Extra Dimensions in Newtonian Gravity
October 29, 2009 7 Comments
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 itcompact 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.
Kaluza-Klein in string theory
Superstring theory is a possible unified theory of all fundamental forces, but superstring theory requires a 10 dimensional spacetime, or else bad quantum states called ghosts with unphysical negative probabilities become part of the spectrum and spoil Lorentz invariance. Fermions are very complicated to work with in higher dimensions, so for the sake of simplicity let’s consider bosonic string theory, which is Lorentz invariant and ghost-free (albeit tachyonic) in d=26.
A particle trajectory only has one parameter: the proper time along the path of the particle. Going from particles to strings adds a new parameter: the distance along the string
and that’s what makes the outcome of Kaluza-Klein compactification far more interesting in string theory than it is in particle theory.
If we compactify x25 on a circle of radius R, we get the usual Kaluza-Klein quantized momentum in that direction
We want gravity in the theory, so we need to look at closed strings. A closed string can do something that a particle cannot do: get wrapped around the circle in the compact dimension.
A closed string can be wrapped around the circle once, twice, or any number of times, and the number of times the string is wrapped around the circle is called the winding number w. The string oscillator sum in the x25 direction changes by a constant piece in a way that is consistent with the periodicity of the closed string and the compact dimension
The string tension Tstring is the energy per unit length of the string. If the string is wound w times around a circular dimension with radius R, then the energy Ew stored in the tension of the wound string is
The mass of an excited string depends on the number of oscillator modes N and Ñ excited in the two directions of propagation around the closed string, minus the constant vacuum energy. Kaluza-Klein compactification adds the quantized momentum in the compact dimensions, and the tension energy from the string being wrapped w times around the compact dimension, so that the total squared mass becomes
A very crucial feature of this mass equation is the symmetry under
This is what makes string theory so different from particle theory. The theory doesn’t really distinguish between the quantized momentum modes, and the winding modes of the string in the compact dimension. This creates a symmetry between small and large distances that is not present in Kaluza-Klein compactification of a particle theory.
This symmetry is called T-duality. T-duality is a symmetry that relates different string theories that everyone thought were completely unrelated before T-duality was understood. T-duality preceded the Second Superstring Revolution.
The theory gains extra massless particles when the radius R of the compact dimension takes the minimum value possible given the above symmetry of T-duality, which is just the string scale itself
This is another purely stringy effect, not occurring with particles.
The Kaluza-Klein compactification of strings can be done on more than one dimension at once. When n dimensions are compactified into circles, then this is called toroidal compactification, because the product of n copies of a circle is an n-torus, or Tn for short.
When fermions are added to make superstrings, the mathematics becomes more complicated but the structures and symmetries become more rich. The most studied superstring compactification is heterotic string theory compactified on a Calabi-Yau space in six-dimensions (or three complex dimensions).
These general models all have in common that the spacetime is a direct product
where M4 is the four-dimensional noncompact spacetime, and X6 is some six-dimensional compact internal space. This means that the metric on M4 doesn’t depend at all on the coordinates in the internal space. In this case, the gravitational coupling constant that we measure as Newton’s constant GN is related to the gravitational coupling G10 of the full ten-dimensional superstring theory by
where VX is the volume of X6.
In terms of the Planck mass MPlanck, which is the quantum gravity mass scale determined by the gravitational coupling GN, this relationship becomes
where the mass MS is the fundamental mass scale of the full ten-dimensional theory.
In the Kaluza-Klein picture, the extra dimensions are envisioned as being rolled up in compact space with a very small volume, with massive excited states called Kaluza-Klein modes whose mass makes them too heavy to be observed in current or future accelerators.
The braneworld scenario for having extra dimensions while hiding them from easy detection relies on allowing the extra dimensions to be noncompact, but with a warped metric that depends on the extra dimensions and so is not a direct product space. A simple model in five spacetime dimensions is the Randall-Sundrum model, with metric
In this scenario, the three-dimensional space that we experience is a three-dimensional subspace, called a 3-brane, located at f=0, with another 3-brane located at f=p, or y=prc. The full four-dimensional space, or five-dimensional spacetime, is referred to as the bulk. The warping or curving of the bulk gives rise to a cosmological constant, which is proportional to the parameter k.
Since the extra space dimension is noncompact, we would expect the force law of gravity to change. However in this picture, the warping of the brane causes the the graviton to become bound to our brane, so that the graviton wave function falls away very rapidly away in the direction of the extra dimension.
This spacetime also has oscillations in the extra dimension that are the Kaluza-Klein modes, but in this case there is a continuous spectrum of modes. This would seem to rule the model out, except that the Kaluza-Klein modes here are so weakly coupled that they can’t be detected on the brane.
Why would this model be preferable to having compact extra dimensions? In Kaluza-Klein compactification, the Planck mass in the full ten-dimensional superstring
The parameter M is the fundamental mass scale in the full theory in the bulk, and k is about the same size as M. So for krc>>1, the Planck scale measured on our brane would be about the same size as the Planck scale as measured in the full theory. This avoids the situation in the Kaluza-Klein compactification where the Planck mass in four spacetime dimensions depends on the volume of the compactified space, which is hard to control dynamically.
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 Kaluza-Klein 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.
The attribute of superstring theory that looks the most promising for experimental detection is supersymmetry. Supersymmetry breaking and compactification of higher dimensions have to work together to give the low energy physics we observe in accelerator detectors.
Braneworld models in general are very different from superstring Kaluza-Klein compactification models because they don’t require there to be so many steps between the Planck scale and the electroweak scale. The huge difference between the Planck scale and the electroweak scale is called the gauge hierarchy problem.
Supersymmetry is interesting to particle physicists because it can address this problem. But some braneworld models need supersymmetry for the brane geometry to be stable.
If supersymmetry is detected at next-generation particle physics experiments, then the details of the supersymmetric physics will have something to say, hopefully, about any underlying superstring model and whether there is Kaluza-Klein compactification of extra space dimensions into some tiny rolled up internal space, or whether we are all living as the four dimensional equivalent of flies stuck on the wall of a higher dimensional Universe.