The nagging question then, is why the inverse cube law has no possibility of balancing (except the perfect circle case, which is impractical)? The reason is that 1/r^3 grows too fast when you get closer to the star, and falls too fast when you get away from the star.
ac2000 wrote:Thanks for your interesting summary on that topic, quickfur. I found it very interesting.
While reading I was wondering about one thing.
You wrote:The nagging question then, is why the inverse cube law has no possibility of balancing (except the perfect circle case, which is impractical)? The reason is that 1/r^3 grows too fast when you get closer to the star, and falls too fast when you get away from the star.
This is plain and obvious as long as we deal with the same sort of masses in 4D as in 3D.
But I was wondering:
If Newton's gravitational law F= G * ((m1*m2)/r^2) is changed to F= G * ((m1*m2)/r^3)
... then I wonder if it's OK to change something about the denominator part of the fraction (that is change r^2 => r^3)
but continue to calculate with the conventional 3D kind of masses of the stellar objects?
Because assumingly the 4D masses are of a different quality (in a way "heavier" than the 3D masses) and I thought this might be taken into account.
For example the 4D masses might be multiplied with a certain (unknown) constant or squared in a specific way to make up for their 4D extra "heaviness".
If we did this, the fast "growing &decreasing" of the gravitational force would be levelled out and orbitals would be possible in 4D, wouldn't they?
wendy wrote:The usual model for action at a distance, is the radiant model, in that the source radiates a flux of forcekins. These buffer anything that they chance to hit, which either pulls them or pushes them. A source radiates a shell of fluxions, which travel outwards at a constant speed. Since the total flux is proportional to the strength of the source, say GM, the fluxions are spread out ever-thinner as they go further, so eg, S = 2 pi² r³ in four dimensions.
The force is then GM/2pi2 r3, in four dimensions. The same relations hold for electricity and any other radiant force. With light and heat, the inverse cube still holds, but the action is not of vector relation (pull vs push), but of scalar relation (more). So the intensity of a light at 1 foot, is eight times that at two feet, since the candle has to light eight times the volume.
I'm not sure what you mean by "3D kind of mass" vs. "4D kind of mass" here. Mass is directly proportional to the amount of matter in the object, regardless of its dimension, and regardless of its shape. It doesn't matter if the object is a 2D disk, 3D sphere, or 4D hypersphere, or even a tube or some other shape; the total amount of force that is "felt" by the object is directly proportional to how much "stuff" is in it -- that is, multiple of the number of atoms in it.
ac2000 wrote:I'm not sure what you mean by "3D kind of mass" vs. "4D kind of mass" here. Mass is directly proportional to the amount of matter in the object, regardless of its dimension, and regardless of its shape. It doesn't matter if the object is a 2D disk, 3D sphere, or 4D hypersphere, or even a tube or some other shape; the total amount of force that is "felt" by the object is directly proportional to how much "stuff" is in it -- that is, multiple of the number of atoms in it.
I meant by "3D kind of mass" vs. "4D kind of mass" that maybe there could be different kind of atoms in 4D matter. So that not only the whole volume extents in a 4th dimension but every single atom has an extentention in the 4th dimension and that this might somehow make a 4D mass heavier.
I don't know, maybe be it's nonsense.
So, if mass is proportional only to the amount of matter in the object regardless of the dimension, as you wrote, does this mean one can simply calculate the mass of a 4D object if we know what's material it is made of (or the density) and the size?
So it doesn't matter what that value is; the qualitative behaviour of gravity is not affected by the value of M. It's the 1/r^3 part that changes its behaviour.
ac2000 wrote:Thanks for your explanations, quickfur.So it doesn't matter what that value is; the qualitative behaviour of gravity is not affected by the value of M. It's the 1/r^3 part that changes its behaviour.
I still don't understand this. Because on the wiki page about gravitation http://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation it says in the first formula that m1*m2 is divided by r^2 (so r^3 in 4D).
And the bigger the masses are in this formula, the smaller would be the F force, when the multiplied masses are divided by r^2 or r^3, wouldn't they?
So I don't understand where the "number 1" in 1/r^3 comes from which you mentioned?
No, the bigger the masses, the larger the force.
In other words, the object's mass does not change how much it accelerates under the force of gravity (one recalls Galileo's experiment, where he dropped a light object and a heavy object simultaneously from a high tower, and both reach the ground at the same time -- contrary to popular expectation that the heavier object will hit the ground first).
ac2000 wrote:[...]
I don't know if I do fully understand those formulas. I guess it's too much math for my unmathematical brain. I probably should stay away from mathematical things because it's so frustrating. It's very interesting on the one hand but I always fail to really understand it. That's no fault of your explanations which are certainly very well written.
No, don't stay away from mathematical things. The fact you find them frustrating is not because they are inherently hard to understand, but simply that you haven't found the right frame of mind to correctly interpret them, and through no fault of your own: a lot of math is thrown about these days but often inadequately explained, filled with unstated assumptions or otherwise unhelpful over-simplifications. Your frustration is more a sign of inadequate explanation (on my part or others') than any inability to grasp mathematical things.
I find that stubborn persistence in spite of all this eventually yields great insight.
It's a long road from the "wandering stars" of the ancients to today's understanding of elliptical orbits, and even then, I don't think we have fully captured all the intricacies of it yet, though we are pretty close.
quickfur wrote:Which leads to the conclusion that in very high dimensions, the universe would be very cold and dark, because light diminishes too quickly with distance, and so does heat. This applies both to stars and to artificial light/heat sources.
quickfur wrote:Basically, to have an orbit, there are two factors at play: the inward pull of gravity, which draws the orbiting object closer, and the momentum of the object, which wants to fling it out into outer space. An orbit can only happen if these two opposing forces are balancing each other out.
Now, balancing these two forces isn't as simple as it sounds, because the strength of the forces change: the strength of gravity changes depending on how far away you are from the center of gravity, and the momentum changes as a result of the gravity. Think of a falling object: it starts from zero momentum because it's stationary, say when you're holding it out the window of a high tower; as gravity acts on it, it accelerates, so its momentum also increases.
quickfur wrote:The object is moving a little too fast: so it starts moving away from the central star. However, because gravity decreases too fast with increasing radius, it doesn't quite manage to counteract the momentum; the object's distance will gradually increase -- to be precise, for each revolution around its orbital path, its distance will increase by a constant amount. So its path is a spiral, spiralling outwards from the star. This is bad news, because a spiralling path means there is no orbit; it's moving away from the star at a constant rate.
quickfur wrote:OK, I found pat's post that has the link to his paper explaining why 4D planets can't have stable orbits (except the unrealistic perfect circle).
gonegahgah wrote:It dawns on me that all I've described are circular orbits.
For an orbit to be parallel to the surface at opposite sides of the planet at the same distance; it can only be so if the orbit is circular.
If you start at the same distance with higher velocity you will push into a higher orbit initially that becomes elliptical but not parallel at opposite sides of the planet.
Instead you probably require a tangential elliptical orbit to achieve a stable orbit or perhaps otherwise have one that precesses.
Still, surely slight tangential variations would be allowable in 4d instead of just a circular orbit?
Klitzing wrote:"It's more space
in four-space"
Well even when it is much sharper to fall out of a stable orbit into an unstable one,
the probability of collision also should decrease there: there are infinitely times more ways to miss a target!
--- rk
quickfur wrote:Another way to think about it is that the set of possible 3D orbits is like a mountain with a flat top, where the top of the mountain represents stable orbits. While it's possible to get knocked off the top and roll down the mountain (i.e. end up in an unstable path that either crashes into the sun or flies off into space), there's quite a lot of nearby stable orbits that, for the most part, something currently in a present stable orbit is quite likely to still be in some kind of stable orbit even after you knock it around a bit. In 4D, however, it's like a mountain with a sharp, steep, pointed tip, and you're standing tiptoe on top. All it takes is for a tiny nudge and you're tumbling down the mountain. You can hold on to the tip of stable orbits, but it's so much easier to fall off. There's simply no room for error. All it takes is a gust of wind and you're out, the planet is on a collision path with the star, or flying off into space.
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