The problem of Time
The fact that no global space-time coordinates could be introduced without singularities may be one of the causes of the failures of several attempts to quantize general relativity, especially of the attempt of canonical quantization (cf. C. J. Isham, "Canonical quantum gravity and the problem of time," Proceedings of the NATO Advanced Study Institute, Salamanca, June 1992, Kluwer Academic Publishers, 1993, gr-qc/9210011). This may be, in actuality, the true origin of the problem of time, which was posed during 1950's and has been questioning if there is any contradiction between the nonexistence of any physically effective global time and the apparent existence of local time. After many attempts to resolve this contradictory situation, a recent attempt looks like even abandoning any positive efforts by attributing the problem to the unproved undecidability properties of the problem (Stuart Kauffman and Lee Smolin, "A possible solution for the problem of time in quantum cosmology," http://www.edge.org/3rd_culture/smolin/smolin_p2.html, gr-qc/9703026).
Returning to the origin of the problem, i.e. to the idea of relativity theories, a cause of the problem of time seems to lie in associating time to each point which has no positive size. No clocks can reside in a sizeless point. At the stage of special theory of relativity, this difficulty does not appear: Time is associated to each inertial frame which can accommodate actual clocks. At the stage of general theory of relativity, the field equation with the invariance postulate with respect to diffeomorphisms requires one to eliminate the size of the frames in which clocks reside.
A possible solution
A possible solution to this contradictory state is therefore to define time as a notion associated with a system which can accommodate clocks which have actual sizes. Such systems are called "local systems," and their centers of mass are identified with the classical general relativistic points, with their inner structures being equipped with the structures of local systems. These local structures are assumed Euclidean ones tangent to the total 4-manifold, but no connections are imposed among their Euclidean structures. Instead we pose a relation between the inside and outside of a local system, which I will describe below in "A Solution of Unification of Quantum Mechanics and Special Theory of Relativity." With our identification of the centers of mass of local systems with classical particles, the general theory of relativity remains valid as a theory describing the "classical" world among the local systems. In this setting, time recaptures its meaning as a local time inside each local system, defined as a measure of "quantum mechanical" motion inside the local system. As a by-product, quantum mechanical and classical views of the world are reconciled by this formulation at the expense of the field equation.
Sciences and Time
From other viewpoints, sciences are founded on the notion of time. Sciences cannot be called "sciences" if they lose their descriptions of nature according to some time-coordinates. Physics, Chemistry, Biology, History, ..., these are the descriptions of nature or human beings along some time-coordinates. Without the notion of time, these academic areas cannot exist as academic activities.
Time has been, however, a notion whose existence nobody doubts. These academic activities have been assuming the existence of some time-coordinates, and people speak about things as if they move or change following the order prescribed by time. But what is time? Is it an existence in the same sense as the existence of other objective things? Time is not such an existence: Time does not appear until we measure it by some equipments, i.e., by clocks. Time is just a movement of the hands of clocks, and time is not an a priori existence which measures motion. Quite contrarily, just the motion of hands of analogue clocks, or just the change of figures of digital clocks measures time.
The definition of local time is based on this observation in addition to the afore-mentioned physical considerations. It is defined as a local notion effective only in each local system and is equivalent to the quantum mechanical evolution of that local system. By the locality of this notion of time, it is compatible with two basic principles of general theory of relativity in the first place.
The considerations on these problems are described in my recent papers and lectures in the following list. The lecture in the winter semester of 1996-1997 delivered to graduate and undergraduate students of liberal arts is not written in a form available here. However, the main content of the lecture is presented in my paper "Quantum Mechanics and Relativity --Their Unification by Local Time--" cited in the list as "Time IV." These are written in English except for one paper in Japanese, which is mainly concerned with the question "What should be done in Mathematical Sciences?" but on the deeper level, is related with my notion of time as suggested in the following introduction.
(photo -- H. Kitada on a train in Denmark)