June 8, 2012
I wrote two recent articles on Relativity Theory and 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 they 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 and seems closely possible. 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] so it must be pertinent and imminent suo-motu that least action leads to principle of equivalence.
Now Principle of equivalence is unnecessarily complicatedly stated to be ” the gravitational mass and kinematic masses are the same”. I talked about its 3 forms [strong, weak and medium-strong] in one recent articles. Why is it complicated? [and this is what I cannot believe] Its like saying the nose 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, 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 thats the basis of principle of equivalence and its this 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 bot cases. But perhaps it was unnecessary to envisage that they would be different. Because somebody could also argue that speed, acceleration, force, potential or potential energy could also 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.
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 is an extension of special theory as it 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 come up from the fact that one can define a potential also for kinetic energy as one can 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 but space rate of change of potential energy and potential energy divided by mass is the potential] So there is no reason why the two different forms of potential and therefore potential energy 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/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 equivalents to general relativistic effects, this must be the case as kinetic energy and potential energy are equivalents 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 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 ]