures.

We can then think of an object undergoing pe-riodic motion as a (physically realizable) time-measuring clock. Such an object undergoing oscil-latory motion then “measures” the time-parameterof the corresponding (periodic) Borel automor-phism since the period of the motion of such anobject is precisely the period of the correspondingBorel automorphism.

Then, within the present formalism, ameasuringclock is therefore any P-set or an object undergoingperiodic motion.

In the physical world, we do measure distancesand construct clocks in this manner. Then, cru-cially, the present formalism represents measuringapparatuses, measuring rods and measuring clocks,on par with every other thing that the formalismintends to treat.

23

Such considerations then suggest an appropriatedistance function, physical distance, on the fam-ily of all P-sets/objects of the space (B, d). Morethan one such distance function will be definable,depending obviously on the collection of P-sets orobjects that we may be considering in the form ofa measuring rod or measuring clock.This above is permissible since we are dealing

here with a continuum which is a standard Borelspace with Polish topology. Relevant mathemati-cal results can be found in [9].A Borel automorphism of (B,B) may change the

physical distance resulting into “relative motion”of objects. We also note here that the sets invariantunder the specific Borel automorphism are charac-teristic of that automorphism. Hence, such setswill then have their distance “fixed” under thatBorel automorphism and will be stationary rela-tive to each other.On a different note, an automorphism, keeping

invariant a chain of objects separating two otherobjects, can describe the situation of two or morerelatively stationary objects.Effects of the Borel automorphisms of (B,B) on

the physical distance are then motions of physicalbodies. Furthermore, various physical phenomenawill then be manifestations of relevant propertiesof such automorphisms.As an example, a joint manifestation of Borel

automorphisms of the space (B,B) and the associa-tion of a Dirac δ-distribution of an integrated mea-sure with the points of a P-set is a candidate reasonbehind Heisenberg’s indeterminacy relations in thepresent continuum formulation [19]. However, de-tails regarding these considerations are outside thelimits of the present article.But, intuitively, let it suffice to say that as

the size of the P-set gets smaller and smaller we“know” the position of the point-particle (of inte-grated characteristics of a P-set) more and moreaccurately. (But, recall that a P-set is never asingleton subset of B. So, complete positional lo-calization is not permissible.)Now, that P-set “transforms” as a result of our

efforts to “determine any of its characteristic mea-sures” since these “efforts or experimental arrange-ments” are also Borel automorphisms, not neces-sarily the members of the class of Borel automor-phisms keeping invariant that P-set (as well as theclass of its characteristic measures).Hence, any Borel automorphism (as an exper-

imental arrangement) purporting to “determine”a characteristic measure of that P-set changes, ineffect, the very quantity that it is trying to deter-mine. This peculiarity of the present continuumdescription then leads to Heisenberg’s (correspond-ing) indeterminacy relation.

Then, in the present continuum description, itis indeed possible to explain the origin of Heisen-berg’s indeterminacy relations. The present con-tinuum description provides us therefore an originof indeterminacy relations, an alternative to theirprobabilistic origin.

This circumstance is then extremely encourag-ing indeed. Essentially, it tells us that one of thefundamental characteristics, Heisenberg’s indeter-minacy relations, of the theory of the quantum hasan, indeed plausible, explanation in a general rel-ativistic theory of the continuum!

Notice then that, in the present considerations,we began with none of the fundamental consid-erations of the concept of a quantum. But, thepresent continuum formalism unfolds itself beforeus in such a manner that one of the basic charac-teristics of the conception of a quantum emergesout of the present formalism.

Notice also that, in the present continuum de-scription, we have essentially done away with the“singular nature” of the particles and, hence, alsowith the unsatisfactory dualism of the field andthe source particle. Furthermore, we have, simul-taneously, well-defined laws of motion (Borel au-tomorphisms) for the field and also for the well-defined conception of a particle (of integrated mea-sure characteristics).

Any particulate character or undulatory char-acter perceptible in a given physical phenomenonis then attributable to properties of Borel auto-morphisms of the space B, more precisely to aninterplay of Borel automorphisms simultaneouslyacting on the space B.

We refrain here from elaborating further on thisissue. However, intuitively, the particulate naturewould be perceptible in a phenomenon if the clas-sical newtonian concepts (eg., momentum) hold insome useful way. Otherwise, the undulatory na-ture of the (total) field would be perceptible inthat phenomenon. Any phenomenon is, of course,a result of the simultaneous interplay of Borel au-tomorphisms of B here.

All these achievements of the present general rel-ativistic (continuum) formulation cannot be with-out any element of the finality then.

The contention here is then the following: thatthe set of classes of (Lebesgue) measures on P-sets of (B, d) (as various attributes of a physicalparticle) and the group of Borel automorphismsof (B,B) (resulting into dynamics of P-sets) are,both, sufficiently large as to encompass the entirediversity of physical phenomena.

In a definite sense, this approach then providesthe theory of the total field of Einstein’s concep-tion [1]. It is therefore also a continuum theory ofeverything in that sense.

24

The question naturally arises as to where, inthe present formulation, are the various constantsof Nature such as Newton’s constant of gravita-tion, Planck’s constant, constant speed of lightetc. Here, we only note that these constants arisefrom relations of physical conceptions definablein this formulation. In some definite sense, thispresent situation is describable as obtaining Con-stants without Constants.As an example, we need to analyze here as to

how one obtains Newton’s law of gravitation inthe present formalism in order to obtain Newton’sconstant of gravitation. For this, we will need toconsider the “gravitational mass” measure of P-sets, and also analyze the tendency of this P-set tooppose a change in its state of motion, its inertiaas per the conception of Galileo.Any such analysis is, of course, based on the

appropriate subgroup of the group of Borel auto-morphisms of (B,B) as well as the association ofa Dirac δ-distribution (of an “averaged measure”)with a P-set. Such considerations are, of course,beyond the scope of the present article.However, some definite conclusions are ob-

tainable from very general considerations of thepresent formalism and, we now turn to a few suchconsiderations.Now, a Borel automorphism of (B,B) cannot

lead to a singularity of B and, hence, to any kindof “naked singularities” from which some null tra-jectory would reach other points of the manifold.Since the present formalism is logically compelling,this would then mean that the naked singularitieswould not be obtainable in the Physics without afield-particle dualism.Evidently, the Universe also does not originate

or end in any singularity as there also cannot beany such singularity of the space B in this frame-work. But, an Expanding Universe or a portionof B thereof, that may be hotter in the past thantoday but having no origin, singular or otherwise,is thinkable in it.This can be seen from the fact that certain Borel

automorphisms may move the P-sets about in thespace B in such a manner that these P-sets maymove away from each other, while simultaneouslypushing some other P-sets closer to each other.This is a very complicated picture but definitelythinkable nonetheless.Moreover, Borel automorphisms of (B,B) form

a group. Thus, we can always cross any 2-surfaceboth ways. To see this intuitively, let a Borel au-tomorphism of (B,B) produce motion of an ob-ject “into” the given 2-surface. However, the in-verse of that Borel automorphism, producing mo-tion pulling that object “out” of that 2-surface,exists always is the point here.

Therefore, there does not arise any one-waymembrane in the present formalism. It thereforedoes not seem that the concept of a black holeis any relevant to this Physics without the field-particle dualism. As noted earlier, the concept ofa black hole is a child of the “ghostly” equationsof the pure gravitational field.

Now, a P-set, evidently, can be of any size and,hence, objects can also be of any size. Therefore, aone-way membrane (black hole) is not but, ultra-compact objects are of relevance to this Physicswithout field-particle dualism.

As one more example, we consider Schrodinger’scat paradox. In the present formalism, some Borelautomorphism of base space B “causes” the dis-integration of the atom while the “measurement”of this time-instant can take place using the “suit-able structure” on the family of the P-sets of B.Therefore, the time-instant of disintegration of aradioactive atom is “well-defined as a parameter”of that Borel automorphism and, simultaneously,there is also the Heisenberg-type indeterminacy in-volved in its measurement.

There is then some kind of Two-Time formalismin consideration here. The time of disintegrationof a radioactive atom as the parameter of a Borelautomorphism of B can then be different than thatwhich is determinable.

Recall that we began with no considerationsof the quantum conceptions. The disintegrationof a radioactive atom is a quantum considerationand the probabilistic interpretation leads us to a“fuzzy” description of macroscopic system - a cat.That a possible resolution naturally arises here isthen remarkable indeed.

We now turn to some other aspects which wereso beautifully explained and elucidated by vonLaue [3] - those related to conservation principlesin physical theories.

As von Laue expressed [3] it: Mass is nothing buta form of energy which can occasionally be changedinto another form. Up to now our entire concep-tion of the nature of matter depended on mass.Whatever has mass, - so we thought -, has indi-viduality; hypothetically at least we can follow itsfate throughout time. But, this does not hold forthe elementary particles.

These remarks are then understandable in thepresent formalism since the individuality of parti-cles is that of the P-sets.

In the present formalism, a particle is a pointof the P-set of the space B with associated inte-grated measures defined on that P-set. As a Borelautomorphism of the space B changes that P-set,the integrated properties also change and, hence,the initial particle changes into another particle(s),since integrated measures change.

25

Of course, we then need to discover various lawsof such transformations of particles into one an-other in the present formalism. But, it is clear atthe outset that these will crucially depend on thestructure of the group of Borel automorphisms ofthe space B.Von Laue concluded [3] his article with the fol-

lowing relevant remarks: ... Can the notions ofmomentum and energy be transferred into everyphysics of the future? The uncertainty relations ofW. Heisenberg according to which we cannot pre-cisely determine location and momentum of a par-ticle at the same time - a law of nature precludesthis -, can, for every physicist who believes in therelation of cause and effect, only have the mean-ing that at least one of the two notions, locationand momentum, is deficient for a description ofthe facts. Modern physics, however, does not yetknow any substitute for them.We may then conclude the present section with

the following remarks.Here, the notion of energy is then that of the

integrated measure defined on a P-set of the spaceB. The notion of the momentum of a particle (as apoint of the P-set of B with associated integratedmeasures on that P-set) is then that of the appro-priately defined notion of the rate of change of thephysical distance under the action of a Borel au-tomorphism of B, including evidently any changesthat may occur to measures definable on that P-set. Therefore, the notions of energy and momen-tum of a particle are certainly (well-) definable inthe present formalism.Further, none of these two notions is any defi-

cient for a description of the facts since Heisen-berg’s indeterminacy relations are also “explain-able” within the present formalism. This expla-nation crucially hinges on the fact that the pointsof the space B can never be particles since these,as singleton subsets of the space B, are never theP-sets. It is only in the sense of associating themeasures integrated over a P-set that the points ofthe space B are particles.The measurable location of a particle is essen-

tially a different conception and that depends onthe physical distance definable on the class of allP-sets of the space B. The measurable momentumof a particle is also dependent on the notion of thephysical distance changing under the action of aBorel automorphism of B.Therefore, it does seem possible to combine the

virtues of the theory of the quantum and of thegeneral theory of relativity in a formalism thatdoes possess the notions of energy and momen-tum, both. Then, this formalism will further unitemore number of conservation laws than those al-ready united in special relativity.

Now, as seen earlier, physical processes occuras a result of the (combined effects of) the Borelautomorphisms of the space B and these includethe processes of quantum character. Then, it isthinkable that “some suitable statistical descrip-tion” is obtainable by “approximating” the effectsof these Borel automorphisms acting on B to pro-duce the concerned physical processes of quantumcharacter. This, intuitively speaking, can be con-sidered as the probabilistic description of the in-volved physical processes.

Hence, at this point, we also note that, in re-lation to the present formalism, the theory ofthe quantum as represented by Schrodinger’s Ψ-function can be expected to assume a place whichis similar to that of the usual statistical mechanicswithin the realm of the classical newtonian theory.(Details of these considerations are, once again,outside the scope of the present article.)

Provided that the description of physical sys-tems of Nature based on a continuum is permissi-ble, the approach [2] followed here in Section VIIIis also a logically compelling approach as this ar-ticle discussed.

Of course, many mathematical, physical detailsand their implications need to be worked out be-fore we can “test” the above described theoreticalframework vis-a-vis experimentation or astronom-ical observations. But, confidence may be voicedthat it will stand tests of experimentations and/orobservations since it is a logically compelling ap-proach as discussed here.

A comment on the mathematical methods wouldnot be out of place here. Then, we note that themathematical formalism of the ergodic theory iswhat is of immediate use for the physical frame-work of the present section. This much is alreadyclear from the present considerations.

However, it is not entirely satisfactory to use thepresent methods of ergodic theory. One of the pri-mary reasons for this state of affairs is the inabilityof the present methods in ergodic theory to let ushandle, in a physical sense, the P-sets. Some newermethods are then required here.

IX. CONCLUDING REMARKS

In the present article, we outlined the reasonsbehind the logically compelling character of theapproach adopted in Section VIII.

In doing so, we also critically examined the rea-sons behind the “failures” of other approaches toGeneral Relativity based on the equations of thepure gravitational field and Einstein’s makeshiftequations for the matter fields.

26

In particular, the equations of the pure gravita-tional field are “ghostly” in the sense of Newton’sAbsolute Space or in the sense of this approachproviding only the equations for the field withoutany possibility for the equations of motion for thesources of the field. These equations cannot there-fore provide the means to verify or test predictionsin any manner whatsoever.Then, any solution of these field equations of

the pure gravitational field is “ghostly” as well.Any conclusion of a physical nature obtained usingthese equations is therefore dubious.Further, Einstein’s makeshift equations for gen-

eral field are based on an ill-defined concept of theenergy-momentum tensor. This is mainly becausevarious concepts leading us to the notion of theenergy-momentum tensor are based primarily onthe concept of a point-particle which is not definedabinitio in a (dynamical) geometric approach suchas the one postulated and adopted by general rela-tivity. Clearly, a point-particle as a spacetime sin-gularity leads to the impossibility of the definitionof the energy-momentum tensor.The pivotal problem with the solutions of the

makeshift field equations is then that of their phys-ical interpretation in view of some continuum de-scription of physical systems. We cannot considerthese solutions as describing some “fluid” mattersince the conceptions of fluid properties are ill-defined to begin with. Therefore, it follows thatnot all the solutions of Einstein’s makeshift fieldequations are necessarily useful for the continuumdescription of physical bodies. Of particular men-tion is the Friedmann-Lemaitre-Robertson-Walker(FLRW) geometry of the standard model of thebig bang cosmology.(Therefore, we must, unassumingly, also seek

the pardon of all those whose sincere and her-culean efforts provided us the many solutions [4]of Einstein’s makeshift field equations. Clearly,these equations are based on ill-defined considera-tions. Furthermore, the solutions of these highlynon-linear partial differential equations do not fol-low any superposition principle. Therefore, it alsofollows that even if there existed some particularspacetime geometry useful for some specific contin-uum description of physical systems, it would notbe a superposition of other spacetime geometries.Other spacetime geometries are then rendered un-physical in this sense.)However, it also follows that only the smooth

spacetime geometries are the ones that need to beconsidered if the description of physical systems ispermissible on the basis of a continuum. The ques-tion then arises of choosing some particular space-time geometry that can provide us this continuumdescription of physical systems.

Clearly, the answer to this question requires useof some physical principles, not contained withinthe framework of standard general relativity (theframework of Einstein’s makeshift equations) sincethis theory permits many different smooth space-time geometries containing matter fields. This iswhat we considered in Section VIII.

In Section VI, somewhat separated section fromthe earlier considerations, we then considered de-velopments related to the theory of the quantumconception. In this section, we recalled Einstein’sexposition of the relevant developments of the con-cept of a quantum. In particular, we noted the pre-cise nature of problems with the (classical) newto-nian theories. We also noted that the contradic-tion with the laws of newtonian mechanics is ofmore fundamental nature than that with the lawsof Maxwell’s electromagnetism.

The theory of the quantum is based primarilyon the probabilistic conceptions. Schrodinger’s Ψ-function then provides us only the probability ofany physical event. As is well known, this thenleads to an indeterminacy in the simultaneous mea-surement of canonically conjugate (classical) vari-ables. This is the probabilistic origin of Heisen-berg’s indeterminacy relations.

This probabilistic nature of the theory of thequantum leads to paradoxical situations of seriousconcern if every physical system, microscopic ormacroscopic notwithstanding, obeyed probabilisticlaws. We then also considered, in Section VII, var-ious objections of such serious nature related tothis interpretation of the theory of the quantum.In particular, some of the serious objections raisedby Einstein [1] were quoted in details.

In view of the considerations of these objections,it is then possible to adopt the view that the the-ory of the quantum, Schrodinger’s Ψ-function, rep-resents an ensemble(s) of systems.

With this above point of view, the theory of thequantum may then be expected to assume, in re-lation to an appropriate general relativistic the-ory, one based on the principle of general covari-ance, for the case (b) of general fields, a place sim-ilar to that of the statistical mechanics within therealm of the classical newtonian framework. Then,various paradoxical situations arising in the the-ory of the quantum as embodied in Schrodinger’sΨ-function evidently disappear. Einstein [1] had,very clearly, perceived this situation with the the-ory of the quantum.

With this view point, within the theory of thequantum as embodied in Schrodinger’s Ψ-function,there is obviously no possibility of obtaining anynon-probabilistic theoretical framework for the de-scription of physical phenomena. Einstein had alsoclearly recognized this aspect.

27

Therefore, we had to look “elsewhere” for thetheoretical framework (obeying the principle ofgeneral covariance) that would provide us the the-ory of the case (b) of general fields. This is whatEinstein meant in his remarks which were quotedin the context of the relevant discussions.Therefore, on the basis of some few (physically

reasonable) principles (observed to hold in Na-ture), we developed the approach [2] outlined inSection VIII. This theoretical framework thenobeys the principle of general covariance, in around about manner, since the group of Borel au-tomorphisms of the space B is very large.We then saw that this formalism provides us a

clear possibility of explaining the laws of the quan-tum realm while simultaneously treating the con-cept of a particle in a non-singular manner. Italso shows us the possibility of visualizing the the-ory of the quantum as embodied in Schrodinger’sΨ-function to be of similar character to the usualstatistical mechanics.In particular, the (continuum) formalism of Sec-

tion VIII provides [11] a non-probabilistic expla-nation for Heisenberg’s indeterminacy relations.Furthermore, it also allows us satisfactory resolu-tions of different (serious) paradoxical situationsfaced by the theory of the quantum as embodiedin Schrodinger’s Ψ-function.Moreover, it is then also clear that many of the

current ideas in theoretical cosmology need drasticmodifications. Notably, there is no singularity ofthe space(time) in the past as well as in the futurein the present framework.The physical matter in the universe can only be

“rearranged” in the present framework (of SectionVIII) that is a complete field theory. This factcan be expected to have important implicationsand consequences [12] for our understanding of thecosmological phenomena.So also our models of some (galactic as well

as extragalactic) high energy sources need dras-tic changes. There does not arise a black hole or anaked singularity in the present framework. There-fore, our models of astronomical sources cannot bebased on these conceptions.

In summary, Einstein had been one of the propo-nents of the ensemble interpretation of the quan-tum theory. We then recall here that, to em-phasize Einstein’s contributions to developmentsin the theory of the quantum, Max Born wrote[1] about Einstein that: He has seen more clearlythan anyone before him the statistical backgroundof the laws of physics, and he was a pioneer in thestruggle for conquering the wilderness of quantumphenomena. Yet later, when out of his own work,a synthesis of statistical and quantum principlesemerged which seemed to be acceptable to almostall physicists, he kept himself aloof and sceptical.Many of us regard this as a tragedy - for him, as hegropes his way in loneliness, and for us who missour leader and standard-bearer.

However, as a result of our combined efforts,with so many in the past wrestling with and weak-ening to a large extent the difficult problems ofgrasping the Nature, we have been able to see alittle beyond their vision.

It was, of course, extremely difficult for thephysicists of earlier times, Einstein and others in-cluded, to imagine the current path [2], the formal-ism of the Section VIII, in those times, turbulenttimes of vigorous theoretical and experimental ac-tivities. In fact, many of the mathematical concep-tions of Section VIII, specifically those of the er-godic theory, were not even available to them. But,from our present considerations, it then should beclear that Born’s leader, Einstein, had indeed theright intuition all along.

The physically complete framework of the Sec-tion VIII is also in conformity with the relevantideas of Descartes [7, 11]. Clearly, this field-theoretic program of Section VIII is then also: thecomplete description of any (individual) real situa-tion (as it supposedly exists irrespective of any actof observation or substantiation).

Then, a stage can be said to have been certainlyreached in the history of Physics, once again sinceNewton’s times, in that we have a physically com-plete framework in the formalism of Section VIIIfor describing the totality of physical phenomenain Einstein’s sense.

[1] Einstein A (1970) in Albert Einstein: Philosopher-

Scientist (Ed. P A Schlipp, La Salle: Open CourtPublishing Company - The Library of LivingPhilosophers, Vol. VII). See, in particular, Auto-biographical Notes, Reply to Criticisms and otherrelevant essays.Pais A (1982) Subtle is the Lord ... The science

and the life of Albert Einstein (Oxford: Claren-

don Press) and references therein[2] See, for details, Wagh S M (2004) Some fundamen-

tal issues in General Relativity and their resolu-

tion Database: gr-qc/0402003 and referencesthereinSee also, Wagh S M (2004) A Theory of the Total

Field, to be submitted.[3] von Laue M (1970) essay in Albert Einstein:

28

Philosopher-Scientist (Ed. P A Schlipp, La Salle:Open Court Publishing Company - The Libraryof Living Philosophers, Vol. VII).

[4] Kramer D, Stephani H, MacCallum M A H andHerlt E (1980) Exact Solutions of Einstein’s Field

Equations (Cambridge University Press, Cam-bridge) and references thereinC W Misner, K S Thorne and J A Wheeler (1973)Gravitation (W H Freeman) and references therein

[5] Einstein A (1970) in Albert Einstein: Philosopher-

Scientist (Ed. P A Schlipp, La Salle: OpenCourt Publishing Company - The Library of Liv-ing Philosophers, Vol. VII) (Remarks concerning

the essays brought together in this co-operative vol-

ume) p. 666[6] Wagh S M (2002) Classical formulation of Cosmic

Censorship Hypothesis, Database: gr-qc/0201041.[7] Einstein A (1968) Relativity: The Special and the

General Theory (Methuen & Co. Ltd, London)(See, in particular, Appendix V: Relativity andthe Problem of Space.)See, also, Lorentz H A, Einstein A, Minkowski H,Weyl W (1952) The Principle of Relativity: A col-

lection of original papers on the special and gen-

eral theory of relativity. Notes by A Sommerfeld

(Dover, New York)[8] Joshi K D (1983) Introduction to General Topology

(Wiley Eastern, New Delhi) and references therein[9] Nadkarni M G (1995) Basic Ergodic Theory (Texts

and Readings in Mathematics - 6: HindustanBook Agency, New Delhi) and references therein

[10] Parthasarathy K R (1967) Probability measures of

Metric spaces (Academic Press, New York) andreferences therein

[11] Wagh S M (2004) On the continuum origin of

Heisenberg’s indeterminacy relations Database:physics/0404066

[12] Wagh S M (2004) On cosmological considerations

in a theory of the total field (in preparation)[13] Einstein commented [1] on this essay as: Max von

Laue: An historical investigation of the develop-

ment of the conservation postulates, which, in my

opinion, is of lasting value. I think it would be

worth while to make this essay easily accessible to

students by way of independent publication. —

[14] Therefore, just as various theoretical conclusionsbased on the existence of the absolute space of

Newton’s theory were unjustified, destined to befailure from the very beginning, the solutions tothe equations of the pure gravitational field aresimilarly not justifiable. Clearly, a black hole isone of the many “children” of the “ghost” of thefield equations of the pure gravitational field and,hence, is itself a ghost. Similar is the case with a“naked singularity.”Then, anything done employing such “children ofthe ghost” is dubious. For example, the worksof the author in “The energetics of black holesin electromagnetic fields by the Penrose process”(Wagh, S.M. and Dadhich, N.) Physics Reports,vol. 183, p. 137 (1989) (and references therein)are, simply, dubious, since these works use this“ghost” of the equations of the pure gravitationalfield in every possible manner. A naked singular-ity, it too being dubious, is then not useful for anyastrophysical purposes.It is then evident that efforts (for example, see,McClintock J E, Narayan R, Rybicki G B (2004)Database: astro-ph/0403251 and referencestherein) to establish the existence of a black holehorizon on the basis of observations of astronom-ical sources are not justifiable. The correspond-ing observations must then admit explanations ofsome entirely different nature.

[15] See also Section VIII.[16] There certainly are (philosophical) weaknesses of

such an argument. However, we shall not enter therelevant issues here. It suffices for our purpose hereto say that “the space must be indistinguishablefrom physical bodies.”

[17] We define an equivalence relation “∼” such thatx ∼ y iff ℓ(x, y) = 0 where ℓ is the pseudo-metricdistance defined on the space X. Denote by Y theset of all equivalence classes of X under the equiv-alence relation ∼. If A,B ∈ Y are two equivalenceclasses, then let e(A,B) = ℓ(x, y) where x ∈ A

and y ∈ B. The (metric) function e on Y is thecanonical distance.

[18] This is a field-theoretic comprehension, in a defi-nite sense, of the energy-momentum tensor.

[19] Recall that Einstein [1] regarded the correctnessof Heisenberg’s indeterminacy relations as being“finally demonstrated”.