Steinein -
Like you, I followed a link here on my search for a better explanation of the concepts behind GR. I've been scouring the internet for a couple weeks now, linking from one page to the next, and I swear if I watch one more video of a guy dropping a ball in a rocket ship or falling elevator, I'm going to try my own gravity experiment by leaping off the nearest bridge. And don't get me started about bowling balls on rubber sheets - there must be an awful lot of bedwetting bowlers in the physics community.
The problem is that there are lots of grade-school explanations of GR, and a fair number of graduate-level descriptions, but there's not much in between. As far as I can tell from my search, Einstein's thought process for GR was "Some guy in a rocket ship... curved spacetime." The same goes for SR: "Two guys with clocks in trains... E=mc2." It's like the underpants gnomes' business plan. (Actually, the offerings for SR aren't quite so bad - there are at least a couple places that make the connection fairly well. I suspect its greater popularity just means more videos overall, so we're in a monkeys+typewriters=Shakespeare situation.) I'm not sure why this is, but I've seen it with other difficult concepts, as well.
Actually, the explanations here are some of the better ones I've seen, but even here, we seem to jump from rocket balls to geodesics with very little in between. I get that the physics for the guy in the accelerating ship works the same as for the guy standing on the street in Cleveland. And I know what a geodesic is, and why you would use one to describe movement on a "straight" line through curved spacetime. What's missing is the clear, CONCEPTUAL explanation of how we go from one to the other (i.e. no references to Riemannian manifolds!)
However, I think I've kind of pieced together a rough understanding, and so I'd like to post it here, both as an answer to your question, and in hopes that greater minds will chime in and (nicely) straighten out some of my more bizarre logical twists. Here goes...
Let's start with the accelerometer, first mentioned on this page by DaleSpam. Also, again following DS's lead, let's get rid of some extraneous dimensions, so my poor brain doesn't explode. The principles will translate just fine to higher orders, I think. We'll keep the z-axis - the traditional up-down axis that we most commonly associate with gravity. And we'll keep the t-axis, of course. Finally, we are also going to eliminate the Earth, and its tidal effects, n-body diagrams, etc. Let's put our accelerometer on a really huge planet far out between galaxies. Its surface is so big we can consider it a plane, its radius so big that its g is effectively constant over our distances, and it is so massive that our own mass is negligible.
So our simple 1D accelerometer is just a metal frame, with a lead ball suspended between two springs - one at the top and one at the bottom. As we know from a gazillion rocket-ball videos, the accelerometer in free fall matches the accelerometer in free space - no acceleration. The one on the planet surface matches the one in the accelerating ship - yes acceleration. And if you can't believe a ball on springs, who can you believe, right? Seriously, though, we've kind of leap-frogged Einstein here, but he has something to say about this. Specifically, as I understand it, he was answering the question, "Why does the ball have to represent acceleration? Couldn't the ball just be pulled down by the gravitational 'force', the invisible string of classical physics?"
The answer to this is no, and the reason is simply that forces don't work like that. We all know that F=ma, so if I tie one invisible string to M1, and another to M2, and tug on them with the same force, they will accelerate at different rates, dependent on their masses. And I don't know from the strong or weak forces, but I can tell you that two protons, each with the charge of 1 proton, will rocket away from each other; but two bowling balls, each with the charge of 1 proton - yeah, not so much. EM force is also independent of mass.
But gravity doesn't follow F=ma - it pulls on each mass with exactly the amount of force necessary to attain a given acceleration. There's a name for forces that are proportional to mass: we call them accelerations. And so we come back to Einstein. Einstein's specific genius, from what I can see, was the ability to call a spade a spade. Physicists were trying to figure out why light seemed to have the same velocity regardless of the velocity of the observer. Einstein comes along and says, "Well, hey, maybe that's just the fastest anything can go." Obviously a complete load of tosh... except SR has worked, over and over. Okay, then, how about this: gravity doesn't behave like other classical forces, acting instead like an acceleration. Einstein says, "Huh, well maybe that's because it is." Once again, obvious BS, except that it works, both then and now - the more precisely we measure, the better his predictions look (go check out Gravity Probe B).
So Einstein agrees with the accelerometer, and we will too, at least until the LHC spits out its first graviton. But so far, I don't think I've said anything you couldn't pick up elsewhere. Pay close attention, though, because here's where the magic happens.
Imagine that you pick up our accelerometer and take it way, way high up, though still in the approximately constant-g zone of our giant planet, and drop it. There's no air resistance and no starting velocity, so v(t)=gt, at least from the perspective of an observer on the planet surface. As a graph, with t on the horizontal and z on the vertical axis, we are looking at a hyperbolic, with the slope of z(t) ever increasing (or decreasing, whatever - signs don't matter here.)
But according to our accelerometer (and Albert E.) that's not true - there is no acceleration; as far as the accelerometer is concerned, it's floating blissfully through empty space at a constant inertial speed (or no speed, inertial reference frames being relative, after all.) That means that, from the accelerometer's perspective, which according to E. is the real perspective, its z(t) line, called its "world line", represents a constant velocity, and so it should be arrow-freakin'-straight. So we grab hold of that hyperbolic z(t) line, and like a bent metal bar, we yank on it until it's straight, like it should be. But hey, that world line is still tied to the t-z coordinate system we originally drew it on, so when we yank the world line straight, we're putting a bend in one or both axes.
I'll get to what this means in real life, but for now, just take a second and contemplate what this means visually. In order for the world line to be straight, as the accelerometer knows it is, the coordinate system has to be bent. This, at least as I understand it, is the origin of curved spacetime.
Okay, but what does it actually mean? Well, now we run up against my poor math background, but here's what I see. According to the accelerometer, it is moving at a constant velocity, so dz/dt is constant throughout its journey. Picture your basic sloped mx+b line from 8th grade. But the outside observer sees the value of dz/dt increasing at each measurement interval.
Either the dz of the observer is getting bigger for each dz of the accelerometer (picture the vertical z-axis bowing away from the straight mx+b accelerometer world line, so that each time you run a horizontal shadow from the world line to the z-axis, it is ever further along the observer's z-axis than what it would be on the "true" vertical z-axis of the accelerometer's world line. Savvy?) OR the dt of the observer is getting smaller for each dt of the world line (picture the horizontal t-axis bowing up toward the mx+b accelerometer world line, so each vertical shadow from the world line to the t-axis is ever less far than the "true" horizontal dt of the world line) ... OR some combination of both.
Makes your brain hurt, I know, and sounds like total BS, but 1) if you accept the accelerometer's zero-reading in free fall, as Einstein did, this is the natural outcome, and 2) like the time dilation of SR, it WORKS. The effects predicted here have been tested using satellites in various orbits, and the results are consistent with curved spacetime.
This can be extended to explain the accelerating-while-standing-still dart mentioned above, though I'm going to skip the dart and just use my guy on the ground. First, let's look at the simple explanation - the observer on the ground sees the accelerometer accelerating toward the ground. The accelerometer sees the observer, and the ground, accelerating toward it (whether it thinks "Oh, no, not again!" like the bowl of petunias, I will leave up to you.) SOMEBODY is accelerating, the accelerometer knows it's not itself, and Einstein agrees, so who's left?
Looking from a slightly more complex perspective, it helps to remember that GR specifies that gravity curves spaceTIME, not necessarily just space. When we were looking at the zt graph before, I said that either the z-axis, the t-axis, or both, could be curving. So when I tell you that you are accelerating just by standing still on the planet's surface, one way to look at it is that you are experiencing reduced dt's, rather than increased dz's. (Mind you, this may not matter all that much - my understanding of relativistic spacetime, versus Galilean spacetime, is that any of the axes are pretty much interchangeable, but hey, if it makes you feel better...)
Incidentally, this is where geodesics come in - a geodesic is a "straight" line on a curved surface, like me taking an airplane from here to Podunk, Iowa. It's a straight shot... for a given value of "straight". If spacetime is curved, the "straight" line is also curved, just as the Earth's surface between here and Podunk is curved. It feels like straight, because that surface is the only reality I know, but from an additional dimension, it's obviously curved. Where I would disagree with other posters, though, is that I think the free falling accelerometer is the "true" straight line. It is the elevator going at a constant velocity upward that follows a geodesic (paralleling the outward-bowed z-axis).
So as I said, you are accelerating because of reduced dt's, which make your velocity into an acceleration. What's that? You say you don't have velocity because you aren’t' moving? Oh please. Relativity, remember? Moving is a frame of mind, or at least a frame of reference. Our observer and accelerometer may disagree on who is accelerating, but there is no disagreement that their velocities are completely relative. Without the acceleration issue, there would have been no disagreement. So there is a reference frame in which you are moving; continually reduce the length of dt, and your v turns into a. Once again, sounds like BS, but is confirmed by experimentation. Clocks on satellites further away from Earth experience different time than those closer in, but you need a hydrogen maser clock to note the difference.
So there you have it, from rocket-balls, to curved spacetime, to accelerating-while-standing-still. I hope you appreciate it, because I'm about to get pilloried for this.Doug
