Gravity in 2D Universe: 1/r Decrease?

[gonegahgah]

Gravity decreases at a squared rate with distance.
This seems to inversely correspond with the surface area of a sphere as it expands.
As such gravity generally decreases proportionally to 1/r².

Our universe has 3D volume but math explores the idea of 2D universes.
I know it is not a reality that we deal with but I was wondering what rate gravity would decrease in an imaginary 2D universe.

In a 2D universe gravity would expand as a weakening circle instead of as a weakening sphere of influence.
As a circle increases in circumference directly with radius - instead of at a squared rate as does a sphere - does this mean that in an imaginary 2D universe gravity would decrease at a rate proportional to 1/r?

 

[Werg22]

Yeah why not?

 

[gonegahgah]

In a 3D volumetric universe planets form approximately the shape of spheres.
In an imaginary 2D areametric universe planets would form approximately the shape of a circle.

I would imagine that similar stepping would occur for an imaginary 1D linearmetric universe.
That is planets would form the shape of straight lines.
Also continuing this stepping the rate of gravity in an imaginary 1D universe would decrease at rate directly proportional to 1/rº(=1); in otherwords no decrease at all.
Gravity in an imaginary 1D universe would remain constant would it not?

I guess there are analogies to show this.
For 3D: Light spreads out at a decreasing intensity proportional to 1/r² from a globe.
For 2D: Water waves spread out a a decreasing intensity proportional to 1/r¹.
For 1D: Constrain water in a narrow channel with a wave maker and the waves spread away with constant intensity proportional to 1/rº.

So this is all also imaginarily correct for imaginary 1D gravity isn't it?

 

[arildno]

Gravity decreases at a squared rate with distance.
This seems to inversely correspond with the surface area of a sphere as it expands.
As such gravity generally decreases proportionally to 1/r².

Our universe has 3D volume but math explores the idea of 2D universes.
I know it is not a reality that we deal with but I was wondering what rate gravity would decrease in an imaginary 2D universe.

In a 2D universe gravity would expand as a weakening circle instead of as a weakening sphere of influence.
As a circle increases in circumference directly with radius - instead of at a squared rate as does a sphere - does this mean that in an imaginary 2D universe gravity would decrease at a rate proportional to 1/r?

Well, why not explore the well-known results from applying Coulomb's law in 2-D between a negative charge and a positive charge?

 

[Werg22]

The 1D equivalent of a circle isn't a straight line, it's two points.

 

[Office_Shredder]

In 1D the equivalent of a ball is an interval though, which is what I think he was referring to (since planets aren't actually spheres, they're balls)

 

[aniketp]

and 2d equivalent of solid sphere is a disc.

 

[Gear300]

Well...technically the formula could still apply to 3D volume...and from thinking it though...many of our laws are like that. It could be that 3D systems could just as well be projected onto 2D systems (making a 2D universe seemingly equivalent to a 3D universe so that an entity in a 2D universe may not be able to tell its in 2D).

 

[Werg22]

In 1D the equivalent of a ball is an interval though, which is what I think he was referring to (since planets aren't actually spheres, they're balls)

and 2d equivalent of solid sphere is a disc.

Right and right, but in 3D we're speaking about the surface of the sphere, and similarly in 2D we're speaking of the circumference of the disc, a circle. So we ought to be speaking about the end points of the interval in 1D.

 

[gonegahgah]

That all sounds correct and corresponding.
Even 'interval'; though without the time connotations?
Though the terms sphere, circle and line are less technical.

Gear, are you in agreement in your statement? I wasn't quite sure what you've concluded. You're happy that gravity would decrease at 1/r² from the surface of a ball in 3D universe, at 1/r¹ from the circumference of a disc in imaginary 2D universe, and at 1/rº from the end points of an interval in an imaginary 1D universe?

I would also imagine that an imaginary 0D universe wouldn't have space for anything anyway?

I would have to guess that gravity doesn't somehow dissipate out through the other (3D + 1T + 7 = 11D) 7 dimensions found in superstring theory otherwise we would experience its decrease as being 1/r¹º.

 

[Gear300]

That all sounds correct and corresponding.
Even 'interval'; though without the time connotations?
Though the terms sphere, circle and line are less technical.

Gear, are you in agreement in your statement? I wasn't quite sure what you've concluded. You're happy that gravity would decrease at 1/r² from the surface of a ball in 3D universe, at 1/r¹ from the circumference of a disc in imaginary 2D universe, and at 1/rº from the end points of an interval in an imaginary 1D universe?

I would also imagine that an imaginary 0D universe wouldn't have space for anything anyway?

I would have to guess that gravity doesn't somehow dissipate out through the other (3D + 1T + 7 = 11D) 7 dimensions found in superstring theory otherwise we would experience its decrease as being 1/r¹º.

Actually...I don't think I even knew exactly what I was saying...
If gravity weakens as the number of dimensions increases, its sort of like saying that it is distributing over a larger space, thus becoming less dense...but to assume that it weakens is sort of like saying that there is a limit to the "amount of gravity" in the universe regardless of the number of dimensional space.

 

[gonegahgah]

Gear, I would assume, from the formulas for gravity that we use, that gravity only dissipates out through the volume of space we know and not into any of the other dimensions hypothesized by the string theories. If gravity leaked into any of the other 7 dimensions then by extension of the 2D universe idea we would see it decrease at a greater rate than squared. As it doesn't I would assume that it only presents itself in the volume of space as we know it. Whether this idea has been answered or looked at by science itself I don't know.

 

[CaptainQuasar]

This of course is all implemented in many video games. Check out http://armorgames.com/play/2153/aether", for example.

⚛​

 

[Gear300]

Gear, I would assume, from the formulas for gravity that we use, that gravity only dissipates out through the volume of space we know and not into any of the other dimensions hypothesized by the string theories. If gravity leaked into any of the other 7 dimensions then by extension of the 2D universe idea we would see it decrease at a greater rate than squared. As it doesn't I would assume that it only presents itself in the volume of space as we know it. Whether this idea has been answered or looked at by science itself I don't know.

I see...in this concept what is gravity viewed as (space-time curvature, particle, both, etc...)?

 

[gonegahgah]

I guess it could be seen any which way.
ie. that gravity is space-time curvature (that somehow doesn't touch the other 7Ds for whatever reason), as particles/waves spreading out through volume and therefore decreasing through an expanding square (again avoiding the other 7Ds somehow), even as gravity acting instaneously at a distance (though this is a no-no in current science).

It does raise some interesting questions for me that I am trying to explore in my head.
Gravity certainly seems to spread out at a decreasing squared rate in the same fashion as a wave that expands all directions through a volume. I don't believe in gravity waves or gravity particles but there certainly is a correlation which forces me to think about it.

If gravity behaved like little ropes attached to other bits of mass out there then it would not decrease in strength at all; but it does. If you pull a rope then the force you exert at one end is the same force that is felt at the other at the attachment point. This is unlike gravity which weakens in our 3D volumetric universe (unlike in an imaginary 1D universe).

 

[heldervelez]

when we do aproximate solar system gravity simulation we apply all masses concentrated in the center of each planet, and all planets in a 2D plane, and regular Newtonian formulation. it seems to me a 2D world and force still goes proporcional to 1/r^2, as usual.
and simulation gives results according with observations.

http://en.wikipedia.org/wiki/Gauss'...e:_Spherical_surface_centered_at_a_point_mass

 

[gonegahgah]

In those simulations isn't the sun regarded as the gravity source ie. the planet's orbits occur around the sun; not around each other?

 

[heldervelez]

I miss mentioning the sun, just as you did, but it is not important to reasoning about your post.
The nature of the answer on my reply remains with or without the sun.
1/r^2 is OK 1/r is not ok even on a 2D world.

Your new question is unrelated with original question.
Gravitational force is simulated by a N-Body problem, so every mass of system are accounted and, of course, the Sun. Planets orbit around the Sun, satelites around planets, and even the Sun has an orbital motion . It is oscilating around a point that resides in his interior as it translates in his orbit.

If you have insufficient computational power you could use test particles for the most light particles (for ex. objects in Asteroid Belt) , considering that they don't contribute to the field, but only respond to the field.

 

[gonegahgah]

Even though the solar system occurs mainly within a plane it is not a disc like a galaxy is. A galaxy has its mass more evenly spread out. For a solar system most of the mass is concentrated towards the planets because most of the mass apart from the comets and asteroid belt have been swept up by the planets. I might be wrong but that is correct isn't it?

So you still have these balls (sun, planets, moons) which are gravitationally attractive and which are volumetric objects. In this respect they seem to present their attractiveness out through the volume of space in a manner similar to a spreading wave. A wave spreading in volume decreases in intensity proportional to 1/r². That is a fact I would think; and fairly equivalent to what happens for our gravity.

A 2D planet & sun don't have a volume through which to present themselves. They only have an area out which to present themselves through. Waves spreading over an area decrease in intensity proportional to 1/r¹.

A 2D universe is purely imaginary anyway so any answer is imaginary as far as it goes.
The only problem I could see for anyone here is if I am somehow trampling on superstring theory. To be honest I wouldn't know if I were or weren't. They might have some explanation why mass doesn't present itself gravitationally through the other 7 dimensions. Objects might not attract other objects gravitationally through those greater dimensions. I don't know.

Whether gravity would decrease at different proportional rates for different numbers of dimensions is only conjecture really. We know what we experience and that is the only thing we know.

 

[heldervelez]

I was wrong in my comments above. Sorry.
The solar system is in a plane, (much more aproximate than a galaxy is)
but radiation is still 3D. And this was my mistake.
So your original post seems ok.
Math is downgrading where appropriate in the link
where integral of flux thru enclosing area will be integral of flux thru enclosing perimeter.
integral in volume by integral in a surface, etc...
http://en.wikipedia.org/wiki/Gauss's_law_for_gravity

But what about the stability of such a world and the range of forces ?
As you remark in a 1D world a gravitational force is somewhat strange. Is not even a central force.
It seems that we cannot consider that downgrading 1 or 2 dimensions we have an equivalent property/(large scale world constructor). Upgrading dimensionality is ...what ?

 

[robphy]

You might be in interested in earlier discussions of the "dimensionality of space":

https://www.physicsforums.com/showthread.php?p=756405#post756405
https://www.physicsforums.com/showthread.php?t=109090

Concerning gravity...
one way to generalize the dimensionality is by studying the Laplace (Poisson) Equation in various dimensions. Another is to look at the Einstein Field Equations (especially the properties of the Riemann tensor) in various dimensions.

 

[heldervelez]

I do not know if it is usual or not to show some gratefullness (*) in posts.
Anyway tanks to Mr RobPhy for your kind indication of your post https://www.physicsforums.com/showthread.php?p=756405#post756405
It was very usefull to me reading the paper http://arxiv.org/abs/gr-qc/9702052 of Alex TegMark 'On the dimensionality of spacetime'

It seems space only with 3+1 dimensions is predictable and stable
wich is not the case with other dimensions.
I think that 'predictable and stable' are necessary requirements to conceive any 'physical law'.
Beeing so, I wonder why the quest of higher dimensions theories.

(*) English is not my usual language and sometimes I miss the correct spelling. I beg your pardon if sometimes my english is not correct. Also my keyboard sometimes misses the key 'T'.