Gravitational force in different dimensions

[pavi_elex]

How Gravitational force differs in different dimensions.
what it would be for four dimensions, two dimensions and one dimension.
Give me the formula of Gravitational force in n dimension space.
If it is described well (complete derivation) in other web site, send me the link.

 

[George Jones]

Look at chapter 3 (including exercises and problems) from A First Course in String Theory by Barton Zwiebach.

 

[atyy]

(2+1)D:
http://arxiv.org/abs/gr-qc/9503024
Lectures in (2+1)-Dimensional Gravity
Steven Carlip
"Work on (2+1)-dimensional gravity dates back at least to 1963 [1], and occasional articles appeared over the next twenty years [2, 3, 4]. But credit for the recent growth of interest should probably go to two groups: Deser, Jackiw, and ’t Hooft [5,6,7], who examined the classical and quantum dynamics of point sources, and Witten [8, 9, 10], who rediscovered and explored the representation of (2+1)-dimensional gravity as a Chern-Simons theory.*
*The Chern-Simons representation was first pointed out, I believe, by Achucarro and Townsend."

(2+1)D versus (3+1)D:
http://arxiv.org/abs/gr-qc/9905087
An Introduction to Spin Foam Models of Quantum Gravity and BF Theory
John C. Baez
"In particular, general relativity in 3 dimensions is a special case of BF theory, while general relativity in 4 dimensions can be viewed as a BF theory with extra constraints. ... unlike BF theory, general relativity in 4 dimensions has local degrees of freedom."

(3+1)D versus (4+1)D:
http://arxiv.org/abs/hep-th/0608012
Black Rings
Roberto Emparan, Harvey S. Reall
"A black ring is a five-dimensional black hole with an event horizon of topology S1 x S2. We provide an introduction to the description of black rings in general relativity and string theory. Novel aspects of the presentation include a new approach to constructing black ring coordinates and a critical review of black ring microscopics. "

 

[tom.stoer]

For space dimension D=3, 4, ... it's U(r) ~ 1/r^D-2;

Basically this can be understood via solving a Poisson equation for the D-dim. laplacian. Doing this in momentum space one finds a Greens function ~ 1/k². The potential U(r) is the Fourier transform of this Greens function which is ~ 1/r^D-2, therefore the force is div U(r) ~ 1/r^D-1.

This calculation is exact in D-dim. Maxwell theory, but only approx. valid in ART as one has to use the Newtonian limit in order to arrive at the Poisson equation.