A gravitational law that works in higher dimensions

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.

A gravitational law that works in higher dimensions

Postby anderscolingustafson » Sun Jan 04, 2015 7:19 pm

In our Universe the equation for gravity is (GMm)/rn where r is the radius and n is the number of dimensions minus 1. This equation produces stable orbits in 2d and 3d but in 4 or more dimensions it does not produce stable orbits or at least it would be extremely difficult to produce stable orbits using this equation in more than 4 dimensions.

I found that if the equation (GMmsin(alogbr))/rn is used for gravity with r as the number of dimensions and n as the number of dimensions minus one it produces stable orbits in higher dimensions. I'm not sure if it produces stable orbits in any number of dimensions but it produces stable orbits in 4 spatial dimensions. It produces cogwheel shaped orbits. Because this equation would mean that gravity is attractive at some distances and repulsive at others this only allows stable orbits at certain distances with orbits impossible at other distances do to the repulsiveness of gravity. One thing this has in common with the normal law of gravity is that it produces a self similarity between the small scale and the large scale.
Last edited by anderscolingustafson on Fri Jan 09, 2015 5:40 pm, edited 1 time in total.
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Re: A gravitational law that works in higher dimensions

Postby Prashantkrishnan » Fri Jan 09, 2015 5:03 pm

anderscolingustafson wrote:In our Universe the equation for gravity is (GMm)/r-n


I suppose you mean (GMm)/rn
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Re: A gravitational law that works in higher dimensions

Postby anderscolingustafson » Fri Jan 09, 2015 5:40 pm

Prashantkrishnan wrote:
anderscolingustafson wrote:In our Universe the equation for gravity is (GMm)/r-n


I suppose you mean (GMm)/rn


Yes I meant (GMm)/rn.
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Re: A gravitational law that works in higher dimensions

Postby Prashantkrishnan » Sat Jan 10, 2015 2:48 pm

What are a and b in your equation? Do they have the same values in any number of dimensions?

And do you have a proof of this equation or is it a hypothesis that gravitation might follow this equation in four dimensions?
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Re: A gravitational law that works in higher dimensions

Postby anderscolingustafson » Thu Jan 15, 2015 5:10 am

a and b do not depend on the number of dimensions but they could vary from one Universe to another. They control how big each layer of the gravitational field would be in relation to the layer in it and outside of it.

I ran some simulations using this equation for gravity in four dimensions and found that it produced stable orbits. One of the simulations I ran using this equation was of a solar system with two planets and some moons around the planets. The inner planet had one moon and the outer planet had one moon.

I found that for many orbits the outer moon of the outer planet stayed in orbit.



but eventually the outer moon of the outer planet escaped.

4d Solar System with escaped moon.png
(329.5 KiB) Not downloaded yet


Eventually the remaining moon of the outer planet kicked the former moon out of the solar system.



After the solar system loosed it's moon I continued running the simulation for a while and there was no sign of any orbits becoming unstable after that.



The fact that an entire solar system with gravitational interaction between planets and moons is able to stay stable with the exception of one moon shows that this gravitational law produces stability in four spatial dimensions.

For this simulation I used 1.6 as b and 4 as a although I ran some simulations using other values for a and b and they produced stable orbits as well.

I'm not sure if gravity would follow this law in a Universe that has four spatial dimensions but if it did then this law would produce stable orbits in higher dimensions. I was thinking that in any number of dimensions the equation for any force would either be r^(-n+1) or it would be r^(-n+1) multiplied by something with n as the number of dimensions and my equation for gravity has r^(-n+1) multiplied by something.
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Re: A gravitational law that works in higher dimensions

Postby anderscolingustafson » Tue Feb 10, 2015 1:26 am

Actually the equation would be (GMmsin(alogb(r/c)))/rn with c controlling the scale.
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Re: A gravitational law that works in higher dimensions

Postby PWrong » Mon Mar 09, 2015 8:49 am

Why did you choose the form sin(log(r) to look at? That's a very complicated function.

In theory it's possible to calculate the solution for any function of r:
http://eqworld.ipmnet.ru/en/solutions/sysode/sode0309.pdf

However in this case the solution probably can't be expressed in terms of elementary functions (I can't express it at all except as the inverse of an integral).

Your escaping moon might just be an artifact of the way you're running the simulation. If you have a small creeping error term in your program, that might ruin things.
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Re: A gravitational law that works in higher dimensions

Postby Keiji » Mon Mar 30, 2015 6:12 am

Well, it's a simpler function than the one I was trying to come up with over here... I'm glad to see someone else who has gone down this train of thought :)
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Re: A gravitational law that works in higher dimensions

Postby anderscolingustafson » Wed Apr 01, 2015 3:47 am

I wonder what kind of effect this gravitational law would have on accretion considering that its repulsive at certain distances and attractive at other distances.
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Re: A gravitational law that works in higher dimensions

Postby Secret » Tue Sep 22, 2015 10:49 am

https://www.newscientist.com/article/dn ... two-stars/

Something interesting

I wonder, if our calculations in the forum are actually showing such scenario will be very common place for 4D planets, given if my memory serves cockscrew orbits tends to be the norm for planet orbits extrapolated to 4 space

thoughts?
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Re: A gravitational law that works in higher dimensions

Postby anderscolingustafson » Wed Sep 23, 2015 6:02 am

Something I was wondering is whether gravity would need to be modified for galaxies to work in 4d because in 3d stars beyond a certain distance from the center of a galaxy tend to orbit at about the same distance from the center regardless of distance from the center which means that inside a galaxy gravity tends to drop off with the distance instead of with the square of the distance because they are held together with dark matter. In 4d even without modifying gravity stars inside a galaxy would not experience an inverse cube law with respect to the center of the galaxy because they would be inside a halo of dark matter and so I'm thinking that even if we don't modify gravity in 4d galaxies would still be stabilized by the dark matter.
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