Cosmic Singularities
What is a singularity?
Mathematically, a field is a continuous and infinitely large set of infinitely small points. As explained in more detail in the section on Quantum Field Theory, physicists associate each point of a field with one or other physical quality such as matter, force, charge, energy, etc. whereby the field becomes a force-field, an energy-field or whatever. As a natural consequence of the mathematics, utilized in field theories, one often encounters formulas according to which an infinitely large amount of the associated quality is gathered in a single point. Such an entity is called a singularity.
Usually, when encountered in theoretical-mathematical considerations, singularities are troublemakers which physically do not make any sense. Imagine, as an example, an infinitely large amount of positive charge gathered somewhere on Earth. What would happen then? - The Earth would vanish. Every negatively charged particle would be sucked into the singularity while every positively charged particle would be kicked out in space. Since the Earth actually does exist, something must be done for the theory to be consistent. In Quantum Field Theory the problem is solved by means of a tedious but consistent technique called renormalization. In General Relativity (gravitational field theory) the problem is dealt with quite differently - here the singularities are regarded as real physical entities.
A gravitational singularity is an infinitely small source of infinite gravity, which, in the language of General Relativity, corresponds to an entity of infinite curvature - i.e. space-time is curled up completely. Currently, no gravitational singularity has been detected directly. However, the theoretical prediction of singularities is not in any conflict, neither with observational astronomy nor with empirical data obtained in other branches of physics. Actually, it is the consensus among cosmologists, that every black "hole" veils a gravitational singularity residing at its center. It is also the consensus, that this universe originated from a singularity and that, provided it is closed, the universe will end up as a (probably identical) singularity.
What is inside a space-time singularity?
A space-time singularity is characterized by the property that all four dimensions have exactly zero extent - as far as our usual frames of reference are concerned. This property is, in a sense, the cause for the extremely small sizes and time-periods associated with the early epochs of the "Big Bang" Model. However, to state that the four dimensions are of zero extent is just another way of stating that the space-time continuum does not exist in relation to the local frame of reference of the singularities. This again, is just another way of stating that the internal realms of space-time singularities are not of this universe. Nevertheless, suppose we look at things from inside a singularity - i.e. we correlate our usual concepts to the singularity's local frame of reference. How do the dimensions appear relative to this frame then? - To ease matters, let us consider the time-dimension and the three spatial dimensions separately:
Time. - One natural consequence of space-time curvature is time-dilatation - and the stronger the curvature the more profound the dilatation. Now, our usual concept of time is merely a sequence of time-lapses. Imagine a traveler, carrying a clock, who travels through regions of increasing curvature. As the traveler advances, the clock appears to go slower and slower due to the increasing space-time curvature. Consequently, some lapse of time, which in our ordinary frame of reference is measured as a tiny fraction of a second, may by the traveler be measured as e.g. a couple of millennia. In the extreme limit, when our traveler enters the singularity, the clock stops - which means that the corresponding time-lapse stretches to infinity. Thus, while in our ordinary frame the duration of the time-lapse is measured to be exactly zero, the same time-lapse appears as eternity to the traveler.
Space. - This is actually the same story as for time, but with the role of the clock taken over by a device to measure lengths - a stick for example. As the traveler is approaching the singularity, through space-time regions of increasing curvature, the stick continuously appears longer and longer - corresponding to an apparent lengthening of the three spatial dimensions. Consequently, a universe, which is much smaller than an atom in our usual frame of reference, may to the traveler appear as huge as a galaxy. When the traveler has entered the singularity, the stick appears infinitely long - corresponding to an infinitely large space. Thus, while in our ordinary frame the singularity is measured to be infinitely small, it appears infinitely large to the traveler.
There seems to be some contradiction here - so we evidently must conclude that whatever is inside a singularity, our usual concepts of time and space do not suffice to describe it. Nevertheless, according to the "Big Bang" Model, the two seemingly contradictive frames of reference, considered above, were identical at the advent of this universe and, if it is closed, they will merge again at the end. So, since the whole issue is a natural consequence of General Relativity, cosmologists either have to make profound modifications to their preferred theory or learn to handle a frame of reference in relation to which the same entity can be infinitely small and infinitely large as well as have an infinitely short existence and exist for eternity. Maybe it is necessary to do both. - Actually, a variety of attempts to solve these problems sort of define the current frontiers of contemporary cosmology - which we shall deal with next.
