For whatever reason, my mind has kept wandering back to General Relativity when I'm riding the train home from work. I had a moment of clarity last week, and the presence of mind to start writing things down on paper. After a couple revisions to simplify and clarify, plus explaining it to my teenaged daughter in about 20 minutes, I'm ready to post it here.
How General Relativity Works
Pythagoras's theorem lies at the heart of reality.
The square of the speed of light is the square of the energy plus the square of the proper time.
Energy is a shorthand for the velocity of a particle, or the strength of the local gravitational field when we're talking about a point in space. Incidentally, these are vectors and add to produce the final motion of a particle. But for the purposes of this part of the equation above, they're just scalars (directionless numbers). Yes, I know I'm using definitions rather loosely here.
Proper Time is what physicists call locally perceived time. That's time as the subject experiences it. Notice that this can change due to the subject's own speed, and also with the local gravity field. Both speed and gravity distort space time - speed for the particle itself, and gravity for anything that happens to be there. Proper time is best thought of a a (less than one) multiplier to the standard theoretical unit of space-time. The t of space and the t of a particle occupying that space multiply to get the final result of locally perceived space-time, based on both gravity and velocity. (If a particle's t is 0.9, and local space's t is 0.5, then the end result is a t' of 0.45. Meanwhile, the particle is moving forward, while being pulled in the direction of local gravity field. Simple is not the same as easy.)
The speed of light is constant in a vacuum. This explanation works best if we select a system of units that defines c as 1.
Space is Time and Time is Space. Believe it, because it is true. This is the foundation of everything here. When time shortens, so does space. When space shortens, so does time. This follows from the fact that c is a constant. Since c = distance / time, if c is held constant, then the proportion of distance to time must also hold constant.
That's it, really. Everything else follows from these simple relationships. It's not that complicated. I have no idea why it's not taught or explained in these simple terms.
Thinking about this led to my moment of clarity last week. I had the idea, but then had to spend a few days researching what was known already. Frankly, a lot of the explanations I could find seemed self-contradictory. Here's the summary of what I believe is really going on.
Mass is a property of matter. (Photons don't have mass, just energy, but they work in precisely the same way.) The mass of an object is a fixed property. However, the energy it represents can increase in inverse proportion to the decrease in perceived time.
Mass is best thought of as a rotation of a particle. It completes one rotation per unit mass per unit space-time. It can not have fractional units, and must always complete a whole number of rotations. (This is the essence of quantum mechanics - there are only whole units, never partial units. Although, I suppose the units being used could perhaps be half rotations.) This is the frequency of the particle - the number of rotations per unit space-time. Notice that this is already accepted for photons - they don't have a mass, but do have a variable energy, based on their frequency. (OK, so I can't draw all that well. The squiggle over the arrow is supposed to be a sine wave, with the arrow representing the motion of the particle as it is rotating in a circle about the central axis.)
So, we have a particle with a fixed frequency. But we've already shown that length (space-time) changes with speed and local gravity. That means that the speed of rotation must increase as the distance (wavelength) decreases. It is this speed of rotation that is the actually measure of the energy represented by the mass of the particle. At rest, it is identical to the mass. At near the speed of light, or in a high gravity field, it increases markedly. Physicists don't like to refer to the mass changing (because it doesn't), so they refer to the total energy as the relativistic energy of an object.
It's easiest to think of things in terms of density. As gravity increases, space gets denser. As an object's speed increases, its energy becomes more dense (in comparison to a theoretical t=1 background).
The object itself doesn't notice any of this - because from it's point of view, nothing happens. When physicists talk about objects in a reference frame, they mean objects at the same level of absolute density. That's the normal, every day life for you and me. We're all in (approximately) the same level of density of space-time. From it's own perspective, based on the local proper time, mass is constant, speeds and lengths are normal, and the speed of light is still, as always, the speed of light, and a constant.
Notice that there are some interesting, theoretical consequences to all this. An object must move to exist, even if it moves only the smallest amount possible. If it didn't move at all, with a speed of zero, you get a division by zero error and undetermined mass. It also wouldn't move through time, as it didn't move through space. This would seem to rule out ever actually reaching absolute zero. Fortunately, there is unlikely to be any spot in the universe where there is no gravity, and no realistic way to create a particle with no inherent motion.
This rotation concept also helps explain (at least to me) the particle - wave nature of objects.
Conjecture - The rotational velocity of a massive particle is what causes gravity. The faster it rotates, the greater the gravity field created. How this actually would work is beyond my abilities. But it certainly seems reasonable, and is consistent with how other things work - especially electromagnetism. (This concept explains how relativistic energy creates gravity, without changing the rest mass.) Gravitational force would be measured by the absolute speed of rotation, not the locally perceived speed.
Other thoughts - What, if any, difference would there be in particles that rotate in different directions? Is there a maximum possible rotational velocity related to c, which is to say, is there such a thing as the maximum possible energy of a photon, or the maximum possible mass of a particle? And what happens if you then accelerate that particle and place it in a high gravitational field, increasing its energy?