Once Before Time - Martin Bojowald [55]
Loop quantum gravity is even more complicated than general relativity; one should not expect that simplifying assumptions can be dropped. But this is no reason to give up. If one could describe at least the highly symmetrical examples in quantum gravity, an investigation of the singularity problem would be possible; even these simple classical solutions challenge general relativity sometimes. An analysis of space-times in quantum gravity satisfying the cosmological principle, a subset called quantum cosmology, is already worthwhile. How can such a simplification be performed? It looks obvious if symmetry is demanded for a smooth rubber band such as general relativity’s space-time. But what remains of the loops of quantum gravity? A symmetrical weave should become more orderly, as in figure 13 compared to figure 12, but even such a regular lattice is not homogeneous: Its links differ from the nothingness in between. A spatial weave woven from its links would be completely blurred by symmetry requirements as strong as homogeneity. One could worry that no trace of the lattice properties of quantum gravity, including the quantum jumps of volume indicating crucial effects in a small universe, would be left.
Fortunately, and perhaps surprisingly, it turns out that the most important properties do in fact continue to exist even if homogeneity and isotropy of space are imposed. Such a symmetrical situation does not allow single loops in a space made uniform, but the change in time, especially of the volume, still occurs in smallest jumps. There is, moreover, a loopless State of Hell in which the volume vanishes, much as at the big bang singularity. In 2000 and the years following, the mathematical methods developed in the 1990s allowed me not only to formulate such isotropic geometries in the newly minted loop quantum cosmology, but also to analyze their temporal evolution as defined by Thiemann.
An overarching quantum theoretical investigation of the singularity problem thus came within reach. Still, the concrete application did not arise directly, for even the equations with the highest possible symmetry initially appeared too complicated. The breakthrough happened with a little help from chance, as I can directly report from my own experience, in an example that illustrates how external influences can act on scientific developments. Progress does not happen as logically as it may sometimes seem from the outside.
13. A regular lattice of space-time.
After I had imposed the first set of symmetry assumptions to simplify the equations, they remained too intractable for an analysis of quantum cosmological evolution. A further possibility of simplifications did suggest itself, but seemed to me to be mathematically invalid—for good reasons, I thought at that time. This point was a decisive one, for the equations without concrete applications would be of little value. Realizing homogeneity and isotropy in spite of the not at all symmetrical form of loops may have posed a mathematically interesting issue, but on its own it does not show the way to new physical phenomena. There was an additional difficulty: Since the 1960s a theory of quantum cosmology has already existed—founded by John Wheeler