On the nature of motion

To get the right answers, you must first ask the proper question. I started this journey wanting to know about relativity. After years of reading through pop science bafflegarble, I started gaining an understanding of the actual issues. After years of that, and many false starts, I finally came to understand that the true question, apparently unasked by science, was “What is motion?”

You’d think such a simple, fundamental question would have been asked and answered decades if not centuries ago. You would be incorrect. Scientists have asked and answered the questions “How do objects move?” and “Why does motion change?” But the central question remained. What is motion in and of itself? What causes it? What is the mechanism?

The answer, like many things in science, to be at once quite simple and devilishly complex. Motion is caused by a gradient in the potential energy field. A gradient in the field causes a self-perpetuating gradient in a particle. The particle’s energy causes gradients in the field.

The particle’s gradient in blue. The field’s gradients in red. Flat Euclidean space is at the top.

We interpret the gradient’s velocity at any point by taking the sine of the angle compared to the flat spacetime of the horizontal axis. 0 degrees is stationary (sin 0° = 0). 90 degrees is the speed of light (sin 90° = 1).

But what does that really mean? What is velocity but motion in a direction at a speed? How does the speed of light enter into this?

Velocity, speed, motion, they’re all essentially the same thing: change in distance per change in time. Distance and time. Time and distance. Wait - doesn’t the gradient also have something to say about distance and time?

Indeed it does. The cosine of the gradient’s angle gives the contraction of both space and time. They are always balanced so that the speed of light remains a constant 1. But so what? Why is this important?

Distance is time. Time is distance. It’s not space or time, it’s space and time. Spacetime.

The speed of light, 299,792,458 meters per second, is a conversion factor.

When you are sitting still reading this, you are still moving through time at the rate of one second per second. (We’ll ignore the earth’s rotation, etc.) The gravitational force near the earth’s surface (the angle of the curve) is approximately 9.8 meters per second per second. How much of that is due to motion through space, and how much is motion through time? (Remember, your subjective motion through time is not your objective motion through time.)

Well, you’re not moving (relatively speaking), so all of it is due to the time factor.

That figure of 9.8 meters per second is entirely due to the temporal gradient. None of it is due to the spacial gradient. Even if you were falling instead of sitting still, the spacial gradient is so small as to be nearly irrelevant. 9.8 divided by 299,792,458 is about 0.000,000,003. In the course of normal existence, you can’t tell the difference between that and nothing. It’s lost in the weeds.

Newtonian mechanics works very well in normal, every-day situations.

Relativistic mechanics is for when you can’t ignore the tiny difference. It’s even more important when you are moving so quickly, or being accelerated so strongly, that the spacial component is no longer tiny.

Light from a distant star passing close to the sun is shifted by twice the amount Newton predicted. This is because light travels at, well, the speed of light, so the spacial effects equal the temporal effects. Plus, with radio telescopes, we can measure this effect rather precisely and very close to the sun.

In general, the spacial effects of a force are important if the force is either large or exerted over a very long distance and/or time.

The anomalous perihelion of Mercury (43 arc-seconds per century) is another phenomenon explained by general relativity. See also the two body problem.

The differences caused by the finite speed of the propagation of change in the field are also present, but different. You have to account for the direction and speed the sun is traveling around the galaxy and the amount of time its change of position takes to reach an orbiting planet. The curve contracts and steepens before the sun, and extends and flattens behind it.

And then there’s the frame dragging caused by the sun not actually being a point object, but a rather large, not completely spherical, rotating body.

Physics is hard.

The difference between energy and work

Work and energy have the same units: the joule. Joules are a derived unit of measure in the SI system, being comprised of kg⋅m²/s². Why do we have two different terms for the same thing? What’s the difference?

The potential energy field, source of gravity and motion. The wellspring from which all others flow.

Work is bound. It’s being used. It exists in actuality.

Energy is free. It’s available to be used. It exists in potential.

Work is change. Energy is the ability to cause change.

Work deforms the field. Energy is the deformation of the field.

Here’s one of the secrets of science: That free, potential energy? It never gets used up. It’s always there, whether you use it or not, because of the nature of the field. That’s not to say that whatever’s using the potential energy won’t have an effect on the source doing all the work. It probably will. All those gravitational waves LIGO has detected? Their passages through the earth, each energetic enough to squeeze reality for a brief moment, didn’t use up any of their energy at all. The earth and the moon have held each other in a gravitational embrace for billions of years, at no cost in energy. The atoms in my desk hold up the monitor and keyboard through electromagnetic repulsion, fighting against the gravitic attraction of the entire earth at every moment, at no cost in energy.

If energy actually got used up, all the matter in the universe would have evaporated billions of years ago.

Energy and work are not conserved. They are continuously recreated. Energy density, on the other hand, is strictly conserved. You can’t ever withdraw more energy than the universe provides, and you can’t use less than nothing. The potential energy field pictured above shows zero energy at the top. It has a bottom, far, far below, at some unimaginable yet finite amount of energy. You can’t go above the top. You can’t go below the bottom. You must color within the lines.

Which to choose?

The bounded potential energy field model is both powerful and simple enough to model either Newton or Einstein’s laws of motion and gravity through choosing between basic assumptions.

1. Newton:

   1. Changes to the field propagate instantly.

   2. Angles (velocities) add.

   3. Every particle’s perception of time and space are equivalent.

2. Einstein:

   1. Changes to the field propagate at a finite speed.

   2. Gradients add.

   3. The cosines of angles represent every particle’s perception of time and space.

That’s it. Those are the only changes you need to go from one model to the other.

Which should you choose? That depends on what sort of system you are modeling. For most every day purposes, the Newtonian model suffices. Either way, action arises naturally as a fundamental principle.

The difference between Newton and Einstein

We start, as always, with the energy conserving inverse square function, this time set to represent a stationary particle with a mass of 10.1

Notice that our particle is not exactly point-like. It has a radius of 1.2 The blue dashed line represents its internal gradient, in this case set to zero. Its velocity (sine) is zero (the minimum), while its perception of time and space (cosine, the Lorentz alpha factor) is one (the maximum).

Now, imagine some blue particle being attracted to our red test particle. It has a mass of two, and is moving relatively slowly to the left (its gradient line is slightly lower on the left side). Notice that this blue particle has little effect on the more massive red particle, even at this short range.3 (In future diagrams, I shall hold the red particle stationary through magical means for simplicity.)

Having a lower potential energy on the left side, the blue particle moves in that direction. The gradient created by the red particle across the blue particle adds to this gradient, accelerating the blue particle toward the red. But by how much?

Newton would integrate the force over distance and add it directly to the slope of the particle. This is simple, straight forward, and relatively easy. It is also wrong. If you directly add the slopes, you can get velocities above the speed of light - the speed at which the potential energy field can change. That would tear reality, not to mention bend the gradient curve beyond the 90 degree mark. This way lies madness.

Einstein, remembering that we are working with energy, would add the gradients. Holding the higher (right) end fixed at the mass (-2), decrease the lower (left) end by the amount the red curve changes across the blue particle as it moves. Don’t forget to take into account the time it takes for the blue particle to cross a given amount of space. (Action is the principle of motion here.)

If the blue particle has a gradient of -0.1 (-2.1 left, -2.0 right), and the red particle is exerting a force gradient of -0.1 across the blue particle, then the total gradient across the blue particle becomes -0.2 (-2.2 left, -2.0 right). Notice that would place the energy level at -2.1 at the center of the particle. (Is this why kinetic energy is ½mv²?)

Let us continue the leftward motion of the blue particle. Notice the closer you get to the red particle, the greater the force gradient becomes, so the faster the blue particle accelerates. Let us say that over the next time span, the gradient is -0.4. So the blue particle updates its internal gradient to account for this to -0.6 (-2.6 left, -2.3 center, -2.0 right).

This is a ham-fisted approximation.

Now we must remember that the velocity of a particle is the sine of its internal gradient, while its perception of time and space is the cosine. At this point in spacetime, the blue particle has a gradient of -0.6. Ignoring the sign (which indicates direction), arctan 0.6 is about 30.96 degrees. That makes the velocity (measured in fractions of c) about 0.5145 (154,241,947 m/s) and the alpha contraction about 0.8575. (That’s a gamma factor of about 1.166, for those who prefer it.)

No matter how much it is accelerated, a massive particle can never reach the speed of light, because its gradient will never reach a vertical pitch. Its total energy will be the mass plus the center point of its gradient line (the kinetic energy). Photons emitted in the forward direction will have increased energy (blue shift), while photons emitted backwards will have decreased energy (red shift). Photons emitted 90 degrees to the direction of motion will have have no energy shift, as they originate from the central (total) energy level.4

Addendum:  Newton's solution is approximately correct when the slope is small and thus velocities low, because of the "small x" rule of sines.  When the slope x is small (less than 0.5, 0.5236 radians, 30 degrees), sin(arctan x) ≈ arctan x (or in radians, sin x ≈ x).  The smaller the slope, the closer the congruence.  At a slope of 0.01, the angle is 0.0099996667 radians, the sine of which is 0.0099995000, which is a "mere" 2,997,774.7 meters per second.  The difference between that and 0.01c is only 149.88 meters per second.

Copenhagen interpretation delenda est!

1

Ten what? I don’t know. Somebody smarter than me can probably scale this whole thing so a vertical unit is the reduced Planck constant, h.

2

One what? I don’t know. Somebody smarter than me can probably scale this whole thing so a horizontal unit is a Planck length.

3

Even this fairly small blue gradient across the red particle represents a massive acceleration in normal, everyday life. A 30 degree slope is half the speed of light.

4

An observant reader will note that total energy is not conserved. Energy density is. Our physics professors lied to us, hand-waving away the conversion of potential to kinetic energy. After the conversion, the potential energy is still there, waiting for something else to use it! This is the fundamental difference between (free) energy and (bound) work.