Once Before Time - Martin Bojowald [17]
What exactly would happen at a singularity? To see this, one must study the mathematical equations of general relativity, for they describe the structure of space and time. According to the continuous form of space-time, represented mathematically by differential geometry, those are differential equations. That is, Einstein’s equations determine how the form of space-time near a point must behave on account of the matter present, or more precisely, on account of its energy density and pressure. Thus the equations correspond to a continuum picture, visualizing space-time as a curved and wrinkled sheet, albeit in four dimensions. In contrast to a real fabric woven from threads with gaps in between, the sheet of general relativity has no structure whatsoever when it is viewed on the smallest distance scales: hence the use of differential equations, which describe the change of space-time under the smallest shifts of position in it. These smallest changes are not atomic, but smaller than any possible size: They arise in a mathematical limiting procedure to conceptualize the continuum picture.
A differential equation can be visualized by its velocity field, in which an arrow is planted at every point to indicate the direction and size of the rate of change. In general relativity, the ground of the field in the simplest cases is a plane on which each point, by its two coordinates, determines time and the expansion of the universe at that time. A solution to the differential equation is a curve in the plane required to follow the direction of the velocity arrow at each point. One can often represent the solutions graphically, as in figure 3, but there are also mathematical methods to directly represent them in the form of a function, or numerical techniques to find such curves with computers.
The graphic construction shows that the velocity field is not sufficient to determine a unique solution; it indicates, after all, just the direction to move at each point. First one must know where to start: A point, the initial condition, must also be chosen. If this is done, one obtains a unique solution in most cases, but the form of initial conditions can, depending on the problem analyzed, take more complex forms than a single point.
In cosmology, the velocity field of cosmic expansion is determined by matter in the universe. Small changes of space-time caused by the smallest shifts in position are then given by the size of the energy density and pressure of the surrounding matter. As the initial condition we can posit a state resembling what we see now in the universe, a mathematical configuration with expanding space and diluting matter content. This leaves different possibilities for future development: If the amount of matter is small, below a certain critical density, the universe will expand forever and become more and more diluted. But amounts of matter above the critical density would exert a stronger gravitational force, which, in the distant future, can first stop the expansion of the whole universe and then make it collapse on itself. In this case, there will be a time when the universe turns around, like a rock thrown up in the air, and then contracts. To decide which case will be realized for our universe, we must determine the exact amount of matter—not an easy task. We can estimate and