Semi-Legit Physics: The Wave Interpretation of Relativity
So, while I try to motivate myself to finally learn Tensor mathematics, some other thoughts; semi-legit physics. These aren’t -entirely- crackpot physics, they are, perhaps imperfect, ways of conceptualizing more mainstream physics. They’re models; sometimes useful, sometimes misleading.
I’ll begin with something I think is very close to correct, in the sense that I don’t think it significantly misleads you in any major ways to think of things this way.
Relativity is, broadly, a description of a topology. Special relativity is description of a very specific topology, general relativity is a generalization of that topology. Consider that topology as a standing wave - a change in the amplitude of some quantity which varies with distance from the origin, which we can relate to the idea of curvature (or the rate of change thereof, depending on what you’re focusing on).
This topology is in spacetime; in terms of spacetime, we might conceptualize the amplitude of relativity as the degree in rotation of a given coordinate between time and space; because lightspeed gives the conversion factor, you get a very good deal on space for a relatively small amount of time. That is, in a certain sense, you’re rotating some of your time into space.
There are a few ways of thinking about this, perhaps the most natural is to think about the traditional “depression in a plane” picture of relativistic gravity, that characteristic bowl shape. There are a couple of very important things to notice, thinking of this purely as a two dimensional object: First, if you compare the surface area of the sphere to the surface area of the bowl, you’ll discover that the surface area of the bowl is larger. A gravity well is “bigger on the inside”. Second, if you compare the circumference of each circle in that bowl and add them up together, you get the surface area of the circle, not the surface area of the bowl - the additional surface area is coming from the mysterious third dimension that the bowl is depressed into.
It’s not wrong to interpret that mysterious third dimension as time, and while there isn’t really a nice visualization of this I can provide, it may make intuitive sense that the surface area gained in “space” is lost from “time”.
But it’s more interesting than this. Take that flat plane, and draw a line perpindicular to the surface; it will be helpful if you have it point up, rather than down. Now, consider what happens to that line if you drew it perpindicular to the surface somewhere in the bowl; the surface isn’t flat, so that line gets rotated slightly, such that it is now “leaning” in the direction of the center of the bowl.
Your perpindicular line is now pointing a little bit straight up - in time - and a little bit sideways - in space. This is the direction “the future” is pointed at that location, from our perspective in flat space. From the perspective of somebody occupying that location, they, locally, are in flat space, and it is -us- whose “future” is pointed slightly into space instead of purely into time.
So, consider rolling a ball across this surface. It is rolling with respect to time, necessarily - which means that, just as the ball is moving across space, it is also moving through time. In the fairly boring fashion - we’re constantly moving into the future. While it is rolling across a flat plane, it’s moving forward in space, and let’s call it upward in time, forming a nice straight line.
When it hits the depression, where “forward in time” no longer points straight up, this straight line, from a purely space-based perspective, appears to bend into a curve, as the ball’s trajectory forward in time is now crossing some of what is, from our perspective, space. (Remember, from the ball’s perspective, it is always on a locally flat surface - it’s us whose future is no longer straight up.)
We can thus think of curvature as, instead, an entirely different phenomenon - rotation. It is a local rotation of spacetime, where the extent of the rotation is given by the angle between space and time.
(Now, this is all slightly misleading, because time does not in fact point “upward”, but “inward” - I’ll cover this at the end.)
So, armed with curvature-as-rotation, we are now armed to describe general relativity as a wave. The simplest interpretation here is a one-to-one mapping, using the amplitude of the wave to represent the amount of rotation.
But we’re not done, because, if you pause for a second to think about it, we actually just stumbled across something really interesting: A topological description of motion through space. And it’s worth pausing here to think about this, because it will better arm us for the wave interpretation itself.
See, if we just let that little perpindicular line -stay- oriented slightly askew, then, assuming only that the ball travels forward in what it thinks of as time, we get motion through space. If its perpindicular line is pointed slightly up and slightly to the left, relative to us, then we will see it, as it moves forward in time, also drifting to the left.
Now this is actually really interesting, because, if we kind of draw a box around the ball, and think of this box of space moving with the ball inside, then, if we draw the perpindicular lines, and examine what this rotation means for the space comprising the box, we will observe that this means that space and time are rotated (relative to us, the observer). Not rotating, rotated. Some of space is rotated into time, and some of time is rotated into space.
As with gravity, this means this box is “bigger on the inside”, but particularly on the axis of motion, whose dimensions are those being rotated. So from our perspective, the ball is occupying a smaller space than we would otherwise expect. (Sort of, we’ll come back to this later.) That means - the ball will appear shortened. That is Lorentz Contraction.
Because some of time was rotated into space, and because some of the ball’s forward motion through time is now redirected into space (from our perspective), the ball will also appear to be, if we mounted a clock on it, progressing more slowly through time.
But, armed with a conceptualization of this phenomenon as a -rotation-, maybe it is more obvious that, from the ball’s perspective, -we- are the ones who are shorter, and moving more slowly through time; if we orient ourselves such that the ball’s skewed time axis is properly aligned upward again, and look back at where we were standing, we will see that - hey, time is rotated.
It is slightly easier, in my opinion, to see how rotation can be relative; just reorient yourself so that a line is pointing “up”, and it should be obvious how all the lines that you were previously oriented to are now pointing at an angle.
This framework may also make it clear that velocity doesn’t “cause” Lorentz contraction, or time dilation; nor do Lorentz contraction or time dilation -cause- velocity, or acceleration. Rather, they’re all the same thing. This should satisfy us quite deeply, because the most fundamental description of how one thing causes another thing must, at its root, be a description of how one thing is the same thing as that other thing - if you have not reached that point, your description is insufficiently fundamental, and is incomplete. At some level, all causal interactions must form an identity. (If they didn’t, there is some mapping which is missing; some step where A becomes B, with no intermediate behaviors, which represents a discontinuity in the mapping - a gap. A point where your description is incomplete.)
But perhaps more fascinating, to me, is what happens if you pause to think about how acceleration gives rise to velocity. And here, I think, is where the wave description begins to really shine.
Let us consider the waves of two objects - a planet, and an infalling object. Let’s say it’s our ball from before. Consider the wave of the planet - observe that the amplitude varies. The amplitude increases as you approach the planet, and decreases as you move away from it. The same is true of the ball.
Now consider the interaction between these waves as the ball falls into the planet. As it accelerates inward, we know, given Lorentz Contraction, it is in some sense shortening as its velocity increases; its wave is changing shape. Let’s make a note of this and set it aside for later.
There’s another thing happening, as these two waves interact. Remembering that a gravity well is bigger on the inside, because the surface area of the depression is greater than the surface area of the flat plane, an outside observer will see the ball shortening due to gravity, as well. But it isn’t flattening in a nicely symmetric way, as it would from velocity; the part nearest the planet is shortened a little bit more than the part farthest from the planet, as gravity varies with the inverse square law, meaning curvature increases faster the closer you get to the planet.
So, again ignoring the contraction from velocity for the moment, but keeping it in the back of our head, the ball continues falling until it hits the surface of the planet, and we’ll pretend it just hits the planet and stops in a perfectly elastic collision, transferring all its energy to the planet. It is no longer in motion - and from our perspective, its wave is shortened in a very assymetric way.
It may seem silly at first to ask this, but I think it is illuminating: Suppose, for a moment, that our perspective is correct, and the ball -is- shortened in this asymmetric way. Not in a figurative way, but a literal way. Where did the rest of the wave go?
Well, I think the answer may be obvious when you realize what the ball gained in exchange for this forming asymmetry, and what it lost when it collided with the planet, and what it would need to regain in order to escape the planet once more. Velocity. Which is itself just a form of contraction.
But - the ball was shortened asymmetrically. In some sense, the velocity that was gained must also have some kind of asymmetry. And - it does. The velocity has an *orientation*. It is pointing in a particular direction.
Suppose the planet is actually just a clump of dark matter, and the ball instead passes through the center, proceeds through the gravity well, and escapes the other side. Just as, when it was falling inward, there was a particular asymmetry (gravity is slightly stronger on the leading edge of the ball than the trailing edge of the ball), as it is escaping, we get the exact opposite asymmetry.
Considering the ball as a wave, we can express this as some kind of transformation. In this case, I expect the transformation to be equivalent to the interaction between the wave representing gravity, and the wave representing the ball, in spacetime.
Velocity can also be expressed as some kind of wave transformation; it may be easier to picture it as a scalar transformation, but I think it is in a particular sense useful to imagine it as the “missing” length of an asymmetrically transformed wave. In order for that wave to be symmetric again, it must escape the gravity well; in order to escape the gravity well, it needs velocity, which is then used to “rebuild” the symmetric wave shape. Velocity is, in some sense, a piece of a wave which has been torn out, and which preserves the properties it had as a piece of that wave, such as local rate of change of amplitude - this is what gives rise to the asymmetry of velocity with respect to Lorentz contraction, those preserved properties.
Once we have this information in hand, we can make an interesting observation: If acceleration can be regarded as the interaction between two waves, acceleration is, in some sense, some particular -shape- of wave, which exhibits a particular form of asymmetry on the waves being accelerated.
Likewise, velocity, as a wave transformation, can itself be regarded as some particular shape of wave, albeit perhaps an entirely uninteresting one.
When two objects collide, what is happening is that their waves are interacting in a way which transforms both - but we can also kind of imagine them trading those pieces of velocity torn away from some waves, not necessarily their own.
Since velocity is energy, and velocity is also a piece of a spacetime wave, we can thus observe that, in some nontrivial sense, spacetime and energy are the same thing. Spacetime-mass-energy, thus, must be conserved.
This model of viewing relativity has some interesting properties. Personally, I find it far more intuitive than other ways of viewing relativity, but certain unintuitive parts of relativity in particular suddenly become quite obvious - Rindler Coordinates, for example, pop up fairly trivially from this model.