Book Description

The large separation between the weak scale {approx} 10{sup 3} GeV and the traditional scale of gravity--the Planck scale with M{sub PI} {approx} 10{sup 19} GeV--is one of the most puzzling aspects of nature. The origin of this large ratio, as well as its stability under radiative corrections, demands explanation. This is known as the hierarchy problem. One theoretical means of solving this problem is to introduce Supersymmetry. Alternatively one may hope to address the hierarchy by exploiting the geometry of space time. Specifically, recent theories involve the idea that the 3-spatial dimensions in which we live could be a 3-spatial-dimensional ''membrane'' embedded in a much larger extra dimensional space, and that the hierarchy is generated by the geometry of the additional dimensions. Such ideas have led to extra dimensional theories which have verifiable consequences at the TeV scale. Our knowledge of the weak and strong forces extends down to scales of {approx} (100 GeV){sup -1} (or of order 10{sup -15} mm). On the other hand, we have almost no knowledge of gravity at distances less than roughly a millimeter, as direct tests of the gravitational force at the smallest distances are based on torsion-balance experiments, which are mechanically limited. It is thus conceivable that gravity may behave quite differently from the 3-dimensional Newtonian theory at small distances. This leads to the possibility that matter and non-gravitational forces are confined to our 3-dimensional subspace, whereas gravity may propagate throughout a higher dimensional volume. In this case, the gauge forces are trapped within our 3-dimensional space, unaware of the extra dimensions, and maintain their usual behavior. Gravity, on the other hand, would no longer follow the inverse-square force law at distances smaller than the size of the extra dimensions, as the gravitational equivalent of Gauss' Law mandates that the gravitational field spreads out into the full spatial volume. Since Newton's Law must be reproduced at large distances, gravity must behave as if there were only three spatial dimensions for r & 1 mm. This is achievable either by compactifying all the extra dimensions on circles, where the geometry of these dimensions is thus flat and the topology is that of a torus, or by using strong curvature effects in the extra dimensions. In the first case, Arkani-Hamed, Dimopoulos, and Dvali (ADD) [1,2] used this picture to generate the hierarchy by postulating a large volume for the extra dimensional space, building on earlier ideas in Refs. 3,4. In the latter case, the hierarchy can be established by a large curvature of the extra dimensions as demonstrated by Randall and Sundrum (RS) [5,6]. It is the relation of these models to the hierarchy which yields testable predictions at the TeV scale.