Autor: Eloisa Lima | Publicado: 2011-09-20T15:24:00-07:00

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TOWARDS A UNIFIED THEORY OF FORCES 

Pieter C. Wagener BA, MSc, MA, MSc, LLM, PhD, DipData, DipLL 

Department of Physics 
Nelson Mandela Metropolitan University, Port Elizabeth 6000, South Africa. 

 

Abstract 

 

A theoretical model is proposed that unifies the forces of gravitation, electromagnetism and the 
nuclear (strong) force. The classical electron radius, Dirac's Large Numbers and an explanation 
for electron spin follow from the model. The Mach relation of cosmology is derived and 
applications to modern astrophysics are indicated. These include an explanation for the 
accelerating expansion of the universe and a force equation for the higher order gravitational 
properties of binary pulsars. 

______________________________________________________________________________ 

 

Vers une théorie unifiée des forces 

Pieter C. Wagener BA MSc MA MSc LLM PhD DipData DipLL 
Département de Physique, Université Métropolitaine Nelson Mandela, Port Elizabeth 6000, 
Afrique du sud. 

 

Résumé 

 

Un modèle théorique unifiant les forces de gravitation, l’électromagnétisme et la force nucléaire 
(forte) est proposé. Suivent le rayon classique de l’électron, les grands Nombres de Dirac et une 
explication du spin de l’électron à partir du model. La relation de Mach en cosmologie est 
dérivée et des applications d’astrophysique moderne sont indiquées. Ceux-ci incluent une 
explication pour l’expansion accélérée de l’univers et une équation de force pour les propriétés 
gravitationnelles d’ordre supérieure des pulsars binaires. 

______________________________________________________________________________ 

 

Hacia Una Unificada Teoría de las Fuerzas 

Pieter C. Wagener BA MSc MA MSc LLM PhD DipData DipLL 
Depto. De Física, Universidad Metropolitana Nelson Mandela, Puerto Elizabeth 6000, Sur 
África. 

 

Extracto 

Se propone un modelo teórico que unifica las fuerzas de gravitación, el electromagnetismo y la 
(fuerte) fuerza nuclear. Del modelo siguen el clásico radio del electrón, los Grandes Números de 
Dirac y una explicación de la revolución del electrón. Se deriva la relación de la cosmología 
Mach y se indican aplicaciones a las modernas astrofísicas. Éstas incluyen una explicación de la 
acelerante expansión del universo y una ecuación de fuerza para las propiedades gravitacionales 
de mayor orden de los pulsares binarios. 

______________________________________________________________________________ 


 

Para uma teoria unificada de forças 

Pieter C. Wagener BA MSc MA MSc LLM PhD DipData DipLL 

Departamento de Física, Universidade Metropolitana Nelson Mandela, Porto Elizabeth 6000, 
África do Sul. 

 

Sumário 

 

Um modelo teórico é proposto que unifica as forças de gravitação, do electromagnetismo e da 
força (forte) nuclear. O radius clássico do elétron, os Números Grandes de Dirac e uma 
explicação para a rotação do elétron seguem a partir do modelo. A relação da cosmologia do 
Mach é derivada e as aplicações à astrofísica moderna são indicadas. Estes incluem uma 
explicação para a expansão de aceleração do universo e uma equação da força para as 
propriedades gravitacionais de uma ordem mais elevada de pulsars binários. 

______________________________________________________________________________ 

 

Ansaetze zu einer vereinten Theorie der Kraefte 

Pieter C. Wagener BA MSc Ma MSc LLM PhD DipData DipLL 

Department of Physics, Nelson Mandela Metropolitan University, Port Elizabeth 6000, South 
Africa. 

 

Zusammenfassung 

 

Es wird ein theoritisches Model vorgeschlagen, dass die Kraefte der Gravitation, des 
Elektromagnetismus und der starken Nuklearkraft vereinigt. Der klassische Elektronenradius, 
Diracs Grosse Zahlen und eine Erklaerung der elektronischen Drehung erscheinen in dem 
Model. Der Mach-Zusammenhang der Kosmologie folgt daraus und Anwendungen in der 
modernen Astrophysik werden angedeutet. Diese schliessen ein die beschleunigte Expansion des 
Universums und die Kraftgleichung fuer die hoehere Ordnung der Gravitationseigenschaften der 
binaeren Pulsare. 

______________________________________________________________________________ 

1. Introduction 




Modern physics distinguishes amongst four fundamental forces, i.e., forces that act differently 
under different conditions. They are the forces of gravitation and electromagnetism, the nuclear 
force and the weak force. Physicists have been successful in devising a model that incorporates 
the latter three forces, but the incorporation of gravitation remains elusive. A plethora of theories 
has been developed over the past forty years to include gravity, with string theory as the most 
popular recent candidate.1 There are also supersymmetry and quantum gravity, but none of these 
models appear to be successful. 

 

There is a firm belief among physicists that a single model of forces should exist, and the pursuit 
of such a model has been described as a search for the Holy Grail of physics. This model should 
be a single consistent mathematical formulation giving the properties of the four fundamental 
forces. 

 


Such unification is desirable from a philosophical viewpoint. A fundamental tenet of mystical 
schools is that creation forms a unity, with the different forms the corporeal manifestations of a 
single, creative spirit. Our understanding of creation is driven by striving for such simplicity and 
beauty. 

 

Another fundamental principle of the mystical schools is that of “As above, so below.” Within a 
scientific context it implies that the very large should be reflected in the infinitesimally small. An 
example of this would be a model that describes both planetary motion and the motion of 
electrons in an atom. 

 

We approach the problem by starting with a model of gravitation. In a previous paper in this 
journal2 we presented a model, derived from Newton’s ideas, which satisfies the three classical 
tests for a theory of gravitation. 

 

At this stage we need to clarify our use of the terms ‘model’ and ‘theory.’ By a model we 
understand the body of mathematics that describes a system, and by theory we also include the 
postulates and assumptions on which the mathematical body is constructed. A model is 
constructed from postulates, assumptions, and the appropriate mathematical tools, and yields 
specific results or predictions. A theory tells us how that model is constructed. 

 

Our paper is mainly descriptive and we shall limit its mathematics to an essential minimum. We 
shall avoid calculus where possible. The mathematical detail can be found in Ref. 3. However, 
no discussion on a unified theory can be sensible without some mathematics. As stated by 
Richard Feynman (1918-1988):4 “People who wish to analyze nature without using mathematics 
must settle for a reduced understanding.” For the scientist an elegant mathematical formulation 
represents its version of beauty. 

 

Correct mathematics is one of the requirements for a model of physics. The other requirements 
are valid postulates or presumptions, consistency of argument, and agreement with observation. 
We aver that the proposed model satisfies all these requirements. 

 

We must emphasize that our model is strictly based on standard, conventional physics, and 
specifically on the principles of Hamiltonian dynamics and special relativity theory. 

 

2. The Newtonian model 




Our Newtonian-type model of gravitation is based on a Lagrangian [see eq. (17) of Ref. 2], 

 , (1) rRevcmL/220)(+-=

where 

0m 

= gravitational mass of a particle, 

c 

= speed of light, 

R 

= Schwarzschild radius = 2GM/c2 , 

G 

= gravitational constant, 

M 

= mass of a massive, central body, 




v 

= speed of the test particle relative to the central body, 

R 

= distance of the test particle from the central body. 



 

A Langrangian is a compact mathematical expression from which we can derive the properties of 
a system, such as the energy and the forces acting on the system. It will also give the equations 
of motion of a particle, i.e., how it moves under the forces acting on the system. We call the 
actions due to a force the dynamics of a system. 

 

In the rest of the text ‘the Lagrangian’ shall refer to the Lagrangian of (1). 

 

The energy E of a particle can be derived from the Lagrangian (see Ch.3 of Ref. 3) as 

 

 2/
20.rRecmE=, (2) 

where 22/1/1cv-=.. 

It will be convenient, and more general, to write the above equation in terms of the gravitational 
potential F = GM/r = Rc2/2r. The above equation then becomes 

 

 

 2/2202.cecmEF= . (3) 

It is also useful to define a variable gravitational mass, 

 , (4) 20/.mm=

so that (2) and (3) can be rewritten as 

 

 . (5) 2/22/2crRemcemcEF==

 

It must be noted that m decreases with velocity. 

 

For the systems described in Ref. 2, the Lagrangian describes two types of motion. The first one 
is that of a particle with non-zero mass moving about the central body. In the case of a planet, 
such as Mercury, it describes, not only an elliptical path for the planet, but also a precession of 
the ellipse. 

 

The second type of motion is for a particle of zero mass, i.e., for a photon, or particle of light. In 
this case the Lagrangian predicts a hyperbolic path, which results in the bending of a light ray by 
the sun. 

 

Both the above predictions conform with observation. These two phenomena are dynamical in 
origin. 


 

With the foregoing in mind we can state our first postulate: 

 

Postulate 1: The dynamics of a system of particles subject to gravitational forces is determined 
by the Lagrangian of (1). 

 

3. Gravitational redshift 




The third test is that of gravitational redshift. This effect implies that the frequency of a photon 
will decrease with increasing strength of a gravitational field. For example, the frequency of a 
photon moving closer to a star will decrease, i.e., its frequency is shifted towards the weaker, or 
red side, of the spectrum. In order to accommodate this effect in the Newtonian model we make 
an important assumption. To understand this assumption we first look at the tenets of Special 
Relativity. 

 

4. Special relativity 




Special relativity (SR), as formulated by Einstein (1879-1955) in 1905, can be represented by the 
metric, 

 , 222222drdtcdcds-==t

where 

 

 ds = invariant Minkowskian metric, 

 t = the proper time as measured on a clock (the proper clock) 

 carried by the test particle, 

 t = the coordinate time as measured on the different clocks 

 (the coordinate clocks) as the test particle passes them. 

 

If we substitute , the above equation can be written as dtdrv/=

 

 dtd=t., (6) 


the time-dilation formula of SR. 

 

Since the proper time is measured at one position on one body it will be the same for all 
observers, whether they are fixed at various points in the coordinate frame or moving with the 
proper clock. We call such a quantity an invariant. The energy E in (2) is similarly an invariant. 

 

In Fig. 1 the coordinate clocks A and B are fixed in some reference frame, whereas the proper 
clock moves from one coordinate clock to the other. SR states that the time interval 12ttt-=., 
as measured on the proper clock, will be shorter than the time interval 12ttt-=. between the 
two coordinate clocks. For infinitesimal small intervals the difference is given by (6), where v is 
the relative velocity between the proper and coordinate clocks. 


 

 coordinate clock A coordinate clock B 

 


 

Figure 1: Proper clock passing coordinate clocks. 

 

A fundamental requirement of SR is that the relative velocity between the coordinate and proper 
clocks must be constant, which also implies motion along a straight line. The description of such 
motion, as well as any others, is called kinematics. Kinematics only tells us how a particle moves 
in space with time, without referring to the forces giving rise to the motion. 

 

SR is a pure kinematical theory. The early challenge of gravitation theory was to incorporate the 
results of SR into the dynamics of gravitation. Einstein's first efforts were along this line.5 

In this paper we shall show how such incorporation can be achieved. 

 

5. The red-shift formula 




SR is not only a kinematical theory; it is also a theory of electromagnetism. This is indicated by 
the title of Einstein’s paper: Zur Elektrodynamik bewegter Körper. Incorporating SR into our 
theory of gravitation will then provide the link between gravitation and electromagnetism. For 
this we need the red-shift formula. 

 

Frequency is the inverse of time, so that the inverses of (6) give the Doppler shift for frequency: 

 p...=, (7) 

where . and .p are respectively the frequencies of a coordinate clock and a proper clock. By 
substituting for . from (2) we find 

 

 rRpeA2/-=.., (8) 


where 

 20/cmEA=. (9) 


Since both A and .p are invariants we may define another invariant frequency as 

 

 pA..=0. (10) 

Equation (8) can then be written as 


 , (11) rRevv2/
0-=

 )2/1(0rR-˜., (12) 

which gives the observed values for gravitational redshift. 

 

One of the perplexities of astrophysics is the relation between Doppler shift and gravitational 
redshift. The above relations clarify this relation. 

 

Multiplying (11) by Planck’s constant h on both sides gives an energy equation, 

 rRehE/
0~-=., 

where 

 .hE=~ (13) 

has the form for the energy of a photon. A significant aspect of the equation is that it contains the 
quantum mechanical h and the gravitational G in one equation. 

 

Since both E and h.0 are invariants, they are related by a constant. Since these invariants do not 
have any absolute values at this stage, we may set the constant equal to unity. 

 

Therefore 

 rReEE/~=. (14) 

With the foregoing in mind we now formulate our second postulate: 

 

Postulate 2: Special Relativity is valid instantaneously and locally at all points in the reference 
system of the central massive body. 

 

This is also a standard assumption in relativity theory. It implies that the time-dilation formula of 
(6) is valid locally at all points in space. The time difference between the proper clock and the 
coordinate clocks varies with ., or equivalently, with the velocity. But, since . is related to the 
gravitational potential according to (3), the time differences also depend on the gravitational 
potential at the point where the measurement is made. 

 

6. The photo-electric effect 




Einstein received the Nobel Prize for his 1905 paper on the photo-electric effect. This relation 
relates the kinematical energyE~ of SR to the frequency of a photon: 

 

 .hcmE==2~~, (15) 

where 

 0~~mm.=. (16) 


The quantitym~
is the electromagnetic mass of a particle, and it varies relative to its rest mass 
0~maccording to (16). The difference between the gravitational mass m and the electromagnetic 
mass0~mis fundamental to our theory. 

 

The above relations forE~, as well as the link between electromagnetism and the kinematics of 
SR, persuades us to associateE~with the electromagnetic energy of a system. 

 

From (14), (15) and (16) we get 

 
rRrRecmcvecmE2/2222/
20~
/1~=
-
=, (17) 

or, in terms of, F

 
22/
22/
20~
/1~cceEcvecmEFF=
-
=. (18) 

The corresponding Lagrangian for the above energy can be derived as 3 

 
2/2220/1~cecvcmLF--=. (19) 

 

7. The Hydrogen atom 




Expanding (18) gives 

 
K+F+F+
F+F++=
4220220220020204/~2/~
2/~~2/~~
cvmcmcvmmvmcmE 
(20) 

Consider the first three terms on the right-hand side (RHS) of the above equation: 

 
. (21) F++=020202/mvmcmE

In order for the motion of the system to be bounded, the last term must not only be inversely 
proportional to r, but also negative. To ensure this we let 

 rem/~20-=F , (22) 

where e is an arbitrary constant. 

 

This is a significant result as the constant e eventually turns out to be the charge on an electron. 
Its existence is a mathematical requirement for bound motion. Alternately, a positive sign in (22) 
results in unbounded, repulsive motion. 


 

Since , (22) can also be written as rRc2/2=F

 erecm/~220-=, (23) 

where 

 2/Rre=. (24) 

In this case R is the Schwarzschild radius of the proton. 

 

Eq. (23) has the same form as the expression for the classical electron radius. In standard physics 
this expression is an ad hoc defined quantity, whereas in our model it is derived from the 
properties of the system. 

 

Eq. (17) can now be written as 

 rreecmE/20~=. (25) 

8. Hydrogen spectrum 




In mathematical physics we use a perturbation method when a quantity, such as the energy 
above, is expressed in a convergent series. To apply this method to E above, we need to express 
the energy in terms of the momentum. From SR, 

 

 224202~cpcmE+= , (26) 

where vmprr~= is the classical momentum of a particle. 

 

Substituting this expression in (18), we get 

 

 . (27) 2/2/142022)(cecmcpEF+=

We take the finite radius of the proton into account by changing r to egrr-, where g has a value 
between 0 and 1. The potential function can then be written as 

 , (28) ))/(exp()/exp(2eegrrrc-=F

which can be expanded as 

 L+-+++=F332222)4/1(1)/exp(
rrwrrwrrceee, (29) 

where 

 )2/1(+=gw. (30) 

With this form for the potential, and using 220~ercme-= from (23), (27) can be expanded as 


 

 

L4443444214342143421434213214342143421321+-+
-+++--+=
765432103322023042022220202230420220)4/1(~
~8~2~
~2~8~2~
EeEEeEeEeEEErrwcmrcmreprmrpwrcmrwrmrpcmprempcmE 
(31) 

Each of the braced terms has a specific physical meaning, and each one can be quantized 
according to the unperturbed Bohr theory (see section 8.3.3 of Ref. 3). We then find the 
following expressions: 

 

200~cmE= 

: 

rest mass energy 

L,2,1,/21=-=nnREe 

: 

Bohr energy 

nknknREeL,2,1,431222=..
.
..
.--=
a 


: 

“relativistic correction” of 
the Bohr- Sommerfeld model 

..
.
..
.-=
nknREe12323a 


: 

orbital magnetic energy 

kncmekncmwknRwEe32408320432412~1~12h±===aa 

: 

quantum mechanical spin 
energy 

20240425~
~221...
.
...
.
...
.
...
.
±=
mmrcmevE 

: 

radiative reaction 



 

The constant 2/~202cmRea= is the Rydberg constant,p2/h=h, and is the fine-
structure constant. The value , which corresponds to the quantum values of electron 
spin, arises from truncating the series after E6, which implies setting E7 and higher order energies 
equal to zero. In physical terms this implies a limit to the resolution of our observational 
apparatus. 
ceh/2=a
½w=±

 

The sum of the first three terms gives the fine-structure spectrum of the hydrogen atom: 

 

 ..
.
..
.
..
.
..
.-+-.++
nknncmEEE431121~/)(
22220210aa 
. (32) 

The sum of the remaining terms E3 and E4 represents the total spin-orbit energy. This energy is 
related to higher order effects such as the Lamb shift. Details about these effects can also be 
found in Ref. 3. The calculated value of the Lamb shift differs by an order of 10 from the 
observed value. However, since the Lamb shift is a third order perturbation of the hydrogen 
spectrum, the discrepancy could possibly be resolved by a second order correction to the 
effective nuclear radius. 

 


For k=1 the expression for E4 is equal to the Darwin term of the Dirac theory.6 

 

9. The Large Number Coincidences 




Dirac postulated that the large dimensionless ratios (~1040) 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).7 Some of 
these relations are derivable from (24). 

Taking R as the Schwarzschild radius of the proton, Rp = 2GMp/c2, we rewrite (24) as 

 ,~2202cGMcmep=- 

 or 

 .1~
02=-
mGMep 
(33) 

 

 

Defining the relationship between the gravitational mass Mp of a proton and its electromagnetic 
rest mass pm0~, as 

 pDpmNM0~= , (34) 

where ND is a dimensionless number, we can write (33) as 

 DpNmmGe=-
002~~ . (35) 

Taking the absolute value of the above equation gives the basic relationship of the LNH. Further 
LNH derivations are derived in Ref. 3. 

 

10. A unified force model 




One can derive the forces acting on a particle by applying the Euler-Lagrange equations to the 
Lagrangian of the system. From the Lagrangian of (19) we find a Lorentz-type force equation 
(see 7.4 of Ref. 3): 

 


 ][~
0HvEkmprrr&r×+= , (36) 

where for 

 Gravitation: 1-=k, 

 Electromagnetism: 1+=k. 

 

The same equation gives either planetary or atomic motion, where the vectors Er 
and Hrare 
respectively given by 

 

 222ˆˆ
rcrrrGMrEe==
r 
, (37) 

 332)()(
rrvrrcrvGMHerrrrr×=
×= . (38) 

 

11. Interim summary 




We have shown in a descriptive way that the Newtonian-type Lagrangian (1) gives both the 
gravitational and electromagnetic equations of motion. In particular, we can derive the paths of 
planets and the spectrum of the hydrogen atom from this fundamental Lagrangian. The model is 
further confirmed from the fact that both the classical electron radius and Dirac’s Large Number 
Relations are inherently part of the model. 

 

The distinguishing feature from other attempts to unification is the distinction between the 
gravitational mass m and the electromagnetic mass m~ 
of a particle. This distinction is discussed 
further in section 14. 

 

The Lorentz-type force offers a description of higher order gravitational properties such as 
geodetic precession and frame dragging of binary pulsars. 

12. The nuclear (strong) force 




Any theory of the nuclear, or strong interaction, must satisfy certain basic requirements 
concerning the properties of the force. They are briefly (see Ch.17 of Ref. 8): 

 

(a) The force is charge independent, 


(b) it only acts over a range ~ 10-13 cm, 


(c) the form of its potential is )/, 

exp(/2qrrrQ--
(d) where the coupling constant h/2Q could range from 1 to 15, 


(e) qris related to the mass pmof a pion by cmh 

rqp/~.


 

At the gravitational energy equation (5) can be rearranged in a unique form as:3 rR˜

 . (39) /exp()/(2RrrRmcE˜


Repeating the same procedure for the electrodynamic energy (25) we find that at , err˜

 

 /exp()/~(2eerrrrcmE˜ . (40) 

 

Both (39) and (40) are accommodated in the model for a deuteron as depicted in Fig. 2. We also 
use for . qm%m%

 

 


 

Fig. 2: Model of a deuteron. Two protons are separated at a distance R from each 
other. A particle of mass qm~
and charge –e moves in a figure eight pattern about 
each of them, alternately at a radius of r = rq =|re| from each proton. 

 

For this case we write (23) as 

 eerecm/~220-= , (23) 

where 0~
emis the electromagnetic rest mass of the electron. 

Substituting (23) in (40) gives 

 [])~//(exp1~
~2022022cmerrcmecmEee-...
.
...
.
-˜ . (41) 

Defining 

 eqrr= , (42) 

(41) can be written as 

 )/exp(~2qqrrrrcmE--˜, (43) 

 )/exp(
2qrrrQ--=, (44) 

where Q2 is defined as 

 

 qqrErcmQ~~22==. (45) 


Eq. (44) has the form of the Yukawa potential. The mass qmm~~= is then equal to the pion mass 
. pm

 

Property (e) for the pion mass can be derived from the above relations by assuming property (d) 
as a boundary condition (see Ch.7 of Ref. 3). 

13. The weak force 




The weak force cannot yet be incorporated explicitly in the model as no classical equivalent of 
this force exists. This force gives rise to radioactivity, in particular ß-decay. It acts over distances 
r < R = 2re and we therefore expect that our Lagrangian (1) must be adapted for this range. 

14. Possible implications for cosmology 




Einstein was deeply influenced by the ideas of Ernest Mach (1838-1916), in particular that the 
local properties of matter should be determined by the properties of the distant, massive stars. He 
wrote:9 

But in the second place the theory of relativity makes it appear probable 
that Mach was on the right road in his thought that inertia depends upon 
a mutual action of matter. For we will show in the following that, 
according to our equations, inert masses do act upon each other in the 
sense of the relativity of inertia, even if only very feeble. What is to be 
expected along the lines of Mach's thought? 

The principle implies that changes in the distant universe will cause changes in local properties, 
such, for example, as the mass of an electron. We shall not be able to measure this change as it is 
a relative change, and our measuring apparatus will be changed equivalently. This effect has 
become known as the Mach effect. 

 

Dennis Sciama (1926-1999) proposed an ad hoc formulation for this effect as10 

 

 . (46) /2MLcG.

where 

 

L = radius of the universe, 

M = mass of the universe . mass of the distant stars. 

 

This relation can be found (see Ch.10 of Ref. 3) by applying the energy relation of (2) to the 
system of Fig. 3: 

 


 

 

Fig. 3: Mutual gravitational interaction between a central mass M1 and the distant 
stars of total mass M2. 

 

The potential at due to is and the potential of the shell atis 
. Furthermore, since and are in relative motion, the value of . 
will be the same for both of them. Applying (2) then gives 
2M1MLcRLGM2//2111==F1MLcRLGM2//2222==F1M2M

 

 . /exp()/exp(122221LRcMLRcME==

 

Since L > R2 >> R1 we can realistically approximate the exponential to first order in R2/L. After 
some algebra we then get R2 ˜ L, which gives the Mach relation 2GM2/Lc2 ˜ 1. 

 

One of the mysteries of modern astrophysics is the accelerating expansion of the universe. This 
can be explained in terms of the conversion of radiation energy into mass. 

 

We note from (4) that m decreases with increasing velocity, implying that mass is converted into 
kinetic energy. Alternatively, if the masses m of two bodies are increased through the conversion 
of radiation energy into mass, then the masses will repel each other according to (5). For 
example, if the mass m of a test body in the vicinity of a central mass M is increased, then the 
distance r between them will increase equivalently to keep E constant. The rate of repulsion will 
depend on the rate of conversion of energy into mass. This conversion is the opposite to the 
conversion of mass into energy as is found in nuclear fission. 

 

At the Big Bang the universe consisted of radiation only. As this radiation was converted into 
mass, the masses repelled each other. This process is still continuing, and according to our 
model, accounts for the accelerating expansion of the universe. 

 

Higher order gravitational effects are becoming increasingly relevant in the study of binary 
pulsars.11 12 Astrophysicists are looking for confirmation of the Einstein-Mach effects predicted 
by General Relativity theory. These effects include geodetic precession and frame dragging, or 
Thirring effect. We are presently looking at applying (36) to calculate these effects. 

15. Conclusion 




The equivalence of planetary and atomic motion is in accord with the axiom of “As above, so 
below.” Similarly, Mach’s Principle also appeals to the mystical scientist. Changes in the 


universe cause changes in the locally small. Since we are part of this change, we are not aware of 
it. This also casts light on the meaning of absolute time. Absolute time is the time interval 
between two coordinate clocks fixed in the reference frame of the universe. This interval will 
change according to the properties of the universe, or equivalently, to those of the distant stars. 

 

A fundamental feature of the model is the distinction between the dynamics and kinematics of a 
system. This is reflected in the two postulates. The Lagrangian of (1) specifies the dynamics of a 
system. It describes a gravitational force that causes particles to move, specifically in accelerated 
motion. The way the particles move is described by the kinematics of the system. There is no 
reference to forces. Kinematics is described by special relativity, which, at speeds much less than 
that of light, gives the classical Newtonian equations of motion. Special relativity also describes 
the electrodynamics of a system, which depends on the velocities of the particles. The velocity of 
a particle changes according to the magnitude of the force acting on it, but our postulate 2 
stipulates that the velocity remains constant within infinitesimally small intervals of time and 
space. This is a standard assumption in physics. It allows the global validity of special relativity, 
and therefore also that of the Maxwell equations of electrodynamics at all points in space under 
the influence of gravitation. 

 

The electrodynamical force therefore results from the motion induced by the gravitational force 
determined by the Lagrangian. The two forces are linked by the relations of the LNH. 

 

Postulates 1 and 2, therefore, form the foundation of a unified model of gravitation and 
electrodynamics. 

 

The model is also characterized by two different types of masses: a gravitational mass m and an 
electromagnetic massm~
. These two masses are related by the relations of the LNH. Different 
types of masses for the electron are not unusual in modern physics. Quantum electrodynamics, 
the theory of the electron and photon, is plagued by conceptual problems of mass 
renormalization (see p. 217 of Ref. 13). It must still be seen to what extent our model could 
resolve the ambiguities of this procedure. 

 

We have mentioned that m decreases with velocity, and it is well known that increases with 
velocity. The first effect is consistent with our closed particle model of gravitation. As velocity, 
or kinetic energy, increases, it must be at the expense of mass, which is converted into energy. 
This does not apply to electrodynamic systems that are not closed: energy is applied from an 
outside source, and part of the energy is converted into mass to impede the acceleration of a 
charged particle. This is equivalent to the Lenz effect. 
m%

 

An intriguing aspect is the possibility of demonstrating gravitational repulsion in the laboratory. 
The theoretical mechanism is explained in section 14. A suggestion of how this can be applied 
appears in the work of Roger Jennison.14 

 

The paper shows that there is possibly only one force in nature, that of gravitation. The 
electromagnetic force arises from the kinematics induced by the gravitational force, and the 
strong force is a unique form for gravitation at a separation equal to the Schwarzschild radius of 
a nucleon. No expression has yet been found for the weak force, but we can deduce that it would 


have a form for the gravitational force in a region smaller than the Schwarzschild radius of a 
nucleon. 

 

The proposed Lagrangian is general enough to be applied, in principle, to all dynamical systems. 
Equivalently, the Lorentz-type force of (36) offers more direct applications. It should simplify 
the present mathematical formulations to describe the higher order gravitational effects of binary 
pulsars. 

 

The proposed model has a rigid structure with a consistent mathematical formulation. Change the 
form of the classical electron radius and the perihelion precession of Mercury is affected. All 
assumptions are well established in physics and all predictions, with the exception of the Lamb 
shift, agree with observation. The discrepancy of the Lamb shift should be resolvable with a 
closer look at the effective nuclear radius of the hydrogen atom. Taking all these conditions into 
consideration we must conclude that the proposed model satisfies the basic requirements for a 
unified model of forces. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


REFERENCES 

 

 

1. Brian Greene, The Elegant Universe: Superstrings, Hidden Dimensions and the Quest for the 
Ultimate Theory (Vintage, 2000). 

 

2. P.C. Wagener, “Newton and the international year of physics,” The Rose+Croix Journal 2 
(2005): 63-77 – http://www.rosecroixjournal.org/issues/2005/articles/vol2_63_77_wagener.pdf. 

 

3. P.C. Wagener, Principles of a Theory of General Interaction, PhD thesis, University of South 
Africa, 1987. Published as a book in January 2004. 

 

4. Richard Feynman. Quoted in http://www.imdb.com. Search for ‘Feynman.’ 

 

5. A. Pais, ‘Subtle is the Lord ...’ (Oxford, UK: Oxford Univ. Press, 1982). 

 

6. Randal C. Telfer, Everything you always wanted to know about the hydrogen atom but were 
afraid to ask. http://www.pha.jhu.edu/. May 1996. 

 

7. E.R. Harrison, Physics Today 12 (1972):30. 

 

8. Robert Eisberg and Robert Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei, 
and Particles (New York: John Wiley & Sons, 1974). 

 

9. A. Einstein, The Meaning of Relativity (Princeton: Princeton Univ. Press, 1955). 

 

10. D.W. Sciama, The Physical Foundations of General Relativity (Doubleday, 1969). 

 

11. Michael H. Soffel, Relativity in Astrometry, Celestial Mechanics and Geodesy (New York: 
Springer-Verlag, 1989). 

 

12. F. Curtis Michel, Theory of Neutron Star Magnetospheres (Chicago: University of Chicago 
Press, 1991). 

 

13. Walter T. Grandy, Introduction to Electrodynamics and Radiation (New York: Academic 
Press, 1970). 

 

14. R.C. Jennison and A.J. Drinkwater, “An approach to the understanding of inertia from the 
physics of the experimental method,” J. Phys. A: Math. Gen. 10 (1977): 167-179.