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Newton's Model
It was Newton who first showed that the properties of Nature can be described through
mathematical formulae. The mathematical structure developed by Newton is based on
four independent quantities--i.e. space, time, matter and energy. It is significant
that even today the description of Nature is given only through these four quantities.
Newton was so impressed by the events in the functional universe that he was completely
lost in its dynamics. His concepts of space, time, matter and energy gave him a
model which described Nature, especially at macrolevel, with amazing accuracy. However,
his model failed miserably in the atomic scale.
The greatest success of his mathematical model is the discovery of the gravitational
force. Newton visualized the gravitational force as a force existing between any
two material particles by virtue of their masses. The absence of a gravitational
repulsive force was a great puzzle to Newton. Since one talks of attractive and
repulsive forces in the case of electric charges and magnetic poles, Newton always
wondered at the absence of a gravitational repulsive force. Such puzzles and wonders
made him realize the limitations of his model. This is clearly seen when he said,
"I am like a child playing on the shore with pebbles and shells while the whole
ocean lies before me". In retrospect, one can observe yet another severe restriction
on Newton's model. While Newton was concerned with findings the causes for the events
occurring in the functional universe, his model did not deal with the grand cause
for the genesis of the functional universe.
Einstein's Model
A careful study of the limitations in Newton's model made Einstein introduce certain
significant refinements in the concepts of space, time, matter and energy. Based
on his refined concepts, Einstein built a new model. He had shown that space, time,
matter and energy are no longer independent quantities, but one can be converted
into the other. The "absoluteness" of Newton was replaced by the "relativeness"
of Einstein. Because of these refinements Einstein could get a better and broader
perspective than Newton. This is clearly seen in his concept of gravitational force
which is drastically different from Newton's concept. According to Einstein, gravitational
force arises not out of the interaction between two masses but out of the interaction
between mass and the surrounding space. Thus, Einstein elevated space-time from
a passive role to a very active role. Einstein developed a set of equations to govern
the interaction between the mass and its surrounding space-time structure. The equations
formed a basis for his general theory of relativity. If Newton's formula f=ma is
one major landmark in the history of Science, Einstein's concept of the interaction
between the mass and the surrounding time-space structure gave him a much wider
perspective than Newton. For example, Einstein's theory predicted the perihelion
of Mercury with perfect accuracy which Newton's gravitational theory could not.
The success of Einstein's field equations prompted him to tackle certain cosmological
questions like, "Does the universe possess a limit?" "How did this universe come
into being? "Is there a Unified Force from which all the presently known forces
could be derived?"
Prior to Einstein, scientists were of the view that the universe as a whole was
incomprehensible and hence questions like the above could not be tackled by mathematical
models. However, by using his field equations Einstein got some partial answers
to some of the above questions. This limited success of Einstein was a breakthrough
in the field of cosmology and it opened up avenues for many cosmological theories.
Emboldened by his limited success, Einstein remarked, "the most incomprehensive
thing about the universe is that it is comprehensive".
The success of his theory here and there did not make Einstein heady or blind. While
he was happy over his success, he was equally concerned with the limitations of
his model. The absence of gravitational repulsion which puzzled Newton continued
to puzzle Einstein. At one point in time, Einstein felt the necessity to introduce
the gravitational repulsion and did not have sufficient basis to support it. Hence,
he withdrew it hurriedly from his subsequent theories. Another aspect which kept
Einstein worrying throughout his life was the concept of Unified Force. Einstein's
attempt to understand Unified Force within the framework of the functional universe
did not succeed. Though on the surface of it Einstein's theory yielded some positive
results, at the core Einstein himself realized that the theory was not that sound.
This realization made Einstein remark, "In my field equations, the left hand side
is of fine marble but the right hand side is of perishable wood". (Perhaps the time
has come to replace the perishable wood.)
Einstein's theory should be considered as a refinement over Newton's theory. Einstein's
theory is built on the merits and demerits of Newton's theory. When questioned whether
he had proved Newton Wrong, Einstein's reply was "....No, if I have seen anything
new that is because I see the world standing on the shoulders of Newton".
Einstein intuitively felt that in the cosmological ladder of discovery we are standing
on the lower rungs. Finer theories are yet to come and they should come. This is
clearly seen in his famous statement, "When I compare all our theories to the mighty
Reality, they are all trash. However they are the most precious ones we have today".
Einstein took Newton further: Now we need someone to take Einstein further.
Vethathiri's Model
It is Nature's nature to reveal its secrets only to a select few spread over centuries
and around the world. These few chosen and gifted individuals set new trends which
the majority study and investigate. The selection of these individuals by Nature
is the result of an opportune moment which finds its outlet in these chosen few.
Such individuals have a special power to attune their minds to be in unison with
the cosmos and while in that unison, revelation comes to them.
In ancient days, when science was not developed, the revelators were generally called
philosophers. But in the modern scientific era, such chosen individuals could simultaneously
be philosophers and scientists. It is our firm view that Yogiraj Vethathiri Maharishi
of the World Community Service Centre is one such chosen individual. By revelation,
he not only lives in bliss, but tries to lift the broad masses of the population
to his level.
Vethathiri's philosophy is an all-embracing super-theory. In Vethathiri's Model,
space is accorded a supreme position and paramount importance. It is precisely because
of this, that this model is able to visualize the secrets of Nature which other
models could not. No doubt, Einstein talks of space in his General Theory of Relativity,
but he was so concerned with the functional universe that he did not accord space
its due place.
Since Vethathiri's model is based on the properties of the Static-Silent State,
its canvas is extremely vast. While on one hand it deals with cosmology, on the
other hand it deals with the most fundamental particle--primary energy particle.
Its broad sweep encompasses not only the all-pervading universal magnetism but also
the bio-magnetism functioning in living beings. Its canvas is vast enough to include
not only the consciousness of the human mind, but also its link with the total consciousness
for self-realization. It is our earnest hope that Vethathiri's Model is a theory
for the 21 st Century-a theory with intuitive science and profound philosophy. This
model reminds us of the famous saying of Einstein, "Science without religion is
lame and religion without science is blind".
Gravity
A New Approach To Gravity
According to Newton, space is a passive background in which material objects move.
Einstein’s General Theory of Relativity redefined space from a passive background
to an active participant in the dynamics of the Cosmos. In essence, Einstein realised
a quality of space which Newton did not. Going further, the following qualities
of space are proposed here as a set of postulates:
- Space is characterised with an inherent self-compressive force whose intensification
results in the formation of a spinning quantum of space which we call a fundamental
particle. Such free fundamental particles are present throughout the physical universe
in the so-called vacuum, and constitute what we call the free particle medium.
- Every particle due to its spin motion produces a spreading wave around it through
space and the force of this spreading wave impinging on another particle tends to
increase the distance between them. This interactive repulsion between any two particles
is proportional to the spinning speed of the particles. With regard to the postulate
1) above, we are to state that when we propose a single state and develop the formation
of structures from that state, it is logical that this state contains a built-in
mechanism to produce the fundamental particles of which further structures are made.
This built-in mechanism is postulated here as the self-compressive nature of space.
The self-compressive nature of the space tends to decrease the distance between
any two particles and thus the inherent self-compressive force of space itself is
gravity. Postulate may be seen as a kind of distant reference to the repulsive force
introduced by Einstein through his cosmological constant [13]. The above concept
of gravity can be contrasted with already existing theories of gravity as follows:
When Newton observed two masses coming together (the legendary apple and earth)
he placed the cause for the observed effect on the masses and hence, he proposed
the Universal law of gravitation: Every particle attracts every other particle with
a force directly proportional to the product of their masses and inversely proportional
to the square of the distance between them. When Einstein formulated his Special
Theory of Relativity, he found that Newton’s theory of gravity is not compatible
with his Special theory. Newton’s concept of instantaneous action at a distance
implied Absolute or Universal time which Einstein’s Special theory had abolished
in favour of personal or relativistic time. Einstein was aware of this ‘conflict’
in 1907 itself while he was still at the patent office in Bern but did not take
serious note of it until he was at the German University in Prague in 1911. A few
years later, Einstein and Grossmann put forward the idea that gravitational force
is a manifestation of curvature in space-time. Stated another way, according to
Einstein, gravity is the effect of mass on space. By contrast, here we consider
gravity as the effect of space on mass. The role of space in the dynamics of cosmos
has been progressively upgraded. As already stated, Newton considered space as a
passive background; Einstein considered the same as an active medium for the gravitational
interaction between two systems and attributed a mechanical property (curvature)
for the same. In this paper gravity is considered as the inherent property of space
itself. The significance of our formalism lies in its assertion that the gravity
is not the property of the particle but is an inherent quality of the space itself.
Application To Planetary Motion
Before we proceed to present an application of our concept of gravity, for the sake
of ready reference and comparison, we present a brief account of Newton and Einstein’s
formalisms regarding the study of Planetary orbits.
Newton's Formalism:
The planetary motion has been extensively studied in classical mechanics as a two-body
problem in a central force field. Describing the motion of a test particle around
the sun by the two co-ordinates r and f, a Lagrangian is written and from that,
Lagrangian equations are formed.
Solving these equations, a differential equation between r and f is obtained as,
d2u/dØ2 + u = -(1/q2u2) F(r), with u=1/r (1)
where q is the orbital angular momentum of the test particle.
From Newton’s law of gravity, we have F(r)= - k/r2 (where k is a constant involving
the gravitational constant G) and hence eqn (1) becomes [14],
d2u/dØ2 + u = k/q2 (2)
Equation 2 represents a conic. Under certain initial conditions, the Conic becomes
a closed ellipse. The perigee and apogee turning points of the elliptical orbits
can be a well illustrated in a graph between the effective potential (Veff) and
the distance r.
The effective potential is given by the relation. Veff = - k/r + q2 / 2m r2 (2a)
Einstein’s formalism:
The study of planetary motion, in Einstein’s formalism, becomes a theory13) in a
curved space-time structure, the mass(M) of the sun being the cause of the curvature.
The motion of a planet in such a curved space takes place along its geodesic given
by the equation,
d2 Xa / ds2 +{abg} dXb/ds dXg/ds = 0 (3)
The Christoffel symbols {abg} are determined by the Schwarschild metric expression
given by,
ds2 = (1- 2GM/r) dt2 - (1- 2GM/r)-1 dr2 - r2dø2 - r2 sin2ødØ2 (4)
From eqn(3) follow four relations for a =1,2,3,4; using these relations the equation
for the orbital path of the test particle around the sun is obtained as,
d2u/dØ2 + u = k/q2 + 3ku2 (5)
The non-linear term 3ku2 prevents the ellipse from being closed and hence gives
rise to the perihelion effect. Eqn (5) is solved by the use of an approximation
and the perihelion effect (µ GM) is determined [13,15].
Vethathiri formalism:
The study of planetary motion presented in this paper proceeds from the second postulate
(Sec.2), by using which we derive an expression for the force which keeps a test
particle along its orbit around the sun. We consider a test particle (p) at a distance
r from the sun with an orbital velocity v.
We discuss in this situation the net force on (p) due to gravity and repulsive forces.
Let C be the gravitational force on p directed towards the sun, due to the self-
compressive nature of the space. Due to the spin motion of the sun and its constituent
particles there occurs a radial outward flow of energy from the sun. This radial
outward flow exerts a repulsive force on the test particle.
The energy emanating from the surface (r=R) of the sun per second flowing through
unit area at a distance r from the centre of the sun is directly proportional to
1/r2.
Consequently the momentum received by the unit area in unit time and hence repulsive
force on the test particle may be written as Rs/r2, where Rs is a parameter related
to the sun.
As stipulated by Postulate(2), the free particles which are present between the
sun and the test particle (p) spin with relatively high speed and give out spreading
waves repelling each other. Due to the compressive force (C - Rs/r2) on (p), the
column of the free particle medium between the sun and the test particle is under
compression.
Since the free particles (as well as any other material particles) present in the
column are repelling one another due to their spin, this compression is counteracted
by a resistance. As a result, if K is the mean repulsive force on the test particle
per unit length of the column, Kr gives the effective repulsive force on the particle.
Hence the net centripetal force on the particle is (C - Rs/r2 - Kr).
In other words, the net force F on the particle is given as:
F = C - (Rs / r2) - Kr (6)
We follow the standard convention that negative value of F means centripetal (radially
towards the sun) and positive F means centrifugal (radially outward).
The expression obtained here for the centripetal force acting on a test particle
has three terms: the first one is a constant term contributed by the space; the
second one contributed by the sun varies inversely with r2 and the third one contributed
by the free particle medium around the sun varies proportional to r. The expression
for F leads to a potential
V(r) = (Cr + Rs/r - 1/2 Kr2)
so that
F = -¶V/¶r.
The r-dependence of the expression for F in eqn.(6) has certain interesting features.
According to Newton’s law of gravity, the gravitational force binding any two systems
vanishes only at infinity.
But here, the force F as given by eqn (6) vanishes for three values of r of which
the two positive values are of physical significance. The variation of F with r
is given in the graph I. The Lagrangian (L) of the test particle is, given as,
Since L is independent of Ø, the angular momentum q=r2(dØ/dt) is a constant of motion.
Using Lagrangian equation of motion along with eqn. (6,7) we obtain:
d2u/dØ2 + u = 1/q2u2 ( C - Rsu2 -K 1/u )
where
u=1/r. = Bu-2 - Du-3 - A (8)
B = C/q2 ; D = K/q2; A = Rs/q2 (9)
d2y/dØ2 + y = B (y-A)-2 - D(y-A)-3 ,
where
u+A=y. (10)
Each term on the R.H.S. of equation (8) deserves special explanation. When compared
to Newton’s formalism (eqn.2) the constant term A here gets a negative sign.
This is understandable because the role of the sun regarding its effect on the test
particle is reversed in Vethathiri formalism. According to Newton the sun attracts
the particle whereas our postulates imply that the sun repels the test particle.
The term Bu-2, though non-linear, differs vastly from the corresponding term of
Einstein, i.e. 3ku2. This is because our view of space is drastically different
from that of Einstein. According to Einstein, the curvature of space is distance
dependent. But in our formalism the compressive force of the space on a given particle
is independent of distance; in fact, here the question of distance dependence of
C does not arise at all. It is significant to note that the term Du-3 which is highly
non-linear has no counterpart in Newton’s and Einstein’s formalism. This is because
the role of the free particle medium around the sun has not been taken into account
in their formalisms.
Perihelion Effect:
Considering Eqn.(10), let us assume, y = a where a is very small.
d2a/dØ2 + a = (B/A2) (1+ 2a/A) + (D/A3) (1+ 3a/A) (11)
d2u/dØ2 + n2a = w (12)
n2 = { 1 - 2B/A3 - 3D/A4 }; w = ( B/A2 + D/A3 ) (13)
Assuming the solution to be a = w/n2 + E CosnØ where E is a constant,
we get 1/r = { w/n2 - A } + E CosnØ (14)
Eqn (14) represents a Conics. For eqn (14) to represent an ellipse, (w/n2 - A) has
to be positive which requires, as a simple calculation shows 8g + 9h > 6 where
h=2B/A3; g=3D/A4 (15)
Eqn (15) imposes a restriction on the relative values B,A and D. r repeats its value
when
nØ = 2pØ = 2p/n = 2p + 2pCq4/Rs3 + (3pk/Rs4) q6 (16)
The last two terms on the R.H.S. will prevent the elliptical orbit from being closed
and hence represent the perihelion effect. It may be noted that the two terms representing
the perihelion effect arise due to two different reasons: The first one is due to
the interaction of the particle (p) with the space and is similar to the perihelion
effect obtained by Einstein and the second one is due to the interaction of the
particle (p) with the free particle medium, which has not been considered in Einstein’s
formalism.
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