Once Before Time - Martin Bojowald [18]
3. Most physical systems are mathematically given by specifying their rates of change everywhere, formulated as a differential equation. The rate of change can graphically be represented by arrows such that a solution to the problem is a curve always taking the direction of the arrows. In this example, the velocity field of a vortex is shown, whose solutions are circles around the center point.
The future is always uncertain; we should therefore have a look at what happened in the past. Owing to its expansion, the younger universe was smaller than it is now, and very long ago it was hotter than any kind of matter existing on Earth or even in stars. Under such extreme conditions, the behavior of matter is little known, but general consequences for the universe only weakly depend on it: A detailed analysis of Einstein’s equations shows that some of the spatial or timelike distances must be unimaginably small at the moment of the big bang (and in black holes). One can see this in the velocity field of figure 4, which pushes every solution curve to a point of vanishing volume. The energy density of matter—the ratio of the total energy to the occupied volume—becomes infinite: One divides by zero when an extension, and thus the volume as a product of extensions, vanishes. Here we have the physical dilemma that no matter can persist at infinite density; also, such a divergence is mathematically so problematic that it leads to the breakdown of Einstein’s equations. Even if we tried to ignore the problem of infinite density, the equations would tell us that the form of space-time changes by an infinite amount even at the tiniest change of position (or time). This cannot possibly be a useful coherent structure: Space-time is torn asunder at the singularity.
Singularities constitute a serious problem, a theoretical misdemeanor that will eventually force us to abandon general relativity as a fundamental description, or to extend it. The tremendous consequence that space and time reach an end in their theoretical description does not mean a physically predicted beginning or end of the world. Even though the mathematical equations show that a point is reached at which all distances vanish, the equations themselves then lose all their meaning. The theory becomes unreliable at this point and simply can no longer be used for predictions; it leads us to this singular place near the abyss, but then leaves us alone with the question of its meaning and of what lies beyond. From then on, we have to look for a new guide.
4. The differential equation of cosmology is shown here by its velocity field as it follows from Einstein’s equation. The volume of the universe changes horizontally; vertically, the density of the matter content grows. The closer one comes to the left border, where the universe volume vanishes, the longer the arrows become. Every solution curve along the arrows is inescapably drawn to the border. Moreover, the matter density grows without bounds and becomes infinite for vanishing volume at the left border.
Just like Newton’s theory, Einstein’s gravitational law is flawed, but much more seriously so. While Newton was uneasy with the animalistic tendencies of his theory, the shadow looming over general relativity is decidedly more ominous. In the words of the physicist John Wheeler, general relativity contains the seed of its own destruction—which is a condition for greatness: “All great things perish on their own, by an act of self-elimination.”9 Indeed, general relativity is unique not just in its power but also in telling us its own limits in a strong, undeniable way. Singularity problems of this