Once Before Time - Martin Bojowald [20]
But not even this hardy state can prevail against gravitational pressure when mass is increased further. From then on, no known force could counter the rise of gravity in this cold war of forces. Equilibrium is no more; all chunks of matter do nothing but attract each other, and they collapse to a black hole, another manifestation of the singularity. In its interior, the unrestrained collapse of all infalling matter generates ever higher densities and temperatures, and then at last—through matter’s influence on space and time—the end of time itself, once all the matter of the former star has collapsed. “What an ending! What an appalling ending!”13
Here we have the physical reason for the existence of singularities in general relativity: the attractive nature of gravity without powerful repulsive forces. That bold leap of relativity beyond Newtonian physics, fueled by the changeable structure of space and time, turns out to be a very risky one. Space-time becomes a dynamic object, subject to mathematical equations. Unfortunately, only in exceptional cases do these equations have solutions defining the universe at all times. Most solutions, including realistic ones, lose their validity at some point, so space-time as described by them comes to its own end, too. Such singularities exist because general relativity, while eliminating Newton’s action at a distance, does not change the purely attractive nature of the gravitational law. Moreover, matter attraction is allowed to influence even space and time by the daring innovation of general relativity. It rather fiendishly unleashes gravity on space and time, which now must tolerate the whims of matter without any safeguard by a counteracting force to halt the full collapse. To solve this not only aesthetic but also decidedly fundamental problem, we must extend the theory once more, and take a leap beyond.
There are comparable cases in which a theory leads to singularities, which can be eliminated by suitable extensions. Liquids such as water, for instance, can often be described quite precisely by assuming a continuous distribution, ignoring the composition of molecules. One might use this approximation for the flow through a pipe. But for small flow rates, and if the pipe ends in a faucet, this description breaks down: The continuous distribution decays to single droplets at the end of the faucet. From the viewpoint of the continuum description, the breaking apart appears as a singularity, for some quantities such as the surface tension would diverge at the time of separation; continuum equations then lose their meaning. In this case we know what causes the problem: When a droplet splits off, nothing really violent is happening. We merely have to consider the molecular nature of matter; and if the gravitational tension surpasses the cohesion of water molecules, water can split into separate drops. One has to extend the continuum theory by a more fundamental theory taking into account the molecular nature of matter. Such a theory is more realistic but also mathematically more involved. Its key advantage: It does not lead to singularities in its solutions and can describe, for example, the separation into droplets.
There are significant differences between gravity and this water analogy, in particular due to the more intimate involvement of space and time in general relativity. But this analogy conveys an important lesson: There is no avoiding an extension of general relativity. Singularities show the limits of the theory, and to understand their role one must find an encompassing theory that regularly describes the would-be singularities.
Only then can we check whether limits such as those in black holes or at the big bang are truly physical limits of space and time, or merely limitations in our theoretical description. Here we have a situation similar to