88Chapter 3ON MANY-BODY PROBLEM ANDSCHRODINGER’S WAVE FUNCTIONThe discussion of the Schwarzschild singularity does not apply to any gravitational field actually known to existanywhere in the universe. However, like Aesop’s fables, it is useful.... [20]

STEVEN WEINBERGGravitation & Cosmology3.0 IntroductionThe Newtonian system needs a basis for concepts such as velocity,acceleration, etc. that is, some framework, relative to which these concepts are welldefined. It is indeed a matter of great difficulty to discover and to distinguish thetrue motions of particular bodies from the apparent, because the parts of theimmovable space in which those motions are performed, do by no means comeunder the observation of our senses. Newton was aware of the difficulty of definingabsolute space, as were Euler and other mathematicians. We are confronted withthe question: How can one give a meaning to the concept of velocity, if you do nothave a space to refer? Therefore, Newton introduced the concept of absolute spaceand absolute time. At the same time, Newton recognized clearly that only relativequantities could be directly measured. Unlike his relationist contemporariesHuygens, Leibnitz, etc. he was convinced that a scientifically useful notion of motioncould not be based on relational quantities. Instead, he sought to demonstrate howabsolute quantities could be deduced from relative observations. A good substitute

for the universal time interval is given by the proper time interval = 1 −

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or equivalently = 1 + , where dash denotes derivative with respect to τ.Similarly we can use the proper length which satisfies = 1 − orequivalently = 1 + as the substitute for absolute spatial distance.These relations are known as the time- dilation and length contraction, prevalentbefore the advent of relativity. It may be noted that is restricted by thecondition| | < whereas ′ is unrestricted. When / and ′/ are very small, wemay use the local space-time co-ordinates as a good approximation to the absoluteco-ordinates.

However, if motion is relative and everything in the world is in motion, as itis in the Cartesian philosophy, a question arises according to Newton: how can oneever set up a determinate theory of motion? In this Chapter, we first examinevarious criticisms against the Newtonian theory and continue with the discussion ofthe many body problem on Newtonian premises by using the centre of mass co-ordinates and the principle of least time. This leads to a confirmation of Newton’sinverse square law. Some of the proper time metrics of general relativity areexamined in the light of the modified Newtonian-theory we have presented inChapter 2. By examining the Compton-shift analysis, it is shown that the momentumconsists of a mechanical part and a quantum-mechanical/electromagnetic partℏ so that = ± ℏ . Further, an interpretation of Schrodinger’s wavefunction as the quantum-mechanical/electromagnetic energy density is examined.

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3.1 Criticisms on Newtonian mechanicsThe Newtonian law of gravitation was not well received even in Newton’sown University of Cambridge. The inverse square law in its elementary formimplies elliptic orbits for planets. Scientists consider this as a serious defect of thetheory. Many astronomers believed that the law was not reconcilable with theobserved motions of heavenly bodies. They admitted that the planetary orbits areellipses as an approximation. By the end of the seventeenth century, departure fromelliptic motion was confirmed by astronomical observations. The inequalities wereof two kinds. First, there were disturbances, which sighted themselves after a time,to have no cumulative effect, known as periodic inequalities. Secondly, there weredepartures that have a cumulative effect known as secular inequalities; the bestknown of them are in the orbits of Jupiter and Saturn. Newtonian law cannot beapplied in its static form to have a satisfactory explanation. However, we shallprove in section 3.6 that the inverse square law is always valid, by the use of centreof mass coordinate system; then Newton’s theory implies a non-elliptic orbit as inGTR. Marquis de Laplace in 1784 was able to assert that the great inequality of

Jupiter, is not a secular inequality, but an inequality of long period, in fact 929years. Laplace showed that the mean motions of planets could not have secularaccelerations because of their mutual attractions. After the triumphant conclusionof Laplace’s research on the great inequality of Jupiter and Saturn, there is still theoutstanding unresolved problem, which formed a serious challenge to theNewtonian theory, namely the secular acceleration of the mean motion of the moon.

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The inequality is not secular but periodic, though the period is immensely long, infact of millions of years. In the following paragraphs (i) to (iv) we examine thecriticisms against Newtonian theory described in the book ‘Einstein Studies-Vol. 6’. [5](i) In 1710, in his book, Principles of Human Knowledge, Berkeleycommented that in space, the motion of two globes with a common centre of masscannot be conceived by the imagination. But if we suppose that the sky of the fixedstars is created, suddenly from the conception of the approach of the globes to thedifferent parts of the sky, the motion will be conceived. Berkeley argued furtherthat philosophers, who have a greater extent of thought and jester notions of thesystem of things, discover even the earth itself to be moved. There is nomathematical, quantitative justification in Berkeley’s comments.(ii) In 1882, in his book, The Science of Mechanics, Mach proposed twoequations to support his dissent with Newtonian mechanics. He pointed out thatone cannot conclude from the flattening of the earth due to its rotation about itsaxis, that absolute space exists; all that one can conclude is that the effect isassociated with the rotation of the earth relative to the rest of the matter in theuniverse. Newton’s experiments with the rotating pail of water, simply informs thatthe relative rotation of water with respect to the sides of the bucket/pail producesno noticeable centrifugal forces. The distant heavenly bodies have no influence onthe acceleration, but they have on the velocity [5]. He claims: “try to fix Newton’sbucket and rotate the heaven of fixed stars and then prove the absence of centrifugalprocess”. John D Norton [5 ] retorted that the challenge of Mach is futile, as the two

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cases are just one case, described differently. It is well known that the centrifugalforces due to rotation of earth and gravitational mass of the earth cannot beseparated experimentally and for this reason, tables of the acceleration due togravity at various points on the earth’s surface often include the contribution of thecentrifugal force due to the rotation of the earth. According to the opinion ofEinstein, these forces are called gravitational forces originating from the rotation ofthe distant masses relative to the rotating system of co-ordinates. But Newton usedthe words ‘fictitious’ forces instead of the words ‘gravitational’ forces. The choicebetween ‘fictitious’ and ‘gravitational’ is a matter of taste only and is irrelevant;besides the critics did not substantiate anything logically or quantitatively. On theother hand the verbal exercises about the rotating pail are just equivalent tointerpretations of the second term in the Newton-Lorentz force law of the Chapter2; namely= + ×

In 1912, Einstein is said to have produced [5] the first definite result in favourof relativistic theory of gravitation in contrast to Newton’s theory: a new type of‘force’ analogous to electromagnetic induction. Further, he is said to have obtainedthe result that the presence of a mass shell of mass M increases the inertial massof a point mass at the centre to + (M G R⁄ ) which is same as = [1 +( ⁄ )]. We have shown, in Chapter 2, that this is always true. But the aboveassertion of Einstein is in contradiction with his own principle of equivalence whichstates that inertial mass and gravitational mass are equal. But Newton never

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treated them at par. If they are equal, how can the mass + containing both(i) inertial part and (ii) gravitational part be in agreement with the principle ofequivalence? According to the modified Newtonian dynamics, this question doesnot arise, as that principle is not necessary. Hence, the inference is that GTR is atheory of contradictory ideas, contradictory to the special theory as well.(iii) In the book it is described thus: Mach points out in his book referredabove, that the distance between two bodies moving purely inertially, satisfies the

differential equation = − … (1) where is a constant. Hecontinues by saying that the direction and velocity of a mass in space remainconstant and we may employ the expression: the mean acceleration of the mass with respect to the masses , , ⋯ at distances , , ⋯ is zero, or ∑∑ =0 … (2) Equation (1) is a kinematical equation and equation (2) has the appearanceof a dynamical equation. From the definition of weighted arithmetic mean, we knowthat ∑∑ = R′ represents the average value of , the various distances of the mass-bodies , , ⋯ from the test mass . Without loss of generality we can assumethat is situated at the centre of mass of the system containing , , ⋯ . Since∑ is a series of positive non-decreasing terms, it is convergent only if the numberof terms is finite. Therefore, the number of mass-bodies of the relationist universeis finite and the centre of mass exists. Therefore, equation (2) implies = 0 orR = A + B and hence R → ∞ as → ∞. Without the assistance of Newtonianprinciples, the conclusion is that the average distance of the mass-bodies from their

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centre of mass tends to infinity i.e. the outer mass bodies fall off to infinity one byone. By comparing R′ with the definition of the radius of gyration K whereK2 = ∑∑ we see that R′ in this discussion, is merely a crude form of the radius ofgyration and hence (2) does not contain any new physical principle. The firstequation can be rewritten as:

( ⁄ )− = 1 . . 2 − = 2 = 2 = ( ) . . ( ) − 2 − = 0.Integrating this we have ( − ) = C . . − = . . . = −

. . =. . = . = . Integrating again, − C = +

∴ = (C ⁄ ) + ( + C ) . Therefore, → ∞ as → ∞, the same conclusion as inthe case of equation (2). This shows that (2) which has the resemblance of adynamical equation, is devoid of any physical or mathematical significance.(iv) Mach’s equations (1) and (2) are mathematically flawed, since they arescalar equations, whereas a vector equation with three components is needed tospecify the motion of a particle. In the same book J.B. Barbour made the followingcomments:‘we must consider those perennial bogeymen, the so-called manifestlyanti-Machian solutions of general relativity, especially matter freespace-times and above all empty Minkowski space. We can go a long

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way to exorcizing these bogeymen, if we hold fast to the followingprinciple. Any solution of pure geometro-dynamics, is not to beanalyzed as a matter free structure in which test particles have inertiaor as a structure that has a disconcerting resemblance to Newton’sabsolute space and time. For us, as opposed to mathematicians “paidby their math department”, to find Einsteinian solutions, that processcan never end’.In this context it is meaningful to examine the opinions of Einstein on Mach.Einstein was fascinated by the Machian arguments till about 1920 and hoped toincorporate Mach’s ideas into his theory of gravitation. This hope was not realizedin the end; there are still several anti-Machian solutions in general relativity[20].Einstein had lost his enthusiasm in the principle he proposed as the Machianprinciple. In his autobiographical notes, Einstein [5] writes:‘Mach conjectures that, in a truly rational theory, inertia would haveto depend upon the interaction of the masses, precisely as was truefor Newton’s other forces, a conception which for a long time Iconsidered as in principle, the correct one. It presupposes implicitly,however that the basic theory should be of the general type ofNewton’s mechanics: masses and their interactions as the originalconcepts.’

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3.2 Proper Time in the Presence of Gravitational MassThe co-ordinate singularity at the horizon of the Schwarzschild solution hasstimulated work towards a general method of determining whether a truesingularity exists and there is not even a fully satisfactory definition of a truesingularity [18]. We shall examine the truth of this statement of Bernard F Schutz andthe statement of Weinberg quoted in the beginning.Books on GTR have descriptions of Schwarzschild metric, in the presence of acentral field of a mass M, given by

= [1 − (2MG R⁄ )] − 1 − (2MG R⁄ ) R − R− R sinBy taking = 1 we get the simple form

= [1 − (2MG R⁄ )] − 1 − (2MG R⁄ ) R − R − R sin (3.2.1)WhenR = 2MG, the coefficient of R has a singularity and there is said to bepossibility of a black hole of radius 2 MG. To get rid of this singularity, the usualexercise in GTR is to make the transformationR = 1 + MG2 (3.2.2)or equivalently = 12 [R − MG + (R − 2MGR)½)] (3.2.3)In addition, the Robertson-Walker/Friedman metric is stated to be

= 1 − 2MGR − ( ) 1 − + + sin (3.2.4)

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where = 0 or ± 1In this section, we examine these metrics, in the light of the modifiedNewton-Lorentz Theory. In the absence of gravitational/electromagnetic fields, theMinkowski’s metric of proper time, is

= − R − R ( + sin ) (3.2.5)A comparison between equation (3.2.1) and (3.2.5) reveals that the propertime is the same in both inertial and non-inertial frames and anygravitational/electro-magnetic field does not affect the proper time. This showsthat the proper time can serve as substitute for the absolute time.From equation (2.6.8) of Chapter 2, we have

R = ℎ Exp −MGRBy R = ℎ we may write

= Exp −MGR (3.2.6)Also from the equations (1.5.28) and (1.5.29) for length contraction and time-dilation, we have

= or R ≈ R .When becomes Exp , R becomes Exp R∴ R = Exp MGR (3.2.7)

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∴ In the central field of a mass M, the radial part R as well as the local time areaffected. Accordingly, we can write Minkowski’s metric in the presence of acentrally symmetric gravitational field of mass M in the form= Exp −2MGR − Exp 2MGR R − R ( + sin ) (3.2.8)

This metric has no singularity at R = 2MG or any other positive value. On theother hand, the metric (3.2.8) reduces to the Schwarzschild metric (3.2.1) byusing the approximationsExp −2MGR ≈ 1 − 2MGR and Exp 2MGR = 1Exp −2MGR ≈ 1 − 2MGR

Similarly the Eddington-Robertson form of the metric can be obtained bywriting Exp ≈ 1 − + 2M2G2R2 . Since equation (3.2.8) has no singularityat R = 2MG , the conclusions of GTR based on the metrics (3.2.1) and/or (3.2.4) orany such metrics, are invalid according to the modified Newton-Lorentz theory.Next we shall derive the transformations (3.2.2)/ (3.2.3) from the Newton-Lorentztheory. We can rewrite equation (3.2.8) in the isotropic form

= Exp −2MGR( ) − Exp[2 ( )]( + + sin ) (3.2.9)by using the transformationR = Exp ( ) (3.2.10) (a)Exp MGR R = Exp ( ) (3.2.10)(b)

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Equation (3.2.10) imply = Exp MGR RR= 1R 1 + MGR + M G2R + ⋯ RIntegrating = R Exp − MGR − M G4R ⋯ (3.2.11)From (3.2.10) and (3.2.11) we have ( ) = − + ⋯ (3.2.12)Now R has to be eliminated by finding R as a function of in the form R = R( )From (3.2.11) we have ≈ R, for large values of and R.Using this initial approximation on the RHS of (3.2.11) we find the nextapproximation

≈ R Exp − MG − M G4 ≈ R 1 − MG − M G4 + M G2. . ≈ R 1 − MG2 or R ≈ 1 − MG2. . R ≈ 1 + MG2 (3.2.13)

This is the transformation (3.2.2) of GTR, from which we obtain (3.2.3) on inversion.Substituting the value of R = 1 + in (3.2.12) we have ( ) as a series ofpowers of . By using these values in (3.2.9) we get the isotropic form without anysingularity.

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Next we shall find the transformation which transforms (3.2.8) into theRobertson-Walker form= Exp − 2MGR( ) − Exp 2 ( ) 1 − + + sin (3.2.14)

Comparing (3.2.8) and (3.2.14) we haveR = Exp ( ) (3.2.15)(a)and Exp MGR R = Exp ( )√1 − (3.2.15)(b)From (3.2.15) we have Exp MGR RR = √1 −Integrating both sides

1R 1 + MGR + M G2R + ⋯ R = √1 −log R Exp − MGR − M G4R − ⋯ =

⎩⎪⎨⎪⎧ log when = 0log 1 − √1 − when > 0

− log 1 + √1 − when < 0⎭⎪⎬⎪⎫

except for the constant of integration∴ R Exp − MGR − M G4R − ⋯ = ⎩⎪⎨

⎪⎧ when = 01 − √1 − when > 01 + √1 − when < 0⎭⎪⎬

⎪⎫

except for the constant of integration

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It is possible to find ( ) by the method of finding ( ), as described above.Thus we get the Robertson-Walker form (3.2.14). In order to write (3.2.14) in theform of Friedman metric, we replace Exp [2 ( )] in it by an arbitrary factor ( )depending on the local time, as is done in general relativity. Reference [16] containsan exposition of another metric known as Kerr metric, but the author ends with thecomment: ‘we do not know what the source is for the Kerr metric or even whethersuch a metric can be realised in nature’. Thus, it is possible to get the metrics ofgeneral relativity, with or without any singularity.From the above analysis, it is clear that the metrics in general relativity canbe constructed by inflating or deflating the metric coefficients G in the Minkowski’smetric for proper time, according to the requirements of the user. Thus, generalrelativity prescribes singularities, which are non-existent according to Newton-Lorentz theory. These justify the comments of Bernard E Schutz and StevenWeinberg [18, 20]

3.3 Removable singularity of potential function-centre of massFirst we will resolve the singularity of the potential function representingpotential energy per unit mass/unit charge. The potential due to a spherical pointmass M at a distance r from its centre of mass is given by = MG⁄ . Consider thepotential at due to the presence of two masses and as given by

= | − | + | − | (3.3.1)

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where we took G = 1, and , the locations of and . For convenience, wemay take = 0, = 0, = 0, = 0 and = , =∴ = − + − (3.3.2)Let = and ′ =∴ = ( + ) − ( + )( − )( − )∴ = ( + ) ( − )( − )( − ) − A (3.3.3)where A = ( − )( + )Equation (3.3.3) shows that the potential due to two masses will vanish at thecentre of mass = . The conclusions are that, (i) for any system of particles, theorigin of potential (zero potential) should be at the centre of mass; (ii) for a singlepoint mass M the potential at its centre of mass must be zero. This assertiondemands that the potential function for a single mass M, defined in chapter 2 mustbe chosen in the form = [ ]| | when | − | = ≤ radius of the sphericalmass, where [M] is the effective mass within the spherical region, i.e. within| − | = . Outside this region we may take = | | as usual. Similarly for acharge , we define = [ ]| | , when | − | = ≤ radius of thespherical charge. Outside this region we may take = | | , | − | >

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radius of the electron, being the mass of the electron. Clearly both and tendto zero when the radius of [M] and [ ] tend to zero.3.3.1 Centre of Mass of Many-Particle SystemSince gravitational field intensity is not like tension in a string, it can be consideredas force per unit area. Therefore, instead a of single line of force field connectingand , we consider a ‘tube’ of field lines consisting of the same lines joining thesemasses. This may be called a ‘tube’ of field lines. The shortest tube joining andwill contain the centre of mass G of and . For a system of particles, thecentre of mass always moves with constant velocity so that the sum of forces ofinteraction among them is zero. For the study of motion of in the field of andwe consider as one system and { , } as another system with centre ofmass at G . The motion of depends on the tubes of field lines emanating fromG1and terminating at . The shortest tube connecting G and will contain theoverall centre of mass G of the whole system { , , }. Hence the motion ofeach of , and can be studied by considering the tubes of field linesconnecting each particle to the centre of mass of the sub-system not including theparticle under study. We conclude that the potential should be redefined byconsidering centre of mass as the zero-point of potential.

GFig 3.1

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3.4 A correction to the classical solution of the Two-body problemIn the analysis of Bohr atomic model, the hydrogen nucleus consisting of asingle proton, is assumed stationary while the orbital electron revolves around it.What must happen is that both the nucleus and the electron revolve around theircentre of mass. Books on Quantum Mechanics, dealing with hydrogen atom, explainthat there is a versatile way to reduce the two body problem to an equivalent one-body problem if the potential depends only on the separation | − | of the twoparticles. In such cases, the Hamiltonian is

H = 2 + 2 + V(| − |) (3.4.1)In Feynman Lectures in Physics Vol. I Ch. 7.4, we read that ‘because the earthpulls the moon as the moon pulls earth, due to gravitation, the earth goes in a circle

Fig 3.2

G

G

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around a point which is inside the earth, but not at its centre’. The moon does notjust go around the earth; the earth and the moon both go around a central position.All these descriptions reveal the fact that each particle of a system of -particles moves as if acted upon by a central force field emanating from the centre ofmass of the system. Let R be the position vector of the centre of mass of a system ofmasses.

The K.E. of the system of particles can be simplified as below:KE = 12

= 12 + −= 12 + 12 −

. . K. E. = 12 M + 12 (3.4.2)where M = ∑ and is the position vector of the particle relative to thecentre of mass. For the two-particle system, the equation (3.4.2) becomesK. E. = 12 M + 12 + 12 (3.4.3)

G

OFig: 3.3

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where R1 and R2 are position vectors relative to the centre of mass G1.Define = − = − and = | |. Since R = R ,each = R11−1 = R22−1 = R1 + R21−1 + 2−1 = 1 2(R1 + R2)1 + 2 = (R1 + R2) =where μ is the reduced mass.∴ R = and R =∴ R + R = + = ( + ) = M∴ R + R = M = = −V( ) (3.4.4)where V( ) is the potential energy.Also = −∴ = − = −+ = ( − ). . = − = ( − )

∴ = − = −= + − 2 ∙

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= + − ∙ − ∙= + − − ∙ − − ∙= 1 + + 1 +. . = + (3.4.5)

By using equation (3.4.3), (3.4.4) and (3.4.5), the Hamiltonian (3.4.1) for the two-body problem can be written in two equivalent forms,H = 12 M + 12 + 12 − R − R (3.4.6)(a)or H = 12 M + − M (3.4.6)(b)

In both (3.4.6) (a) and (3.4.6) (b), the Kinetic Energy of the centre of mass isM . This shows that the centre of mass, moves as if a free particle of mass M;it may be considered the origin of an inertial frame of the system of particles.Relative to the centre of mass, the Hamiltonian becomes:H = 12 − M (3.4.7)(a)or H = 12 + 12 − R − R (3.4.7)(b)

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(3.4.7)(a) has the appearance of the Hamiltonian of a particle of mass μ in thepotential field M/r. Hence, the apparent trajectory of (3.4.7)(a) is a conic with theheavier body as a focus; this is the solution of two bodies, given in text books.But when = the argument fails; there is no ‘heavier’ body. This shows thatthe correct solution of the two-body problem can be found only by using the C.M.system, represented byH = 12 + 12 − R – R (3.4.8)This Hamiltonian can be split in to two:

H = 12 − R (3.4.9)H = 12 − R (3.4.10)We need not solve both (3.4.9) and (3.4.10) since = − and andlie on a straight line through G . From (3.4.9) we can infer that moves along acertain conic and hence is moving along a dependant conic with focus at thecentre of mass viz. G . We can extend this method for each one of the n-bodies, byconsidering the fact that each body moves as if acted upon by a tube of field of forcesconnecting the body towards the centre of mass of the sub-system obtained byexcluding the body under consideration. In this case, the reduced mass for willbe = and similarly for etc. In all the cases, the individual bodywill have a conic as the trajectory; but there is dependence between the conicsbecause of + ⋯ ⋯ + = , , ⋯ , being the force fields towards the

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centre of mass from the individual particles. Therefore, each trajectory is liable toperturbation due to the presence of the remaining bodies. In addition, we have notdefined the potential of each mass-particle relative to the centre of mass; this can bedone by considering equations (3.4.9) and (3.4.10). The potential at is andsimilarly for the potential at etc. Clearly when = 2, = . But without thesepotentials, we can find the orbit by using calculus of variations. In the Chapter 2, wehave shown that the path of a particle in the two-body system is not exactly anellipse, but a moving ellipse with its perihelion moving slowly; this is due toperturbation. For the many-body problem also we shall prove that the sametrajectory will be obtained. The assumptions made are (i) each body is acted uponby a central force (ii) each body moves according to the principle of stationary time.In order to use these principles we derive3.5 The Work Energy TheoremWork done, W = ∙ = ∙ (3.5.1)But, ∫( ⁄ ) ∙ = ∫ ∙ = ∙ − ∫ ∙= ∙ − ∙= − + C′= − exp( 2⁄ ) + C′= − (1 − ⁄ )exp( 2⁄ ) + 2 by choosing C′ = 2

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= + 12 + 38 + … (3.5.2)≅ + 12 (3.5.3)

∴ W = 12 ( − ) (3.5.4)that is, change in Kinetic Energy = work done (3.5.5)Since the force field is central, we may take = ( ) where I is a unit vector alongthe radial direction from centre of mass of the sub-system excluding the test mass.∴ ∙ = ∙ = ( )When A and B are points on the trajectory very near each other, the force may beapproximated by a constant value to∴ ∫ ∙ = ∫ = ( − ). . W = ( − ) (3.5.6)

Equations (3.5.4) and (3.5.6) imply12 ( − ) = ( − )this can be expressed as

= + (2 ⁄ )( − ) = ( − )or = − (3.5.7)

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3.6 Principle of Least Time – Many Body Problem and the Inverse Square LawIn 1924, Louis de Broglie proposed that matter possesses waves as well asparticle properties. The wave length λ of a particle of momentum is given by= ℎ | |⁄ or = ℏ . In books on Quantum Mechanics, a simple wave isrepresented by = Exp. ( ℏ⁄ )( ∙ − ℏ ) where A is normal to the wavefront, | | = 2 ⁄ is the wave number or propagation constant. A superposition ofany such waves form a wave group or wave packet, moving with the velocity of theparticle. Fermat’s principle of least time is about the wave velocity/phase velocityof individual waves. It states that the time taken for a ray of light from a point Ato a neighbouring point B is least

. . = 0 (3.6.1)We shall extend this principle to the wave groups, since this assumption isconsistent with existing theory and serves as an alternative for the inverse squarelaw, described in the next section. So we state the principle of least time: A body ofthe n-body problem moves from A to B, taking the path of least time between A andB subject to the field constituted by the n bodies.

. . = 0where = −

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PropositionThe principle of least time/stationary time implies the inverse square law ofgravitation.To prove the proposition we find the trajectory of bodies in the Many Body Problemby solving I = 0 whereI = ∫ ( + )½ = 1 + ( − ) ½ (3.6.2)The Euler-Lagrange equation( ⁄ )( ⁄ ) − = 0⁄ (3.6.3)with = (1 + ′ ) ( − )⁄ gives( ⁄ )( ′⁄ ) = 0 since integrand as independent of∴ = constant⁄

( ) 1 + ( − )⁄ = A∴ = A 1 + r ( − ). . ′ [ − A ( − )] = A ( − )

= A ( − )[ − A ( − )]1 = − A ( − )[A ( − )] = [A ( − )] − 1 (3.6.4)

Let = 1 or = −1 ∙

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. . 1 ∙ = −1∴ 1 = 1 (3.6.5)∴ 1 = A ( − ) − 1∴ + = 1A ( − ) (3.6.6)Differentiating write θ2 + 2 = 1A −1( − ) −1cancelling 2 and letting 12A = MGℏ

+ = 12A ( − ) = MGℏ ( − ) (3.6.7)When = 0 this represents the conic

+ = MGℏwith a focus at the other end of the tube of force field of the particle.If ( ) is the law of force field for a central motion, then by a formula from classicalmechanics, we have

+ ≡ ( )ℏ (3.6.8)with the usual notations. Comparing (3.6.7) and (3.6.8) we have

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( )ℏ = MGℏ ( − )∴ ( ) = MG( − ) (3.6.9)This represents the inverse square law; but is the distance of the body from thecentre of mass of the sub-system excluding the test body. Since − 1 < , thedistance − must be the distance from the overall centre of mass of the wholesystem including the test mass . Thus Newton’s inverse Square law is always trueand the law of force is

( ) = MG[ ] (3.6.10)Where [ ] is the radial distance of the test mass from the common centre of mass.This equation (3.6.10) justifies us to use the centre of mass coordinates and definethe correspondingpotential of chapter 2 as = MG[ ] (3.6.11)Hence the principle of least/stationary time implies the inverse square law. When≠ 0 we proceed as follows: Expand the RHS of (3.6.7) by using binomial series∴ + = MGℏ (1 + 2 + 3 + ⋯ )

+ (1 − 2 ℎ ) = ℎ (1 + 3 + ⋯ ) (3.6.12)By letting = (1 − 2MG ℎ ) , (3.6.12) can be re-written as

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+ = MGℎ 1 − 2MG ℎ (1 + 3 + ⋯ . ) (3.6.13)Discarding higher powers , …, we can rewrite (3.6.13) in the form of equation(2.6.10) of Chapter 2,

+ = ℎ + 4 (3.6.14)This shows that the modified Newton-Lorentz theory gives non-elliptic orbit as inGTR. It is also obtainable from the Schwarzschild metric= [1 − (2MG R⁄ )] − 1 − (2MG R⁄ ) R − R − R sin (3.6.15)by redefining the parameters.In the modified Newtonian dynamics, we are justified to use the retarded-potentials and retarded fields relative to centre of mass. On the other hand themetric (3.6.15) has the special disadvantage that it has a singularity at = 2MGwhich is non-existent according to Newton-Lorentz theory. Further, it does notconsider Ampere’s vector potential. Hence GTR cannot be considered as ageneralization of modified Newton-Lorentz theory.There have been many attempts to imbed the theory of electromagnetisminto the framework of an extended theory of general relativity by geometrization.However no unified theory of electromagnetism and gravitation had been developedwhich is convincing and satisfactory as the gravitational theory of general relativity.Weyl’s theory of electromagnetism has a high degree of consistency and elegancebut it has led to no prediction of any new physical phenomena, which could beobserved as confirmation of Weyl’s theory. On the contrary, in an appendix to

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Weyl’s exposition of his unified field theory, Einstein raised some very seriousobjections to it on empirical physical grounds and concluded that Weyl’s theory is incontradiction to experience.3.7 Derivation of Momentum-Velocity Relation: = + ℏ and mass-energy relation: E = + ℏ where = exp( /2 )In the Compton-effect experiment, it was observed that, when a photon of energyℏ = , strikes an electron of rest mass , both will be deflected, and if theformer makes angle and the latter an angle with the initial direction of photon,then by law of conservation of energy( − ) = ℏ ( − ) = ( − ) (3.7.1)where , are the initial and final momentum of the photon and is the mass ofthe electron in motion.∴ − = ( − ) (3.7.2)Also by the law of conservation of momentum, if is the momentum of the electron,then

cos + cos =sin = sin

∴ = ( − ) + 2 (1 − cos ) (3.7.3)But – = ( – ) exp

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≈ 1 − 1 + + 2= 1 − 2 − 3. . – ≈ 1 − 2 (3.7.4)

From (3.7.2) = + ( − )∴ − = [ + ( − )] – ( − ) − 2 (1 − cos )

= + 2 ( − ) − 2 (1 − cos )= + 2 ( − ) − 2 (1 − cos ) by (3.7.2)

But − = [1 − exp (− /2 )] ≈ ( /2 )∴ − = + 2 ( /2 ) − 2 (1 − cos ). . − = + − 2 (1 − cos ) (3.7.5)

As a first approximation we may take ≈ ∴ LHS of equations (3.7.4) and(3.7.5) are equal.∴ RHS must be approximately equal.∴ 1 – 2 ≈ + − 2 (1 − cos θ)∴ 2 (1 − cos ) ≈ +

≈ + 2

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= exp∙ 2 + 2≈ 1 + 2 + 2= 1 +≈ exp= (3.7.6)

using (3.7.6) in (3.7.3) the latter equation becomes= ( − ) +

which is of the form = + ℏ (3.7.7)i.e. momentum consists of a mechanical part and a quantum-mechanicalpart/electromagnetic part ℏ ; so we can write

∗ = ± ℏ = ± A= ± (3.7.8) (a)= ± (3.7.8)(b)

Equation (3.7.4) can be rewritten as+ ℏ + 12 − ( + ℏ ) =

which is of the form E − = or E − =

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where E = + ℏ and ℏ ω = ℏ + 12∴ Energy consists of an inertial part and a quantum-mechanicalpart/electromagnetic part ℏ . It is thus possible to take E = − ℏ = −ℏ( / ) log ). . E = − ℏ (3.7.9)

and = + ℏ = + ℏ (log ψ). . = + (3.7.10)

3.8 Application of Newton-Lorentz Theory in Quantum MechanicsFrom equation (2.3.31), the Lorentz Force Law is = E + × B

for a mass particle and = E + × B for motion of an electronFor an electron (having both mass and charge), we use (3.7.8) (a). Hence,we replace by ± for the electron and ± for the mass particle.Equivalently we replace by ± for electron and by ± for amass particle in the above law (2.3.31). Dropping the subscripts 1 and 2, we have

( ± ) = + ± × for electron and (3.8.1) (a)− ± = + ± × for mass particle (3.8.1) (b)

Equating real parts ( ) = + × for mass particle

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= + × for electron. . − ≈ + × for mass particle (3.8.2) (a)

and ≈ ( + × ) for electron (3.8.2) (b)by assuming ≈ 0Equating imaginary parts of (3.8.1), we have

= × . . = × for electron (3.8.3) (a)and = − × . . ≈ − × for mass particle (3.8.3) (b)Now equation (3.8.3) (a) for the electron becomes, by using polar co-ordinates

= + J × B( × J)= + J × B , Since =

= B − J+ + J = − B J+ B

∴ = B (3.8.4)and = − B (3.8.5)

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Here we have changed the usual formula for the radial and transversecomponents of acceleration of a particle by the following considerations. Theapparent position of fast revolving leaves of a fan can be ‘everywhere’ within(0, 2 ). Similarly an electron in the shape of a spherical shell moving about a meanposition has the appearance of a cloud having both radial and transverse velocitiesand hence the formulae for acceleration of a particle are no longer applicable henceare replaced by and . In other words is not localized but existing

simultaneously at all points on a sphere. That is, = and = , for electron motion.

Thus we get equations (3.8.4) and (3.8.5). On integration, equation (3.8.5) gives

= − B + C= − + C where B =

∴ B = (− + C)∴ Equation (3.8.4) becomes = (– + ). . (D + ) = C . Solving this differential equation we get

= cos( + ) + C = R + cos( + ) (3.8.6)where λ, ε, R are constants of integration = − sin( + ) (3.8.7)

∴ = − cos ( + ) (3.8.8)

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Equations (3.8.6) to (3.8.8) indicate a simple harmonic motion about a meanposition R . The solution of equation (3.8.2) (a) is the same as the one we havefound in section 2.6, the equation for planetary motion. Thus, the solution ofequation (3.8.1) (a) consists of superposition of an elliptic motion and a timedependent simple harmonic motion.A quantum mechanical partitioning of space around a mass/chargeWe shall take R = ( + 1) and = , where is Bohr radius∴ R = ( + 1) (3.8.9)or equivalently we may write = ( + 1) , in order to have similarity withBohr’s principle. This partitioning of space will be used in the next section. (In thetheory of hydrogen atom Bohr took = ). From equations (3.8.7) and (3.8.8)we have

= + =∴ = and = = ℏ (where ℏ is Planck constant) inaccordance with De Broglie hypothesis. Hence, ℏ can be interpreted as quantum-mechanical angular momentum, as phase velocity and as wavelength ofelectron cloud.3.9 Resolution of Singularity of Field Energy

In this section, we attempt to remove the singularity of field energy due to anelectron/mass particle, by using Bohr’s principles. In electromagnetism, we havethe equation that the electromagnetic field energy of a point charge of radius is

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given by [7] U = 8 . Similarly, the energy of a charged sphere of radius isalso given by U = Q28 0 . Therefore, total energy in the field and the energywithin the charge are each equal to . When tends to zero, the conclusion isthat there is an infinite amount of energy in the field of a point charge or chargedsphere. Applying Bohr’s theory of hydrogen atom, we can remove the apparentsingularity stated above. We can re-write the above formula in the formU = Q28 0 = Q28 0 1( + 1)∞

1 since 1( + 1)∞1 = 1

= Q28 0 = Ε∞1

where Ε = 8 , and is our notation for from Chapter 2, Ε is the energylevel at a radial distance = ( + 1) , = or an integral multiple of andis the Bohr radius. But Bohr uses = instead of ( + 1) . Thus, thefield energy at from a charge is Ε and the total energy in the field is U = 8 .We can extend this result to mass particle of radius . The field energy due to amass particle of mass M may be taken U = . When → 0. We replace M by[M] and Q by [M].

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3.10 An Interpretation of Schrodinger’s Wave Function:From Chapter 2 we have noted that the four potentials ( , ) is ameasure of the energy momentum felt by a charge in the four-field ( , ) of asource charge Q. Similarly, ( , ) is a measure of the energy-momentum felt bya mass-particle in the four-field ( , ) of mass M. From equation (3.7.8) andequation (3.7.9) we can infer that the total momentum and energy of a chargedparticle consists of a (i) mechanical/inertial part (ii) quantum mechanicalpart/electromagnetic part.For an electron in an atom, and represent a measure of the energydue to (i) inertial mass and (ii) electrostatic potential of the nucleus. Letdenote mass density so that is energy density; therefore energy momentumdensity due to mass content of the moving electron in the orbit is ( , ). If ischarge density of the electron then the energy momentum density due to charge ofthe moving electron is ( , ) = ( , ), with the notations of Chapter 2.By the equation of continuity and the gauge condition1 ( ) + div ( ) = 0 (3.10.1)

1 ( ) + div ( ) = 0 (3.10.2)Combining these equations

( + ) + div ( + ) = 0 (3.10.3)(a). . ∗ + div ∗ = 0 (3.10.3)(b)

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where ∗ = + , (3.10.4)(a)and ∗ = + = + (3.10.4)(b)are all densities.By Newton’s law of motion,

∗ = −∇( ∗ ) = −∇( + ) (3.10.5)Taking divergence, div ∗ = − ∇ ( ∗) (3.10.6)

By defining L* and H* as Lagrangian and Hamiltonian densities,we have L∗ = ∗. – H∗, S∗ = L∗ = ∗. − H∗∴ ∇S∗ = ∗and − S∗ = H∗ = ∗∴ ∇ S∗ = div ∗ = − ∗ = − − 1 S∗ = 1 S∗

∴ ∇ − 1 S∗ = 0From (3.10.3)(b), by differentiating partially w.r.t.

∗ + div ∗ = 0 (3.10.7). . ∗ + div ∗ − ( . ∇) ∗ = 0. . ∗ – ∇ ∗ = div [( . ∇) ∗]

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. . 1 ∗ − ∇ ∗ = div[( . ∇) ∗] (3.10.8)Again from (3.10.5), ∗ + ( . ∇) ∗ = − ∇ ∗∴ ∗ + [( . ∇) ∗] = − ∇ ∗ = ∇ (∇ ∙ ∗)

= [∇ ∗ + ∇ × (∇ × ∗)] (3.10.9). . ∗ − ∇ ∗ = − [( . ∇) ∗] + ∇ × (∇ × ∗). . 1 ∗ − ∇ ∗ = − 1 [( . ∇) ∗] + ∇ × (∇ × ∗) (3.10.10)

As an approximation, the RHS of (3.10.9) and (3.10.10) are negligible. Therefore,(3.10.9) and (3.10.10) become the wave equations.1 ( ∗ ) − ∇ ( ∗ ) ≈ 0 (3.10.11)where ∗ = + = + or ∗ = ∗ + ∗and 1 ∗ − ∇ ∗ ≈ 0 (3.10.12)Both real and imaginary parts of (3.10.11) and (3.10.12) satisfy the homogeneouswave equation. To solve (3.10.11) by the method of separation of variables,we substitute ∗ = ( , , )T ( ) (3.10.13)where ( , , ) = ( , , ) + ( , , ), into (3.10.11). Taking imaginaryparts we get

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1 ( , , ) T′′ = (∇ )T. . ∇ = T T Let each = −

∴ (∇ + ) = 0 (3.10.14)and T + T = 0 (3.10.15)The last equation has solutions T = A Exp(± ) and then |T| = A∴ = ( , , )A Exp(± ) (3.10.16)(a)and = A| ( , , )| (3.10.16)(b)∴ ( , , ) has the dimension ofTherefore, the wave function ( , , ) represents quantum mechanical energydensity.

From classical dynamics, we have T + V= constant = E where T is the kineticenergy and V is potential energy. Take V( ) as the potential energy per unit chargeat the electron due to the nucleus of the atom. By using the mass velocity relation= Exp and substituting T = ( − ) , the energy equation givesExp 2 − + V( ) = E (3.10.17)

∴ Exp 2 = 1 + E − V( )∴ = Exp ( ⁄ ) = ∙ 2 log 1 + E − V( ) 1 + E − V( )

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= 2 E − V( ) − 12 E − V( ) + ⋯ 1 + E − V( )Letting = ℏ impliesℏ = 2 E − V( ) 1 − E − V( )2 + ⋯ 1 + E − V( )∴ = 2ℏ [E − V( )] 1 + 32 E − V( ) + ⋯ (3.10.18)By using this approximation in equation (3.10.14) we get∇ + 2ℏ [E − V( )] 1 + 32 E − V( ) + ⋯ = 0 (3.10.19)

By omitting second and higher powers of, E − V( ) we get the Schrodinger’sequation∇ + 2ℏ [E − V( )] = 0 (3.10.20)

Thus, the Schrodinger’s wave function satisfying (3.10.14) and (3.10.20)has the dimension of quantum mechanical energy density. Substituting the solutionof Schrodinger’s equation (3.10.20) in to (3.10.16) (a), we get the imaginary part ofthe solution of equation (3.10.11). Similarly, we can find the real part of thesolution of equation (3.10.11).3.11 Conclusions

The many-body problem can be treated by using the concepts of Newton’sinverse square law, centre of mass co-ordinates and retarded potentials. Since theconcept of vector potential is absent in GTR, whereas it is a part of the modified

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Newtonian dynamics/STR, and the Schwarzschild metric of GTR is a special case ofthe metric (3.2.8) of Newtonian theory, the GTR cannot be considered as ageneralized version of STR/modified Newtonian dynamics. Further GTR stipulatesthat there exists no preferred frame of reference. But according to Newton-LorentzTheory, the centre of mass frame is a preferred frame of reference in the study ofmany body problem.