A Note on Potential Energy

Introduction

In my main paper on physics, A Computational Model of Time-Dilation, I presented a theory of gravity that is consistent with the General Theory of Relativity (see Section 4), in that it implies the correct equations for time-dilation due to gravity, despite having no notion of space-time at all. I briefly discussed Potential Energy (see Section 4.2), hoping even then to address it formally in a follow up paper that would include other topics, but the economic pressure to focus on A.I. has led me to spend the vast majority of my time on A.I., not physics. As a result, I’ve decided to informally present my theory of potential energy, below, though I nonetheless plan to address the topic somewhat more formally in the work I’m now starting in earnest on thermodynamics, at the intersection of A.I. and physics.

Pressure

Imagine you have a wooden bookshelf. Even though the bookshelf is not visibly moving, in reality, there is some motion within the wood at very small scales, due to heat and intramolecular forces. Even if you cut into a plank, you still won’t see any motion inside of the wood at our scale of observation, because again, the motion is due to forces that are so small, you can’t see them with your eyes.

Now imagine you place a book on top of the shelf. Gravity will act on the book, and cause it to exert pressure on the surface of the bookshelf –

You know this must be the case, because it’s something you can perceive yourself, by simply picking up the book, and letting it lay in your hand.

Physicists discussing gravity typically ignore this aspect of pressure and focus instead on the notion of potential energy, saying that the book has some quantity of potential energy, because it is in the presence of a gravitational field, that would cause it to fall, but for the bookshelf.

We can instead make perfect physical sense of this set of facts without potential energy by assuming that the book is again actually moving at some very small scale, pushing the surface of the wood down, and in response, the intramolecular forces within the wood, which are now compressed past equilibrium, accelerate the surface of the wood back up, creating an oscillatory motion that is simply not perceptible at our scale of observation. In crude terms, the force of gravity pushes the book and the wood down, and the forces within the wood push the wood and the book back up, creating a faint oscillatory motion.

Note that this set of interactions doesn’t require potential energy at all, since if you push the book off of the shelf, it falls due to gravity, accumulating new kinetic energy. Therefore, the notion of potential energy is not necessary, at least for the purpose of explaining the accelerating effects of gravity.

The term I’ve used to describe this phenomenon is net effective momentum, which is simply the average total vector momentum of a system, over some period of time, or space. Returning to the example above, the net effective momentum of the book on the shelf would be roughly zero over any appreciable amount of time, since for every period of time during which the book moves up, there’s some period of time over which the book moves down, by a roughly equal distance, at roughly the same velocities, in each case, producing an average momentum that is very close to zero.

Now consider a baseball that’s just been hit by a bat –

Initially, the ball will likely be wobbly and distorted, if examined carefully over small periods of time, though its macroscopic velocity will probably stabilize quickly, and therefore, so will its net effective momentum, despite the internal tumult of the ball that certainly exists during its entire flight. The purpose of the net effective momentum is to capture the macroscopic motion of the ball, despite the complexities of the underlying motions.

As it turns out, my entire model of physics is statistical, and assumes that even elementary particles have complex motions, and the notion of net effective momentum is simply a macroscopic version of the definition of velocity that I present in Section 3.3 of my main paper.

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