Printer Friendly

A unified theory of interaction: gravitation and electrodynamics.

A theory is proposed from which the basic equations of gravitation and electromagnetism are derived from a single Lagrangian. The total energy of an atom can be expressed in a power series of the fine structure constant, [alpha]. Specific selections of these terms yield the relativistic correction to the Bohr values of the hydrogen spectrum and the Sommerfeld-Dirac equation for the fine structure spectrum of the hydrogen atom. Expressions for the classical electron radius and some of the Large Number Coincidences are derived. A Lorentz-type force equation is derived for both gravitation and electrodynamics. Electron spin is shown to be an effect of fourth order in [alpha].

1 Introduction

In a previous article [2] in this journal we presented a classical Lagrangian characterizing the dynamics of gravitational interaction,

L = - [m.sub.0]([c.sup.2] f [v.sup.2]) exp R/r, (1)

where we denote:

[m.sub.0] = gravitational rest mass of a test body moving at velocity v in the vicinity of a massive, central body of mass M,

[gamma] = 1/ [square root of 1 - [v.sup.2]/[c.sup.2]],

R = 2GM/[c.sup.2] is the Schwarzschild radius of the central body.

The following conservation equations follow:

E = m[c.sup.2] [e.sup.R/r] = total energy = constant , (2)

L = [e.sup.R/r] M = constant, (3)

[L.sub.z] = [M.sub.z][e.sup.R/r] = [e.sup.R/r] [m.sub.0][r.sup.2] [sin.sup.2] [theta] [??] (4)

= z component of L = constant,

where m = [m.sub.0]/[[gamma].sup.2] and

M = (r x [m.sub.0] v) (5)

is the total angular momentum of the test body.

It was shown that the tests for perihelion precession and the bending of light by a massive body are satisfied by the equations of motion derived from the conservation equations.

The kinematics of the system is determined by assuming the local and instantaneous validity of special relativity (SR). This leads to an expression for gravitational redshift:

v = [v.sub.0][e.sup.R/r2], ([v.sub.0] = constant), (6)

which agrees with observation.

Electrodynamics is described by the theory of special relativity. If the motion of a particle is dynamically determined by the above Lagrangian, then a description of the kinematics of its motion in terms of special relativity should yield equations of motion analogous to those of electrodynamics. This, in principle, should allow the simultaneous manifestation of gravitation and electrodynamics in one model of interaction.

We follow this approach and show, amongst others, that electrical charge arises from a mathematical necessity for bound motion. Other expressions, such as the classical electron radius and expressions of the Large Number Hypothesis follow.

The total energy for the hydrogen atom can be expressed in terms of a power series of the fine structure constant, [alpha]. Summing the first four terms yields the Sommerfeld-Dirac expression for the total energy. For higher order terms the finite radius of the nucleus must be taken into account. This introduces a factor analogous to "electron spin".

Details of all calculations are given in the PhD thesis of the author [1].

2 Gravitation and Special Relativity

Einstein's title of his 1905 paper, Zur Elektrodynamik bewegter Korper indicates that electrodynamics and SR are interrelated, with SR giving an explanation for certain properties of electrodynamics. Red-shift is such a property, combining both gravitation and electromagnetism in a single formulation, and should provide us with a dynamical link between these two phenomena. To do this, we substitute the photoelectric effect,

hv = [??][c.sup.2], (7)

where [??] = [gamma][[??].sup.0] and [[??].sub.0] is the electromagnetic rest mass of a particle, into (6). This gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where E is another constant of energy and [??] = [??][c.sup.2] is the total energy of the theory of special relativity.

Let us compare this expansion with the expansion of (2) for the gravitational energy,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The negative sign of the second right hand term in (9) ensures attractive, or bound, motion under gravitation. In order for the motion determined by (8) to be bounded, the third right hand term must similarly be negative and inversely proportional to r. To ensure this we let

[[??].sub.0][c.sup.2] = -[e.sup.2]/[r.sub.e], (10)

where [e.sup.2] is an arbitrary constant and

[r.sub.e] = R/2. (11)

Eq.(8) can then be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

As we shall see for the hydrogen atom, e represents the electron charge, [r.sub.e] represents the classical electron radius and (11) yields some of the numbers of Dirac's Large Number Hypothesis.

The choice of a positive sign in (10) gives repulsive motion. Such a freedom of choice is not possible for the gravitational energy of (9).

2.1 Hamiltonian formulation

Confirmation of the above conclusions can be found by examining the predictions for the hydrogen spectrum. We follow a classical approach based on the principles of action variables [3].

Using the identity [[gamma].sup.2] = 1 + [[gamma].sup.2][v.sup.2]/[c.sup.2] to separate the kinetic and potential energies in (8), a corresponding Lagrangian can be found:

L = - [[??].sub.0][c.sup.2] [square root of 1 - [v.sup.2]/[c.sup.2]] exp([r.sub.e]/r.)]. (13)

We obtain the conjugate momenta:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

The associated Hamiltonian can be derived from the formula H = [summation] [[??].sub.i][p.sub.i] - L as follows

H = [[[[??].sup.2.sub.0][c.sup.4] exp([r.sub.e]/r) + + [c.sup.2] ([p.sup.2.sub.r] + [p.sup.2.sub.[theta]/[r.sup.2] + [p.sup.2.sub.[phi]/[r.sup.2] [sin.sup.2] [theta])].sup.1/2]. (17)

From the canonical equations

[[??].sub.i] = [partial derivative]H/[partial derivative][q.sub.i], (8)

we find the following conservation equations:

[L.sup.2] [equivalent to] [M.sup.2] exp([2r.sub.e] /r) = [p.sup.2.sub.[theta]] + [p.sup.2.sub.[phi]]/[sin.sup.2] [theta], (19)

[L.sub.z] [equivalent to] [M.sub.z] exp([r.sub.e]/r) = [p.sub.[phi]], (20)

where L.sup.2] and [L.sub.z] are constants and

M = (r x [??[v), (21)

is the total angular momentum of the orbiting particle.

It should be noted that (12), (19), (20) and (21) have respectively the same forms as for the gravitational equations (2), (3), (4), (5), but with m = [m.sub.0]/[[gamma].sup.2] replaced by [??] = [gamma][[??].sub.0] and R by [r.sub.e] = R/2.

3 The hydrogen spectrum

In order to determine an expression for the energy levels of the H-atom, two different approaches can be followed: (i) Analogously to the Wilson-Sommerfeld model, one can apply the procedures of action angle variables, or (ii) perturbation theory, where the contribution of each energy term is evaluated separately.

To generalize our discussion we shall, where appropriate, use a general potential [PHI] = [Rc.sup.2]/2r = [r.sub.e][c.sup.2]/r.

3.1 Method of action angle variables

The theory of action angle variables originated in the description of periodic motion in planetary mechanics [4, Ch.9]. From that theory Wilson and Sommerfeld postulated the quantum condition:

For any physical system in which the coordinates are periodic functions of time, there exists a quantum condition for each coordinate. These quantum conditions are

[J.sub.i] = [??} [p.sub.i] [dq.sub.i] = [n.sub.i] h, (22)

where [q.sub.i] is one of the coordinates, [p.sub.i] is the momentum associated with that coordinate, [n.sub.i] is a quantum number which takes on integral values, and the integral is taken over one period of the coordinate [q.sub.i].

Applying these quantization rules to the conjugate momenta of (14), (15) and (16) gives [3]

[L.sub.z] = [M.sub.z] exp([r.sub.e]/r) = [n.sub.[phi]] [??] , (23)

L = M exp([r.sub.e]/r) = ([n.sub.[theta] + [n.sub.[phi]]) [??] = k [??], (24)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (25)

where [n.sub.[theta]], [n.sub.[phi]], k and [n.sub.r] have the values 0, 1, 2, ...

To determine the atomic spectrum we need to evaluate the integral of (25). Because of the finite radius of the nucleus we choose an arbitrary effective nuclear radius of [gr.sub.e]. The potential term in the exponentials is then written as

exp(2[PHI]/[c.sup.2]) = exp([2r.sup.e]/r - [gr.sub.e]), (26)

so that

exp(2[PHI]/[c.sup.2]) =

= 1 = 2 [r.sub.e]/r + 2 [r.sup.2.sub.e]/[r.sup.2](g + 1) + 3 [r.sup.3.sub.e]/[r.sup.3] g(g + 1) + ... (27)

For convenience we also define a parameter f such that

f = 2(g + 1). (28)

We shall subsequently see that the value of g, or f, is related to the concept of electron spin.

Approximating (27) to second order in [r.sub.e]/r, substituting this approximation in (25) and integrating gives

[E.sup.2.sub.m] = 1 - [[alpha].sup.2]/[[n - k + [square root of [k.sup.2] + f [[alpha].sup.2]].sup.2], (29)

where [E.sub.m] = E/[[??].sub.0][c.sup.2], n = [n.sub.r] + k and [alpha] = [e.sup.2]/[??]c is the fine structure constant. This expression is simplified by expanding to fourth order in [alpha]:

[E.sub.m] [congruent to] 1 - [[alpha].sup.2]/[2n.sup.2] [1 + [[alpha].sup.2]/n (1/4n - f/k)]. (30)

The corresponding Sommerfeld/Dirac expressions are respectively

[E.sup.2.sub.m] = [(1 + [[alpha].sup.2]/[[n - k + [square root of [k.sup.2] - [[alpha].sup.2]].sup.2]).sup.-1], (31)

and

[E.sub.m] [congruent to] 1 - [[alpha].sup.2]/[2n.sup.2] [1 + [[alpha].sup.2]/n (1/k - 3/4n)]. (32)

where k = j + 1/2 for the Dirac expression, and j = 1/2, 3/2, 5/2, ... ... (n - 1)/2.

The difference between the energy given by our model [E.sub.W], as given by (30), and that of the Sommerfeld-Dirac model, [E.sub.D], as given by (32), is

([E.sub.D] - [E.sub.W])/[[??].sub.0][C.sup.2] = [[alpha].sup.4]/[2n.sup.3] [1/k (f + 1) - 1/n]. (33)

We shall show below that this difference corresponds to the energy associated with the "spin-orbit" interaction of our model.

4 Perturbation method

We use this method as applied by Born and others [3, Ch. 4].

To apply the perturbation method we need to express the energy [??] in terms of the momentum:

E = [([p.sup.2][c.sup.2] + [[??].sup.2.sub.0][c.sup.4]).sup.1/2] exp([PHI]/[c.sup.2]), (34)

where p = [??]v. Again, taking the finite radius of the nucleus into account, we choose for the potential,

exp([PHI]/[c.sup.2]) = exp([r.sub.e]/(r - [gr.sub.e])), (35)

so that the potential term can be written as

exp([PHI]/[c.sup.2]) = 1 + [r.sub.e]/r + w [r.sup.2.sub.e]/[r.sup.2] + ([w.sup.2] - 1/4) [r.sup.3.sub.e]/[r.sub.3] + ..., (36)

where

w = (g + 1/2) = (f - 1)/2. (37)

With this form for the potential, and using [[??].sub.0][c.sup.e][r.sub.e] = -[e.sup.2], (34) can be expanded as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... (38)

Applying the unperturbed Bohr theory to each braced term, we find the following quantized expressions:

4.1 [E.sub.0]: rest mass energy

The first term on the right is the rest mass energy, which we denote by [E.sub.0]:

[E.sub.0] = [[??].sub.0][c.sup.2]. (39)

4.2 [E.sub.1]: Bohr energy

The next two terms represent the unperturbed Coulomb energy of the hydrogen atom, which we indicate by [E.sub.1]:

[E.sub.1] = [p.sup.2]/[2[??].sub.0] - [e.sup.2]/r. (40)

According to the method of the Bohr theory

[E.sub.1] = -[R.sub.e]/[n.sup.2], n = 1, 2, ... (41)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (42)

4.3 [E.sub.2]: relativistic correction

The third term is denoted by [E.sub.2]. It can be shown that [1]

[E.sub.2] = -[p.sup.4]/8[[??].sup3.sub.0][c.sup.2], (43)

= - [[alpha].sup.2][R.sub.e]/[n.sup.3] [1/k - 3/4n]. (44)

This is the "relativistic correction" of the Bohr-Sommerfeld model [3, [section] 33]. This energy term is similar to that contained in the Dirac expression of (32). The sum of [E.sub.0], [E.sub.1] and [E.sub.2] gives an expression identical to that of Sommerfeld and similar to that of Dirac.

It is well-known that Sommerfeld's result was fortuitous as the effect of spin-orbit coupling was ignored in his model. This effect is incorporated in the Dirac model. In our model we shall see below that [E.sub.3] is an orbit-interaction term and that [E.sub.4] is related to 'electron spin'. These two terms, missing in the Sommerfeld model, can now be added to [E.sub.0] + [E.sub.1] + [E.sub.2] of the Sommerfeld energy expression.

4.4 [E.sub.3]: orbital magnetic energy

We denote the fourth term by [E.sub.3]:

[E.sub.3] = [p.sup.2][r.sub.e]/2[[??].sub.0]r. (45)

Applying the unperturbed Bohr theory, we find from (40):

[E.sub.3] = ([E.sub.1] + [e.sup.2]/r)[r.sub.e]/r

= [r.sub.e]([E.sub.1]/r + [e.sup.2]/[r.sup.2]). (46)

Using (41) and the average values [3, p144],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (47)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (48)

as well as

[r.sub.e]/[a.sub.0] = [a.sup.2], (49)

we get

[E.sub.3] = [[alpha].sup.2][R.sub.e]/[n.sup.3] (2/k - 1/n) = [[alpha].sup.4[[??].sub.0][c.sup.2]/2[n.sup.3] (2/k - 1/n). (50)

The physical interpretation of [E.sub.3] is that it is the energy due to the magnetic interaction of an electron moving in orbit about a proton. This can be seen as follows.

Substituting p = [??]v and [r.sub.e] = -[e.sup.2]/[[??].sub.0][c.sup.2] into (45) gives

[E.sub.3] = - [e.sup.2][v.sup.2]/2r[c.sup.2] [([??]/[[??].sub.0]).sup.2]

[approximately equal to] - [e.sup.2][v.sup.2]/2r[c.sup.2] in the non-relativistic limit. (51)

It corresponds to the classical form of the magnetic energy due to orbital motion, as given by (70) below:

4.5 [E.sub.4]: "electron spin"

[E.sub.4] = [wr.sup.2.sub.e][[??].sub.0][[c.sup.2]/[r.sup.2], (52)

= [we.sup.4]/[[??].sub.0][c.sup.2][r.sup.2].

Applying (48) gives

[E.sub.4] = w2[[alpha].sup.2][R.sub.e]/[n.sup.3]k = w[[alpha].sup.4][[??].sub.0][c.sup.2] 1/[n.sup.3]k. (53)

We consider the significance of the factor w. We note that the potential energy expression (36) can be truncated after the quadratic term in [r.sub.e]/r by letting [w.sup.2] - 1/4 = 0. As such, truncation can be considered as the limit to the resolution of the apparatus used for spectral observation. With this condition, we find that

w = [+ or -] 1/2 (54)

gives the spectrum due to all interactions up to second degree in r/[r.sub.e]. Therefore, from (42) and (53):

[E.sub.4] = [+ or -] 1/2 [e.sup.8[[??].sub.0]/[[??].sup.4][c.sup.2] 1/[n.sup.3]k. (55)

The above expression for [E.sub.4] corresponds to the quantum mechanical result for the energy due to electron spin. Except for the quantum numbers, Eisberg and Resnick [6, Example 8-3] find a similar result for the energy due to spin-orbit interaction.

The equivalence of (55) to the result of Eisberg and Resnick also confirms the implicit value [g.sub.s] = 2 in [E.sub.4].

In this study [E.sub.4] corresponds to the energy due to quantum mechanical spin only. Combining [E.sub.3] and [E.sub.4] gives the corresponding total spin-orbit energy.

For k = 1 the expression for [E.sub.4] is equal to the Darwin term of the Dirac theory. In the Dirac theory the Darwin term has to be introduced separately for l = 0 states, whereas in our model [E.sub.4] already provides for l = 0 through the degeneracy (l = 0,1) associated with the k = 1 level. In summary, 'electron spin' represents a second order contribution [r.sup.2.sub.e] /[r.sub.2] to the total energy of the atom.

The above reasoning also applies to higher orders of approximation. Expanding (35) to fourth degree in [r.sub.e]/r gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (56)

The coefficient of [r.sup.4.sub.e]/[r.sup.4] is zero if ([w.sup.2] - 1/4) w = 0, or

w = 1/2, -1/2, 0. (57)

A next higher resolution to [r.sup.3.sub.e]/[r.sup.3] therefore introduces an additional value of w = 0, giving a triplet symmetrical about this value.

For a comprehensive survey of the conceptual developments surrounding electron spin we refer to the text by Tomonaga [7].

4.6 [E.sub.5]: radiative reaction

[E.sub.5] = w [p.sup.2][r.sup.2.sub.e]/2[[??].sub.0][r.sup.2] (58)

= [+ or -] 1/2 [[alpha].sup.4] [R.sub.e] [1/[n.sup.5]k - 1/[n.sup.3][k.sup.3]]. (59)

substituting p = [??]v in (58) gives

[E.sub.5] = [+ or -] 1/2 [v.sup.2][e.sup.4]/2[[??].sub.0][c.sup.4][r.sup.2] [([??]/[[??].sub.0]).sup.2]. (60)

In the non-relativistic limit, [??] [approximately equal to] [[??].sub.0], the above term corresponds to the last RHS term of (69), i.e. the classical energy resulting fromradiative reaction. Its value is too small (~, [10.sup.-8] eV) to affect the values of the fine-spectrum.

4.7 Summary

[E.sub.0] = [m.sub.0][c.sup.2] : rest mass energy,

[E.sub.1] = -[R.sub.e]/[n.sup.2] : Bohr energy,

[E.sub.2] = -[[alpha].sup.2][R.sub.e]/[n.sup.3] [1/k - 3/4n]: relativistic correction,

[E.sub.3] = [[alpha].sup.2][R.sub.e]/[n.sup.3]/[2/k - 1/n] : orbital magnetic energy,

[E.sub.4] = w 2[[alpha].sup.2][R.sub.e]/[n.sup.3] k : electron spin energy,

[E.sub.5] = w[[alpha].sup.4] [R.sub.e] [1/[n.sup.5] k - 2/[n.sup.3] [k.sup.3] : Radiative reaction,

where w = [+ or -] 1/2.

The sum of the energy terms [summation], [E.sub.i] = [E.sub.0] + [E.sub.1] + [E.sub.2] + [E.sub.3] + [E.sub.4] + [E.sub.5] is:

[summation] [E.sub.i]/[[??}.sub.0][c.sup.2] = 1 - [[alpha].sup.2]/2[n.sup.2] [1 - [[alpha].sup.2]/n(f/k - 1/4n)], (61)

which, as expected, is the same as (30).

Each term in (38) can be related to a standard electro-dynamic effect. It is significant that although (38) does not explicitly contain any vector quantities, such as the vector po tential A, this potential is implicit, as shown in the discussion of [E.sub.3] and the comparison with (69).

An explanation for the difference (33) between the spectrum of the proposed model and that of Dirac-Sommerfeld can be seen as follows:

Consider the sum

[E.sub.3] + [E.sub.4] = [[alpha].sup.2][R.sub.e]/n.sup.3] [2/k (w + 1) - 1/n] (62)

or, since w = (f -1)/2,

[E.sub.3] + [E.sub.4] = [[alpha].sup.2] [R.sub.e]/[n.sup.3] [1/k (f + 1) - 1/n. (63)

The above equation corresponds to (33), the difference between the Sommerfeld-Dirac expression and that of our model. The expression (30) therefore already incorporates the spin-orbit interaction.

The energy [E.sub.3] + [E.sub.4] therefore represents a perturbation to the Sommerfeld-Dirac values. The only candidate for this perturbation is the Lamb-shift. For the (n, k) = (2,1) level and for w = - 0.5 the value of [E.sub.3] + [E.sub.4] is 4.52 83178 e-5 ev. The Lamb-shift for this level is 4.37 380 19 e-6 eV, which is an order 10 smaller. It would be overly ambitious to find the observed Lamb-shift from the present simple model. At this degree of spectral resolution one would have to look at a modification of the effective nuclear radius to r - [[alpha].sub.1][r.sub.e] - [a.sub.2][r.sup.2.sub.e]

4.8 Comparison with classical electromagnetic energy

In order to compare the results of this study with those of conventional electromagnetic theory, we give a brief summary of the energy relations of classical electrodynamic theory.

The Hamiltonian describing the interaction of an electron with fields H and E is given by [8, p.124]

[H.sub.classical] = e [PHI] + [(p - e/c A).sup.2] /2[??], (64)

where [PHI] and A are respectively the electrostatic and vector potentials of the system.

It is important to note that A and [PHI] do not merely represent the external fields in which the particle moves, but also the particle's own fields. This implies that the force of radiative reaction is automatically included.

The corresponding classical Lagrangian is

[L.sub.classical] = [p.sup.2]/m[??] - e [PHI] + e/c A x V. (65)

For an electron moving under the influence of a magnetic field,

H = e (v x r)/[cr.sup.3], (66)

a vector potential A can be found as

A = 1/2 (H x r) = ev/2cr. (67)

Substituting this expression for A and using p = [??]v, yields

[(p - e/c A).sup.2] = [p.sup.2] - [e.sup.2][v.sup.2][??]/[c.sup.2] r + [e.sup.4][v.sup.2]4[c.sup.4][r.sup.2]. (68

Since the Hamiltonian of (64) does not contain t explicitly, we may equate it to the total energy. Consequently, substituting (68), and e[PHI] = - [e.sup.2]/r, in (64) gives the classical energy

[E.sub.classical] = - [e.sup.2]/r + [p.sup.2]/2[??] - [e.sup.2][v.sup2]/2[c.sup.2]r + [e.sup.4][v.sup.2]/8[??][c.sup.4][r.sup.2]. (69) (69)

The third RHS term is the magnetic energy due to the orbital motion of the electron:

[E.sub.orbital] = [[mu].sub.l] x H = - [g.sub.l][e.sup.2][v.sup.2]/2r[c.sup.2], (70)

where [[mu].sub.l] = magnetic moment, [g.sub.l] = Lande g factor = 1, and M and H are parallel to one another. This energy corresponds to that of [E.sub.3] above.

The fourth RHS term of (69) represents radiative reaction, which corresponds to our [E.sub.5] as given by (60).

The standard relativistic Hamiltonian is given by:

[H.sub.relativistic] = [[[(p - q A/c).sup.2][c.sup.2] + [[??].sup.2.sub.0][c.sup.4]].sup.1/2] + q [PHI]. (71)

The Hamiltonians of (64) and (71) must be compared to ours of (17).

It is well-known that the Bohr model for the atom fails because of radiative reaction; in our model this loss is compensated for by the additional and associated potential term, [E.sub.4], This term can also be interpreted as a modification of Coulomb's law. It is significant that this energy term can also be interpreted as arising from electron spin.

It is also significant that the Sommerfeld relativistic correction term, [E.sub.2], does not appear in either (69) or (71).

We can consider the electromagnetic energy arising from the Hamiltonians of (64) and (71) as approximations to that of our Hamiltonian of (17).

We also note that the energy derived from the Hamiltonian of (64), which is normally derived from a Lagrangian containing the vector potential A, appears as an approxima tion to our model, which does not explicitly contain a vector potential. A vector potential arises in our theory because of the variation of mass according to (12).

5 The large number coincidences Dirac postulated that the large dimensionless ratios (~ [10.sup.40]) of certain universal constants underlie a fundamental relationship between them. A theoretical explanation for these ratios has not yet been found, but it became known as Dirac's Large Number Hypothesis (LNH). [9] Some of these relations are derivable from (11).

Taking R as the Schwarzschild radius of the proton, [R.sub.p] = 2G[M.sub.p][c.sup.2], we rewrite (11) as

- .sup.2]/[[??].sub.0][c.sup.2] = G[M.sub.p]/[c.sup.2]

or - [e.sup.2]/G[M.sub.p][[?].sub.0] = 1. (72)

Defining the relationship between the gravitational mass Mp and the electromagnetic rest mass [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the proton as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (73)

where [N.sub.D] is a dimensionless number, we can write (72) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (74)

which, if the absolute value is taken, is the basic relationship of the LNH.

6 Lorentz force

The force equation for a particle, mass [??] and velocity v is found by applying the Euler-Lagrange equations to (13). This gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (75)

Defining

E = [??] [r.sub.e][c.sup.2].[r.sup.2], (76)

H = [r.sub.e]v x r/[r.sup.3], (77)

we can write (75) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (78)

For v [much less than] c, [??][r.sub.e][c.sup.2] [right arrow] [[??].sub.0][r.sub.e][c.sup.2] = [e.sup.2] and then (75) approaches the classical Lorentz form.

7 Unifying gravitation and electromagnetism

Equation (16) of reference [1] can be combined with (78) in one formulation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (79)

where for

Gravitation : k = -1,

Electromagnetism : k = 1.

The same equation gives either planetary or atomic motion, where the vectors E and H are respectively given by

E = [??] GM/[r.sup.2] = [??] [r.sub.e][c.sup.2]/[r.sub.2], (80)

H - GM(v x r)/[c.sup.2][r.sup.3] = [r.sub.e]v x r/[r.sub.3], (81)

8 Summary

Gravitation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Electromagnetism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

9 Nuclear force

In a subsequent article we shall show that equations for the nuclear

force, such as the Yukawa potential, can be derived by considering the forms of both the energy equations (2) and (8) at r [approximately equal to] R/2 = [r.sub.e].

Submitted on May 25, 2008

Accepted on June 03, 2008

References

(1.) Wagener P C. Principles of a theory of general interaction. PhD thesis, University of South Africa, 1997, An updated version is to be published as a book during 2008.

(2.) Wagener PC, A classical model of gravitation. Progress in Physics, 2008, v. 3, 21-23.

(3.) Born M. The mechanics of the atom. Frederick Ungar Publ. Co., New York, 1960.

(4.) Goldstein H. Classical Mechanics. Addison-Wesley, Reading, Mass., 1959,

(5.) Eisberg R.M. Fundamentals of modern physics. John Wiley, New York, 1961,

(6.) Eisberg R. and Resnick R. Quantum physics of atoms, molecules, solids, nuclei and particles. John Wiley, New York, 1985.

(7.) Sin-Itiro Tomonaga. The story of spin. Univ Chicago Press, 1997.

(8.) Grandy Jr, W. T. Introduction to electrodynamics and radiation. Academic Press, New York, 1970,

(9.) Harrison E. R. Physics Today, December 30, 1972,

Pieter Wagener

Department of Physics, Nelson Mandela Metropolitan University, Port Elizabeth, South Africa

E-mail: Pieter.Wagener@nmmu.ac.za
COPYRIGHT 2008 Progress in Physics
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2008 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Wagener, Pieter
Publication:Progress in Physics
Article Type:Report
Geographic Code:6SOUT
Date:Oct 1, 2008
Words:4675
Previous Article:International injustice in science.
Next Article:On emergent physics, "unparticles" and exotic "unmatter" states.
Topics:

Terms of use | Privacy policy | Copyright © 2022 Farlex, Inc. | Feedback | For webmasters |