Extra dimensions in Newtonian gravity
Theoretical physicists didn't start studying higher
dimensional theories of fundamental physics until after the modern era
of 20th century quantum mechanics and relativity had begun. But the
physical consequences of extra dimensions of space can be worked out
in Newtonian physics and it is there that we actually find the first
and most important observational constraint on the number of space dimensions
in our Universe.
The first thing we need to know about an extra dimension
is: is it compact or noncompact?
An example of a noncompact dimension is the infinite line of real numbers
that makes the axis of a rectangular coordinate system, say the x axis.
The line has a onedimensional volume that is infinite. An example of
a compact dimension would be rolling the x axis into a closed circle
of radius R, which then has a finite volume of 2pR.
The length of the infinitesimal line element in spherical
coordinates in D noncompact dimensions is
where dW_{D1} represents
the D1 angular terms in the metric. The gravitational potential F(r)
solves the Laplace equation with a point source, which generalizes in
D dimensions to
The force F(r) is proportional to the gradient of
the potential F(r), so therefore
the force must vary with distance from the source as G_{D}/r^{D1},
where G_{D} is Newton's constant, which determines the strength
of the gravitational coupling, as measured in D space dimensions. (Remember,
in Newtonian gravity time isn't being treated as a dimension yet.)
Extra noncompact dimensions would change the force
law of gravity away from being the inverse square law that has been
and still is measured experimentally. This would drastically alter the
behavior of planets, because it's only in an inverse square potential
that the equations of motion of Newtonian gravity predict stable closed
orbits. So astronomers and physicists can set limits on possible extra
dimensions without even going to fancy accelerators, by watching the
orbits of planets and satellites.
A compact extra dimension has a completely different
effect on the Newtonian force law. In a Ddimensional space with one
dimension compactified on circle of radius R with an angular coordinatea
that is periodic with period 2p,
the line element becomes
The force law derived from the potential that solves
the Laplace equation becomes
So if we added an extra compact space dimension to our three existing
noncompact space dimensions, then D=4, but D2=2, so the force law is
still an inverse square law. The Newtonian force law only cares about
the number of noncompact dimensions. At distances much larger than R,
An extra compact dimension can't be detected gravitationally by an altered
force law.
The effect of adding an extra compact dimension is
more subtle than that. It causes the effective gravitational constant
to change by a factor of the volume 2pR
of the compact dimension. If R is very small, then gravity is going
to be stronger in the lower dimensional compactified theory than in
the full higherdimensional theory.
So if this were our Universe, the Newton's constant
that we measure in our noncompact 3 space dimensions would have a strength
equal to the full Newton's constant of the total 4dimensional space,
divided by the volume of the compact dimension.
That's an important detail, because the size of the
gravitational coupling constant is what determines the distance scale
of quantum gravity. So the distance scale of quantum gravity has to
be very carefully defined in theories with compactified extra dimensions.
The KaluzaKlein idea
Why would anyone consider a theory with extra dimensions?
Because this turns out to provide a convenient mathematical framework
for unifying gravity with electromagnetism and the other known forces.
The first consideration of this idea occurred in the 1920s in separate
work by Theodore Kaluza and Oskar Klein.
Consider a 5dimensional spacetime with space coordinates
x^{1},x^{2},x^{3},x^{4} and time coordinate
x^{0}, where the x^{4} coordinate is rolled up into
a circle of radius R so that x^{4} is the same as x^{4}+2pR
Suppose the metric components are all independent of x^{4}.
The spacetime metric can be decomposed into components with indices
in the three noncompact directions (signified by a,b below) or with
indices in the x^{4} direction:
The four g_{a4} components of the metric look like the components
of a spacetime vector in four spacetime dimensions that could be identified
with the vector potential of electromagnetism with the usual field strength
F_{ab}
The field strength is invariant under a a reparametrization of the
compact x^{4} dimension via
which acts like a U(1) gauge transformation, as it should if this is
to act like electromagnetism. This field obeys the expected equations
of motion for an electromagnetic vector potential in four spacetime
dimensions. The g_{44} component of the metric acts like a scalar
field and also has the appropriate equations of motion.
In this model, something miraculous happens: a theory with
a gravitational force in five spacetime dimensions becomes a theory
in four spacetime dimensions with three forces: gravitational, electromagnetic,
and scalar.
When the wave equation is solved in this spacetime, the
periodic boundary conditions in the compact x^{4} dimension
lead to integer eigenvalues for the momentum in that direction
This quantized momentum acts as the charge for the vector potential
A_{a}. The spectrum of the fourdimensional theory therefore
includes an infinite number of charged particles with mass
where n is an integer.
If R is very small, then the masses of these KaluzaKlein
modes are very large even when n is small. So that means we'd need very
high energy to create these particles in an accelerator experiment.
If R is very large, then the KaluzaKlein modes starts to form a continuous
spectrum.
But those are not the only new states in the KaluzaKlein
spectrum. The g_{44} component of the metric propagates as a
massless scalar field f(x^{a})
in the noncompact dimensions. This would result in a
new long range force not observed in Nature. So there has to
be a way for this field to become massive, and quite a lot of work has
gone into trying to find a good answer to that question.
As in the Newtonian limit, the Newton's constant measured
in four spacetime dimensions is again derived from the full gravitational
coupling constant in the fivedimensional theory, divided by the volume
(in this case a circumference) of the compact dimension.
KaluzaKlein compactification like this has been extended
to many dimensions, and to supergravity theories. The theory of eleven
dimensional supergavity with the extra seven space dimensions
compactified on a seven dimensional sphere was a very popular candidate
before 1985, and now it is part of the bigger theory that encompasses
and relates all string theories, called Mtheory.
Next:
Superstrings and braneworlds>>
