KaluzaKlein compactification
In the original KaluzaKlein theory from the 1920s,
had five spacetime dimensions, but the extra space dimension was rolled
up into a tiny circle, existing everywhere but too small to see with
existing instruments. The vibrations of the gravitational field in the
rolled up extra space dimension would look to observers like vibrations
in an electromagnetic field and a scalar field in the remaining three
space and one time dimensions.
So the underlying five dimensional spacetime only
has one kind of force: a gravitational force. But the four dimensional
spacetime that is seen at large distances appears to have three kinds
of forces: a gravitational and an electromagnetic and a scalar force.
This is one way of making a unified theory  derive all the other forces
from higherdimensional modes of the graviton.
In string theory, KaluzaKlein compactification of
the extra dimensions has one important difference from the particle
theory version. A closed string can get wound several times around a
rolled up dimension. When a string does this, the string oscillations
have a winding mode. The winding modes add a symmetry to the theory
not present in particle physics. A theory with a rolled up dimension
with size R turns out to be equivalent to a theory with a rolled up
dimension of size L_{s}^{2}/R with the winding modes
and momentum modes in the extra dimension exchanged. (L_{s}
is the string length scale.)
This is a very important symmetry because it equates
theories with very small extra dimensions to theories with very large
extra dimensions. This symmetry is called T
duality. It shows that seemingly different
different string theories can turn out to be the same theory looked
at in a different way.
In superstring theory, KaluzaKlein compactification
has to be done on six space dimensions at once. One proposed model has
been to take the heterotic string theory and make six space dimensions
form a small compact space called a CalabiYau
space. The geometry and topology of the CalabiYau space determine
the symmetries and spectrum of the particle theory measured at low energy
or large distance.
There have been many interesting models, but if a
realistic model had been found, we wouldn't need to discuss the next
option: braneworlds.
Braneworlds
Pretend you lived on your computer screen and could
only move on that two dimensional surface. The computer exists in three
space dimensions but you can only move on a two dimensional subspace
made by the screen, so the spacetime that you experience would look
like three dimensions (two space plus time) rather than four.
That's sort of the idea in a braneworld higher dimensional
theory. Our observed four dimensional spacetime is like the computer
screen, a subspace of some bigger space that
we can't see because all matter and forces are constrained to move (mainly)
on our subspace, or brane (as in membrane).The total space is
called the bulk and the subspace or brane on which we would live is
called the brane.
Light is made of electromagnetic radiation, and in
a braneworld model, the charges and the fields should propagate only
on the brane. So there wouldn't be any way to probe the extra dimensions
in the bulk by using light, even if the extra dimensions were large.
Gravity is the force that determines the shape of
spacetime. Therefore, at least in principle, gravity should propagate
in all dimensions equally. That means we should be able to detect large
extra dimensions by looking for suspicious behavior in the gravitational
force.
But in certain braneworld models, gravity can actually
be confined or bound to our brane so that it doesn't propagate very
far in the bulk. That makes the extra dimensions harder to detect using
gravity.
Braneworld models are conceptually different from
compactified KaluzaKlein models because they don't attempt to derive
nongravitational forces from the gravitational oscillations in the extra
dimensions. On the contrary, if the extra dimensions are large, the
gravitational oscillations have to die out quickly in those directions,
so that we can't detect them. There are still KaluzaKlein modes of
oscillation in the extra dimensions, but because they couple through
gravity, and gravity is mostly confined to the brane, they are effectively
invisible to our world on the brane.
How could they be observed?
One problem with theoretical models of gravity and
particle physics is that before they can make unique testable predictions
of new physics, they have to be worked on so that they don't contradict
any existing theoretical or experimental knowledge. That can be a long
process, and it's not really over for superstring theories or for braneworld
models, especially not braneworld models derived from superstring theories.
In superstring theory with KaluzaKlein compactification,
there are several different energy scales that come into play in going
from a string theory to a low energy effective particle theory that
is consistent with observed particle physics and cosmology:
