Newton’s theory of gravitation has been outdated by relativity theory explaining specific phenomena like perihelion precession of Mercury, light deflection and very recently the detection of gravitational waves. But the disappearance of the obvious gravitational force and the variation of time are arguable concepts difficult to directly prove. Present methodology is based on hypotheses as expressed in a previous article: a universal time and an inertial mass variable according to the Lorentz factor (which could not be envisioned at Newton’s age). Because this methodology is mainly stood on Newtonian mechanics, it will be called neo-Newtonian mechanics. This theory is in coherence with the time of the Quantum Mechanics. In Newtonian mechanics, all forces, including gravitational force, are deducted from the linear momentum. Introducing the variable inertial mass, the result of the demonstration is an updated expression of the net force at high velocity: F = γ 3 mga. If such a factor in γ 3 can look a bit strange at first sight for a force, let us remind that the lost energy in a synchrotron is already measured in γ 4. Next article will be on the perihelion precession of Mercury within neo-Newtonian mechanics.
For a century, hypothesis of a variable time is laid down by the special theory of relativity. This hypothesis can explain many Nature observations, experiments and formulas, for example, the demonstration of the Lorentz factor. Because of such good explanations, the hypothesis of a variable time has been validated. Nevertheless, it remains some paradoxes and some predictions which are difficult to measure directly, as a reversible time, an “imaginary” time or even the time variation itself.
And in developing his ideas about the consequences of the equivalence principle between gravitational mass and inertial mass, Einstein leads to a new vision of gravitation which is to replace that of Newton: the general theory of relativity. The most important aspect is the disappearance of gravitational force concept. For Einstein, the motion of a body is not determined by strength, but by the configuration of space-time [
But the absence of gravitational force and a variable time according to the reference frame remain concepts difficult to directly prove.
The question is: is it possible to explain such phenomena within another theory, i.e. using gravitational forces and a universal time? It is what we will try to do in this article.
A universal time would give in coherence with the universal time of the Quantum Mechanics.
The basis has been laid down in a previous article [
Now in this article, we will check the consequence of these hypotheses on force expression in general (which includes resultant gravitational force).
The linear momentum p is by definition the product of the mass of a body by its velocity [
It is a general formula, the mass m is the inertial mass (it is not the gravitational mass).
So the linear momentum can be written more precisely
With
According to neo-Newtonian demonstration [
with
Chart 1. Comparison of hypotheses.
By property of the net force F according to the second Newton’s law of motion:
Because gravitational mass is constant:
so
And due to Equation (4):
So, with Equation (9):
And again due to Equation (4):
And so, with Equation (13)
By definition, the acceleration a is:
So the updated property of the net force is:
This expression in
Let us remind the synchrotron radiation. The cyclotron is used for particles, and the synchrotron is used for particles at velocities close to light celerity. The loss of energy per turn by synchrotron radiation can be mea- sured as follows [
formula which can also be written:
or
And let us remind a work is a force by a length, and a length is a velocity by a time. So
Then, the work
This
・ Planet revolution (Mercury is the fastest planet of our solar system).
・ Particle in a cyclotron.
・ Particle in a synchrotron.
Let us check the value of
It confirms
- Variation of
- Expression of the net force with
That means that, at very high velocity,
- For a same variation of velocity (or acceleration), the net force will be slightly higher than traditionally ex- pected.
Chart 2. Value of γ3 according to the velocity.
- For a same net force, the variation of velocity (or acceleration) will be slightly lower than traditionally ex- pected and at usual velocity,
First, we remind results of a previous article: Lorentz factor can be demonstrated without using a variable time, but using a variable inertial mass. Such a hypothesison time, called neo-Newtonian theory, is in coherence with the Quantum Mechanics.
Then in present article, consequence of this hypothesis has been checked on net force expression. Deducted and demonstrated from the linear momentum, net force is so expressed to:
This
Application of such a neo-Newtonian hypotheses on the perihelion precession of Mercury (the faster of the planets of our solar system), the deflection of light or the Doppler Effect will be done in next articles.
I would like to thank the reviewers for their advice about the looking of my article.
Olivier Serret, (2016) Net Force F = γ3ma at High Velocity. Journal of Modern Physics,07,656-661. doi: 10.4236/jmp.2016.77065