Początek i koniec wszechświata w zamkniętym modelu Friedmana

Abstrakt

How to define space-time singularities is a serious problem in general relativity. Schmidt's b-boundary construction was commonly regarded as leading to the best (and very elegant) definition of singularities: space-time is said to be singular if it contains at least one b-incomplete curve. Unfortunately, Bosshard (1976) and Johnson (1977) demonstrated that the b-boundary of the closed Friedman universe consists of the single point. This means that the initial and final singularieties (i.e., the beginning and the end of the Friedman world) coincide. So far no remedy to cure this disaster was known. In the present paper, the author reviews the results obtained by himself and W. Sasin (1994) which clarify the problem. The closed Friedman space-time together with its singularities is not smooth manifold but it turns out to be a structured space which is a strong generalization of the manifold concept. Consequently, if the problem is considered within the correct category, i.e. the category of structured spaces, the patologies dissapear. If one remains within the space-time everything is all right. It is only if one attempts to „touch” the singularity (to extend the differential structure from the inside of space-time to the singularity) when the beginning and the end of the universe coalesce. From the perspective of the Demiurg creating the closed Friedman universe the cosmic history is but a single point. The methodological significance of this result is briefly discussed.