Douglas Adams, actually, describing how the Infinite Improbability Drive worked.

IIRC the next part of the phrase goes "The Heart of Gold told space to get knotted..."

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HRH Joshua Abraham Norton II, Emperor of the United States, Defender of Mexico and the Dominion of Canada

HankSolo: "Is there a way to measure [the velocity of an object in curved spacetime]?"

I think that the answer is Yes! with one condition attached: that you must define a refrence frame in which the velocity is to be calculated, because spacetime itself offers no absolute, universal reference frame.

There's a good page with nice animated graphical descriptions of how a spacetime velocity is calculated on this page byAndrew Hamilton of University of Colorado.

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Einstein and many others have postulated that space is finite but unbounded; it has no 'edge'. You can easily visualize a finite but unbounded 2-dimensional surface: it's just the surface of an egg or balloon or beachball. No edges but definitely finite. I don't know that anyone can really visualize a finite but unbounded 3-D + time surface, like Einstein's finite-but-unbounded universe. It can be described mathematically, though, just like we can describe the surface of a 2-D sphere using formulas for the surface area and so forth.

HankSolo, I kind of expect from reading your excellent posts that you would really like to be able to directly grasp, intuitively and concretely, what this 3-D + time curvature really 'looks like' but I don't think that's possible for our minds. The universe is not forced to make itself intuitively comprehensible to us, unfortunately. (Or, to invert that, nothing in our biological history prepared our minds to intuitively grasp concepts like spacetime curvature or quantum mechanics.)

I think that the abilities of our minds to visualize things, to grasp things in a concrete and intuitive way, are really limited by our biological history. Those abilities of our minds aren't universally applicable: they don't really work for all the properties or phenomena in the universe. We can only intuitively visualize things for which we have been pre-programmed by evolution and by our experience; and the universe has properties and behaviors which go beyond us. (I deeply regret that!) Fortunately, our other mental abilities--abstract thought, mathematics and logic in particular, give us ways to describe the universe which do not depend on direct, concrete visualization.

Not to say intuition and visualization are useless in physics! Sheesh no! But when a theoretical physicist makes an intuitive jump, I suspect it is sometimes--perhaps most often--an intuition about the "shape" or balance or "beauty" of abstractions like equations and not the result of a direct and concrete visualization of, say, an 11-dimensional manifold used in superstring physics. And analogies like rubber-sheet spacetime and 2-D finite but unbounded surfaces are useful as long as they are understood to be analagies and not the shape of reality itself, whatever that is.

The funny thing being is that everyone will say a theory on gravity is completly wrong, but never be able to point out anything conclusively right.

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When I took physics classes 30+ years ago, I learned there are 3 words to help me understand physics: energy, energy, and energy. The reason your surfer moves is that the water transfers its energy to him, moving him forward. When he is "in back" of the wave, no more energy. AFAIK, according to current theory, there is no transference of energy between matter and space-time. (That's the reason we are able to see galaxies billions of LY away. The photons do not lose (transfer) any energy to space-time). So your analogy of water and space-time does not work.
I’ve always thought of the attraction of 2 objects as a result of Newton’s first law: “Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it”, applied to 4 dimensional space-time. To compare it to the rubber sheet analogy, when the object moves down the pinched sheet, it is not because the object is attracted by gravity, but because there is nothing stopping it from moving along a curved 4 dimensional path.. Space-time does not absorb the energy of an object, so the object will continue moving through space-time. Matter produces “the pinch” which warps space-time. AFAIK, the mystery is "Why does matter curve space?".
If there are 2 objects approaching each other, each object curves space-time around itself. In 3 dimensional space, if the speed and angle in space are just right, they will orbit each other. In 4 dimensional space-time, each will follow the curved 4 dimensional path the other object produces.

I think that part of the problem is that you're using the term spacetime, but I don't think you're understanding it quite correctly. You're talking about it being curved, but I think that you're only really visualizing the space part being curved, when in fact the time part curves as well. Here are a couple images to help you see how curved spacetime can cause an object which is at rest to start moving. They aren't terribly accurate mathematically, but I think they're sufficient to get the point across.

First, here's a spacetime diagram showing the world line of an object appearing at some arbitrary time t = 0. It's at rest, relative to some reference frame that we've chosen. As is typical, time is the vertical axis, and we're only bothering to look at a single space dimension. As time passes, we move up the vertical axis, following a specific path. The particularly important thing to realize is that the object traces out a path here even though it's at rest.




Now at some later time, some other object suddenly appears (you can tell that these objects are appearing out of nowhere because their past world lines don't extend indefinitely far back; it's artificial, but makes life convenient in drawing these images). It has an intense gravitational field, such that it warps spacetime around it. Our test object continues to follow a world line that it sees as straight, but you can see that this line now moves toward the larger object and will eventually intersect.



In one sense, the test object continues straight on the same world line. There's no force, it's just that its future path has been curved from the direction it would travel if the second object weren't there. In another sense, we see a mysterious gravitational attraction that causes the object to start moving when it was at rest before. Does this help make sense of what's going on here?

Thanks for the link. Uh oh, I see a lot of math there.... Mathematics was my favorite subject throughout school (and I still believe that math is the only truth we can count on in the universe), but I've lost my knack, and it sure looks formidable now. But it's looking more and more like I have to freshen up on it if I want to get a better understanding of these things. In the meantime I can only speculate on the general ideas.

How about looking at Einstein's universe as a donut? That would seem to make some sense to me, but there is still a "middle" even though it is outside the actual donut. So you can make the argument that there is no middle, as well as the argument that the donut emanated and spread from this empty middle.

Thanks for the compliment. And you are absolutely right. I have some innate "need" to directly grasp this concept, along with a few others, and I find it hard to just accept any of it until I can understand it. If I can learn the truth about gravity, UFO's, and the inner workings of the brain before I die, I will die a happy man. Oh, and I think my family figures in there somewhere too! :wink:

The perfect culmination for me would be: Designing a next-generation computer processor that will work like our brain (hardware-wise not just software-wise), and have it travel and explore other solar systems, while harnessing gravity as its propulsion system and thereby bypassing the light-speed barrier. And then making billions of dollars, retiring, and enjoying the world with my wife and kids.

Not too lofty of a goal, huh? I have more of a chance with the computer chip than anything else... Spent most of my first 35 years partying, when I should have been at the drawing board. As you said, there is a lot that is out of our grasp intellectually. We can model it mathematically, but not imagine it. But I don't think that these limitations need to be on us forever. Understanding and grasping the physical nature of the phenomena of gravity and the universe, may just be a matter of perception. After all, we are living in this universe, so it is obviously there physically. And we can make predictions based on it, so it follows known mathematical principles. We just need to figure out how to see it and harness it. And then the sky's the limit knowledge-wise and technology-wise.

Absolutely. I haven't really thought of it this way (traveling through time). But the diagram should actually show the warpage as an hourglass shape, and it would not intersect or cross over. That's the difference that I see, and it would still fit my analogy. It may get very close to the downward line (perhaps close enough for the two objects to crash, in which case there would be a totally new set of forces), but if it doesn't crash it would return to its normal path after the downward object passes. Right? So the lines for the two objects would resemble an hourglass. Again, the test object has to be motionless, otherwise it would go into orbit around the massive object (or get flung elsewhere). So I have to adjust the analogy that although it moves through time, it's the lack of motion through 3D space that would cause the hourglass shape. That's the way I'm seeing this picture. What do you think?

It wouldn't be an hourglass shape unless the object appears and then disappears. The big object to the right suddenly shows up, and from then on, stays put, as seen by the fact that it's world line continues straight upward. It's not a "downward" object that passes; there is no downward in this picture since the vertical axis is time. The only motion through space that we see in this picture is the small test object that moves to the right because of the influence of the heavy object that appeared. So stationary world lines will continue to curve toward it indefinitely into the future.

You're right. I was mistaken in thinking the right line was moving down. Duh (to myself). I have to think about this. Thanks.