Extra DimensionsThe line has magnitude in one way, the plane in two ways, and the solids in three ways, and beyond these there is no other magnitude because the three are all.

Aristotle, from On Heaven

Hamidreza Sepangi

► The practicality of Aristotle’s observation is difficult to argue against; it seems to be our everyday experience that there are no more than three spatial dimensions. We only need three numbers to specify the location of a point in space.

►The possibility of extra dimensions is not even considered in Euclid’s Elements while Ptolemy went so far as to offer a proof of the non-existence of extra dimensions in his treatise on distance.

►Kepler entertained the idea that nature’s apparent preference for three dimensions had to do with the holy trinity, while more modern geometers have offered arguments in favor of what our intuition tells us, including the extra-ordinary stability of planetary orbits and atomic ground sates in 3-dimensional space.

► Yet, despite the reasoning of the common sense, many people have been interested in the idea that the world is a fundamentally higher-dimensional arena.

► Extra dimensions have even been considered, in the nineteenth century, to answer questions like “where do angels live,” and so on. A certain sect of the Bolshevik party even tried to use the idea of extra dimensions to present a novel brand of spiritualism and then use it to gradually convert Russian peasantry to the tenets of socialism.

The 4th dimension► The story of extra dimensions

began, arguably, with Bernhard Riemann’s habilitation lecture on June 10th 1854. This was a theme that his supervisor, Carl Friedrich Gauss, had preferred amongst a number of other themes that Riemann had proposed. The lecture titled “On the hypothesis which lie at the bases of geometry.” It was in this lecture that the notion of a curved manifold, or “ many-fold,” made its debut.

► Despite the initially hesitant reception, Riemann’s ideas concerning n-dimensional manifolds gradually became accepted in scientific circles.

Bernhard Riemann (1826-1866)

► Other important figures in presenting the notion of extra dimension were William Clifford and Hermann von Helmholtz.

► A gifted mathematician by the name of Charles Howard Hinton even tried to visualize a four-dimensional hypercube.

A representation of Hinton's four- dimensional hypercube

► Hinton argued that a being that could move freely in the fourth dimension could appear or disappear at will by temporarily leaving our 3-dimensional world and re-entering at some other location, which was seen as a plausible explanation of the behavior of ghosts and other creatures!!

► Even legal consequences of extra dimensions surfaced in the nineteenth century. The London trial of Henry Slade is an example in which the existence of an extra dimension was an integral part of the defense. Testimony on Slade’s behalf was offered by William Webber, J. J. Thompson and Lord Rayleigh. Their efforts was in vein; Slade was eventually found guilty!!

The unification of time and space

► The first popular novel concerning extra dimension was published in 1895; the “Time Machine” by H. G. Wells. Well’s conception of an extra dimension was closest to what was soon to become an accepted truth in physics.

► Einstein’s special theory of relativity, inspired by the structure of Maxwell’s theory of electromagnetism, was the undisputed proof of what we now consider as the fourth dimension.

► This destroyed the Newtonian notion of a universal time.

Well's imagination and extradimensions notwithstanding, mostphysicists still believe time travel to beeither impossible or highly impractical

► In 1909, Hermann Mikowski put forth a geometrical interpretation of Einstein’s theory. He added a fourth dimension, ict, to the familiar three dimensions of Aristotle and Euclid, showing that the predictions of special relativity could be understood in terms of an extended space-time manifold.

Various geometric properties ofMinkowski spac-etime are illustrated inthis cartoon. Time runs verticallyupwards and the (two) spatial directionsare orthogonal. The figure outlined ingrey is the light cone of an observer inthe center of the plot.

► It was Einstein who accepted the idea of space-time and constructed a generally covariant 4-dimensional theory of relativity. Fortunately, Einstein did not need to invent everything himself, for it was at this time that the idea of a n-manifold, put forward by Riemann, came out of the realm of pure mathematics.

► Einstein used it in 1915 to propose the general theory of relativity

Einstein writing down the vacuum fieldequations of general relativity

Unification of gravity and electromagnetism

► About the time when the general theory of relativity was introduced, there were two well-founded theories in physics: gravity and electromagnetism. Strong and weak forces were still decades away.

► Attempts to unify gravity and electromagnetism bore fruit in 1919 through a theory presented by Theodore Kaluza.

Gravity manifests itself as the curvature ofSpace-time induced by massive objects

► In 1919, Kaluza sent Einstein a preprint—later published in 1921–that considered the extension of general relativity to five dimensions. He assumed that if the space-time is endowed with an extra dimension, then electromagnetism and gravity could be unified in an extension of general relativity to 5 dimensions where the resulting 15 field equations naturally break into a set of 10 equations representing gravity, four describing electromagnetism, and one, the wave equation of a scalar field. Kaluza postulated that both effects could

be understood in terms of the same 5-dimensional geometric framework.

Kaluza-Klein theory

►The metric

► Field equations

► But there were problems with Kaluza’s theory, the least of which was the nature of the fifth dimension. Where is this extra dimension and why can’t we see it? Oscar Klein suggested that the extra dimension is compact, that is to say, it is as small as the Planck length, 1.6 x 10^-35 meters. This construction is called compactification of the fifth dimension.

Theodor Kaluza (1885-1945)

Dimensionality and unification of fundamental forces

►As the years passed, physicists realized that Kaluza-Klein theory cannot possibly be the theory of everything. The strong and weak interactions needed more degrees of freedom than Kaluza-Klein could offer.

The strong force is what holdstogether quarks in atomic nucleiThe weak force mediates theradioactive decay of various types ofparticleshttp://userweb.

► The question is: how many dimensions do we need to unify strong and weak forces with gravity and electromagnetism? The answer is at least eleven which was shown by E. Witten in 1981.

► There are two types of fields representing fundamental forces: fermions and bosons. Kaluza-Klein theory can only accommodate bosons and says nothing about fermions. Since both are needed, the idea of supersymmetry was introduced. This symmetry states that the fundamental theory of physics should be invariant under an exchange of identity between fermions and bosons. The resulting theory is called supergravity. But this theory is not free of difficulties!

The weak force mediates theradioactive decay of various types ofParticles.

► String theory gradually came about as the most promising theory which unifies the fundamental forces. It is a 10-dimensional theory where both bosons and fermions live together. In string theory, elementary particles

and field are realized as the oscillations of fundamental one dimensional objects.

The 5th dimension and Space-Time-Matter (STM) theory

► In the early 1990’s, space-time-matter theory started to appear. This is a 5-dimensional theory that attempts to realize Einstein’s old dream of transforming the “based wood” of the stress-energy tensor in his field equations into the “pure marble” of geometry.

► In conventional relativity, it is the distribution of matter in the form of the energy-momentum tensor that determines the geometry through the Einstein’s field equations: In STM theory, all the matter in the

universe is viewed to be a manifestation of higher-dimensional geometry

►In STM theory, we postulate that the 4-dimensional universe is embedded in a higher dimensional vacuum manifold.

►An observer measuring the matter content of the universe using its curvature, geometrical artifacts from this embedding appear to be real matter. That is why this theory is sometimes called “Induced Matter Theory.”

Brane-World Theory► An extremely popular

alternative theory involving extra dimensions is the brane-world scenario.

► Our 4-dimensional universe (brane) is embedded in a n-dimensional manifold (bulk).

► Ordinary matter cannot leave the brane and wander into the bulk space.

►Gravity however, can escape into the bulk.►The theory has the Z2 symmetry, meaning

that each half of the bulk is exactly like the other.

►This setup inspired Randall and Sundrum to construct a 5-dimensional model where there is only one extra dimension, which can be taken to be either non-compact or compact with macroscopic radius.

Implications of the Randall-Sundrum model

►3-dimensional Newtonian gravity is recovered at large scales.

►The model explains the problem of hierarchy in particle physics. The huge disparity between the fundamental forces.

►Brane-world scenarios have left a great influence in the way people look at the universe and try to understand its origin and dynamics.