Given two lengths of distance from a planet's surface, as depicted in the diagram below; in a flat space-time, A and B are of equal length. Suppose instead we are viewing these two lengths from a great distance away, such that the effect of the planet's gravity on us is negligible; from our observations, the two lengths are equal, by which I mean that light moves across A in the same time, from our perspective, that it takes for that light to move across B.
In General Relativity, A is longer than B. We can think of this in a number of ways, but for now, just accept this notion as-is. Now, by necessity, an observer in the middle of A, timing the light crossing A, would measure a longer time than an observer in the middle of B who happened to be timing the light crossing B; if A is longer than B, then, in a sense, time is also "longer" in A; a clock at the middle of A ticks slower than an identical clock in the middle of B.
Now, usually, when considering questions of gravity in GR, we think in terms of curvature - but this is not, in fact, the only way of conceptualizing general relativity. We can also think in terms of rotation.
To illustrate this idea, we will begin by thinking in terms of a Newtonian force operating on an object - but we're going to keep in mind the modified distances. Now, this is an incorrect way of thinking about it, and is only used to demonstrate a way of thinking about it.
Now, think about the distance from the planet in terms of meters. And also, think about the distance from the planet in terms of seconds. For the distance from the planet in terms of meters, it should be obvious that the near (to the planet) side of the object experiences slightly greater gravity than the far side of the object. For our object, this is a tidal force; the only interesting thing to notice is that, for oblong objects, this can generate a very slight torque, causing the object to spin (but not very far).
Now, consider the distance in time - and observe that there is, in fact, a kind of torque there, as well. But what does it mean for an object to rotate in time?
Well, one potential answer might be found in Penrose-Terrell rotation - what is generally considered a purely visual phenomenon in which objects moving close to the speed of light appear to be rotated in particular ways. This is, I submit, not merely a visual phenomenon, but rather a fairly straightforward way of understanding velocity itself. Which is not to say it is the correct way of understanding things, but I will argue that it is -a- correct way of understanding things.
Now, the force, as described, isn't real. But suppose, for a moment, that the force, as described, doesn't need to be real. We can imagine the object, rotated in space-time by this imaginary Newtonian force. Can we extend our imaginations to imagining spacetime itself, rotated by this imaginary Newtonian force?
We don't really need to; that's curvature. Remember this graph?
A is longer than B. Now, an observation: A is longer than B. A second in A is "longer" than a second in B. So far so good, right? But if we measure a length of distance in A, compared to B, we get more meters. If we measure a length of time in A, compared to B, we get -fewer- seconds. The idea of length is misleading in a particular way.
We could ask, where the meters came from, and where the seconds went to - but once we formulate this question, the answer becomes a bit obvious, particularly in the context of thinking about space-time itself being rotated. The seconds became meters; part of time "rotated in" to become part of space, and part of space "rotated out" to become part of time, and because the speed of light is the conversion factor and heavily favors space over time, we gained meters, and lost seconds.
Let's consider velocity as rotation again.
Now, motion, like rotation, is relative. (Indeed, once we think of motion as rotation, it becomes very difficult to see it as anything but relative). And one problem we immediately run into is the fairly straightforward question - if this is rotation, why can't we just rotate 90 degrees, and hit lightspeed?
One answer is that, when we consider motion as rotation, we need to consider it as hyperbolic; if this statement doesn't make any sense to you, go ahead and skip this paragraph. The idea that motion-as-rotation is hyperbolic is both right and wrong; it is right, in the sense that motion, considered from a particular perspective, must be a hyperbolic rotation; you can't, after all, ever actually reach C. Also, we can usefully model all this behavior mathematically, and it's clearly hyperbolic. It's also wrong, because it's only true in a particular set of coordinate systems. I won't get into this much more than that, because this isn't an approach to thinking about the problem I find interesting.
Another answer is that, well, we can. That's what happens when we fall into a black hole, after all. The reason it is difficult is entirely to do with our perspective, and the nature of time; accelerate hard enough, for long enough, and you'll pass that ninety degree mark, at least from the perspective of everyone else; from your own perspective, you can't rotate at all. What everyone else will see, of course, is a mass accelerating until it accumulates enough inertial energy to turn into a very fast-moving singularity.
If you stop and think about it, this implies something very interesting about the curvature of fast-moving objects, which is both straightforwardly obvious, from the mass-energy tensor, and also strangely divorced from the way we usually actually think about fast-moving objects. If velocity is rotation, a fast-moving object is rotated; if the gravity well of a fast-moving object is rotated, it is increased. Gravity and velocity are, basically, the same phenomenon; rotation of space-time. The only real question is what is rotated, and how much, relative to what else.
Energy, both potential and realized, dissolve into rotation. That's not too interesting. The interesting thing is that, once you start conceptualizing gravity and velocity as rotation, you can conceptualize something else as rotation: Mass itself. That's somewhat more complex, but it ties into the general Crackpot Physics grand unified theory going on here. It's a boring, and probably somewhat disappointing, grand unified theory; everything is just rotation of space-time? Yep. Well, that or rotation of rotation, which is where the grand unification comes from.
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