The Elegant Universe - Brian Greene [128]
A complete table would be infinitely long, since the winding and vibration numbers can take on arbitrary whole-number values, but this representative piece of the table is adequate for our discussion. We see from the table and our remarks that we are in a high-winding-energy/low-vibration-energy situation: Winding energies come in multiples of 10, while vibrational energies come in multiples of the smaller number 1/10.
Now imagine that the radius of the circular dimension shrinks, say, from 10 to 9.2 to 7.1 and on down to 3.4, 2.2, 1.1, .7, all the way to .1 (1/10), where, for our present discussion, it stops. In this geometrically distinct form of the Garden-hose universe we can compile an analogous table of string energies: Winding energies are now multiples of 1/10 while vibration energies are multiples of its reciprocal, 10. The results are shown in Table 10.2.
At first glance, the two tables might appear to be different. But closer inspection reveals that although arranged in a different order, the "total energy" columns of both tables have identical entries. To find the corresponding entry in Table 10.2 for a chosen entry in Table 10.1, one must simply interchange the vibration and winding numbers. That is, vibration and winding contributions play complementary roles when the radius of the circular dimension changes from 10 to 1/10. And so, as far as total string energies go, there is no distinction between these different sizes for the circular dimension. Just as the interchange of fitness-high/valves-low with valves-high/fitness-low is exactly compensated by an interchange of the number of shares held in each company, interchange of radius 10 and radius 1/10 is exactly compensated by the interchange of vibration and winding numbers. Moreover, while for simplicity we have focused on an initial radius of R = 10 and its reciprocal 1/10, the conclusions drawn are the same for any choice of the radius and its reciprocal.3
Vibration number
Winding number
Total energy
1
1
1/10 + 10 = 10.1
1
2
1/10 + 20 = 20.1
1
3
1/10 + 30 = 30.1
1
4
1/10 + 40 = 40.1
2
1
2/10 + 10 = 10.2
2
2
2/10 + 20 = 20.2
2
3
2/10 + 30 = 30.2
2
4
2/10 + 40 = 40.2
3
1
3/10 + 10 = 10.3
3
2
3/10 + 20 = 20.3
3
3
3/10 + 30 = 30.3
3
4
3/10 + 40 = 40.3
4
1
4/10 + 10 = 10.4
4
2
4/10 + 20 = 20.4
4
3
4/10 + 30 = 30.4
4
4
4/10 + 40 = 40.4
Table 10.1 Sample vibration and winding configurations of a string moving in a universe shown in Figure 10.3, with radius R = 10. The vibration energies contribute in multiples of 1/10 and the winding energies contribute in multiples of 10, yielding the total energies listed. The energy unit is the Planck energy, so for example, 10.1 in the last column means 10.1 times the Planck energy.
Vibration number
Winding number
Total energy
1
1
10 + 1/10 = 10.1
1
2
10 + 2/10 = 10.2
1
3
10 + 3/10 = 10.3
1
4
10 + 4/10 = 10.4
2
1
20 + 1/10 = 20.1
2
2
20 + 2/10 = 20.2
2
3
20 + 3/10 = 20.3
2
4
20 + 4/10 = 20.4
3
1
30 + 1/10 = 30.1
3
2
30 + 2/10 = 30.2
3
3
30 + 3/10 = 30.3
3
4
30 + 4/10 = 30.4
4
1
40 + 1/10 = 40.1
4
2
40 + 2/10 = 40.2
4
3
40 + 3/10 = 40.3
4
4
40 + 4/10 = 40.4
Table 10.2 As in Table 10.1, except that the radius is now taken to be 1/10.
Tables 10.1 and 10.2 are incomplete for two reasons. First, as mentioned, we have listed only a few of the infinite possibilities for winding/vibration numbers that a string can assume. This, of course, poses no problem—we could make the tables as long as our patience allows and would find that the relation between them will continue to hold. Second, beyond winding energy, we have so far considered only energy contributions arising from the uniform-vibrational motion of a string. We