Hypergeometry of a Clifford Planet.

[PatrickPowers]

So we are natives of a Clifford planet. How would our geometry evolve?

At first it would be local, just like here on Earth. There was no Earth-wide definition of anything. East and West differed in various localities. There were no time zones until railroads came. There was no north star, so north and south wasn't that obvious. Maybe only civilized people like the Egyptians knew about it.

On a Clifford planet what really matters is phase. That is, where the great circle of rotation [circle] on which you reside is relative to the Sun on its path around the analemma. That is, if the Sun is directly above your circle things are hot, if perpendicular then very cold. This relationship is on a yearly cycle whose severity of temperature change depends on the obliquity of the ecliptic. (On a 4D planet there are two angles of obliquity instead of our one.)

Farmers would naturally see areas of similar phase as being important. The phase is the same everywhere on your circle. There is a direction perpendicular to the circle that also doesn't change the phase. The third direction is perpendicular to both, this maximizes the change in phase. Call them X,Y, and Z. If you want a change in climate, go Z-ward young man. That and their opposites -X, -Y, -Z form the six basic directions.

The strange thing though is that as one moves in the Y and/or Z direction then the path is a curved geodesic. My guess is maybe it would be shaped like the edge of a Pringles potato chip. This makes them tricky to use as a global system of navigation. I suppose the easiest answer is that once there is a demand for such then a system similar to ours arises. Maybe it supplants the old way, maybe it remains a specialized thing for astronomers and navigators, maybe they peacefully coexist.

[gonegahgah]

I don't mean to interrupt the forward movement of your thread Patrick but I wondered if you might even start a little earlier in the process.
Maybe you have thought to do that already and are getting into the challenging part.
It is interesting to consider the genesis.

This would begin with general gases in an area of space that will form the 4D solar system.
These would begin to coalesce and start forming a statistically bi-rotational area of space.
This might have an effect on both the orbital "plane" and the orbital "spin" of the planets.

[PatrickPowers]

I don't mean to interrupt the forward movement of your thread Patrick but I wondered if you might even start a little earlier in the process.
Maybe you have thought to do that already and are getting into the challenging part.
It is interesting to consider the genesis.

This would begin with general gases in an area of space that will form the 4D solar system.
These would begin to coalesce and start forming a statistically bi-rotational area of space.
This might have an effect on both the orbital "plane" and the orbital "spin" of the planets.

My game is assuming that the 4D world has the same physical laws as ours, so it will be familiar. Once you start worrying about planet formation it becomes pretty obvious that this won't work. If the 4D Earth has the same number of atoms as ours and the same law of gravity then gravity on the surface is a million times more. Angular velocity is about a million times more too. With "everyday life" these things can be ignored for the sake of building intuition and for the fun of it but once you get to planet formation it's all about these physical laws so there's no avoiding these issues. My guess is that the earth collapses into white dwarf matter then neutron star matter then a black hole.

[gonegahgah]

A game! Cool!
I'm with you. We have to ignore the 4D gravity elephant.
Sorry, I thought it was for your book.
I was just meaning that this would contribute to the initial Clifford rotation.
Also, the planets would probably orbit individually in all directions that are somewhat consistent with the Clifford rotations rather than in a single circular plane?
That could be cool?

[PatrickPowers]

A game! Cool!
I'm with you. We have to ignore the 4D gravity elephant.
Sorry, I thought it was for your book.
I was just meaning that this would contribute to the initial Clifford rotation.
Also, the planets would probably orbit individually in all directions that are somewhat consistent with the Clifford rotations rather than in a single circular plane?
That could be cool?

It could be. Half of galaxies are flat spirals while the other half are globular where the stars are going every which way. What would a 4D solar system do?

[PatrickPowers]

My guess is that the earth collapses into white dwarf matter then neutron star matter then a black hole.

Checking further, the Newtonian gravity 4D Earth compacts into a white dwarf but stops there. The density goes up by a factor of 200,000 but this only increases the surface gravity by the square root of that, which is about 400. In order to get a neutron star you need an increase factor of about 100,000. Even the most massive stars would not come anywhere near to that.

This makes one wonder how neutron stars are formed in our universe. It's a cataclysmic event. A big star collapses very suddenly. The instantaneous pressure forms the neutron star. Once formed its gravity holds it together.

It is believed that neutron stars never form by gradual accretion. If a white dwarf gains enough mass it restarts fusion. White dwarf matter doesn't expand with temperature so the pressure builds up as the temperature rises until the white dwarf explodes. This is very useful to astronomers because the energy released is always about the same, so from the apparent brightness it's possible to tell how far away the explosion was.

Recently J005311 was discovered. It is believed to have formed out of the merger of two white dwarfs. Usually this results in that explosion but this time did not. They merged and fusion started up again. Once that runs out of fuel it is predicted that J005311 will collapse into a neutron star. A Japanese film maker produced a low budget film with that title J005311 about two despairing losers who get bound up together. Presumably it has a happy ending.

[PatrickPowers]

My guess is that the earth collapses into white dwarf matter then neutron star matter then a black hole.

Checking further, the Newtonian gravity 4D Earth compacts into a white dwarf but stops there. The density goes up by a factor of 200,000 but this only increases the surface gravity by the square root of that, which is about 400. In order to get a neutron star you need an increase factor of about 100,000. Even the most massive stars would not come anywhere near to that.

This makes one wonder how neutron stars are formed in our universe. It's a cataclysmic event. A big star collapses very suddenly. The instantaneous pressure forms the neutron star. Once formed its gravity holds it together.

It is believed that neutron stars never form by gradual accretion. If a white dwarf gains enough mass it restarts fusion. White dwarf matter doesn't expand with temperature so the pressure builds up as the temperature rises until the white dwarf explodes. This is very useful to astronomers because the energy released is always about the same, so from the apparent brightness it's possible to tell how far away the explosion was.

Recently J005311 was discovered. It is believed to have formed out of the merger of two white dwarfs. Usually this results in that explosion but this time did not. They merged and fusion started up again. Once that runs out of fuel it is predicted that J005311 will collapse into a neutron star. A Japanese film maker produced a low budget film with that title J005311 about two despairing losers who get bound up together. Presumably it has a happy ending.

Hmm, looking this again it seems all wrong. I wrote about surface gravity but I don't see how that matters at all. It's the pressure at the center that is of concern, isn't it? I did a quick search to find the amount of pressure that would make matter collapse into either white dwarf matter or neutron star material and couldn't find anything. And none of the formulas and so forth can be trusted in 4D anyway. So I give up for now.

[PatrickPowers]

It turns out I was 100% totally and completely wrong. Here in 3D pressure depends on the square of the density times the square of the radius. In 4D with the off chance I've derived this correctly, with Newtonian inverse square gravity it depends on the square of the density times the cube of the radius.

In the 4D world the radius of a white dwarf is maybe 30,000 times smaller while the density is at most the same. So the pressure in the center of a 4D white dwarf is a maximum of 10^13 less than in 3D. So nothing is ever going to gradually collapse into neutron matter. Collapsing into white dwarf matter seems very far away as Neutron stars might still be created in supernovae due to intense instantaneous pressure.

In 4D something is going to have to be hypermassive indeed in order to collapse into condensed matter.

Even this is may be bogus. An expert informed me that pressure isn't what matters in these kinds of collapses, it's density, and this all has to do with quantum mechanics. I would guess though that with so little pressure there wouldn't be much density. But who knows for sure.

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With inverse cube gravity it depends on the square of the density times the square of the radius. So the pressure in the center of a 4D white dwarf is as a maximum 3*10^8 less than in 3D. That doesn't make any sense, I think using the same gravitational constant is the problem. Still very low so nothing is ever going to gradually collapse into neutron matter in this case either.
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With so little pressure at the center of HyperEarth the core won't liquify, so there will not be much of a magnetic field and compasses won't work.

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Black holes are a different thing entirely. That has nothing to do with either pressure or density : it's about surface gravity not allowing light to escape. In 4D I'd guess that there is no condensed matter but it's a lot easier to get a black hole. But as you can see, I've been wrong before.

[PatrickPowers]

It turns out I was 100% totally and completely wrong. Here in 3D pressure depends on the square of the density times the square of the radius. In 4D with the off chance I've derived this correctly, with Newtonian inverse square gravity it depends on the square of the density times the cube of the radius.

In the 4D world the radius of a white dwarf is maybe 30,000 times smaller while the density is at most the same. So the pressure in the center of a 4D white dwarf is a maximum of 10^13 LESS than in 3D. So nothing is ever going to gradually collapse into neutron matter. Collapsing into white dwarf matter also seems very far away.

In 4D something is going to have to be hypermassive indeed in order to collapse into condensed matter.

Even this is may be bogus. An expert informed me that pressure isn't what matters in these kinds of collapses, it's density, and this all has to do with quantum mechanics. I would guess though that with so little pressure there wouldn't be much of an increase in density. But who knows for sure.

Also possibly bogus is using the same gravitational constant with the same units.

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With inverse cube gravity in 4D pressure depends on the square of the density times the square of the radius and is almost the same as in 3D. So the pressure in the center of a 4D white dwarf is as a maximum 3*10^8 less than in 3D. It doesn't make any sense to me that this is much MORE pressure than we saw with the the inverse square law. Still very low so nothing is ever going to gradually collapse into neutron matter in this case either.
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With so little pressure at the center of HyperEarth the core won't liquify, so there will not be much of a magnetic field and compasses won't work.

Neutron stars are usually created via instantaneous density due to a collapse. Maybe they can exist in 4D, who knows.

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Black holes are a different thing entirely. That has nothing to do with either pressure or density : it's about gravity not allowing light to escape. Surface gravity would be much higher in 4D. So 4D I'd guess that there is less to none condensed matter but it's a lot easier to get a black hole. But as you can see, I've been wrong before.

[PatrickPowers]

Wrong again. What threw me off was that when converting 3D calculus to 4D it makes a big difference what unit is chosen for space. The only way this conversion makes any sense is if the unit is always one atom. Then measurements don't get involved at all. It doesn't even matter what size the atoms are. Doing that, I get the result that if gravity is inverse cubed then pressure in the center of HyperEarth is about the same as Earth. If gravity is inverse squared then you get a ginormous pressure that causes the planet to collapse.