**Table of Contents**

*.......The Elegant Universe*

**THE ELEGANT UNIVERSE,****Brian Greene,**1999, 2003

```(annotated and with added

**bold highlights by Epsilon=One**)

**Chapter 10 - Tearing the Fabric of Space**

The Physics of Wound Strings

Throughout our previous discussion of string motion, we have focused on unwound strings. Strings that wrap around a circular component of space share almost all of the same properties as the strings we have studied. Their oscillations, just as those of their unwound counterparts, contribute strongly to their observed properties. The essential difference is that a wrapped string has a

It is not difficult to understand the origin of this minimum mass. A wound string has a minimum length determined by the circumference of the circular dimension and the number of times the string encircles it. The minimum length of a string determines its minimum mass: The longer this length, the greater the mass, since there is more of it. Since the circumference of a circle is proportional to its radius, the minimum winding-mode masses are proportional to the radius of the circle being wrapped. By using Einstein's

How does the existence of wrapped string configurations affect the

Let's consider the last cataclysmic stages of our variant on the big crunch in the Garden-hose universe. As the radius of the circular dimension shrinks to the Planck length and, in the mold of general relativity, continues to shrink to yet smaller lengths, string theory insists upon a radical reinterpretation of what actually happens. String theory claims that all physical processes in the Garden-hose universe in which the radius of the circular dimension is shorter than the Planck length and is decreasing are absolutely identical to physical processes in which the circular dimension is longer than the Planck length and increasing! This means that as the circular dimension tries to collapse through the Planck length and head toward ever smaller size, its attempts are made futile by string theory, which turns the tables on geometry. String theory shows that this evolution can be rephrased—exactly reinterpreted—as the circular dimension shrinking down to the Planck length and then proceeding to expand. String theory rewrites the laws of short-distance geometry so that what previously appeared to be complete cosmic collapse is now seen to be a cosmic

*minimum*mass, determined by the*size*of the circular dimension and the number of times it wraps around. The string's oscillatory motion determines a contribution in excess of this minimum.It is not difficult to understand the origin of this minimum mass. A wound string has a minimum length determined by the circumference of the circular dimension and the number of times the string encircles it. The minimum length of a string determines its minimum mass: The longer this length, the greater the mass, since there is more of it. Since the circumference of a circle is proportional to its radius, the minimum winding-mode masses are proportional to the radius of the circle being wrapped. By using Einstein's

*E = mc^2*relating mass to energy, we can also say that the energy bound in a wound string is proportional to the radius of the circular dimension. (Unwrapped strings also have a tiny minimum length since if they didn't, we would be back in the realm of point particles. The same reasoning might lead to the conclusion that even unwrapped strings have a minuscule yet nonzero minimum mass. In a sense this is true, but the quantum-mechanical effects encountered in Chapter 6—remember*The Price Is Right,*again—are able to exactly cancel this contribution to the mass. This is how, we recall, unwrapped strings can yield the zero-mass photon, graviton, and the other massless or near-massless particles, for example. Wrapped strings are different in this regard.)How does the existence of wrapped string configurations affect the

*geometrical*properties of the dimension around which the strings wind? The answer, first recognized in 1984 by the Japanese physicists Keiji Kikkawa and Masami Yamasaki, is bizarre and remarkable.Let's consider the last cataclysmic stages of our variant on the big crunch in the Garden-hose universe. As the radius of the circular dimension shrinks to the Planck length and, in the mold of general relativity, continues to shrink to yet smaller lengths, string theory insists upon a radical reinterpretation of what actually happens. String theory claims that all physical processes in the Garden-hose universe in which the radius of the circular dimension is shorter than the Planck length and is decreasing are absolutely identical to physical processes in which the circular dimension is longer than the Planck length and increasing! This means that as the circular dimension tries to collapse through the Planck length and head toward ever smaller size, its attempts are made futile by string theory, which turns the tables on geometry. String theory shows that this evolution can be rephrased—exactly reinterpreted—as the circular dimension shrinking down to the Planck length and then proceeding to expand. String theory rewrites the laws of short-distance geometry so that what previously appeared to be complete cosmic collapse is now seen to be a cosmic

*bounce.*The circular dimension can shrink to the Planck-length. But because of the winding modes, attempts to shrink further actually result in expansion. Let's see why.