The principle of equivalence as Einstein knows it was also how Galileo would have known!!
In a recent (although not quite quite recent) I wrote an article of how the concept of “Principle of Equivalence” is fairly misunderstood and misrepresented. In that or a related article the proof that the PoE (of GR as is usually said) follows from Principle of Least and/or Stationary Action (PoSA) was performed by me. (like a bolt from sky, I thought it, proceeded to prove it and it happened, which is why I am often a fan of making outrageous comments in Physics and most times they turn out true as far as I know, but sometimes grave silly mistakes might occur which obviously I own up but do not stick by, in other words would correct as soon as I would spot them or as soon as I can review and fix them) I also stated that PoE is directly related to Energy Conservation Principle (perhaps for the reason of PoSA, as PoSA is the central formalism of all of Physics, including QMech) That later is also obvious from the fact that PoE is actually a statement that KE and PE are but the equivalent forms of energy. The amount are not equal but equivalence means some part of one (A) is equal to another (B) in that scale, for all parts of one (A) or the other (B).
That means A = B is not necessary but A = cB where c is a rational number and could be fraction. So only a fractional part of A (or B) is same/equal as the total of B (or A) and in this ratio for each part of A (or B) . SO B constitutes of many parts of A so that for each part A = cB holds or A/2 = cB/2 and A/5 = cB/5 and so forth. (if c is a larger than unity quantity then the reverse is true). Each of these A/5 or A/10 parts are equivalent to B/10 but equal to cB/10. (Do I make a correct statement by this proposition? If not known name this theorem by me .. ;))
So A is KE and B is PE or the reverse. Once thats the case (I started from PoSA and reached this fact of KE \equv PE ) the fact that these two can be different or PoE and therefore PoSA and Energy conservation (and all other known Physics laws) will be invalid if this difference of KE and PE is caused by 1. different mass types in KE and PE called inertial and gravitational mass 2. different speed types nobody said inertial and gravitational speed eg.
If the masses can be non-equivalent (therefore by some constant factor non-equals, which is to be re-normalized or re-scaled to equivalence then ) then so can be space, time, speed (ratio of space and time) and energy (which is mass, equivalently but may not necessarily) .. That would invalidate energy conservation, KE – PE equivalence and PoSA , in short everything Physics knows. That means all equivalence we know such as E-B fields, space-time and mass-energy will now be invalid simply because mass and mass of two types (inertial Vs gravitational) [but also two types space, two types of time, two types of E and two types of B etc]
In other words Einstein simply recognize this fact and all equivalence followed.
But my point is Einstein didn’t invent a new law. PoE was already classical (in the sense of Newton/Galileo) .. He merely recognized it in the framework of Relativity special and general. [Relativity is also a classical mechanics in terms of PoSA because waves and particles are separate, in other words one PoSA for particles called equation of motion or particle equation of motion and another PoSA for waves called wave-equation, only in Quantum Mechanics they both are merged and known as wave-equation or Schrodinger equation etc]
So PoE was known by Physicists who were contemporaries of perhaps even Galileo since they knew PoSA on which were based all laws of Physics. But it was known in terms of KE = c. PE (with c a constant factor, not speed of light) They did not have to look deeper eg equivalence of mass and energy. Now hat Einstein did and the form of the laws changed slightly to better accuracy its thought as a new law discovered by Einstein. Thats not true. But he was first to formulate it precisely and into new laws of nature. I am perhaps the first one to recognize this whole concept. (as described in this article)
This is so because PoE follows directly from PoSA.
A few days ago I had promised to write upon my return from travel an article with explanations of two facts I have stumbled across. My face-book posts for two weeks have disappeared so I don’t exactly recall what I wrote, but I remember only one of the concept I wanted to describe.
One of it is another way of seeing the above facts. How to see that PoE is a simple geometric consideration. And yes All Physics is Geometry. The very fact that there are forces is a result of geometry because acceleration is a curvature of the trajectory. Also remember the article I wrote couple weeks ago? which describes particle sizes. It says how particles of zero mass are necessarily also zero energy particles, in other words zero mass particles such as photons can’t have arbitrarily high energy because that would mean they are waves (a set of large number of particles, to produce that energy) so Quantum Mechanics is a natural result of the fact that extreme high energy particles with zero mass is not possible, a trade-off starts where waves of such zero mass particles comes to the rescue. But also this fact is closely emanating from Pythagorean Theorem. If that theorem in geometry were not known Quantum Mechanics was not going to be predicted and Higgs would not be differentiated from photons. What Pythagoras did 2000 years ago has a bearing on advancements in science and technology now. Isn’t Science amazing? [Also see that particle size is imminently connected to phase space where triangles are regularly drawn, Dalitz Plot is a plot of invariant masses squared and so on, doesn’t a D plot look a bit like a Pythagorean triangle? Also whats this particle phase diagrams like CKM matrix etc ]
In any case here is the concept that occurred to me as an insight as to why PoE was known since Galileo’s time. Its plain geometry.
The very fact that there are forces is a result of geometry because acceleration is a curvature of the trajectory. So thats the hint. If any Physicist would have realized this fact lets say in 1720 then he would look at the trajectory of a canon-ball moving like a parabola. The curvature of the trajectory for little variations in time or space then gives the slope and its derivative (speed and acceleration, thats force and equation of motion just from geometric consideration ) . That means its another way of figuring out a law of nature since a particle (canon ball) is observed to traverse a path. (just like a particle in a detector) Automatically then we see a differential equation was obtained whose solution is the trajectory. In other words laws of Physics are intimately based on observation and experiment and theoretical insights are not free from such a-priori knowledge. There is nothing called a gifted theorist if not the envision of such knowledge he/she has understood from the works of those who have left us long ago. And this comes as a package right from the days of high-school into college and univ where we learn and imbibe such into our psyche called intuition. We learn geometry, trigonometry, statistics, physics concepts and so on, so a Physicist who discovers something when he is a grad student, postdoc or a professor isn’t doing anything but reaping a huge benefit of what he has been doing for decades. Science training isn’t a waste for everyone.
So that hypothetical man/woman could have thought “wait a minute, this ball is drifting along rightwards and also its curved or warped downwards which points to the fact that the warping or gradual downward inclination is a result of something pulling it towards itself, the gravity of earth”
But given that there is nothing to reason such a warping or inclination or curvature in trajectory (therefore energy) couldn’t simply be caused by any other force (therefore acceleration) it would mean acceleration and gravitational potential or forces are but the same thing if not equal in magnitude. Thats the principle of equivalence. Things can have the equivalent (if not equal ) curvature due to acceleration (noninertialness or rotational/curving properties) or due to a field of gravity. One can directly see such a principle of nature if equipped enough with Physics insights gathered in the past.
Principle of Equivalence (PoE) have got nothing to do with inertial mass and gravitational mass, thats just a secondary reformulation of the more general principle as explained.