Derivation of Newton’s Law of Gravity from Basic Axioms
G. Alagar Ramanujam, K.Fitzcharles
Vethathiri International Academy, Chennai
and K.Vinod Kumar, Bangalore
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Gravity was the first discovered force of the four fundamental forces, but remains least understood. A careful reexamination of Newton’s and Einstein’s concepts of gravity reveals that mass remains undefined in both. It is precisely for this reason that gravity is still not fully and properly understood. This paper presents an approach to define mass in terms of its causative (kinematic) factors and thereby to derive Newton’s Law of Gravity which hitherto remains only as deduced from Kepler’s Laws of Planetary Motion.
This work is based on the philosophical work of Shri. Vethathiri Maharishi.
The Newtonian mechanics is well-known to be based on four independent entities namely, space, time, matter and energy. In Einstein’s theory of relativity, space and time were unified by the Lorentz transformation and energy and matter were unified by the famous relation E = m c2. Thus while Newton worked with four independents, Einstein worked with two independents (space-time and matter-energy). Following this trend, in this present work Space is identified as the unitary entity with the following properties, expressed as the Basic Axioms of the Model. [1,2,3,4].
1. Space is the all-pervading substance inherent with all potential energy and consciousness; it has the property of self-compression and exerting surrounding pressure on every system.
2. Self-compression results in the formation of spinning quanta of space; the spinning quanta are termed “formative dust”. Due to the spin, every dust (and group of dust formed by surrounding pressure) is a source of a spherical spreading wavefront, exerting outward pressure.
The first axiom above describes the built-in mechanism of space to transform into the fundamental particles of which further structures are made. This built-in mechanism is postulated here as the self-compressive nature of space.
The second axiom deals with the source and nature of the repulsive forces operating between any two systems.
Concept of Inertia
Let us briefly discuss the interpretation of inertia by Galileo, Newton and Mach. Galileo interpreted inertia of matter as the resistance that a body would give to a force applied on it. Following Galileo, Newton obtained the inertia of a body as a constant of proportionality between a force applied on it and acceleration thus produced. Newton considered the mass of a body to be absolute. Mach challenged the absoluteness of the inertia of a body and argued that inertia of the body is a measure of its interaction with ambient matter and hence depends on the configuration of matter surrounding the body.
If ‘C’ is the compressive force exerted per unit surface area of a body of surface area ‘A’, and ‘R’ represents the repulsive flux emanating per unit area of the body due to its spin, then A(C - R) represents the net gripping force acting on the body. The greater the value of A(C- R), the greater would be the difficulty for an applied force to move the body. This difficulty is interpreted classically as the inertia of the body. If m represents the inertia of the body, then
m µ A(C – R), or m = β A(C – R) (1)
Thus, compressive force due to Space and repulsive flux from the body constitute the source of the inertia of the body.
Derivation of Newton’s Law of Gravity
Newton’s law of gravitation: F = GMm/r2
(Where M and m are two masses separated by distance r)
is the great gift of Newton to the world; it has had tremendous impact on our culture and civilization. However, this profound formula has certain limitations too: for example, Newton assumes “action at a distance” which in turn implies infinite velocity. Later, it was challenged by special relativity, which puts a limit on the speed of communication.
A deep look into the formula F = GMm/r2 reveals that it is deduced from Kepler’s empirical laws but not derived from basic axioms, as admitted by Newton himself. The derivation of Newton’s formula from basic axioms remains a great challenge over the past 350 years, and here we have addressed that challenge.
Let C1 be the compressive force per unit area on a spherical object of mass m1
The compressive pressure field at a distance r will be
C(r) = AC1 / 4P r2 (2)
Due to the presence of the object there is a compressive pressure field around it.
Let R1 be the repulsive flux emanating from unit area of the object of mass m1
The repulsive pressure field at the distance r will be
R(r) = AR1 / 4P r2 (3)
The total force on the mass m1 gripping it will be A (C1 -R1).
If m1 is the mass of the object, m1 = β A (C1-R1)
If we keep a mass m2 at a distance r from m1, the force (F2) on m2 can be written as
F2 µ AC1 / 4P r2
µ -- (AR1 / 4P r2)
Hence, F2 = K1 A (C1 – R1) / 4P r2 (4)
F2 = K1 m1 / β 4P r2
By a similar argument, if F1 is the force on m1 due to the presence of m2 then,
F1 = K2 m2 / β4P r2 (5)
By Newton’s action-reaction principle, F1 = F2 which implies
K1 m1 = K2 m2
K1 / K2 = m2 / m1 (6)
This means, K1 = µ m2 and K2 = µ m1
Therefore F1 = F2 = F = µ m2 m1 / β 4P r2
F = G m1m2 / r2 where the constant G = µ / β 4P (7)
A derivation of a well-known formula by a fresh set of axioms has great significance. It demonstrates not only the relative validity of the past but also the enlarged validity of the present. It is gratifying to note that we are able to get back Newton’s law of Gravity as a component of our expression for gravitational force.
A look at our eqn.(4) reveals that it contains the action at a distance concept of Newton through the term C, and the travelling wave concept of Einstein through R. The term C is rather indicative of ‘action at distance’ but with a more profound implication: as the compressive force on a mass is due to space, such a force is ever-present on the mass no matter where it is located in space. Thus there is no “travel” involved for the compressive action; in other words, gravity does not travel. The flux represented by R in eq. (3) is an emanation and thus takes a certain time to travel and reach another mass. It must be stressed here that what is travelling is not compressive gravity but the repulsive flux.
The significance of eqn.(4) of this paper is that it expresses the centripetal force F in terms of the contributing forces. This is in sharp contrast to the approach of Newton wherein he expressed the force F in terms of the masses and to the approach of Einstein wherein he wrote his field equations in terms of parameters of the curvature of space.
To begin from a basic unitary state and to identify that as the source of all physical phenomena essentially represents a unified approach. The Vethathirian Model begins from the radical Beginning – the single entity, space. Space itself transforms into the universe; by self-compression space becomes particles. The spin of a particle produces in space a radial outward repulsive flux, which we call magnetic wave. By the interaction of the compressive and the repulsive forces innumerable systems are formed.
As Newton believed space to be nothing but a passive, empty background, he had no option but to attribute both the properties of inertia and gravity to the mass of the particle. He was forced to introduce two masses for the same particle -- inertial mass and gravitational mass -- the first one as a measure of its resistance to an applied force and the second as its strength to attract other particles. However, in our model (C – R) appears as inertial effect in eqn.(1), and the same appears in eqn.(4) as the gravitational effect of the same particle. As (C–R) changes, there is a change in the inertial effect and there is also a corresponding equivalent change in the gravitational effect. This is our version of the “principle of equivalence”, which Newton admired as a “God-given gift” and which Einstein exploited to formulate his General Theory of Relativity.
Science progresses only through constant review and updating. The necessity to revise or update a given theory may arise from new experimental results or from the demands of aesthetic or philosophic logic. Instances of both of these are abundant in the history of science. Vethathirian Model is another instance of the latter case. Vethathirian Model begins from the radical beginning itself, and hence has a philosophic base. Contemporary science rather begins from the stage of fundamental particles without sufficient knowledge as to the essential nature of these particles. The fact that the Vethathirian Model begins with and consistently proceeds from the radical, primordial state -- the space -- is its strength and gives us a holistic perspective. Only a holistic theory can be profound enough to reveal the fundamental Truth and it is in this context that Vethathirian Model is significant and valuable. It is not a superficial modification of Newton’s or Einstein’s concepts, but a radically different theory which has finally brought monism into scientific thinking and thereby Effect is seen as inherent and inseparable from its Cause.
It all began with Tycho de Brahe, a great experimentalist of the sixteenth century. His lifelong rigorous observations of celestial objects produced an ocean of data. From these heaps of numbers Kepler carved out a set of empirical laws known as Kepler’s laws of planetary motion. Using these laws Newton deduced his law of gravity in terms of masses. Then came Einstein: he was the first to attribute a role for space for understanding gravity. He considered space as the transmitting agency of gravitational interaction between two bodies. From Einstein, we now come to Vethathiri Maharishi who asserts that gravity is the inherent property of the space itself. The journey of mankind to understand the source of gravity reaches a decisive stage in the Vethathirian concept of space.
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4. G.Alagar Ramanujam, Uma Fitzcharles, K.Perumal; Wide Spectrum, Coimbatore, 2004