Once Before Time - Martin Bojowald [54]
All of space can be seen to arise by adding loop upon loop. In this process one has to start from some initial state, or one is led back to it by performing the game backward and removing loop after loop. In the end, only an object without a single loop can remain: the State of Hell encountered earlier. Here we can see for the first time how loop quantum gravity could give insights about the big bang, for the big bang singularity in general relativity is also a state without volume (but with an infinitely high energy density of matter). To see what is going on there, it is not sufficient to remove one loop at a time by hand; instead, one must analyze how the dynamics of quantized space-time can bring this about. As general relativity describes the expansion of the universe by a space-time that solves Einstein’s equations, such as the differential equations illustrated in figure 4, loop quantum gravity has equations of discrete nature describing the dynamic appearance and removal of loops. Starting with an initial configuration and its quantum theoretical wave function, the further development in time as a consequence of successive additions and removals of loops is determined.
At a precise level, the picture of a jumpy universe, structured on the smallest scales by a discrete weave rather than a smooth space-time, replaces that of a rubber band as used in general relativity. The smallest volume changes are given in terms of the Planck length and thus unimaginably tiny. A single quantum jump is imperceptible in the behavior of a large universe as we see it now. At small extensions as realized in the early universe, however, the woven texture is particularly important. Here the universe itself is so small that a continuous change of volume differs greatly from a quantum jump. The universe’s expansion at those times is entirely unlike what was expected in general relativity: Quantum corrections to Einstein’s equations become necessary.
As a consequence of the continuous picture, the singularity problem arose; now, quantum gravity shows strong deviations from the classical theory just in this exact respect. Could this mean a solution to the singularity problem? Can loop quantum gravity save a universe from a singularity, as quantum mechanics stabilizes hydrogen and other atoms?
LOOP QUANTUM COSMOLOGY:
ATOMS SMEARED OUT
An application of loop quantum gravity to the singularity issue, as it appears for instance in cosmology, at first seems problematic. After all, the dynamics of the loops is so complicated that not a single exact solution has been found yet, and nobody seriously expects this to change in the near future. Also, a mathematical analysis of the equations without knowing explicit solutions would have to overcome severe difficulties. But this problem does not arise only in quantum gravity. Einstein’s equations of general relativity are not easy to solve either, yet we know some of their general properties. The most important example is the singularity problem, according to which all solutions, not just those few known explicitly, must, under some realistic conditions—for instance on the form of matter—reach a singularity or have come from one.
Exact solutions to a complicated theory such as general relativity can be found only in special, often highly symmetrical cases. In cosmology, homogeneity and isotropy are usually assumed: Disregarding details such as planets, stars, or even whole galaxies, the universe on a very large scale does not show position-dependent properties. None of our little fits and fights, on small contested planets or between entire merging galaxies, plays any role for the whole universe. Thanks to wide-ranging galaxy maps such as the Sloan Digital Sky Survey (SDSS) described later, this long-assumed “cosmological principle” has by now found impressive support through observations.