49Chapter 2MAXWELL – LORENTZ EQUATIONS FOR MASS-PARTICLES ANDELECTRONSSince electron theory [11] is in agreement with special relativity, the latter cannot produce results, which arenot already contained in the pre-relativistic Lorentz’ electron theory. WOLFGANG PAULI(Theory of Relativity) [17]2.0 Introduction

Newtonian dynamics, electrodynamics and general relativity are theoriesconsidered as different from each other, although there are some common features.In the former two, the inverse square law of Newton and Coulomb are similar. Thus,there are similarities on gravito-statics and electro-statics. Yet, the theories ofgravito-dynamics and electro dynamics are at present dissimilar. Since gravitationalsignals/waves and electro-magnetic signals/waves propagate with approximatelythe same speed c in free-space, it is feasible to combine both the theories into asingle theory. It may seem that there is no analogue of a magnetic field in the case ofcharge-less particles. The fluid motion near a source/sink reveals that a laminarmotion is accompanied by a vortex motion, causing eddies and turbulence. Hence,there must be a rotational field or vortex field, similar to the magnetic field, in thecase of a moving particle. In this chapter, we shall derive the Maxwell-Lorentzequations in Newtonian dynamics as well. This will reduce the differences betweenthe former two theories. The third theory of general relativity, has discordance withthe former two because of the exclusion of vector potentials and the use ofR = 0, R being Ricci tensor,[19] along with variational equation ( ) = 0, whereds is the four metric of proper time. Although the vector potentials are excluded, the

50scalar potential of the Newtonian theory enter the discussion in theSchwarzschild solution of the metric. This shows that the modified Newtoniandynamics, containing the four-potentials of gravitation and electro-magnetism, aswell as the four metric will include all the three theories. It will be proved that evenwithout the four metric, the modified Newtonian theory can explain suchphenomena as:

(i) perihelion motion of the planets(ii) gravitational red-shift(iii) bending of light rays passing nearer to the sun, etc., which were earlierthought to be inexplicable with Newtonian theoryIt is further shown that the metric of general relativity is a disguised form of theproper Lagrangian, except for their dimensions. The approximation dτ ≈ dt will beused frequently. The contents of this chapter is adapted from the published papers

[*3] of the researcher.2.1 Key Words and Notations

Local time interval (dt), proper time interval (dτ), gravitational fieldintensity(E1), electro-static field intensity (E2), gravitational permittivity of freespace ( ), electrical permittivity of free space ( ), gravitational permeability ( ),electrical permeability ( ), four-potentials of gravitational fields ( , A1), four-potentials of electromagnetic fields ( , A2), four-potentials of combinations offields ( , A3), flow density/current density for gravitational particles ( or J1)

51current density for electrons ( or J2). = 1 and = 1, propermomentum, , classical Lagrangian (L) proper, Lagrangian L = L ( / )2.2 Classical Galilee - Newton Laws of Mechanics

Newton’s laws of motion can treat the motion of macroscopic bodies such asvehicles, satellites, planets, etc. Newton’s concepts are based on free space andinertial frames of reference. In all inertial frames of reference, an acceleratedmotion will have the same value for the acceleration and the laws of physics retainthe same form. Newton’s laws are:(i) A body continues to be in its state of rest or uniform linear motion, unless it isacted upon by an external force.(ii) The rate of change of linear momentum of a body in motion is equal to theforce acting on it.(iii) If two bodies of masses m and m exert forces F12 and F21 on each other, thenF12+ F21= 0These laws are invariant with respect to all Galilean transformations.2.2.1 Gravito-statics - Law of Gravitation

The fundamental problem of gravito-statics is to analyze the mutual actionbetween pairs of a system of masses and ⋯ and their effect on a test-massand the trajectory of the test mass; by assuming that all the masses interact witheach other and the interaction between any two masses is completely unaffected bythe presence of the remaining masses. Hence, the principle of super position is

52applicable. Thus the total gravitational force F on m is the sum of F1, F2, ⋯ due to, , ⋯ of the system.i.e. F = F1+ F2 + ⋯ (2.2.1)The various components F1, F2, ⋯ can be found by using Newton’s law ofgravitation (Coulomb’s law for charges) which had been found true by experiments.Law of GravitationNewton’s law of gravitation states that the static gravitational attractive forcebetween two mass particles of masses , , situated at and isF12= G [N] (2.2.2)

andF = −F12= G [N] (2.2.3)where is F12 is directed from to , G is the gravitational constant, is thegravitational permittivity, = , G = and = − . The value ofG = = 6.673 × 10 [N ∙ /Kg2]. By replacing and by unlike chargesQ and Q and by we get the Coulomb’s law of electro statics.2.2.2 Gravitational Field Intensity ( )

The force of attraction due to a particle of mass situated at Q( ′) on a testparticle of unit mass at P( ) is given by

53= −R4 R [N Kg⁄ ] (2.2.4)

where R = − ′ and R is a unit vector along R. It is called the gravitational fieldintensity of at P ( ). The vector field defined by D1 = E1 is called thegravitational flux density. Now by applying the principle of super position werewrite (2.2.1)F = F1+F2 + ⋯ = 14 R R + R R + ⋯

= 4πε RR + RR + … = E (2.2.5)whereE ( ) = 14 R (2.2.6)is called the gravitational field intensity of the system of masses , m ,... andR = − , being the location of . For a continuous mass distribution ofvolume V′, the equation (2.2.6) can be modified toE( ) = 14 RRV' = 14 ρ( ) R VR2V (2.2.7)where = − ′ and R is the unit vector along R.2.2.3 Energy Stored in a Gravitational Field

The potential energy per unit mass or potential due to a mass m at a fieldpoint P( ) is defined by

54= 4 R J/K ] or [N ∙ K (2.2.8)

If a mass is placed at P( ) in the field of then the potential energy atin the field of isW = 4 R = [N ∙ ] or [J] (2.2.9)

For a system of particles , , ⋯ we define the potential energy of the fieldas:W = 12 [ ′ + ′ + ⋯ ] (2.2.10)where

′ = 4 R≠ 1 , ′ = 4 R etc.≠ 2For a system of continuous distribution of masses, (2.2.10) becomesW = 12 ′ V + 12 ′ V + ⋯ (2.2.11)where the region of integration does not contain the volume of the source so as tocut off singularity.

Similarly, we can write the corresponding equations in electro-statics. Thus,there is perfect reciprocity between static gravitational fields and electro-staticfields. This similarity is not found when we compare electro dynamics with theexisting gravitational-dynamics. Therefore, we derive those equations, which haveat present no place in gravito-dynamics, but occurring in electro dynamics.

552.3 Analysis of Maxwell - Lorentz Theory

In this section, we shall derive Gauss’ Laws, non-relativistic Dirac’s equation,Faraday’s Law, Ampere’s Law, Lorentz Force Law and Gravitational WaveEquations.2.3.1 Gauss’ Laws

Let M be the mass of a particle moving with velocity ; let it be atQ ( ′, ′, ′) at some instant of time. Consider a spherical cap of central angle 2 ,cutt off from a sphere centred at M, by a plane at a distance X, so that is normal tothe base of the cap and the radius of the circular rim of the cap is given by= Y + Z and |R| = X + Y + Z , X = − , Y = − , Z = −

Since the diameter through P( , , ) of the rim C, subtends 2 at the centre Q of thesphere, the area of the cap isA = 2 (1 − cos ) (2.3.1)and the specific area of the cap is

= (1 − cos ) (2.3.2)In contrast to equation (2.2.4), but in accordance with electrodynamics, weintroduce the Definition: At each point on the cap, a vector of uniform magnitude inthe direction of the radial line, emanating from the particle at Q will be defined; theflux of this vector field (gravitational field intensity) over the cap is assumedproportional to (i) the mass M of the particle and (ii) the specific area

56. . = E1 ∙ S = 12 M (1 − cos ) (2.3.3)

where , the permittivity of space, is the proportionality factor.∴ D1 ∙ S = 12 M (1 − cos ) (2.3.4)where the integration is performed on the cap and D = E1 is defined as thegravitational flux density.By letting = in (2.3.4), the flux of over the sphere is given by

D1. S = M (2.3.5)But 1 V = M (2.3.6)Hence Div D1= 1or Div E1= (2.3.7)Changing E1 to E = −E1 and 1to , we have Div E = (2.3.7’)This is Gauss’ Law for repulsive/attractive gravitation fields.

In the case of an oblique surface, the surface is divided into elemental areas,the normal components of E1 for each elemental surface, multiplied by the area ofthe element and summed for the whole surface. More generally a vector field can bereplaced by a contracted tensor field. It can be verified readily, that the Newtonianchoice D1 = MR = E1 (2.3.8)E1 = − ∇ and = M4 R (2.3.9)

57satisfy the requirements of the definition, when the particle is not in motion. Butwhen the test particle is in motion, these have to be modified subject to (i) theequation of continuity (gauge condition) and (ii) the variation of mass with velocity.The modified form of equation (2.3.9) is equation (2.3.32) of section 2.3.5.2.3.2. Ampere’s Circuital LawDifferentiating (2.3.4) w.r.t. time we have,

D1 ∙ S = 12 M sin + 12 (1 − cos ) M (2.3.10)The second term on the RHS, containing M is negligibly small, since it depends on. Discarding that termRHS = M sin

= M R (tan )where X = − , = ( − ′) + ( − ′)

= M ( )R3= M R × R3

= 2 MRR3 × = 2 D1 × R= 2 | E1 × (− )|= 2 | × E1|

58= 2 | B1|= 2 |H1|= 2 |H1|= ∮ H1 ∙= ∬(∇ × H1) ∙ S (2.3.11)

where we used B1 = × E1 (2.3.12)= 1 (2.3.13)

and B1 = H1 (2.3.14)and the integration is performed along the rim C of the cap for the circulation andthe spherical cap S for the flux. Besides S can be extended to the whole surface ofthe conical frustum, since the components of D1/H1 normal to the conical part of thesurface is zero. Hence, (2.3.11) holds for the complete surface Σ of the conicalfrustum. By using the divergence theorem,LHS of (2.3.10) = ∭(Div D1) V

= ∭ V= ∭ + Div ( ) V= ∭ (Div D1) + Div J1 V= ∭ Div (J1 + ) V

59= (J + ∂D1∂t ) ∙ S (2.3.15)where the region of integration is the complete surface Σ of the conical frustum andV is its volume. From (2.3.11) and (2.3.15) we have∇ × H1 = J1 + ∂D∂t (2.3.16)This is Ampere’s circuital law.2.3.3. Modification of Static Potentials by Retarded PotentialsWe have equation (2.3.9)

= M R for static potentials.In accordance with the theory of retarded potentials, since m is moving with

velocity , we replace R = | − | by the present distance ′ = R R∙ in thenotations of Chapter 1. Thus, we take

= M4 = M04 − ∙ R (2.3.17)A1 = = M04 R − ∙ R (2.3.18)We shall verify that these choices will satisfy B1 = × E and Div B = 0

60Curl A = M0

R ∙R R ∙R R ∙ R

= − M04 −R – − −R –R − ∙ R

= − M04 ( − ) − ( − )R − ∙ R

= M04 × RR R − ∙ R= × M0R4 R − ∙ R

∴ Curl A1 = × E1 = B1 (2.3.19)by using equation (2.3.12) whereD1 = M0R4π R − ∙ R 2 = E1 (2.3.20)

61This is the generalization for equation (2.3.8) when the mass particle is inmotion. Clearly Div = Div Curl A1 ≡ 0 (2.3.21)Equations (2.3.17) to (2.3.20) contain the generalization of the earlierequations (2.3.8), (2.3.9), and (2.3.12). Thus instead of the single scalar potential, we have the four-potentials ( , A1). The foregoing analysis shows that thegravitational fields of a moving particle of mass M consists of a scalar potentialand a vector potential A1, known as Ampere’s vector potential. These are exactlysimilar to the four-potentials of electro-magnetic theory. In the above analysis, if weassume M to be at rest, and consider a test particle of mass moving with velocityin the field of M, then the fields due to M at m can be obtained from the aboveequations by replacing by (− ). Since a charged particle has both mass andcharge, it has gravitational as well as electromagnetic fields. Hence any externalfour potentials will be the sum or super position of gravitational andelectromagnetic potentials in the form = ± or = ± andA3 = A1 ± A2 or A3 = A1 ± A2 where ( , A2) is the four potentials due toelectro-magnetism. This statement is based on the following considerations;being energy/unit mass has dimension of and being energy per unit chargehas dimension of so that and have dimension of 2 and we write

3 = ± 2 or 3 = ± 2. Similarly, A = 1 12 has dimension of velocityand A = has the dimension of energy times i.e. A has dimension ofmomentum so that A has dimension of velocity. Hence, we may write =

62± or = ± . In this general case, and are the averagevelocities of test mass/test charge, relative to the centre of mass/centre of charge.More about centre of mass will be considered in Chapter 3. Since theHamiltonian for the motion of a particle of mass is H = + , wecan re-write it as H = E = + = ∗ where ∗ = + . Thisshows that mass consists at least of an inertial part and a gravitational part. Hencethe momentum of a charged particle in general can be taken as = + A1 ±A2 or = + A1 ± A2 and the mass energy relation may be taken asE= ∗ , where ∗ = + ± or ∗ = + ± .That ismass/energy consists of (i) an inertial part/mechanical part, (ii) a gravitational partand (iii) an electro-magnetic part/quantum mechanical part. Later from thediscussions in chapter 3, we shall conclude that the choice ± is preferable to ± .We have derived the field equations for a single particle. When we considerfluid motion, an elemental volume dV of a fluid in motion will have a large numberof particles and the space between particles may not be free. Hence the permittivity

1 and permeability will have to be modified. Moreover, interactions andfrictional forces make the modification complex. Applications to fluid dynamics willbe continued in section 2.4.2.3.4 Mass - Velocity Relations

For a free particle, the Lagrangian L∗ and the Hamiltonian H represent theenergy, one in the moving frame of the particle, and the other in the laboratoryframe of the observer. Hence it is possible that both can be expressed in the form

63where m is a function of the velocity of the particle. This is done by using thedefining equations in classical dynamics, namely,L* = momentum = (2.3.22)

and H = (2.3.23)Thus by letting L* = ( ) =in (2.3.22) and taking along -axis,we have ′ ( ) = =∴ = . Integrating

= Exp 2 (2.3.24)By defining H = ∙ L* − L*it is seen that (2.3.23) is satisfied. It can be similarly shown that the Lorentzianmass given by [9, 11]

= 1 − (2.3.25)can be obtained from (2.3.23) and the assumption. = ( ) . Thus thereare two possible mass-velocity relations, given by (2.3.24) and (2.3.25).

642.3.5 Non-relativistic Dirac’s Equation, Faraday’s Law and Lorentz Force LawWe have two possible mass-velocity relations given by (2.3.24) and (2.3.25). Hence,there are at least two possible Lagrangians/Hamiltonians in classical dynamics.

(i) L* = − − VH* = ∙ − + + V

= Exp 2(ii) L = ∙ − + − VH = − + V

=As an approximation, these giveL* = 12 − V = L and H* = 12 + V = H, where V is the potential energy ofthe conservative field, in which the particle of mass m moves with velocity .However, from the analysis of the previous sections, it is possible to replaceV = by mass times four potentials (i) ( , A1) in the simple case ofgravitational field of a single mass M and (ii) ( , A3) in the general case. Weconsider three possible Hamiltonians and Lagrangians.

(i) L* = + − A1 ∙ ,H∗ = ∙ − − + A ∙= Exp ( 2⁄ )

65(ii) L = ∙ − + − A ∙ ,H = − , =(iii) L = ∙ − + − A ∙ (for mass particle)L= ∙ − − + (for electron)H = −= Exp ( 2⁄ )

The choice of (i) is based on the following discussion; choice (ii) is the onetreated in relativity; choice (iii) is the non-relativistic counterpart of choice (ii).In Quantum Mechanics [13] the Hamiltonian operator for the motion of acharge has the form

H = + ∙ − + (1)where the velocity operator = = ℏ [ ,H] = . By replacing by c, by −ℏ ∇and H by ℏ ,

(1) becomes ℏ = ∙ −ℏ ∇ + A − +operating on ψ we getℏ = ∙ −ℏ ∇ + A − + (2)This equation (2) is known as Dirac’s equation where and are 4×4matrices. Since we consider = Exp 22 instead of Lorentzian mass we can finda Hamiltonian by changing A into A and into in (1) and write a

66Hamiltonian according to Dirac’s argument. The sign is chosen to get the LorentzForce Law with the correct minus sign of E = −∇ − A1 . Thus, we have thechoicefor a mass particle H* = ( + A1) ∙ − − (2.3.26)(a)for an electron H∗ = ( − A1) ∙ − + (2.3.26)(b)L* = ∙ H∗ − H* = ∙ − H* = + − A1 ∙ (2.3.27)where = Exp ( 2⁄ )

From the manner in which equation (2) is obtained from equation (1)it isclear that the non-relativistic Hamiltonian (2.3.26)(b) for an electron can beobtained by changing the signs of and in equation (1); then the non-relativisticDirac’s equation obtainable from (2.3.26)(b), after dropping the subscript 1, isℏ = c ∙ −ℏ ∇ − A + − (2.3.28)

where and are four-matrices. A comparison between equations (2) and (2.3.28)reveals that both are non-relativistic equations.The equation of motion corresponding to the Lagrangian L* of equation (2.3.27), is∂L∗∂ = ∇L* (2.3.29). . ( − A ) = ∇( ) + ∇ − ∇(A ∙ ) (2.3.30)

where = Exp( 2⁄ )

67By taking ∇ ( 2) = 0 the equation (2.3.30) becomes

( ) = − A1 − ∇ + ∇(A1 ∙ )≈ − A1 + ∙ ∇ A 1 − ∇ + [( ∙ ∇)A1 + × (∇ × A1)]= − ∇ + A1 + × (∇ × A1). . = E + × B (2.3.31)

This is Lorentz Force Law where E = −∇ − A1 (2.3.32)and B1=∇ × A 1 (2.3.33)Now ∇ × E1 = (∇ × A 1) = − B1 i. e. ∇ × E1 = − B1 (2.3.34)

This is Faraday’s Law. Case (ii) will not be treated here, which is found in books onrelativity. In Case (iii) we get− 1 + + × ( × ) += − ∇ + ∂ + × (∇ × A1) (2.3.35)By making the approximation ≈ 0 equation (2.3.35) reduces to the Lorentz ForceLaw (2.3.31).

If we consider a particle of mass moving relative to m with velocitythen the motion of is given by ( ) = F12 where

68F12 = E1 + × B1. Expanding equations (2.3.32) and (2.3.33), by usingequations (2.3.20) and (2.3.18) we haveE = 14 RR − ∙ R 2 − 4 R − ∙B (2.3.36)Also by using equation (2.3.19) we haveB1 = × R4 R − ∙ R (2.3.37). . F = 14 R − ∙ R 2 R + × × R4 (R − ∙ R/ )

− 4 − ∙ (2.3.38)Interchanging the subscripts 1, 2 and changing signs of and , we haveF = 14 E R − ∙ R 2 R+ (− ) × (− ) × R4 R − ∙ − 4 −R − (− ) ∙ (− ) (2.3.39)Adding , F12 + F21= 0 in agreement with Newton’s third law.

It is clear that if we substitute individual velocities and of m and m , relative tosome other origin, then the sum F12 + F21 may not be zero, in apparentdisagreement with Newton’s third law. From the above analysis it is clear that theLorentz Force Law is the generalization of Newton/Coulomb Inverse Square Law. In

69the general case, and and R12/R21 do not lie in a single plane, and hence it isnot a case of two-body problem.2.3.6 Poynting TheoremThe Poynting vector is defined by S1= E1 × H1 (2.3.40)and = 12 + 12 (2.3.41)is defined as energy density.

Poynting’s theorem[8] states that the work done on a mass-body bygravitational forces is equal to the decrease in energy stored in the field, less theenergy that flowed out. . W = + ∇ ∙ (2.3.42)

Proof : In the Lorentz Force Law given by equation (2.3.31)F = (E1 + × B1) Suppose = V and F = V for a small mass∴ = (E1 + × B1) (2.3.43)By definition of work

W = ∙ or W = ∙ (2.3.44). . − W = (E1 + × B1) ∙ = E1 ∙ ( ) = E1∙ J1 (2.3.45)

Also ∇ ∙ (E1 × H1) = H1 ∙ (∇ × E1) - E1 ∙ (∇ × H1)= −H1 ∙ − E1 ∙ J + 1

70= − ∙ − ∙ J − ∙

. . ∇ ∙ = − 12 ∙ + 12 ∙ − ∙ J= − + W by using (2.3.41) and (2.3.45)

∴ W = + ∇ ∙ which is (2.3.42)2.4 Applications to Fluid dynamics

In free space, we took and as the parameters of free space. In fluidmotion, even within an elemental volume V, there will be many particles thicklypacked together, hence these parameters will have to be modified to some averagevalue ′ and ′ . This is to make allowance for the interaction of fluid molecules,such as friction, gravity, viscosity. Hence and take the roles of and andmay be less than unity.2.4.1 Biot - Savart LawIn the Lorentz Force Law− = ( + × ) (2.4.1)the second term × B1 gives a measure of the force/torque due to thegravitational vortex/flux density B1 and is also a measure of vortex motion, in thecase of fluids in motion. Consider an elemental volume

V = A| | = A | | of uniform cross-section A and length | |. The rate of flow offluid volume is = A = A | | and the rate of mass crossing A per unit time is

71V = A | | = A| | = A |J | = I

∴I = I = V = = V = J1 V (2.4.2)where V = , is the mass contained in V and J1 = (2.4.3)

Denoting vortex/flux density contributed by V situated at Q( ′), at the field pointP( ) by dB1 we haveB1 = × E12 = × R4 R2

= I × R4 R2 (2.4.4)∴ B1 = μ I × R4 R2C (2.4.5)which can be re-written asB1 = J1( ) × R V4 R2V (2.4.6)where C is arc, of which I is a part and V' is the volume of mass forming C.Equations (2.4.5) and (2.4.6) are known as Biot-Savart Law, the former as a line-integral and the latter as a volume integral.

P( )RQ( ′)

I ℓFig. 2.1

722.4.2 Ampere’s Vector PotentialBy A1( ) = 4 I ℓ| − | (2.4.7)

or A1( ) = 4 J1( ) dV| − |V (2.4.8)where V′ is the volume of the flow/current loop C under consideration, we canprove that B1= Curl A1 (2.4.9)Proof:

Consider an elemental fluid flow d at Q ( ′); we seek its contribution to A1 at P.Take a square loop A-B-C-D-A centred at P of sides 2ℎ with two sides AB and CDperpendicular to QP and the remaining two sides parallel to QP.∴ = ℓ cos φ + ℓ sin

Rφ C BADQ

I ℓ PFig. 2.3

R Ρ( )QFig 2.2

C

73where ( , ) is a pair of basis vectors i being along QP and j orthogonal to it in theplane of ℓ and and is the inclination of loop element with QP.Area of square, ∆S = 4ℎ2

= R, QP = R = R = – ′∴ ∆A1 = I4 RAB = 2 ℎ ,CD = −2ℎBC = −2ℎ , DA = 2ℎ

∆A ∙ = I4 2ℎ sinR + ℎ − 2ℎ cos√R + ℎ − 2ℎ sinR − ℎ + 2ℎ cos√R + ℎ= I ℓ sin (−4ℎ )4 (R − ℎ )

lim→ ∮ ∆A1 ∙∆S = lim→ − I ℓ sin φ4 (R2 − ℎ ) = − I ℓ sin φ4 R2. . ∆ A1 ∙∆S = I × R4 R2 where = ×. . ∆(curl A ) = ∆B

∴ B1 = ∇ × A1 which is equation (2.4.9)and Div B1 = Div (∇ × A1) ≡ 0 (2.4.10)Since A1 at each point on the arc C is in the direction of Id and the Pfaffiandifferential form A1 ∙ or ∙ d is integrable iff A1∙ curl A1 ≡ 0 we must have

74A1 ∙ (∇ × A1) = 0 i.e. lines of A1 are orthogonal to lines of B1= ∇ × A1 and henceparallel to C.2.4.3 Induced vortex field on the axis of a flow-loopTake any point P on the flow-loop in the form of a circle and any point Q on its axis.The induced vortex-field BQ due to an elemental flow at P (on the loop of fluidflow), is given by

BQ = I × (−R)4 R = I R ×4 R = I ℓ4 R2

where is a unit normal vector at Q perpendicular to QP. Considering thecontributions of all such elements on the circular loop we see that the componentnormal to QX vanishes. Therefore BQ has only an axial component along QX/XQ.∴ BQ = I cos 2 +4 (R sin ) along QXsince radius of loop is R sin , makes angle + θ with QX and the central angle ofcircular loop is chosen as , measured from a fixed diameter.∴ BQ = − IR sin2 ∙ 24 R2= − I2 R sin2

− P XCQ θ Fig. 2.4

75= − I sin32 (R sin )BQ = − I sin32 (2.4.11)where is the radius of the loop.∴ At the centre of the loop.B = − I2 (2.4.12)The negative sign shows that the circular vortex flow along C, induced by a flowelement at Q, opposes the flow element at Q by a counter-induced vortex fieldBQ = − I sin3 , causing to stop the flow at Q. This situation is analogous to theFaradays’ law and Lenz’s law in elector-magnetism: A changing magnetic fieldinduces a current in a closed loop of wire placed in the field (Faraday’s Law) and thedirection of the induced emf is such that any current that it produces, tends tooppose the change of flux. (Lenz’s law).2.5 Gravitational Wave EquationsFrom section 2.3 we have E3 = −∇ϕ3 − ∂A3B = ∇ × A3where = + and A3 = A1 + A 2. To restrict our attention to thegravitational field, we take ϕ = 0, A2 = 0 so thatE1 = −∇ − ∂A1∂t (2.5.1)

76B1= ∇ × A1 (2.5.2)∴ Div E1 = −∇ − (Div A ). . = −∇2 − – 2

by assuming equation of continuity (gauge condition) :+ Div A = 0 (2.5.3)

and Gauss’ Law.i. e. 1 − ∇ = (2.5.4)Changing to and to we get − ∇ = (2.5.4 )From Ampere’s law∇ × = +and = = ∇ ×∇ × B1 = ∇ × ( ∇ × A1)=∇ (∇ ∙ A1) − ∇2A1∴ + D1 = ∇(∇ ∙ A1) − ∇2A1 (2.5.5)∴ + D1 = ∇ − 1 − ∇2A1 (2.5.6)From (2.5.1) = −∇ − . Multiplying by 12 and rearranging

77−∇ 1 = 1 + 1= 1 + (2.5.7)By using (2.5.7) in (2.5.6) we have

+ = 1 + − ∇i. e. 1 − ∇ = (2.5.8)Changing to and to we get− ∇ = = − (2.5.8′)Equation (2.5.4) and (2.5.8) show that both and A1 satisfy the inhomogeneouswave-equations, as in electro-magnetism.When ϕ and A1 are independent of time, (2.5.4) and (2.5.8) become the Poissonequations [7, 8, 12] for repulsive/attractive gravitational fields.∇2 ± = 0 (2.5.4 )∇ A ± J = 0 (2.5.8′) where J1 = These have solutions ± , ±A1 where

( ) = 4 ( ) VR (2.5.9)A ( ) = 4 J ′ V′

RV′(2.5.10)

From these we write the solutions of (2.5.4) and (2.5.8) in the form:( , ) = 14 ( , ) VR( ) (2.5.11)

78A1( , ) = 4 J ′,

R( )′V′ (2.5.12)

where the integrals are evaluated at the retarded time ; it is clear that distinctelemental volume V' of V′ will have different retarded time during the evaluation.It can be shown that these retarded potentials satisfy (2.5.4) and (2.5.8) and are thegeneralisations of all the previous expressions for ( , A1) given by equations(2.3.9), (2.3.17) and (2.3.18). By substituting equations (2.5.11) and (2.5.12) inE1 = −∇ − A1 and B1 = ∇ × A1we get

( , ) = 14 ( , )RR2 + ( , )RR − J( , )R V (2.5.13)( , ) = 4 J1 ′,

R2+ J1 ′,

R′× R V' (2.5.14)

where = − ′ and R = /R. Similarly we can find the solution in the electro-magnetic case by replacing the subscript 1 by 2.2.6. Derivation of Orbit of Planetary Motion

We shall derive the equation of planetary motion by two methods as follows.2.6.1 By using Newton’s Law of Motion

We have = F1 (2.6.1)As in approximation

79( ) = where = −MR4 R3 = -MI4 R2

. . + = (2.6.2)∴ × ( ⁄ ) = × (2.6.3)But from = Exp( 2⁄ ) we have

= ( ⁄ ) ∙ ( ⁄ ) ∴ (2.6.2) ⇒+ [( ⁄ ) ∙ ( ⁄ )] =

i. e. + ( ⁄ ) × ( × ⁄ ) + ( ⁄ )( ⁄ ) =. . ( ⁄ ) + [ × ( × E )] ≈⁄ = -MI4 R2 (2.6.4)

discarding ≈ 0× = I J KR R 0−M4 R2 0 0 = MR 4 R× ( × ) = I J K

R R 00 0 M 4 ε1R = M R − RJ4 R = M 4 − MR 4 R J (2.6.5)Since = (R − R )I + 1R R2 ⁄ J , (2.6.4) and (2.6.5) implyR − R = -M4 ε1R2 − M 4 2 (2.6.6)

80and R2 R = MR 4 R 2 (2.6.7)The latter can be re-written as(R2 )R2 = M R4 R2 Integrating we getR2 ddt = ℎ Exp (−M 4 R⁄ ) = ℎ Exp(− 2⁄ ) or ℎ Exp −MGR (2.6.8)As an approximation MR2 ≈ H or R2 ≈ ℎ (2.6.9)Letting = 1R We have R = −1 ( ⁄ ) = −ℎ ( ⁄ )R = −ℎ ( ⁄ ) R − R = −ℎ ( ⁄ )ℎ − (1⁄ )(ℎ ) = −ℎ [( ⁄ ) + ]Now (2.6.6) can be re-written as−ℎ + = −M4 − M ∙ (ℎ )4∴ + = Mℎ ∙ 4 + M4∴ + = MGℎ + M4 = MGℎ + M4 (2.6.10)This is the equation for planetary motion. [19]By replacing by , by , M by Q| ⁄ | where ( ⁄ ) is nearly a constant forsmall velocities and replacing G = (4 ) by (4 ) we get the correspondingequation (2.6.10) for the motion of an electron in Electro-magnetic field. Thus the

81modified Newton-Lorentz mechanics can handle motion of both mass particle andelectron. On the other hand, GTR being a theory of gravitation cannot handlemotion of electrons and is applicable to mass particle only. In the next chapter, weshall examine these things in detail.2.6.2 By using Modified Newton-Lorentz EquationWe have − = E + × (2.3.31)where, E = −∇ − A (2.3.32)B1 = ∇ × A1 (2.3.33)where, = ± and A3 = A1 ± A2we shall exclude A2 and for the purely gravitational .∴ E = −∇ − A ≈ −∇ 1 = E1 I and = (− ) × by (2.3.19)Here the minus is prefixed to , since the test particle of mass is moving, whereasin (2.3.9), × is the due to source mass M moving with the velocity .∴ (2.3.31) becomes− = + × (− × ). . − ≈ E + × (− × E )

taking ≈ 0 we get (2.6.4), viz

82∴ − ≈ − × ( × ) which is equation (2.6.4) since = − .By proceeding as in section 2.6.1 we finally get the equation (2.6.10).2.7 Proper Lagrangian/Hamiltonian and Equations of MotionIn the foregoing analysis we have used the local time interval dt, in defining velocity,acceleration etc. On the other hand, Minkowski defined the velocity vector as thefour vector , , , where τ is the proper time. Accordingly, we define theproper Lagrangian/Hamiltonian by multiplying the classicalLagrangian/Hamiltonian with = where we let = ; further wewrote , and and in place of , , respectively. The classicalLagrangian/Hamiltonian is(i) L = ( − ). − ( − ) (2.7.1)

where = |1 − ⁄ | (relativistic case)(ii) L = ( − ). − ( − ) (2.7.2)

where = Exp (non-relativistic case)By multiplying each of these by we have,

= [( − ). − ( − )]

83= − A1 ∙ − ( − ). . = ∙ − A1 ∙ − − (2.7.3)

which can be rewritten in the form LP = (2.7.4)where dot denotes time rate w.r.t proper time. Hence Hamilton’s principle ofstationary action for LP is S = 0 (2.7.5)where S = L = (2.7.6)where = = =

= − A112 = ; = − A122 = ; = − A132 == −( − ) and A1 = (A11, A12, A13 )

We have noted earlier that the potentials are part of mass. Hence g may beconsidered as potentials or curvatures of the external field as well as of the movingtest particle consisting of gravitational and/or electromagnetic parts.The Euler-Lagrange equation corresponding to (2.7.5)/(2.7.6) is

− = 0or

− = 0 (2.7.7)By solving this equation we get the Lorentz Force Law. In the former case (i) the

84Euler − Lagrange Equation gives − = E + × B (2.7.8)and in the latter case (ii) we get− 1 + + × ( × ) += − ∇ + ∂ + × (∇ × A1) (2.7.9)where = , = , = , E = −∇ − A1 , B1=∇ × A 1Comparing LP = (2.7.4)with the Minkowski metric for proper time

= = G (2.7.10)we see that (2.7.4) and (2.7.10) are equivalent, except for their dimensions; L hasthe dimension of energy, whereas has the dimension of length.For a general holonomic system letL = L , , ⋯ , , , ⋯ (2.7.11)and =∴ L = L , , ⋯ , + , ⋯ where dot denotes time rate with respect to the proper time τ∴ LP = ( ) L , , ⋯ , , , ⋯ i. e. LP = LP ( , , ⋯ , , , , ⋯ ) (2.7.12)

85∴ LP = LP + LP

= LP + LP − LP. LP − LP = LP − LP = 0 (2.7.13)By using Lagrange’s equations we haveLP – LP = 0 = 1,2, ⋯ ( + 1) (2.7.14)

By definingHP = L − LP (2.7.15)as the proper Hamiltonian, we have from (2.7.13)

HP = 0 (2.7.16)∴ H is a constant independent of τ but involving .2.8 A Comparison of Maxwell-Lorentz Equations for Mass Particle and Electron

We have derived the equations for mass particle; in the same manner, we canderive the equations for electrons. We have omitted the derivations of theseequations, as they are proved in books on electro-magnetism.

86Descriptionof Law Gravito-magnetic fields Electro-magnetic fieldsGauss Law Div = Div =Faraday’s Law ∇ × = − ∇ × = −Field Intensity = −∇ − = −∇ − ∂Flux Density = ∇ × = ∇ ×Equation ofContinuity + ∇ ∙ = 0 + ∇ ∙ = 0Ampere’s Law ∇ × = J + ∇ × = J +Wave Equationof four potentials

∇ − 1 =∇ − 1 = J

∇ − 1 = −∇ − 1 = − J

Energy Law = 12 ∙ + 1 ∙ V = 12 ∙ + 1 ∙ VLorentz’ ForceLaw = ( + × ) = ( + × )2.9. Conclusions

The presence of four-potentials in LP of the modified Newton/Lorentz theory,and its absence in of GTR makes them different theories. LP has four-potentials

87as the different whereas in GTR, G are arbitrary and G is chosen so thatNewtonian scalar potential is obtained as an approximation. The Ampere’s vectorpotential is completely ignored in relativity. GTR stipulates that are connectedwith gravitation only and not connected with electro-magnetic effects or any otherfour-potentials. But the modified Newton-Lorentz theory includes both gravitationand electro magnetism in a single theory. GTR is a theory of gravitation based onthe equations ( ) = 0, R = 0 where R is the Ricci tensor and is the four-dimensional metric. This theory cannot lead to the Newton-Lorentz force law.There is no basis in assuming that non-constant is due to the presence ofgravitation, and constant is due to its absence. The GTR excludes the possibilityof a unified theory for the motion of mass particles and charged particles.