Once Before Time - Martin Bojowald [61]
As computed by Planck, and confirmed by direct measurements, this increase does not happen. Quantum theory implies that energy is distributed not continuously but over small, discrete packages, the photons of the radiation. As before, small wavelengths—where classical physics would make us believe that the energy density of black body radiation diverges—correspond to large frequencies, or to large energy packages. Since the total amount of energy in the radiation is fixed, it cannot fill the large packages required at small wavelengths. For small wavelengths, fewer packages are filled, and the energy density starts to decrease; for the smallest ones, no energy is present at all. Thus no divergence arises.
Now consider matter in a collapsing universe. With the whole universe shrinking, all scales of distance as well as the wavelike matter it contains will eventually be small. Normally, one would expect densities—obtained by dividing the mass of matter by the volume of a region—to increase when distance scales decrease. But as in the quantum theory of black body radiation, loop quantum cosmology shows that instead, in a sufficiently small universe, less matter energy is distributed over all atomic space: Quantum waves do not find enough room. As in Planck’s description of heat radiation in a box, the classically expected increase of densities all the way to infinity is interrupted by quantum theory, and instead turns to a decrease toward zero.
Now the problem is less threatening. But if the universe keeps collapsing, its equations could still break down even with finite matter densities. However, repulsive forces counteracting the collapse arise. Matter energy, after all, determines the form of space-time and its curvature, in turn responsible for the gravitational force. While the classical form of energy always causes attractive gravity, the turnaround at small lengths, as it happens in a dense, tiny universe, can lead to a redirection of the force, an additional, independent mechanism for a repulsive force provided by loop quantum cosmology. (Most big bang scenarios in string theory are based on such matter-related effects, without specific insights into the structure of space-time.)
Such counterforces do not seem to be a random result, but are a general phenomenon in this kind of quantum gravity. They do not just appear in some special cases—for instance in detailed solutions analyzed numerically in a set of stimulating articles by Ashtekar, Tomasz Pawlowski,9 and Parampreet Singh in 2006—such properties of quantum cosmological space-times can be found in many situations. Still, the phenomena have not yet been investigated completely. And even though they will reappear in our later description of the collapse of black holes, it has not been proven that sufficiently strong counterforces, powerful enough to prevail against any kind of singularity, are a general property of loop quantum gravity. Here, research is still ongoing.
ON THE USE OF MATHEMATICS—AN ILLUSTRATION: CLIMBING THROUGH THE STATE OF HELL
Loop quantum gravity determines states for three-dimensional spaces of different sizes by wave functions describing the usual geometry of areas and volumes in a quantum and fluctuating way. Not all spatial sizes are possible; rather, there is only a discrete set, just as for energies in atomic spectra. When the spatial weave of loops is extended by a new knot, the total volume changes by a fixed amount of a specific value.
In full generality, the allowed values for volume, assumed by such a spatial foam fluctuating on the smallest scales, are difficult to compute. One can see this in figure 15, which shows an example of values of the volume in dependence of a parameter controlling the asymmetry. Although much headway has recently been made by the calculations of Johannes Brunnemann and David Rideout, no complete picture is known for the general case. Even spatial structures that appear uniform on large cosmic scales are fundamentally constructed in an atomic way, just as a material body