Re: Gravity on a torus

On 2007-10-28, pirillo <ultraman2002@xxxxxxxxxxx> wrote:
This discussion is funny to me because apparently all the
string theory guys have solved this already---since they
speak of the description of gravity in an alternative world
say 2 space dimensions,
where one dimension is compactified and they claim to know
what the gravitational potential would be in such a case.

Sure, string theorists sometimes include gravity on the 2D worldsheets
of strings, but most of the time it doesn't have much effect. The way
that gravity is defined in other than 4 dimensions, nowadays, is using
the Einstein-Hilbert action of general relativity (which is the integral
of the Ricci curvature scalar over space-time). In 2 and 3 dimensions,
this version of gravity behaves very differently from what you get by
reducing Newtonian gravity to 2 or 3 dimensions (which is described by
the 1 or 2 dimensional Poisson equation for the gravitational
potential).

In 2 and 3 dimensions, general relativity is in a sense trivial.
Einstein's field equations are R = T, where R is the Ricci curvature
tensor and T is the energy-momentum tensor. In 2 and 3 dimension, the
full Riemann curvature tensor can be altebraically related to the Ricci
tensor R. So, in the absence of matter, R = 0 implies the Riemann
curvature is also zero, which in turn implies that the metric is flat. A
flat metric implies no acceleration due to gravity. This is very
different from lower dimensional Newtonian gravity. Solving the Poisson
equation in in 1 or 2 dimensions still produces a non-trivial
gravitational potential.

For higher dimensional extensions of gravity, Newtonian equations still
work in the weak field limit. However, if some of these higher
dimensions are compact, then the shape behavior of the gravitational
potential as a function of the compactified coordinates strongly depends
on what kind of sources are used for matter. I'm not sure if point
sources or other kinds of sources are used in those cases. It is
possible that some matter distributions are not allowed, because there
will exist no solution to Einstein's equations compatible with them.
One would have to look at this issue in more detail.

Hope this helps.

Igor

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