I cannot believe this … (original Title) 7:30 pm
A thorough understanding of the concept of Principle of Equivalence, a beginning.
8th June 2012, by Manmohan Dash
I wrote two recent articles on Relativity Theory and talked about how some of its most popular and basic ideas are actually not Relativistic in the sense of Einstein’s New Work, but, in the sense of the ideas that were existent much before Einstein and these ideas, simply, might not have had, the name-sake of the currently understood ideas, known in terms of Relativity Theory of Einstein.
I also proceeded to define an equation which states that “The Principle of Least Action leads to The Equation of Motion and the Principle of Equivalence“.
The fact, that, the principle of equivalence follows from the equation of motion of the Newtonian Physics, I proved, in one of the two articles, I wrote. But this later Principle of Least Action leading to equivalence principle I haven’t proved yet although I tried to do this and it seems closely possible.
[I think within a few days of this article, this latter fact also, I have proved]
The proof of Least Action (or stationary action ** ) leading to equations of motion is found in almost all textbooks of classical mechanics [I guess, its certainly in Goldstein, the most famous Classical Mechanics text followed around the world] so it must be, pertinent and imminent, suo-motu that least action leads to principle of equivalence.
Now, Principle of Equivalence is unnecessarily and complicatedly stated to be ” the gravitational mass and the kinematic masses are the same”. I talked about its 3 forms, (strong, weak and medium-strong) in one recent article. Why is it complicated? (and this is what I cannot believe)
Its like saying the nose (仏 − 亻) in a roundabout way (a popular phrase from where I come from). As an analogy consider this: Men and Women are equivalents because their hands are equivalent, there is no generic anatomical differences there, this leads to, are men and women equivalent everywhere? Oh, we know a few places, where they are not.
But then we can define the functions of these organs or anatomical regions and we see that men and women are functionally equivalent but men and bears are not. The latter two cannot perform the same actions and you can easily see “oh yeah sure, its obvious”
[ feminine and masculine bears are equivalent, except carrying a baby, and so are men and women, not equivalent for the carrying of babies, as a function, so far so good, you see the idea and the usefulness of the equivalence, through this analogy ]
But, what if, we define equivalence between men and women to be “the male sexual organs and the female sexual organs perform the same job of reproduction and carnal pleasure“. That would be a specific condition which actually fits in with, the idea of equivalence. Thats what it is, the mass of gravitational and kinematic origin is a specific condition that fits in the idea of principle of equivalence.
The simpler and the more correct and intuitively proximal idea, would be “the kinetic energy and the potential energy are equivalent“, in other words, the effect of these two forms of energy, is, the same, and, they would transpire, in the same way, into, say, other forms of energy, and so on, despite of their different source of origin. Its this property that forms the basis of principle of equivalence and its this property, that leads to, much more profound ideas of Physics, than, the fact that the two kinds of masses are equivalent.
That latter, is, just a specific condition, which, must be also valid, in any case, as you can imagine right away, the mass is a factor in the energy so mass being a fundamentally measurable quantity, it must be same, in both cases, gravitational and kinematic. But, perhaps, it was unnecessary to envisage, that, they would be different. (gravitational and kinematic, mass, would be different, was unnecessary)
Because, somebody could also argue that; gravitational and kinematic speed, g&k acceleration, g&k force, g&k potential or potential energy, could be different, they are same and thats principle of equivalence. But then the argument stopped at the mass, and did not lead to other physical variables, of fundamental nature.
So people who want to test the veracity of the principle of equivalence in its various, degree of strength of definition, as mentioned above, weak, strong, .. , want to test it on mass, which is easier to do, given how everything is formulated, drop two balls , or realize that planetary objects are constantly falling and so forth. But, the difference if there would be, could be ascribed to any fundamental physical variable, happily there isn’t, or they would totally blow up in some way or other and take away all the elegance of the edifice of the Theory of Relativity.
In-fact, one, remarkable aspect of Theory of Relativity, as I understand, intuitively and comprehensively, to the best of my knowledge, is the fact, that, special theory of Relativity is to kinetic energy what general theory of relativity is to potential energy. But then, through principle of equivalence, general theory of relativity is not just potential energy, but also kinetic energy, because, these two are equivalents. So you have; special theory coming just from kinetic energy concepts and then the rest is potential energy concepts. So, general theory (GT), is an extension of special theory, as GT, contains, both kinetic energy processes and potential energy processes.
Also see it this way, the kinetic energy gives you the speed as a central parameter. Then, various transformation properties are determined based on, Lorentz Transformation. eg the mass changes with speed. The principle of equivalence comes up, from the fact, that, one can define a potential also for kinetic energy, as one can, (define potential) for the potential energy. The potential or force is not definable for a uniform velocity, but, for a velocity, that changes with time, so that, acceleration and force are defined. Once there is a force there must be a potential.
[ force is time-rate-of-change of momentum, classically, but, space-rate-of-change of potential-energy and potential energy divided by mass, is, the potential, for our discussion is centered about mass, and say, not, charge. ]
So, there is no reason, why, the two different forms of potential and therefore two different forms of potential energy, the gravitational and the kinematic, should be different at all.
A kinetic energy can be associated with a force or potential and therefore, not, any different from a, potential-energy, that is, they are equivalent, and, in situations, when there is no force or acceleration, the amount, can still be equated, to a potential energy, hence in all cases, kinetic energy is equivalent to potential energy, therefore, acceleration is equivalent to gravity, as, acceleration is kinetic energy and gravity is potential energy.
This also makes it clear, that, special relativistic effects are equivalent to general relativistic effects, this, must be the case, as, kinetic energy and potential energy are equivalent, so also, all other variables, such as potential and acceleration.
So, we should not have stopped at “mass” and one can go home, with the lesson, that, principle of equivalence can be defined in terms of any variable we just discussed. If that does not have the appeal of “mass” because mass can be touted to have come from different sources, but not potential, perhaps you did not realize, Relativity, in its actual flavor.
In that case, you can still talk about, the whole discussion: that, perhaps, all the relativity ideas can spring from simple classical physics ideas. One should be ready, to take pen and paper, enough thinking and calculate stuff.
In any case, once I prove least action leads to principle of equivalence (– which I did, I shall only link) one can have a similar approach, to any ideas of Relativity, and find insightful conclusions, to make for Physics.
That would be, satisfying, to a good passion of understanding; what really Physics is.