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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 one-dimensional 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 dWD-1 represents the D-1 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 GD/rD-1, where GD 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 D-dimensional 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 D-2=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 higher-dimensional 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 4-dimensional 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 Kaluza-Klein 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 5-dimensional spacetime with space coordinates x1,x2,x3,x4 and time coordinate x0, where the x4 coordinate is rolled up into a circle of radius R so that x4 is the same as x4+2pR

Suppose the metric components are all independent of x4. 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 x4 direction:

The four ga4 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 Fab

The field strength is invariant under a a reparametrization of the compact x4 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 g44 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 x4 dimension lead to integer eigenvalues for the momentum in that direction

This quantized momentum acts as the charge for the vector potential Aa. The spectrum of the four-dimensional 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 Kaluza-Klein 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 Kaluza-Klein modes starts to form a continuous spectrum.
But those are not the only new states in the Kaluza-Klein spectrum. The g44 component of the metric propagates as a massless scalar field f(xa) 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 five-dimensional theory, divided by the volume (in this case a circumference) of the compact dimension.
Kaluza-Klein 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 M-theory.

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