The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [199]
7. For the mathematically inclined reader, here is how you do the calculation of the distance—now, at time tnow—that light has traveled since being emitted at time temitted. We will work in the context of an example in which the spatial part of spacetime is flat, and so the metric can be written as ds2 = c2dt2 – a2(t)dx2, where a(t) is the scale factor of the universe at time t, and c is the speed of light. The coordinates we are using are called co-moving. In the language developed in this chapter, such coordinates can be thought of as labeling points on the static map; the scale factor supplies the information contained in the map’s legend.
The special characteristic of the trajectory followed by light is that ds2 = 0 (equivalent to the speed of light always being c) along the path, which implies that , or, over a finite time interval such as that between . The left side of this equation gives the distance light travels across the static map between emission and now. To turn this into the distance through real space, we must rescale the formula by today’s scale factor; therefore, the total distance the light traveled equals . If space were not stretching, the total travel distance would be , as expected. When calculating the distance traveled in an expanding universe, we thus see that each segment of the light’s trajectory is multiplied by the factor , which is the amount by which that segment has stretched, since the moment the light traversed it, until today.
8. More precisely, about 7.12 × 10–30 grams per cubic centimeter.
9. The conversion is 7.12 × 10–30 grams/cubic centimeter = (7.12 × 10–30 grams/cubic centimeter) × (4.6 × 104 Planck mass/gram) × (1.62 × 10–33 centimeter/Planck length)3 = 1.38 × 10–123 Planck mass/cubic Planck volume.
10. For inflation, the repulsive gravity we considered was intense and brief. This is explained by the enormous energy and negative pressure supplied by the inflaton field. However, by modifying a quantum field’s potential energy curve, the amount of energy and negative pressure it supplies can be diminished, thus yielding a mild accelerated expansion. Additionally, a suitable adjustment of the potential energy curve can prolong this period of accelerated expansion. A mild and prolonged period of accelerated expansion is what’s required to explain the supernova data. Nevertheless, the small non-zero value for the cosmological constant remains the most convincing explanation to have emerged in the more than ten years since the accelerated expansion was first observed.
11. The mathematically inclined reader should note that each such jitter contributes an energy that’s inversely proportional to its wavelength, ensuring that the sum over all possible wavelengths yields an infinite energy.
12. For the mathematically inclined reader, the cancellation occurs because supersymmetry pairs bosons (particles with an integral spin value) and fermions (particles with a half [odd] integral spin value). This results in bosons being described by commuting variables, fermions by anticommuting variables, and that is the source of the relative minus sign in their quantum fluctuations.
13. While the assertion that changes to the physical features of our universe would be inhospitable to life as we know it is widely accepted in the scientific community, some have suggested that the range of features compatible with life might be larger than once thought. These issues have been widely written about. See, for example: John Barrow and Frank Tipler, The Anthropic Cosmological Principle (New York: Oxford University Press, 1986); John Barrow, The Constants of Nature (New York: Pantheon Books, 2003); Paul Davies, The Cosmic Jackpot (New York: Houghton Mifflin Harcourt, 2007); Victor Stenger, Has Science Found God? (Amherst, N.Y.: Prometheus Books, 2003); and references therein.
14. Based on the material covered in earlier chapters, you might immediately think the answer is a resounding yes. Consider, you say,