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Dear Drew,
you are raising a very interesting question. Gravity is very well
understood at big scales, well above the diameter of a proton. The theory
describing gravitational phenomena is the general theory of relativity.
However, at small length scales gravity is far from being understood. At
these scales General Relativity fails, since quantum effects begin to
become more and more important. An adequate quantum theory of gravity has
not been formulated so far. But there are a few candidades, and nobody can
yet decide which one might be wrong. The graviton is the "quantum of the
gravitational waves", deformations of space-time propagating in space and
time. Therefore the dimensionality of space-time is crucial. I want to
name three approaches to quantum gravity. First, there is a theory called
**Loop Quantum Gravity**, where space-time is only 4 dimensional.
Secondly, there is approach to gravity based on **non-commutative
geometry**, where space-time is no continuum any more but rather
something like a foam. There the dimensionality might be 4 or higher, with
higher dimensions compactified to very small distances such that they
cannot be "seen". The third approach is
string theory. Space-time is ten dimensional. The
universe we live in is a four dimensional subspace - called D-brane - of
that ten dimensional space. So matter particles are restricted to the
brane. Gravitons, however, can propagate through all ten dimensions of the
space. What we see of gravity is some effective interaction, some
projection onto our brane.

As you can see, gravity is not very well understood so far. There are various different unrelated theories tackling the problem of quantising the theory of gravity, trying to formulate a quantum theory of gravity. Today we cannot decide which one is wrong. And the answer to your question on the dimensionality of the space the graviton exists in (i.e., on the dimensionality of space-time itself) depends on the theory you favour. In case of loop quantum gravity the dimensionality is four, in case of string theory it is ten. However the dimensionality is usually crucial in these theories. If one theory can be formulated in, say ten dimensions, there will be severe problems in eleven or twelve dimensions. And in most cases a formulation will not be possible in other dimensions.

I hope this answer helps

greetings

Michael

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