## 4D Orbits

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

### 4D Orbits

Ok, I'm a bit new to the wonderful world of the 4th dimension. I understand why, and have suspected that in this new dimension energy would decrease as the cube of the distance, rather than the square. So it was no surprise that this was true, however what was a surprise was that this makes all but a perfectly circular orbit impossible, can somebody please explain to me why this happens?
Higher_Order
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### Re: 4D Orbits

Why this happens is not very obvious, but it arises from the way the equations of motion change in a fundamental, qualitative way when the denominator is r3 instead of r2. Basically, you derive the Keplerian equations for orbital motion by writing out the momentum of the planet, and using Newton's equation for gravity to relate it to the mass of the star, and then try to solve for the parameters which gives you a periodic motion. You can find this derivation if you search for this topic somewhere on this forum.

It turns out that anything except r2 gives rise to unstable motion. That is to say, in every dimension perfectly circular motion is possible, but 3D is the only dimension in which slight deviations from the perfect circle will still give a stable orbit (in the form of elliptical orbits). In all other dimensions, the orbit will either degrade (the planet collides with the star eventually) or diverge (the planet will fly out of orbit). In 4D, there are three kinds of planetary motions: (1) the perfect circle, which is stable in theory, but impractical because nothing is a perfect circle in real life; (2) a spiralling "orbit" in which each orbit adds a constant displacement to the distance between the star and the planet (so the planet is moving towards/away from the central star at a constant rate -- which means no long-term orbit is possible); (3) there is no orbit: the planet either flies into the star within a short time, or flies off the star's gravitational influence very quickly.

Intuitively speaking, this is caused by the way the force of gravity degrades with increasing radius. In 3D, the rate of degradation is just right, that the changing force on the planet pulls it into an elliptical orbit. In 4D, the rate of degradation is too fast: gravity weakens too quickly as you leave the star, so things will gradually fly out of the solar system, or conversely, the force of gravity increases too quickly when you approach the star, so you'll get pulled inwards and hit the star.
quickfur
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### Re: 4D Orbits

Oh, and BTW, this topic is in the wrong forum; metaphysics is supposed to deal with non-physics based topics. This one belongs in "higher spatial dimensions".
quickfur
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### Re: 4D Orbits

Ok, that makes sense. Sorry for the late reply, I've been very busy lately.
Higher_Order
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### Re: 4D Orbits

If you quantise the orbit, it would be stable. That's how Bohr stabalised the electron.
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wendy
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### Re: 4D Orbits

wendy wrote:If you quantise the orbit, it would be stable. That's how Bohr stabalised the electron.

Could you please explain that to me (or link to another resource), I've never heard of that before...
Higher_Order
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### Re: 4D Orbits

wendy wrote:If you quantise the orbit, it would be stable. That's how Bohr stabalised the electron.

I'm hard-pressed to find a justification for such a quantization though. Unless we're to believe that the 4D planet exhibits macroscopic wave-like properties, in which case there might be practical difficulties in living on it.
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### Re: 4D Orbits

Stable elliptical orbits are only stable in 3d. In any other dimension, one can only have stable circular orbits. Anything else is going to sling-shot your planet into the sun or deep space, long before life forms thereon. That's why you need something else to stabalise a radiant inverse-biquadratic. In Bohr's atom, one might point to the quantum nature of planck's constant controlling action. Bohr does not specifically specify waves, but the wave model fits neatly.

You can of course, use something like a radiant repelling force, where the carriers decay. This can be used for to create an inverse-force of a higher dimension, which would make the orbits gravitate to a specific distance from the sun (varyingly for different suns and planets), based on some secondary quantity other than mass. There was some talk about a similar one based on the bayonic number in 3D. Planets of different compositions might give a different bayonic mass, which when coupled with gravitational mass, would drive the ones with more bayons further out.
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wendy
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### Re: 4D Orbits

wendy wrote:Stable elliptical orbits are only stable in 3d. In any other dimension, one can only have stable circular orbits. Anything else is going to sling-shot your planet into the sun or deep space, long before life forms thereon. That's why you need something else to stabalise a radiant inverse-biquadratic. In Bohr's atom, one might point to the quantum nature of planck's constant controlling action. Bohr does not specifically specify waves, but the wave model fits neatly.

You can of course, use something like a radiant repelling force, where the carriers decay. This can be used for to create an inverse-force of a higher dimension, which would make the orbits gravitate to a specific distance from the sun (varyingly for different suns and planets), based on some secondary quantity other than mass. There was some talk about a similar one based on the bayonic number in 3D. Planets of different compositions might give a different bayonic mass, which when coupled with gravitational mass, would drive the ones with more bayons further out.

Interesting idea.

Introducing a repelling force has other consequences. For example, an orbit need not be circular, but sinusoidal -- imagine if initially, the planet is moving towards the sun from a distance where gravity overwhelms the bayonic force, then when it reaches the point where the two forces balance out, it has already acquired momentum from the initial acceleration. As a result, it will move past the stable point, with the consequence that the bayonic force now kicks in and pushes it outward. So the planet will bounce back, but by the time it reaches the balance point, it again has acquired momentum from the bayonic force, so it will fly out past the balance point again. At which point the force of gravity starts pulling it back in. So the resulting motion will be harmonic, and the planet will have a sinuisoidal orbit.

A sinusoidal orbit will cause the planet to not only have seasons and climates caused by its double-rotation and its tilt wrt to the sun, but the sinusoidal variation of orbital radius will add an additional periodic warming/cooling cycle to the overall temperature on the planet's surface.

The bayonic force itself will have other consequences, such as matter of higher bayonic mass being driven to the surface of the planet, and so the interior of the planet will be stratified by bayonic mass. Unless, of course, one postulates that repulsive bayonic force only emanates from the sun (maybe as a result of thermonuclear reactions in the sun, say). Then planets will be solid but there will be no moons.
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### Re: 4D Orbits

Ok, I see what you're getting at there, and as quickfur stated before, it seems like it would work.
Higher_Order
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### Re: 4D Orbits

quickfur wrote:[...] A sinusoidal orbit will cause the planet to not only have seasons and climates caused by its double-rotation and its tilt wrt to the sun, but the sinusoidal variation of orbital radius will add an additional periodic warming/cooling cycle to the overall temperature on the planet's surface.
[...]

Also, from what I can tell (correct me if I'm wrong), this sinusoidal motion would be undampened, or only slightly dampened, so most planets are likely to have orbits with significant sinuidoidal amplitude -- their orbital radius will vary by a large amount. So most planets will likely be unsuitable for life, as the change in orbital radius also means a drastic change in surface temperatures.

Another consequence of a bayonic force is that planets need not orbit at all; they could just "float" around the ideal radius where the bayonic force balances gravity, so there can be stationary planets. There can also be planets that have vertical harmonic motion but zero lateral motion, like a bobbing weight on a spring.
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### Re: 4D Orbits

In the book, The Planiverse, 2D orbits that look like flower petals are described using inverse-distance, or 1/r instead of 1/r squared, gravity flux. Also, we do not know what would be the speed of light in a 2D (or 4D) universe and and if it were much lower than the speed of light in ours, then General Relativistic effects that curve space near a mass may become significant enough to include more stable orbits.
damian666
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### Re: 4D Orbits

Higher_Order wrote:Ok, I'm a bit new to the wonderful world of the 4th dimension. I understand why, and have suspected that in this new dimension energy would decrease as the cube of the distance, rather than the square. So it was no surprise that this was true, however what was a surprise was that this makes all but a perfectly circular orbit impossible, can somebody please explain to me why this happens?

Magnets tend to attract with an inverse cube law. If you attempt to get one magnet to orbit another, either they are sucked together very quickly or there is not much effect.

An inverse cube law seems like the best bet, but forces do whatever they like. Like the strong force, which gets stronger with increased distance. Question: does an inverse squared law in 4D lead to an impossible conclusion, or is it just heuristically favored?
PatrickPowers
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### Re: 4D Orbits

PatrickPowers wrote:
Higher_Order wrote:Ok, I'm a bit new to the wonderful world of the 4th dimension. I understand why, and have suspected that in this new dimension energy would decrease as the cube of the distance, rather than the square. So it was no surprise that this was true, however what was a surprise was that this makes all but a perfectly circular orbit impossible, can somebody please explain to me why this happens?

Magnets tend to attract with an inverse cube law. If you attempt to get one magnet to orbit another, either they are sucked together very quickly or there is not much effect.

Mathematically speaking, the difference lies in how the behaviour of the force as distance varies balances the momentum of the orbiting object. Any stable orbit is a delicate interplay between momentum and gravity. Since momentum varies with the square of the velocity, when gravity also obeys an inverse square law this allows certain subsets of trajectories to have momentum and force balance each other out in an equillibrium.

When gravity obeys an inverse cube law, however, there will almost always be an extra linear term in the total force experienced by the object, giving rise to a number of trajectories (in the best cases) that all have the tendency to either lose or gain average orbital distance per cycle, except for the exceptional case of the perfect circle, which in real life almost never happens because the slightest perturbation would nudge it into one of the unstable paths that eventually diverge from the star or collide into it.

An inverse cube law seems like the best bet, but forces do whatever they like. Like the strong force, which gets stronger with increased distance. Question: does an inverse squared law in 4D lead to an impossible conclusion, or is it just heuristically favored?

The reason we tend to think of a 4D universe as having an inverse cube law for gravity is because of the flux conjecture: think of a force as being carried by "force carriers" emitted by the origin of the force. We assume that the strength of the force would be proportional to the density of force carriers at any point in space. All things else being equal, one would assume that there would be a constant number of force carriers emanating from the origin with constant density in all directions. Which means the farther away from the origin you are, the more spread out the force carriers will be. So the density of the force would be inversely proportional to the surface of an n-dimensional sphere of radius r, where r is the distance to the origin. In 3D space, then, this gives rise to the inverse square law. In 4D space, this gives rise to the inverse cube law.

Assuming an inverse square law for gravity in 4D by fiat won't, AFAIK, lead to any contradictions, but it would seem rather arbitrary and hard to explain.
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### Re: 4D Orbits

quickfur wrote:Since momentum varies with the square of the velocity, when gravity also obeys an inverse square law this allows certain subsets of trajectories to have momentum and force balance each other out in an equillibrium.

Oh my goodness, thank you. This statement is great and really helped me. (But, you meant angular momentum, right? I had originally thought you mistyped and meant kinetic energy, which is what led me to a further insight that I will talk about in a couple of paragraphs, but then I remembered that the units work with angular momentum, too.) EDIT << Energy and torque have the same units, whereas angular momentum and action have the same units, which mr_e_man was awesome enough to notice and point out to me. This messes up my already vague "explanation" down below, but it was only ever a vague "explanation" in the first place, and I am still confident that the underlying idea still applies. Angular momentum becomes upgraded in higher dimensions, balancing the increasingly steeper gravities, and stable orbits are possible in all dimensions, not just 3D. Also, I had only just thought of this idea when I made this post, and the details have since changed. I made a topic about it which can be found here: Stable Orbits in Odd Dimensions Only. Anyways, keep all of that in mind when reading the rest, hah, and thanks again, mr_e_man! >>

3D being special finally seems to make some sort of intuitive sense to me; I like the maths, but I like to internalize an "instinct"--something which I refuse to believe is impossible. That being said and at the risk of being far too simplisitic, vague, and hasty, I would like to share what I think I have grasped, including a possible way to generalize stable orbital systems, but only to the ODD dimensions.

First, here is my interpretation of the reasoning why "simple" orbits are only possible in 3D. Elliptical orbits inherit their nicely stable trade between distance and velocity from the exactly perfect balance between (A) the attractive force of gravity being inversely proportional to the square of the distance, because the force is reaching through 2D "slices" of 3D space, and (B) the "escapist" kinetic energy (or angular momentum) of the orbiting body being proportional to the square of its velocity, because... well... Wait.

---

Why is energy proportional to the square of the velocity? After some googling, I found a 13 year old comment on a physicsforums post that was looking for the same sort of gut feeling "understanding" that I want. They provide a purely mathematical basis for how the velocity exponent can be 2 at all (although they mistyped the equations in a few places). First, they start with the assumptions that (1) linear momentum is conserved, which to me simply means that acceleration can never happen without a cause, and that (2) the "amount of cause" put into creating some new velocity is the same from all perspectives. Next, they raise the example of some massive object splitting into two pieces with different masses that move in opposite directions with the goal of finding a valid way to relate the magnitidue of the splitting event (i.e. "amount of cause" or energy) with the velocities of both pieces, which they assume will involve both velocities raised to some value. They continue by working out how some third party observer with another velocity would calculate the magnitude of the splitting event, which allows them to equate the two results and to "solve" for the unknown exponent for the velocities. However, while they "simplify" by using the binomial theorem, I simply kept the equality and plugged in 0, 1, 2, 3, and 4 to see what works. Trivially, 0 and 1 both work, and, as expected, 2 also works, but then 3 and 4 do not work, with 4 expanding on the way 3 fails. I have not checked any further, but I am confident that the pattern of failure holds and also that negative and noninteger values do not work.

Now, how is this supposed to translate into instinct or whatever? "The velocity is squared because that is the only exponent that nontrivially satisfies assumption #2 given assumption #1" seems a little bootstrappy and unsatisfying. Maybe, but I am starting to think that things might be backward here. Energy scaling with the square of the velocity is a consequence of the assumptions that a change to velocity requires a cause and that such causes do not look different from different perspectives. In other words, these assumptions about how velocity works have a consequence regarding the square of the velocity, and it is convenient to refer to that consequence as energy. This may not be satisfying to anyone else, but it seems to click with me. Perhaps more insight can be gained if we look at one of the fragments from the example and picture splitting it to try to get even more velocity in the same direction, and then again for the next fragment, and so on. This will necessarily produce diminishing returns, but that this process results in precisely a square law is not exactly apparent to me.

---

Anyways, I was also considering that thinking in terms of angular momentum may be more helpful. It has the same units as energy after all, and I interpret it representing the amount of energy stored in the orbit. And then I remembered that in 4D there are 2 ways to simultaneously rotate, which made me think about orbitting 2 ways simultaneously. If an object is going around something using the XY plane, it still has the option to go around it using the ZW plane. Then it hit me.

It should have 2 orthogonal angular momenta! In fact, it makes sense to describe it as having 1 "double-angular" momentum that scales with the 4th power of the velocity, given that "single-angular" momentum scales with square of the velocity. This means that 4D still is not sufficient to achieve a stable orbit because gravity still scales with the inverse 3rd power of the distance, but 5D works!

In 5D, a double-angular momentum that scales with the 4th power of the velocity should perfectly balance a gravitational attraction that scales with the inverse 4th power of the distance, and stable orbits would exist as double-elliptical paths. The closest I can come to visualizing this in 3D is the spiral path around the surface of a donut whose defining circles are actually ellipses.

In 7D, a triple-angular momentum that scales with the 6th power of the velocity should perfectly balance a gravitational attraction that scales with the inverse 6th power of the distance, and stable orbits would exist as triple-elliptical paths. I definitely cannot visualize that.

In 9D, a quadruple-angular momentum that scales with the 8th power of the velocity should perfectly balance a gravitational attraction that scales with the inverse 8th power of the distance, and stable orbits would exist as quadruple-elliptical paths.

And so on!

---

I know I made a mathematical jump with the square to 4th power thing, but I am pretty confident about the orthogonal angular momenta thing. The doubly incremented exponent just seems obvious from there, but, I admit, seeming is clearly not the same as mathematically rigorous proofing.

What is really fascinating to me is that this should translate to kinetic energy in general, which would mean that doubling velocity in 5D does not just quadruple the kinetic energy; it increases it by a factor of 16. In 7D it would increase by a factor of 64, and in 9D it would increase by a factor of 256. Holy crap.

What are your thoughts? I would love for someone to say "omg, you're right! woowow!" but that's probably a bit much, haha. I'll settle for why exactly I'm wrong. ): But my gut says I'm right. (:
Last edited by d023n on Mon Oct 01, 2018 11:29 pm, edited 1 time in total.

d023n
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### Re: 4D Orbits

How exactly does an object have "double" angular momentum? I.e., for it to work the way I think you're trying to describe, that means the object must be travelling (orbiting, etc.) in two different directions simultaneously. I have a hard time grasping what the actual path of the object would look like for it to satisfy such a strange constraint.
quickfur
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### Re: 4D Orbits

quickfur wrote:How exactly does an object have "double" angular momentum? I.e., for it to work the way I think you're trying to describe, that means the object must be travelling (orbiting, etc.) in two different directions simultaneously. I have a hard time grasping what the actual path of the object would look like for it to satisfy such a strange constraint.

A 5D object would have double-angular momentum sort of like how a solitary object in 3D can have spin angular momentum in some plane (i.e. it is spinning around its center of mass) while simulataneously having a kinetic energy in some linear direction. In a certain sense, points within this 3D object do travel in 2 different directions simultaneously (i.e. circularly in the plane of rotation and linearly), but they do not actually split into separate points taking distinct paths. Each point travels in a single path that is the combination of both motions.

Similarly, a solitary 5D object could have spin angular momentum in one plane, a separate spin angular momentum in an orthogonal plane, and a linear kinetic energy in any direction. Again, each point within the 5D object can be thought of as travelling in 3 different directions simultaneously (i.e. circularly in one plane of rotation, circularly in the other plane of rotation, and linearly), but they do not actually split into 3 separate paths. Each point of the 5D object travels a single path that is the combination of all 3 motions. For example, if the center of mass of some already moving 5D object starts at the origin in a coordinate system whose coordinates look like (X,Y,Z,W,V) and moves linearly along the X-axis 1 unit while spinning pi radians in the YZ-plane and pi/2 radians in the WV-plane, if the object were sufficiently large, there could be a point that moves from coordinate (0,1,0,0,1) to (1,-1,0,-1,0). The point will have only traced one path that is longer than the simple sum of pi+pi/2+1 and would look like whatever a path on the surface of a translated 2-torus looks like, which I also have a hard time grasping, haha.

A 5D planet around a 5D star should behave similarly, although we can ignore the linear motion of the system along the leftover 5th axis, and it would be more appropriate to refer to it having 2 orbital angular momenta. This would then mean that the orbital path, when projected down into either plane of rotation should look elliptical instead of perfectly circular, meaning that the actual single orbital path should move along the imaginary surface of a 2-torus whose defining circles are actually those 2 ellipses.

I can imagine a situation where the projection of the orbital path onto one of the planes of rotation is nearly a perfect circle though, while the projection onto the other plane of rotation is extremely eccentric. While the component of the orbital velocity in the plane of rotation with the circular projection would remain almost constant, the component of the orbital velocity in the other orbital plane would change dramatically, meaning that the actual unified orbital velocity of the planet would change dramatically, too. Returning to the imperfect visual I mentioned in my previous post of the donut in 3D, this would translate to one of the donut's defining circles (let's say, the larger one) being nicely circular while the other is an exaggerated ellipse (so, the tube of the donut), meaning that it might look like a flattened donut or a tall thin one, depending on how the true orbital shape is projected into 3D. Meanwhile, the relative sizes of the projections of the orbital path into each of the planes of rotation would determine how the orbital path spiralled around the donut's surface, although the asymmetrical projection into 3D would also distort the actual path and so also the orbital velocity around the 3D donut's surface. Worst of all, there would be no good way to represent where the orbitted star should sit. I... don't even know where to try to start.

Anyways, I hope that cleared things up. Thank you for taking the time to read my longwinded explanations. ^_^

d023n
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### Re: 4D Orbits

A four dimensional planet in a circular orbit, spinning on a clifford rotation (both modes of rotation equal), is still perfectly capable of having seasons.

The main cause of seasons on our earth is tilt: the earth's rotation is at an angle to the solar orbit, and thus when the northern hemisphere is closer, the season there is summer, while the southern hemisphere is in winter. The orbit, and the relation of the perihelion to the solstices contribute a lot less. If the three align, then this is enough to trip an ice age, but at the moment, perihelion is in february, and the poles are pointing around december/janruary.

In four dimensions, tilt is still possible, and you get a full range of seasons, rather than two opposites.

Let's do some calculations. A planet in double-rotation, will by the condition of equipartition of energy, tend to have an equal rotation on both axies. This would make every point rotate around the centre without tension. This means that the fixed stars rise on the same part of the sky, and set at the same part. The whole sky is fibulated by great circles, and each star follows its own fiber.

The vault of the sky is a sphere, or half-glome. The stars rise in the east half of the sky, and set in the west. The middle equator between east and west marks the point where the stars cumulate, or come to their highest point in the sky. A star that rises at 30, will be rotated by 90 degrees as it rises, and it will then cumulate at 30 degrees from the horizon. Then it will continue to be rotated in the sky, and set at 210 deg, which is directly opposite.

We can now draw a 'sphere of cumulation'. This sphere has the obserber at the 'south pole', and the zenith at the north pole. A ray drawn from the point of cumulation of the star will strike this sphere as if the north pole is 90 (ie directly overhead), and the south pole is 0 (horizon-hugging stars). The star that strikes at 30 will make an angle 1/3 of the way up the sphere, ie 60 degrees from the south pole (horizon).

Because every point on this sphere represents a great circle in the sky, and everyone has the same great circles, this sphere will be the same for everyone, except where the horizon and zenith points will vary. It is, as we shall see, the shape of lattitude on this sphere.

Now the sun moves against the fixed stars. This means that the ray from the earth to the sun points at various points of the sky (the zodiac), as the year happens. If we suppose the earth has a tilt to the sun, then it means that the sun moves through a circle of points on the lattitude space, rather than a fixed point. If we suppose further a tilt of 23.5 degrees, as in the earth, then we would get various seasons.

You have a sphere representing lattitude. A circle of diameter 94 degrees is drawn on the surface, and this is divided to the twelve months. This sphere does not itself move, but the sun crawls around this circle. So when it is in August, the sun is on the 'august' mark. Now we observe that every point on the sphere is a great circle on the earth and in the sky. One of these points becomes the horizon, the diametric opposite is the zenith. It doesn't matter how the ball stands, you will see that the highest point in the great circle is mid-summer and the lowest is mid-winter. Seasons.

If we now draw a matching circle 94 degrees from the opposite end of the sphere, we get the artic torus. The first circle is the tropic torus. The 'poles' are points of equal climate, ie no seasons. So the great circle near the tropic torus is the summer-equator, and its orthogonal is the winter-equator.

The climates are given by the year-integration, and so one takes the summer-circle as the equator on the earth, and the winter-equator as the polar points. It is pretty much if one imagined the world as it is, and rotated it in four dimensions, keeping the equator constant. The two poles would unite, and the climate gets colder as you move away from the sun.

So while our earth is like a slice of this circle (with a cold and warm half), this is the full thing. You have not just time zones, but season-zones, and autumn-loving animals could migrate with the autumn.

A greater portion of the world would be temperate. The actual fractions might be had by the versine of the angle (from 180 to 0). That is $$\mbox{versine} \theta = 1 - \sin 2 \theta$$. The distance the polar axis is split into is what we're seeking. A greater portion of the 4d world is temperate.
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wendy
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### Re: 4D Orbits

A number of people do study the higher dimensional physics. It is even possible to deal with additional dimensions of 'extent' and 'depth', of dimensions L^{n-1} and L.

The Moment of X is a product of the coordinate of X over the elements of X, ie $$\xi X = \int r dX$$, where r is the coordinate of dX. Generally, you can see that moment changes if you shift the coordinates, by a measure zX, where z is the shift in zero and X the total amount of the quantity. If the quantity has a zero sum, ie $$\int dX = 0$$, then the moment can be a vector. Things like boats float and do not roll around in the water, because the centre of mass is directly under the centre of the displaced water. This creates +m at 0, and -m at -h, gives a total moment of +mh. The trick is to increase the distance of h, so that the moment is bigger. The dimensions are XL

Momentum is the rate of change of moment. That is, the moment at t1 less the moment at t0, divided by the time interval t1-t0. Momentum is not subject to different coordinates, ie d(z)/dt = 0. It is affected by an inertial frame of reference, ie where Z = z0 + v.t. The dimensions are XL/T

Force is the rate of change of momentum. The sizes of the forces are not affected by inertial frames of reference. The dimensions are XL/T²

Energy is the measure of force against distance. The measure of force by time is impulse. The dimensions of this energy is XL²/T²

So the dimensions of energy are not a function of solid space, but two counts of time and two counts of displacement (travelled length).
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### Re: 4D Orbits

The rotation of an orbiting body has little (or no) impact on its global path around the star being orbited. You can't stabilize an unstable orbital path by making the planet rotate differently. You can't simply combine the two angular momenta willy-nilly, i.e., the angular momentum around the star is not equivalent to the angular momentum of the planet rotating around its axis / in its rotational plane. The two do not affect each other, unless the planet breaks up into orbiting particles or orbiting dust accumulates into a spinning planet, in which case you could talk about the total angular momentum of the star system. I don't see any way of obtaining a stable 4D orbit in this way without in some way changing the inverse cube law.
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### Re: 4D Orbits

quickfur wrote:
PatrickPowers wrote:
Higher_Order wrote:Ok, I'm a bit new to the wonderful world of the 4th dimension. I understand why, and have suspected that in this new dimension energy would decrease as the cube of the distance, rather than the square. So it was no surprise that this was true, however what was a surprise was that this makes all but a perfectly circular orbit impossible, can somebody please explain to me why this happens?

Magnets tend to attract with an inverse cube law. If you attempt to get one magnet to orbit another, either they are sucked together very quickly or there is not much effect.

Mathematically speaking, the difference lies in how the behaviour of the force as distance varies balances the momentum of the orbiting object. Any stable orbit is a delicate interplay between momentum and gravity. Since momentum varies with the square of the velocity, when gravity also obeys an inverse square law this allows certain subsets of trajectories to have momentum and force balance each other out in an equillibrium.

When gravity obeys an inverse cube law, however, there will almost always be an extra linear term in the total force experienced by the object, giving rise to a number of trajectories (in the best cases) that all have the tendency to either lose or gain average orbital distance per cycle, except for the exceptional case of the perfect circle, which in real life almost never happens because the slightest perturbation would nudge it into one of the unstable paths that eventually diverge from the star or collide into it.

An inverse cube law seems like the best bet, but forces do whatever they like. Like the strong force, which gets stronger with increased distance. Question: does an inverse squared law in 4D lead to an impossible conclusion, or is it just heuristically favored?

The reason we tend to think of a 4D universe as having an inverse cube law for gravity is because of the flux conjecture: think of a force as being carried by "force carriers" emitted by the origin of the force. We assume that the strength of the force would be proportional to the density of force carriers at any point in space. All things else being equal, one would assume that there would be a constant number of force carriers emanating from the origin with constant density in all directions. Which means the farther away from the origin you are, the more spread out the force carriers will be. So the density of the force would be inversely proportional to the surface of an n-dimensional sphere of radius r, where r is the distance to the origin. In 3D space, then, this gives rise to the inverse square law. In 4D space, this gives rise to the inverse cube law.

Assuming an inverse square law for gravity in 4D by fiat won't, AFAIK, lead to any contradictions, but it would seem rather arbitrary and hard to explain.

Yes I understand this. I am pointing out that two of the basic forces -- the weak force and the strong force -- don't obey the flux conjecture. That means only 50% of the basic forces obey it. So it would not be a big surprise if 4D gravity didn't obey it either. It's also worth mentioning that searches for a force carrier for gravity have been so far fruitless.

What I was looking for was "if you had the inverse square gravity law in 4D then you could build a perpetual motion machine." Or something like that, where it leads to an absurdity.
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### Re: 4D Orbits

quickfur wrote:The rotation of an orbiting body has little (or no) impact on its global path around the star being orbited. You can't stabilize an unstable orbital path by making the planet rotate differently. You can't simply combine the two angular momenta willy-nilly, i.e., the angular momentum around the star is not equivalent to the angular momentum of the planet rotating around its axis / in its rotational plane. The two do not affect each other, unless the planet breaks up into orbiting particles or orbiting dust accumulates into a spinning planet, in which case you could talk about the total angular momentum of the star system.

I wasn't suggesting that the spin angular momentum of the orbitting 5D planet is needed to stabilize its orbital angular momentum. While it would be double-rotating around a line passing through its own center of mass, and so have 2 spin angular momenta components, the system of planet and star together would itself also be double-rotating around its center of mass with the planet behaving as a single point that is double-orbitting with 2 orbital angular momenta components. These would be combined into a single 4D angular momentum vector that still defines a single curved 1D path, and the total kinetic energy of the planet relative to the solar system's center of mass would scale with the 4th power of its orbital velocity.

quickfur wrote:I don't see any way of obtaining a stable 4D orbit in this way without in some way changing the inverse cube law.

I agree, which is why 5D is needed because gravity would obey an inverse 4th power law.

In general, odd dimensions are needed because they have inverse even power laws that can perfectly balance the addition of extra rotations (and so angular momenta) every 2nd dimension.

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### Re: 4D Orbits

d023n wrote:the system of planet and star together would itself also be double-rotating around its center of mass with the planet behaving as a single point that is double-orbitting with 2 orbital angular momenta components. These would be combined into a single 4D angular momentum vector that still defines a single curved 1D path, and the total kinetic energy of the planet relative to the solar system's center of mass would scale with the 4th power of its orbital velocity.

.

That is incorrect. What you aren't getting is that the 2 planes of rotation are for rigid rotations. An orbit is not at all a rigid rotation.

It might be possible to have two planes of rotation of a solar system. That is, the Sun is in the center while each planet is in either one of two planes of rotation. But it is not obvious whether or not this would be stable, and it might be a big project to answer that question. (It's not obvious that our own Solar System is stable. In the 19th century there was a huge project to demonstrate this.)

It might be possible that multiple body systems could have true 4D rotations. But unless I've gone completely wrong, two-body systems are always confined to a 2D plane. It's because the acceleration vector is always pointing at the center of mass, so the orbit is confined to a 2D plane defined by that acceleration vector and the velocity vector of the planet. The center of mass lies in that plane.

On the other hand I have never tried to really prove this, so there might be some way around it. The center of mass and the Sun aren't the same, maybe that makes a difference.
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### Re: 4D Orbits

PatrickPowers wrote:It's because the acceleration vector is always pointing at the center of mass, so the orbit is confined to a 2D plane defined by that acceleration vector and the velocity vector of the planet.

Wow. That.. is a glaringly obvious point. How did I not see that? What's worse is that I can't see any way around it. Dang.

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### Re: 4D Orbits

The way to figure out what happens with gravity in 4D would be to extend general relativity to 4D. If there is an obvious way to do this, that is... I don't understand tensors and am weak in differential equations, so I can't offer much help there.
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### Re: 4D Orbits

d023n wrote:
PatrickPowers wrote:It's because the acceleration vector is always pointing at the center of mass, so the orbit is confined to a 2D plane defined by that acceleration vector and the velocity vector of the planet.

Wow. That.. is a glaringly obvious point. How did I not see that? What's worse is that I can't see any way around it. Dang.

Hold on a moment.

In 5D, gravity scales with the inverse 4th power of the distance. I was trying to balance this by suggesting that kinetic energy (and angular momentum) scaled with the 4th power of the velocity, and my motivation for doing so was that the orbital path was somehow governed by 2 planes of rotation, and so 2 distinct sources of motion that together contributed to a single path through a now 3D region surrounding the center of mass.

But if the acceleration vector is always pointing at the center of mass, the velocity vector can do nothing but chase it around in a 2D plane, restricting the orbital path to that plane.

The end, right?

Well, what if the acceleration vector is not the vector that is always pointing at the center of mass? What if, instead, there is a jerk vector that is always pointing at the center of mass? The acceleration vector would chase the jerk vector, the velocity vector would chase the acceleration vector, and the actual orbital path would occur within a 3D region defined by these 3 vectors.

Of course, the obvious question is why. Why would the jerk vector always point at the center of mass instead of the acceleration vector? Well, why does the acceleration vector always point at the center of mass in 3D? I think that the degree of rotation allowed by the dimensionality of the spherical symmetry is the cause here, rather than the effect. In 2D and 3D, the 1-spherical (2-ball surface) and 2-spherical (3-ball surface) symmetries are only capable of single rotation, meaning that velocities can only change 1 way. However, in 4D and 5D, the 3-spherical (4-ball surface) and 4-spherical (5-ball surface) symmetries are capable of double rotations, meaning that velocities can change in 2 orthogonal, independent ways.

In other words, the vectors do not determine the degree of rotation. The degree of rotation determines the vectors.

This would mean that in 6D and 7D the jerk vector no longer points at the center of mass. Instead, because of a third plane of rotation, there would be a jounce vector always pointing at the center of mass (although I prefer to refer to it as snap). The jerk vector would chase the snap vector, the acceleration vector would chase the jerk vector, the velocity vector would chase the acceleration vector, and the actual orbital path would occur within a 4D region defined by these 4 vectors.

In 8D and 9D, the crackle vector would always point at the center of mass and be chased by the snap, jerk, acceleration, and velocity vectors; and the actual orbital path would occur within a 5D region defined by these 5 vectors.

In 10D and 11D, the pop vector would always point at the center of mass and be chased by the crackle, snap, jerk, acceleration, and velocity vectors; and the actual orbital path would occur within a 6D region defined by these 6 vectors.

And so on.

Finally, only the odd spaces would achieve a proper balance between kinetic energy (angular momentum) and gravitational attraction.

---

Now then, I am back to wondering if this would work out. Any thoughts? ^_^

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### Re: 4D Orbits

" Well, why does the acceleration vector always point at the center of mass in 3D? "

That's a good question. We know that information travels at the speed of light. So why does Newton's law of gravity works so well? It takes eight minutes for light to get here from the Sun. Shouldn't the Earth's gravitational acceleration vector point at the place the center of mass was eight minutes ago? But that doesn't work. The vector points almost exactly at the place the Sun is now.

I don't quite understand why, but general relatively works out to be almost exactly the same as Newton's law. Just think how disappointed Mr. Einstein would have been if the result were exactly the same. There would have been no way to test his theory.
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### Re: 4D Orbits

PatrickPowers wrote:" Well, why does the acceleration vector always point at the center of mass in 3D? "

That's a good question. We know that information travels at the speed of light. So why does Newton's law of gravity works so well? It takes eight minutes for light to get here from the Sun. Shouldn't the Earth's gravitational acceleration vector point at the place the center of mass was eight minutes ago? But that doesn't work. The vector points almost exactly at the place the Sun is now.

In the example of our planet and sun, I think that the spatial curvature induced by the mass of the sun also captures the velocity of the sun. To get the acceleration vector induced by the sun on the Earth to no longer point at the common center of mass, some force would need to be exterted on the sun to change its velocity, and it would then take the eight or so minutes for the vector to be updated. Interestingly, the velocity of the sun is changing as it orbits the center of mass of the Milky Way, meaning that the acceleration induced on the Earth by the sun is never directed perfectly at the sun-Earth center of mass.

---

However, my question was not meant to address this detail. However, I am glad you brought it up. But first, my original intent was a bit more abstract. In 3D, why do acceleration vectors correlate with gravitational sources at all? In other words, why does the 2-spherically symmetric curvature of 3D space toward a central point cause objects in the surrounding space to change their velocities toward that point? Well, what does space being curved in a 2-spherically symmetric way toward a central point really mean? One way to think about it is that, given some time interval, every point in the 3D space is displaced some amount, according to its original distance relative to the central point, toward that central point. (For points already closer to the central point than the distance they are displaced, well, being able to visualize a 3D rubber "sheet" that is poked orthogonally along a 4th axis would be helpful for that.) This should hopefully convey an image of space flowing toward the central point, where closer means faster.

Now, what happens if we also rotate the space around this central point? In 3D, there is only 1 way to do this at a time, which has some interesting effects on points not in the plane of rotation containing the central point, but something more important happens for the points in that specific plane. Without rotation, their paths are straight to the central point, and their flow rates are determined by the inverse square law. With rotation but without inward flow, their paths are perfect circles, and projecting these circles onto the line that is perpendicular to the non-rotated path (and in the plane of rotation) shows motion that also appears to obey the inverse square law. This means that rotating pi/2 radians in the same amount of time that the point would have taken to fall into the central point leads to a perfectly circular path despite the inward flow. There is more to say here, but I am straying from my original intent. The point is that, in 3D, this inward flow can only be altered by a simple rotation. This restricts stable orbits to a 2D region (and leads to the elliptical orbital path because of the inverse square thing).

In 4D and 5D, on the other hand, the inward flow can be altered by a double rotation. However, just as the rotation in 3D promoted the originally straight paths of the points to stable elliptical paths in a 2D region, motion in a second plane of rotation further promotes those now unstable elliptical paths into stable "doubly-elliptical" paths in a 3D region. The changing motion of one plane of rotation changes the changing motion of the other plane of rotation, which is to say that the independent acceleration vectors push each other away from the central point, leaving a jerk vector that is still always pointing at the central point. I say "still" here because there is already a jerk vector in 3D that always points at the center of mass; moving in an elliptical orbit alters the orbital distance which changes the gravitational acceleration, which is a jerk.

This also brings me back to the detail about the acceleration vector of the Earth not lagging the Earth-sun center of mass by eight minutes, except that it does lag a tiny amount because the Earth-sun system is changing its velocity around the Milky Way. Our acceleration around the sun is itself being accelerated. The difference between this and orbits in 5D is that the center of mass of the Earth-sun system is not the same point as the center of mass of the Earth-sun-galaxy system, while in 5D the source of misalignment is double rotation around the same point. Motions in 5D, however, would still be affected in an analogous way, with jerk vectors lagging their center of mass because of external influences and a finite information propogation rate.

---

Wow, this is long. I need to wrap up and generalize everything.

The ability to access N planes of rotation in space of dimension 2N means that a spherically symmetric curvature toward a central point causes objects in the surrounding space to change their Nth time derivative of position toward that point. However, the presence of an additional axis in space of dimension 2N+1 means that a spherically symmetric curvature toward a central point is proportional to the inverse 2Nth power of the distance from that point, which balances the induced change of the Nth time derivative of position toward that point.

In essence, additional planes of rotation are "peeling" the time derivatives of position away from the center of mass, revealing higher ones. In 1D, gravity is just a velocity because no orbits exist to balance how space curves. In 2D, gravity is an unstable acceleration because simple orbits cannot balance how space curves. In 3D, gravity is a stable acceleration because simple orbits do balance how space curves. In 4D, gravity is an unstable jerk because "double-orbits" cannot balance how space curves. In 5D, gravity is a stable jerk because "double-orbits" do balance how space curves. In 6D, 8D, and 10D, gravity is an unstable snap, crackle, and pop, respectively, because "triple-, quadruple-, and pentuple-orbits" cannot balance how space curves, respectively again. In 7D, 9D, and 11D, gravity is a stable snap, crackle, and pop, respectively, because "triple-, quadruple-, and pentuple-orbits" do balance how space curves, respectively again. Et cetera.

Once again, I appreciate the time and effort anyone puts into reading and responding to this. ^_^ Validation would be great, but precise details about how this is all wrong is welcome as well.

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### Re: 4D Orbits

You can only reduce a mass to a point, without changing the field, in one and three dimensions. In other dimensions, the centre of gravitation and the centre of mass are separate points.

As to the angular momentum of several modes of rotation, these are separate modes, having the dimension of ML²/T, as they do in three dimensions. If one supposes Bohr's model, then the energy of orbit is quantised at multiples of h/L, where L is the length of the orbit.

In four and higher dimensions, the vector of rotation is different to just the orthogonal. In four dimensions, there is a 2-space orthogonal to the vector

Flattening of the sphere is at the rate of w²r/g, where w is the angular speed of rotation, and r the radius.
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### Re: 4D Orbits

wendy wrote:If one supposes Bohr's model, then the energy of orbit is quantised at multiples of h/L, where L is the length of the orbit.

In four and higher dimensions, the vector of rotation is different to just the orthogonal. In four dimensions, there is a 2-space orthogonal to the vector

Flattening of the sphere is at the rate of w²r/g, where w is the angular speed of rotation, and r the radius.

It sounded to me like you were saying what I have been saying, I think, until I got to this stuff. What is Bohr's model in this context and why would we suppose it? I saw that you mentioned it in a previous comment, but I still do not understand what the justification for it is. Quantum orbits seems a bit ad hoc, especially when it looks like stable paths within "orbital regions" would arise in odd dimensions simply because of n-spherical symmetries (a 3D region in 5D, a 4D region in 7D, an 5D region in 9D, and so on).

Next, what do you mean by vector of rotation and by there being a 2-space orthogonal to it in 4D?

Lastly, why is the sphere flattening?

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