Force of Gravity in a 2 Dimensional Universe

[starfish99]

I know that Edwin Abbott more than 100 years ago wrote about a hypothetical universe with only two dimensions. I wonder if the force of gravity could exist in such a universe(please no general relativity). How would the force of 2 masses vary with distance? Could there be stable orbits and would they be part of conic sections even though cones do not exist in this universe(circle, ellipse,parabola, hyperbola)? Would there be an analogue in the 2D universe to Kepler's Laws in our universe?

 

[WannabeNewton]

For a 2 dimensional world, the Newtonian potential would be logarithmic as opposed to $1/r$. To see this, consider a localized point charge in this 2 dimensional space; Poisson's equation in the vacuum region of the point mass will just be $\nabla^{2}\varphi = 0$. The isotropy of this system in the 2 dimensional space implies that $\varphi = \varphi(r)$, where we are using polar coordinates. Evaluating the Laplacian in polar coordinates, we then find that $\frac{\mathrm{d} }{\mathrm{d} r}(r\frac{\mathrm{d} \varphi}{\mathrm{d} r}) = 0$ i.e. $r\frac{\mathrm{d} \varphi}{\mathrm{d} r} = \alpha = \text{const.}$ so $\frac{\mathrm{d} \varphi}{\mathrm{d} r} = \frac{\alpha}{r}$ implying $\varphi = \alpha \ln r + \beta$. The additive constant is arbitrary and can be set to zero. The multiplicative constant can be found using Gauss's Law and comes out to $\alpha = GM$ giving us $\varphi = GM \ln r$. Hence the gravitational field of the point charge in the 2 dimensional space is $g = -\nabla\varphi = -\frac{GM}{r}\hat{r}$, which differs by a power from the gravitational field of a point mass in 3 dimensional space.For more details and for a discussion of Keplerian trajectories as well as stable orbits in a 2 dimensional world, see: http://www.dwc.knaw.nl/DL/publications/PU00012213.pdf

 

[starfish99]

Thanks WannabeNewton