In 1919, Theodor Kaluza (1885-1945) wrote a paper that would have a deep and lasting impact on theoretical physics. In this paper, Kaluza showed that if you assume that there is one extra dimension of space, and you further assume that everything is independent of the extra fifth dimension, then you can integrate out the extra dimension to recover a four-dimensional theory which consists of Einstein's theory of gravity, plus Maxwell's theory of electromagnetism, together with a scalar field which he incorrectly set equal to a constant. Thus, by starting with Einstein's theory of gravity in five dimensions, you can recover Einstein's theory and Maxwell's theory of electromagnetism in four dimensions simply by assuming that nothing depends on the fifth dimension.

Now, most people are under the impression that Kaluza did not think about compactifying the extra dimension. This is not true. In the original paper he does mention the "cylindrical condition" by which he meant the condition that the extra dimension is compactified to a circle. However, he did not say much about the size of the extra dimension. In fact, Kaluza seems almost apologetic about the fact that he had introduced this extra dimension of space, and he often sounds like he wants to think of the extra dimension as a mathematical trick. Luckily, in 1926, Oskar Klein (1894-1977) came along and talked about the size of the circle. He pointed out that it was natural to assume that the circle is very small.

It is clear that during this time other people had a hard time adjusting to the idea of a fifth dimension. Einstein was the referee for the Kaluza paper, which he received in 1919 but did not accept for publication until 1921. We may deduce from Einstein's paper with Bergmann that he was only happy with the idea in the following form:

1. The extra dimension is real.

2. The extra dimension is small.

In other words, Einstein and Bergmann adopted the viewpoint of Klein. These basic assumptions were key ingredients in Einstein's later attempts to construct a Unified Field Theory.

For many years, the Kaluza-Klein idea was more or less a curiosity. The construction of
the Standard Model of particle physics gave theoretical physicists plenty of things to work on.
However, this changed when people started to think seriously about quantizing the gravitational
field. This is because in most of the modern approaches to quantizing gravity, we are led
inevitably to the conclusion that there exist extra dimensions of space. Indeed, in
eleven-dimensional supergravity, the low-energy limit of M-theory, there are ten dimensions of
space, and one dimension of time. Of course, we need to explain why these extra dimensions are
not observed, and to do this we typically invoke the old ideas of Kaluza and Klein. We assume
that the extra dimensions are curled up on such a small length scale that we are unable to detect
them. String theorists assumed that the extra dimensions were curled up, or compactified, on the
order of the Planck length, L_{pl}, 10^{-33} cm, which is the distance at
which gravity becomes equal in strength to the other forces.

In 1990, Ignatios Antoniadis (Ecole Polytechnique) suggested that the extra dimensions
required by string theory might be as large as the electroweak length scale L_{ew},
about 10^{-17} cm. In 1998, particle theorists Nima Arkana-Hamed, Savas Dimopoulos
(both then at Stanford), and Gia Dvali (then at Trieste) published a paper titled "The Hierarchy
Problem at New Dimensions at the Millimeter". Their theory, named after the author’s initials, is
ADD theory.

The hierarchy problem is a long standing problem in particle physics. Simply stated, it is why is the Planck scale so much smaller than the electroweak scale. The most common explanation of this is supersymmetry, which says that all fermions have a supersymmetric partner that’s a boson, and all bosons have a supersymmetric partner that’s a fermion. Before supersymmetry breaking, the scales would have been much closer, and so there would have been no hierarchy problem. Another theory put out to explain the hierarchy problem was technicolor which stated that what we call fundamental particles, quarks and leptons, actually are composed of constituent particles.

ADD theory provides a novel solution to the hierarchy problem. Let’s say there was a large extra dimension. Our universe would be embedded in a higher dimensional spacetime. Our universe is called the brane, and the higher dimensional spacetime is called the bulk. Our universe could be viewed as analogous to a D-brane from particle physics. According to superstring theory, spin-1 gauge bosons, such as the photon, gluons, and W bosons, as well as quarks and leptons, are open strings, which means their end points are stuck on D-branes, which in this case, would be our universe. The graviton, a spin-2 particle, is a closed string, meaning it is a little closed loop that is not attached to a D-brane. According to this theory, gravitons are then not attached to our universe, and can wander off into other dimensions. The result would be that gravity would be diluted as it spread throughout the entire higher-dimensional spacetime. It would therefore appear far weaker. From the point of view of our brane, it would appear weaker than the other forces that are restricted to our brane. You have the same result if you think of gravity as the curvature of space in which such case, it would exist throughout all of space. Models where the universe is a surface moving in some higher-dimensional spacetime are called brane-world scenarios.

Another example of a brane world scenario was developed in 1999 by Lisa Randall (Princeton) and Raman Sundrum (then at Boston University). They took a different approach to the hierarchy problem. Their theory states that the real home of gravity is a brane different then ours. This otherwise inaccessible brane is separated from ours by an extra dimension not much bigger than the Planck length. Gravity feels weak to us because it’s mostly confined to this other brane. The gravitational effect of this other brane on the metric of spacetime would exponentially attenuate gravity as a function of the tiny separation between the branes. In order for this theory to work, gravity would have to be mostly confined to the other brane, and they achieved this using the curvature of the extra dimension. Their insight was to use the fact that the curvature of spacetime itself is something that can confine even gravitons. Following this intuition, they constructed an explicit model where there is a single infinitely large extra dimension of space, and yet a massless graviton is confined to the brane-world by the effects of the curvature of the bulk spacetime.

The discovery of this model is very important because it resolves conceptual questions which have been around since Kaluza and Klein wrote their papers. After 80 years of living with the idea of extra dimensions, we must finally accept the fact that some of these dimensions may be quite large. We’ve come full circle, and now view extra dimensions similar to how Kaluza did in 1919. In some ways, the brane-world can be viewed as similar to the D-branes in particle physics. According to this view, what we call the Universe could be thought of as a giant D-brane.

If the extra dimensions were compactified on the Planck scale, as had been previously
assumed, they would be outside the reach of experiment. The Planck length is 10^{-33}
cm. Examining such tiny distances would require probing energies of the order of the Planck
mass, M_{pl} = [h bar]/L_{pl}c ~ 10^{19} GeV. This is far beyond
the capabilities of any conceivable accelerator. However, according to ADD theory, the extra
dimensions might be curled up on scales as large as a few millimeters. This would make it possible
to detect departures from Newtonian gravity with tabletop experiments. Also the Large Hadron
Collider (LHC) will provide experimenters with 10 TeV (10^{4} GeV) protons. This
should provide evidence for the large extra dimensions, if they exist.

If there are n extra dimensions curled up with diameters R, if you were to look at scales smaller than R, you would see the generalized Newtonian potential.

V(r) = (G*_{n} m_{1} m_{2})/(r^{n + 1}) for r << R

between test masses m_{1} and m_{2}, where G*_{n} is the
appropriate gravitational constant for n extra dimensions. Gravity must spread out in all the
dimensions. The extra dimensions make the gravitational force grow stronger with decreasing
separation. If you were looking at scales larger than R, you would see a Newton-like potential.

V(r) = (G*_{n} m_{1} m_{2})/(R^{n }r) for r >> R

Newton’s constant G that we’ve been measuring at separations larger than R is really

G*_{n}/R^{n} =
L_{ew}^{2}(L_{ew}^{2}/R)^{n}

Therefore gravity is intrinsically comparable to the other forces. It’s only because it’s spread throughout the other dimensions that it appears weaker to us. The compactification size of the n curled-up dimensions is given by

R^{n} =
L_{ew}^{n}(L_{ew}/L_{pl})^{2} ~
L_{ew}^{n} x 10^{32}

If n = 1, R would be on the scale of the Solar System which would have an obvious effect on Solar System dynamics. If n = 2, R is about a millimeter, and thus in reach of tabletop experiments.

One group of people doing such experiments are the Eot-Wash group at the University of Washington. They specially designed a hypersensitive torsion balance. The pendulum is suspended by a torsion fiber above a uniformly rotating attractor. The gap between them can be as small as 100 micrometers. Ten holes in the pendulum and 20 holes in the attractor serve as negative test masses. Their deployment is such that only a short-range gravitational anomaly would produce significant torque pulses as the attractor rotates. Pendulum twists are monitored by a laser beam and mirrors.

Whenever there is a recent advent in modern physics, everyone reinterprets their field in light of the new theory. You had this with supersymmetry, superstrings, D-branes, M-theory, and Maldacena. Brane world cosmology is no exception. Some people have taken advantage of this to explain all kinds of things that never crossed the minds of those who invented it. For instance, what if our brane isn’t the only brane in the bulk? There could be several branes, perhaps an infinite number. These would be analogous to other universes. This would provide another mechanism by which the anthropic principle could operate. You could use brane worlds to explain the matter-antimatter asymmetry of the universe. Baryons have a baryon number of 1, and antibaryons have a baryon number of -1. Why don’t we have a net baryon number of 0? If our brane had overlapped another brane shortly after the Big Bang, particles or antiparticles could have been exchanged between them. We could have ended up with a net positive baryon number.

I’ll illustrate how most people have jumped on the brane world bandwagon with the following example. Pre-big-bang cosmology was an attempt to reconcile inflationary cosmology with superstring theory. With the recent advent of brane world cosmology, which was developed to solve the hierarchy problem and where the Universe is considered to be a giant D-brane, we now have to reconcile brane worlds with pre-big-bang cosmology. Recently, Antonio Riotto of Switzerland has managed to do that. He manages to explain some of the contradictions in these different views of the universe. If the string scale is much smaller than the Planck mass, you should have a period of inflation on our brane driven by the large extra dimensions, so you have to recover pre-big-bang cosmology where you have a growing Hubble constant, called the Hubble rate, and a growing dilaton field.

D-branes are classical solutions that exist in many recent string theories. A Dp-brane is a
configuration that extends along p spatial dimensions and is located in all other spatial transverse
directions. From the point of view of superstring theory, a D-brane is a soliton on which open
string endpoints are attached, and whose mass or tension is inversely proportional to the string
coupling g_{s}. Therefore D-branes become light in those regions of the moduli space
where the string coupling is large. This simple property has led to a new interpretation for
singularities of various fixed spacetime backgrounds in which strings propagate and provided a
generic and physically intuitive mechanism for smoothing away cosmological singularities in the
strong coupling or large curvature regime of the evolution of the early universe.

With the weakly coupled heterotic string, there is a fundamental relation between the
string scale M_{s}, the four-dimensional Planck mass M_{p}, the value of the
dilaton field φ, and the gauge coupling α_{g}.
(M_{s}/M_{p})^{2} ~ α_{g} ~ e^{φ}.
The string scale is about 10^{18} GeV. However, in other regions of the moduli space
the string scale can be much smaller, and it’s possible the string scale is not far from the TeV
region. If this is the case, the ratio M_{s}/M_{p} could be the result of large
compactified dimensions through a scaling relation like M_{p}^{2} =
(M_{s}^{n + 2})(V_{n}) where V_{n} is the volume of
the n large extra dimensions. For M_{s} ~ TeV and n > 2, the radius of the large
dimensions could be as large as a millimeter. Large extra dimensions are allowed only if the SM
fields are confined to a membrane orthogonal to the large extra dimensions. This cosmological
model is consistent with the current view of string theory and D-branes.

There are many interesting cosmological implications of this scenario. In the presence of
matter with energy density p, the evolution of the universe is controlled by the equation
H^{2} = (a’/a)^{2} = 8πp/3M_{p}^{2}, where a is the
scale factor and H is the Hubble rate. This implies that for an expanding universe, p and
H^{2} are proportional. They would increase or decrease together in time. This is true
only if M_{p} is a constant. In the weakly coupled string theory where all the
compactified dimensions have size ~ M_{s}^{- 1}, M_{p} is
controlled by the dilaton, and its possible to have growing H while p decreases. This is the basis
of pre-big-bang (PPB) cosmology where you have superinflationary solutions for the scale factor
a having a growing Hubble parameter and a growing dilaton field. Therefore, in weakly coupled
string theory it is possible to generate a scenario containing a dilaton-driven PBB inflation.

When the tension becomes negative, branes become unstable, and at least in the regime of strong coupling, it becomes easier and easier for gravitational background to produce further massless branes. On the other hand, when the transverse sizes of the branes becomes larger than the horizon, the branes presumably break down and decay, but branes are continuously created. The rate of brane creation due to these quantum effects may be so fast as to balance the dilution of the brane density due to the expansion. You could therefore have a phase of constant brane density and an exponentially expanding Universe. This would be a phase of brane-driven inflation reminiscent of the old idea of string-driven inflation.